A convergence theory for saddle functions
Hédy
Attouch;
Roger J.-B.
Wets
1-41
Abstract: We develop a convergence theory called epi/hypo-convergence, for bivariate functions that essentially implies the convergence of their saddle points. We study the properties of this limiting process in particular. We characterize the limit functions associated to any collection of bivariate functions and obtain a compactness theorem for the space of saddle functions. Even when restricted to the univariate case, the results generalize those known for epi-convergence. In particular, we show that the analysis of the convergence process via Yosida approximates must not be restricted to the convex case.
Some research problems about algebraic differential equations
Lee A.
Rubel
43-52
Abstract: Twenty-four new research problems are posed, and their background and partial solutions are sketched. Many of these problems are in the (somewhat unexpected) area of interaction between algebraic differential equations, topology, and mathematical logic.
Criteria for solvability of left invariant operators on nilpotent Lie groups
Lawrence
Corwin
53-72
Abstract: We define a special nilpotent Lie group $N$ to be one which has a $1$-dimensional center, dilations, square-integrable representations, and a maximal subordinate algebra common to almost all functionals on the Lie algebra $\mathfrak{N}$. Every nilpotent Lie group with dilations imbeds in such a special group so that the dilations extend. Let $L$ be a homogeneous left invariant differential operator on $N$. We give a representation-theoretic condition on $L$ which $L$ must satisfy if it has a tempered fundamental solution and which implies global solvability of $ L$. (The sufficiency is a corollary of a more general theorem, valid on all nilpotent $N$.) For the Heisenberg group, the condition is equivalent to having a tempered fundamental solution.
Controlled boundary and $h$-cobordism theorems
T. A.
Chapman
73-95
Abstract: In this paper two theorems are established which are consequences of some earlier approximation results of the author. The first theorem is a controlled boundary theorem for finite-dimensional manifolds. By this we mean an ordinary boundary theorem plus small $ \varepsilon$-control in a given parameter space. The second theorem is a controlled $h$-cobordism theorem for finite-dimensional manifolds with small $ \varepsilon$-control in a given parameter space. These results generalize the End Theorem and the Thin $h$-Corbordism of Quinn.
On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs
Curtis
Greene;
Thomas
Zaslavsky
97-126
Abstract: The doubly indexed Whitney numbers of a finite, ranked partially ordered set $ L$ are (the first kind) ${w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L}$ with ranks $i,j\}$ and (the second kind) ${W_{ij}} =$ the number of $({x^i},{x^j})$ with ${x^i} \leqslant {x^j}$. When $L$ has a 0 element, the ordinary (simply indexed) Whitney numbers are ${w_j} = {w_{0j}}$ and ${W_j} = {W_{0j}} = {W_{jj}}$ . Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of $ k$-dimensional faces for any $k$, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope $P$ inside the visible boundary as seen from a distant point on a generating line of $P$. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley's theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly $q$ sources (generalizing Rényi's enumeration of permutations with $q$ "outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph's having a unique source.
The word problem for lattice-ordered groups
A. M. W.
Glass;
Yuri
Gurevich
127-138
Abstract: Theorem. There is a finitely generated one relator lattice-ordered group with insoluble (group) word problem.
Highly connected embeddings in codimension two
Susan
Szczepanski
139-159
Abstract: In this paper we study semilocal knots over $f$ into $\xi$, that is, embeddings of a manifold $ N$ into $E(\xi)$, the total space of a $ 2$-disk bundle over a manifold $M$, such that the restriction of the bundle projection $p:E(\xi) \to M$ to the submanifold $ N$ is homotopic to a normal map of degree one, $f:N \to N$. We develop a new homology surgery theory which does not require homology equivalences on boundaries and, in terms of these obstruction groups, we obtain a classification (up to cobordism) of semilocal knots over $f$ into $\xi$. In the simply connected case, the following geometric consequence follows from our classification. Every semilocal knot of a simply connected manifold $ M\char93 K$ in a bundle over $M$ is cobordant to the connected sum of the zero section of this bundle with a semilocal knot of the highly connected manifold $K$ into the trivial bundle over a sphere.
Harmonic maps and classical surface theory in Minkowski $3$-space
Tilla Klotz
Milnor
161-185
Abstract: Harmonic maps from a surface $S$ with nondegenerate prescribed and induced metrics are characterized, showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the Riemannian case. Moreover, holomorphic quadratic differentials are shown to arise as naturally on surfaces of constant $ H$ or $K$ in ${M^3}$ as on their counterparts in $ {E^3}$. The connection between the sine-Gordon, $\sinh$-Gordon and $\cosh$-Gordon equations and harmonic maps is explained. Various local and global results are established for surfaces in ${M^3}$ with constant $H$, or constant $K \ne 0$. In particular, the Gauss map of a spacelike or timelike surface in ${M^3}$ is shown to be harmonic if and only if $ H$ is constant. Also, $ K$ is shown to assume values arbitrarily close to ${H^2}$ on any entire, spacelike surface in $ {M^3}$ with constant $ H$, except on a hyperbolic cylinder.
