Cosmic approximate limits and fixed points
J.
Segal;
T.
Watanabe
1-61
Abstract: We define a condition for approximate inverse systems which implies that the limit of the system has the fixed point property. Moreover, this condition is categorical in the approximate shape category. We investigate the class of complex projective $n$-space like continua with respect to the fixed point property by means of this condition. As a further application we show that the hyperspace $ C(X)$ of nonempty subcontinua of an arc-like or circle-like Hausdorff continuum $X$ has the fixed point property. We also prove that $ {2^X}$ and $C(X)$ have the fixed point property for any locally connected Hausdorff continuum $ X$.
On the K\"unneth formula for intersection cohomology
Daniel C.
Cohen;
Mark
Goresky;
Lizhen
Ji
63-69
Abstract: We find the natural perversity functions for which intersection cohomology satisfies the Künneth formula.
Homotopy invariants of nonorientable $4$-manifolds
Myung Ho
Kim;
Sadayoshi
Kojima;
Frank
Raymond
71-81
Abstract: We define a $ {{\mathbf{Z}}_4}$-quadratic function on ${\pi _2}$ for nonorientable $4$-manifolds and show that it is a homotopy invariant. We then use it to distinguish homotopy types of certain manifolds that arose from an analysis of toral action on nonorientable $4$-manifolds.
The level manifold of a generalized Toda equation hierarchy
Yoshimasa
Nakamura
83-94
Abstract: The finite nonperiodic Toda lattice equation induces a linear one-parameter flow on a space of rational functions. The level manifold of the Toda equation hierarchy is shown to be a product of lines. Our main results establish a generalization of this Toda hierarchy which will be called the cyclic-Toda hierarchy. It is proved that the cyclic-Toda hierarchy is completely integrable and its level manifold is diffeomorphic to a disjoint union of cylinders.
The classification of spinors under ${\rm GSpin}\sb {14}$ over finite fields
Xiao-Wei
Zhu
95-114
Abstract: The spinors of a $ 14$-dimensional vector space $V$ are studied with respect to the group $ \operatorname{GSpin}_{14}$ of the $14$-dimensional vector space $V$ over finite fields ${{\mathbf{F}}_q}$. Results are given as follows: (1) the decomposition of the space of spinors into $ \operatorname{GSpin}_{14}$-equivalence classes or "orbits" over ${{\mathbf{F}}_q}$, (2) the structure of the fixer of $ \operatorname{GSpin}_{14}$ for each orbit as an $ {{\mathbf{F}}_q}$-group.
$v\sb 1$-periodic homotopy groups of exceptional Lie groups: torsion-free cases
Martin
Bendersky;
Donald M.
Davis;
Mamoru
Mimura
115-135
Abstract: The $ {v_1}$-periodic homotopy groups $ v_1^{ - 1}{\pi _ {\ast} }(X;p)$ are computed explicitly for all pairs $ (X,p)$, where $ X$ is an exceptional Lie group whose integral homology has no $p$-torsion. This yields new lower bounds for $p$-exponents of actual homotopy groups of these spaces. Delicate calculations with the unstable Novikov spectral sequence are required in the proof.
Noncommutative matrix Jordan algebras
Robert B.
Brown;
Nora C.
Hopkins
137-155
Abstract: We consider noncommutative degree two Jordan algebras $\mathcal{J}$ of two by two matrices whose off diagonal entries are from an anticommutative algebra $\mathcal{S}$ . We give generators and relations for the automorphism group of $ \mathcal{J}$ and determine the derivation algebra Der $ \mathcal{J}$ in terms of mappings on $ \mathcal{S}$ . We also give an explicit construction of all $\mathcal{S}$ for which Der $\mathcal{J}$ does not kill the diagonal idempotents and give conditions for nonisomorphic $\mathcal{S}$ 's to give isomorphic $\mathcal{J}$ 's.
Fourier multipliers on Lipschitz curves
Alan
McIntosh;
Tao
Qian
157-176
Abstract: We develop the theory of Fourier multipliers acting on ${L_p}(\gamma )$ where $\gamma$ is a Lipschitz curve of the form $\gamma = \{ x + ig(x)\}$ with $\left\Vert g\right\Vert _\infty < \infty$ and $\left\Vert g\prime\right\Vert _\infty < \infty$ . The aim is to better understand convolution singular integrals $B$ defined naturally on such curves by $\displaystyle Bu(z) = {\text{p.v.}}\int_\gamma {\varphi (z - \zeta )u(\zeta )d\zeta }$ for almost all $ z \in \gamma$ .
Boundaries of Markov partitions
Jonathan
Ashley;
Bruce
Kitchens;
Matthew
Stafford
177-201
Abstract: The core of a Markov partition is the nonwandering set of the map restricted to the boundary of the partition. We show that the core of a Markov partition is always a finitely presented system. Then we show that every one sided sofic system occurs as the core of a Markov partition for an $n$-fold covering map on the circle and every two sided sofic system occurs as the core of a Markov partition for a hyperbolic automorphism of the two dimensional torus.