On the oscillation of differential transforms of eigenfunction expansions
C. L.
Prather;
J. K.
Shaw
187-206
Abstract: This paper continues the study of Pólya and Wiener, Hille and Szegö into the connections between the oscillation of derivatives of a real function and its analytic character. In the present paper, a Sturm-Liouville operator $L$ is applied successively to an infinitely differentiable function which admits a certain eigenfunction expansion. The eigenfunction expansion is assumed to be "conservative", in the sense of Hille. Several theorems are given which link the frequency of oscillation of $ ({L^k}f)(x)$ to the size of the coefficients of $f(x)$, and thus to its analytic character.
Convolution theorems with weights
R. A.
Kerman
207-219
Abstract: Analogues of Young's Inequality and the Convolution Theorem are shown to hold when the ${L_p}$ and $L(p,q)$ spaces have underlying measure defined in terms of power weights.
Toeplitz operators on bounded symmetric domains
Harald
Upmeier
221-237
Abstract: In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space ${H^2}(S)$ associated with the Shilov boundary $ S$ of a bounded symmetric domain $D$ in $ {{\mathbf{C}}^n}$ of arbitrary rank. The Jordan triple system $Z \approx {{\mathbf{C}}^n}$ associated with $D$ is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent $ e \in Z$ induces an irreducible representation ("$e$-symbol") of the $ {C^{\ast} }$-algebra $\mathcal{T}$ generated by all Toeplitz operators $ {T_f}$ with continuous symbol function $f$.
A strong type $(2,\,2)$ estimate for a maximal operator associated to the Schr\"odinger equation
Carlos E.
Kenig;
Alberto
Ruiz
239-246
Abstract: Let $ {T^{\ast} }f(x) = \sup_{t > 0}\vert{T_t}f(x)\vert$, where $({T_t}f)\hat{\empty}(\xi) = {e^{it\vert\xi \vert^2}}\hat f(\xi)/\vert\xi {\vert^{1/4}}$. We show that, given any finite interval $I$, $\int_I {\vert{T^{\ast} }f{\vert^2}\;dx \leqslant {C_I}\int_{\mathbf{R}} {\vert f(x){\vert^2}\;dx} }$, and that the above inequality is false with $2$ replaced by any $p < 2$. This maximal operator is related to solutions of the Schrödinger equation.
Analyticity on rotation invariant families of curves
Josip
Globevnik
247-254
Abstract: Let $\mathfrak{G}$ be a rotation invariant family of smooth Jordan curves contained in $\Delta$, the open unit disc in ${\mathbf{C}}$. For each $\Gamma \in \mathfrak{G}$ let ${D_\Gamma }$ be the simply connected domain bounded by $\Gamma$. We present various conditions which imply that if $f$ is a continuous function on $\Delta$ such that for every $\Gamma \in \mathfrak{G}$ the function $f\vert\Gamma $ has a continuous extension to $ \overline {{D_\Gamma }}$ which is analytic in $ {D_\Gamma }$, then $ f$ is analytic in $ \Delta$.
Some prime elements in the lattice of interpretability types
Pavel
Pudlák
255-275
Abstract: A general theorem is proved which implies that the types of PA (Peano Arithmetic), ZF (Zermelo-Fraenkel Set Theory) and GB (Gödel-Bernays Set Theory) are prime in the lattice of interpretability types.
The $S\sp{1}$-transfer in surgery theory
H. J.
Munkholm;
E. K.
Pedersen
277-302
Abstract: Let ${S^1} \to X \to Y$ be an ${S^1}$-bundle of Poincaré spaces. If $ f:N \to Y$ is a surgery problem then so is the pullback $\hat f:M \to X$. We define algebraically a homomorphism ${\varphi ^!}:{L_n}({\mathbf{Z}}{\pi _1}(Y)) \to {L_{n + 1}}({\mathbf{Z}}{\pi _1}(X))$ and prove that it maps the surgery obstruction for $ f$ to the one for $ \hat f$.
Qualitative differentiation
Michael J.
Evans;
Lee
Larson
303-320
Abstract: Qualitative derivates and derivatives, as well as qualitative symmetric derivates and derivatives, are studied in the paper. Analogues of several results known for ordinary derivates and derivatives are obtained in the qualitative setting.
Maximal abelian subalgebras of von Neumann algebras and representations of equivalence relations
Colin E.