Removing point singularities of Riemannian manifolds
P. D.
Smith;
Deane
Yang
203-219
Abstract: We study the behavior of geodesics passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed. We show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point.
Capacitability theorem in measurable gambling theory
A.
Maitra;
R.
Purves;
W.
Sudderth
221-249
Abstract: A player in a measurable gambling house $\Gamma$ defined on a Polish state space $ X$ has available, for each $x \in X$, the collection $\Sigma (x)$ of possible distributions $\sigma$ for the stochastic process ${x_1},{x_2}, \ldots$ of future states. If the object is to control the process so that it will lie in an analytic subset $A$ of $H = X \times X \times \cdots$, then the player's optimal reward is $\displaystyle M(A)(x) = \sup \{ \sigma (A):\sigma \in \Sigma (x)\}.$ The operator $M( \bullet )(x)$ is shown to be regular in the sense that $\displaystyle M(A)(x) = \inf M(\{ \tau < \infty \} )(x),$ where the infimum is over Borel stopping times $ \tau$ such that $ A \subseteq \{ \tau < \infty \}$. A consequence of this regularity property is that the value of $M(A)(x)$ is unchanged if, as in the gambling theory of Dubins and Savage, the player is allowed to use nonmeasurable strategies. This last result is seen to hold for bounded, Borel measurable payoff functions including that of Dubins and Savage.
The regular module problem. I
T. R.
Berger;
B. B.
Hargraves;
C.
Shelton
251-274
Abstract: In the study of induced representations the following problem arises: Let $H = AG$ be a finite solvable group and ${\mathbf{k}}$ a field with $ \operatorname{char}{\mathbf{k}}\nmid\; \vert A\vert$. Let $V$ be an irreducible, faithful, primitive $ {\mathbf{k}}[AG]$-module. Suppose $H$ contains a normal extraspecial $r$-subgroup $R$ with $ Z(R) \leq Z(H)$ and that $ A$ acts faithfully on $ R$. Under what conditions does $A$ have a regular direct summand in $V$? In this paper we consider this question under the hypotheses that $ G = MR$, where $ 1 \ne M$ is normal abelian in $AM$, $A$ is nilpotent, $(\vert A\vert,\vert MR\vert) = (\vert M\vert,\vert R\vert) = 1$ , and $R/Z(R)$ is a faithful, irreducible $AM$-module. We show that $A$ has at least three regular direct summands in $V$ unless $\vert A\vert$, $\exp (M)$, and $r$ satisfy certain very restrictive conditions.
Hilbert's tenth problem for rings of algebraic functions in one variable over fields of constants of positive characteristic
Alexandra
Shlapentokh
275-298
Abstract: The author builds an undecidable model of integers with certain relations and operations in the rings of $S$-integers of algebraic function fields in one variable over fields of constants of positive characteristic, in order to show that Hilbert's Tenth Problem has no solution there.
R\'esolvant g\'en\'eralis\'e et s\'eparation des points singuliers quasi-Fredholm
J.-Ph.
Labrousse;
M.
Mbekhta
299-313
Abstract: C. Apostol et K. Clancey (Trans. Amer. Math. Soc. 215 (1976), 293-300), ont introduit la notion de "projection spectrale généralisée". Cette notion permet, en particulier, de séparer les ensembles finis de points singuliers dans le domaine semi-Fredholm $({\rho _{s\phi }}(A))$ d'un opérateur $ A$ borné dans un Hilbert H. Dans ce travail, on se propose de généraliser ce résultat au domaine quasi-Fredholm de $A({\rho _{q\phi }}(A))$, et pour cela, nous donnons une nouvelle représentation triangulaire du type d'Apostol. D'autre part on construit, pour un opérateur fermé à domaine dense dans $H$ , un résolvant généralisé vérifiant l'identité de la résolvante et analytique dans le domaine régulier de Fredholm de $A(\rho _\phi ^r(A))$ sauf éventuellement sur un ensemble au plus dénombrable de points situés prés de la frontière de ${\rho _\phi }(A)$.
Adjoint transform, overconvexity and sets of constant width
François
Bavaud
315-324
Abstract: The properties of the adjoint transform (associating to a set the intersection of all disks of given radius centered in the set) are systematically investigated, in particular its relationship with the overconvex, the parallelisation and completion of sets. Sets conjugate by the transform can be characterised in a new way as the union or the intersection of all completions of the reference body. New relationships satisfied by their areas and perimeters are derived. Two applications in problems of random intersection of disks are finally treated.
Distortion theorems for Bloch functions
Xiang Yang
Liu;
David
Minda
325-338
Abstract: We establish various distortion theorems for both normalized locally schlicht Bloch functions and normalized Bloch function with branch points. These distortion theorems give lower bounds on either $ \vert f\prime(z)\vert$ or $ \operatorname{Re} f\prime(z)$; most of our distortion theorems are sharp and all extremal functions identified. The main tools used in establishing these distortion theorems are the classical form of Julia's Lemma and a new version of Julia's Lemma that applies to certain multiple-valued analytic functions. As applications of these distortion theorems, we obtain known lower bounds for various Bloch constants and also establish improved lower bounds on a number of Marden constants for Bloch, normal and Yosida functions.
Uniqueness of radial solutions of semilinear elliptic equations
Man Kam
Kwong;
Yi
Li
339-363
Abstract: E. Yanagida recently proved that the classical Matukuma equation with a given exponent has only one finite mass solution. We show how similar ideas can be exploited to obtain uniqueness results for other classes of equations as well as Matukuma equations with more general coefficients. One particular example covered is $\Delta u + {u^p} \pm u = 0$, with $p > 1$. The key ingredients of the method are energy functions and suitable transformations. We also study general boundary conditions, using an extension of a recent result by Bandle and Kwong. Yanagida's proof does not extend to solutions of Matukuma's equation satisfying other boundary conditions. We treat these with a completely different method of Kwong and Zhang.
Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem
Tzong-Yow
Lee;
Wei-Ming
Ni
365-378
Abstract: We investigate the behavior of the solution $u(x,t)$ of $\displaystyle \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}} {{\partial ... ...hi (x)} & {{\text{in}}\;{\mathbb{R}^n},} \end{array} } \right.$ where $\Delta = \sum\nolimits_{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2}$ is the Laplace operator, $p > 1$ is a constant, $T > 0$, and $ \varphi$ is a nonnegative bounded continuous function in ${\mathbb{R}^n}$. The main results are for the case when the initial value $\varphi$ has polynomial decay near $x = \infty$. Assuming $\varphi \sim \lambda {(1 + \vert x\vert)^{ - a}}$ with $\lambda$, $a > 0$, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution $u(x,t)$ are answered in terms of simple conditions on $\lambda$, $a$, $p$ and the space dimension $n$.
Examples of capacity for some elliptic operators
Jang-Mei
Wu
379-395
Abstract: We study $ L$-capacities for uniformly elliptic operators of nondivergence form $\displaystyle L = \sum\limits_{i,j} {{a_{ij}}(x)\frac{{{\partial ^2}}} {{\parti... ...rtial {x_j}}} + } \sum\limits_j {{a_j}(x)\frac{\partial } {{\partial {x_j}}};}$ and construct examples of large sets having zero $L$-capacity for some $L$ , and small sets having positive $ L$-capacity. The relations between ellipticity constants of the coefficients and the sizes of these sets are also considered.
Connected simple systems, transition matrices, and heteroclinic bifurcations
Christopher
McCord;
Konstantin
Mischaikow
397-422
Abstract: Given invariant sets $A$, $B$ , and $C$ , and connecting orbits $A \to B$ and $B \to C$, we state very general conditions under which they bifurcate to produce an $A \to C$ connecting orbit. In particular, our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits, and for which the Conley index of the invariant sets is the same as that of a hyperbolic critical point. Our proof depends on the connected simple system associated with the Conley index for isolated invariant sets. Furthermore, we show how this change in connected simple systems can be associated with transition matrices, and hence, connection matrices. This leads to some simple examples in which the nonuniqueness of the connection matrix can be explained by changes in the connected simple system.
Some integrable subalgebras of the Lie algebras of infinite-dimensional Lie groups
J.
Leslie
423-443
Abstract: This paper gives a proof of Lie's second fundamental theorem in the context of infinite dimensional Lie groups; that is, we define a class of Lie subalgebras of the Lie algebra of a large class of infinite dimensional Lie groups, say $G$ , which can be realized as the Lie algebras of Lie subgroups of $G$ .
On $q$-analogues of the Fourier and Hankel transforms
Tom H.
Koornwinder;
René F.
Swarttouw
445-461
Abstract: For H. Exton's $ q$-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson's $q$-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are $q$-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskiĭ. As a specialization we get ($q$-cosines and $q$-sines which admit $q$-analogues of the Fourier-cosine and Fourier-sine transforms. We also get a formula which is both an analogue of Graf's addition formula and of the Weber-Schafheitlin discontinuous integral.
An analogue of Siegel's $\phi$-operator for automorphic forms for ${\rm GL}\sb n({\bf Z})$
Douglas
Grenier
463-477
Abstract: If $\mathcal{S}{P_n}$ is the symmetric space of $n \times n$ positive matrices, $Y \in \mathcal{S}{P_n}$ can be decomposed into $\displaystyle Y = \left( {\begin{array}{*{20}{c}} 1 & 0 x & I \end{array}... ...)\left( {\begin{array}{*{20}{c}} 1 & {{T_x}} 0 & I \end{array} } \right),$ where $W \in \mathcal{S}{P_{n - 1}}$ . By letting $v \to \infty$ we obtain the $\phi $-operator that attaches to every automorphic form for $G{L_n}(\mathbb{Z})$, $f(Y)$, an automorphic form for $G{L_{n - 1}}(\mathbb{Z})$, $f\vert\phi (W)$.