Sutherland
321-337
Abstract: We associate to each pair $ (\mathcal{M},\mathcal{A})$, (with $ \mathcal{M}$ a von Neumann algebra, and $ \mathcal{A}$ a maximal abelian subalgebra) a representation $ \alpha$ of the Takesaki equivalence relation $ \mathcal{R}\,(\mathcal{M},\mathcal{A})$ of $ (\mathcal{M},\mathcal{A})$ as automorphisms of a ${{\text{I}}_\infty }$ factor. Conversely each such representation $\alpha$ of $ \mathcal{R}$ on $ (X,\mu)$ as automorphisms of $ \mathcal{B}\,(\mathcal{H})$ determines a von Neumann algebra-abelian subalgebra pair $ S^{\prime}\,(\mathcal{R},\alpha) = (\mathcal{N},\mathcal{B})$ where $ \mathcal{N}$ is the commutant of the algebra of "self-intertwiners" for $ \alpha$ and $ \mathcal{B} = {L^\infty }(X,\mu) \otimes 1$ on ${L^2}(X,\mu) \otimes \mathcal{H}$. The main concern is the assignments $(\mathcal{M},\mathcal{A}) \to \mathcal{T}\;(\mathcal{M},\mathcal{A}) = (\mathcal{R}\,(\mathcal{M},\mathcal{A}),\alpha)$ and $(\mathcal{R},\alpha) \to S^{\prime}(\mathcal{R},\alpha)$, and in particular, the extent to which they are inverse to each other--this occurs if $\mathcal{R}$ is countable nonsingular and $ \alpha$ is (conjugation by) a projective square-integrable representation (cf. [8]), or if $ \mathcal{A}$ is a Cartan subalgebra (cf. [5]), among other cases. A partial dictionary between the representations $(\mathcal{R},\alpha)$ and pairs $ (\mathcal{M},\mathcal{A})$ is given--thus if $ \mathcal{R}$ is countable nonsingular and $\alpha$ is what we term replete, $ S^{\prime}(\mathcal{R},\alpha)$ is injective whenever $ \mathcal{R}$ is amenable, and a complete Galois theory generalizing that for crossed products by discrete groups is available. We also show how to construct various pathological examples such as a singular maximal abelian subalgebra $ \mathcal{A} \subseteq \mathcal{M}$ for which the Takesaki equivalence relation $ \mathcal{R}\,(\mathcal{M},\mathcal{A})$ is nontrivial.
Recursivity in quantum mechanics
John C.
Baez
339-350
Abstract: The techniques of effective descriptive set theory are applied to the mathematical formalism of quantum mechanics in order to see whether it actually provides effective algorithms for the computation of various physically significant quantities, e.g. matrix elements. Various Hamiltonians are proven to be recursive (effectively computable) and shown to generate unitary groups which act recursively on the Hilbert space of physical states. In particular, it is shown that the $n$-particle Coulombic Hamiltonian is recursive, and that the time evolution of $n$-particle quantum Coulombic systems is recursive.
The initial trace of a solution of the porous medium equation
D. G.
Aronson;
L. A.
Caffarelli
351-366
Abstract: Let $u = u(x,t)$ be a continuous weak solution of the porous medium equation in ${{\mathbf{R}}^d} \times (0,T)$ for some $T > 0$. We show that corresponding to $u$ there is a unique nonnegative Borel measure $ \rho$ on ${{\mathbf{R}}^d}$ which is the initial trace of $ u$. Moreover, we show that the initial trace $\rho$ must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then $ \vert x{\vert^2}$ as $\vert x\vert \to \infty $.
Global invariants for measured foliations
Steven
Hurder
367-391
Abstract: New exotic invariants for measured foliations are constructed, which we call the $\mu$-classes of a pair $(\mathcal{F},\mu)$. The dependence of the $ \mu$-classes on the geometry of the foliation $ \mathcal{F}$ is examined, and the dynamics of a foliation is shown to determine the $\mu$-classes in many cases. We use the $ \mu$-classes to study the classifying space $ B{\Gamma_{S{L_q}}}$ of foliations with a transverse invariant volume form, and we show the homotopy groups of $B{\Gamma _{S{L_q}}}$ are uncountably generated starting in degrees $q + 3$. New invariants for groups of volume preserving diffeomorphisms also arise from the $ \mu$-classes; these invariants are nontrivial and related to the geometric aspects of the group action. Relations between the $ \mu$-classes and the secondary classes of a foliation are exhibited.
Selfadjoint representations of polynomial algebras
Atsushi
Inoue;
Kunimichi
Takesue
393-400
Abstract: In this paper we will investigate the selfadjointness of unbounded $ ^{\ast}$-representations of the polynomial algebra. In particular, it is shown that the notion of selfadjoint representation is equivalent to that of standard representation in the case of the polynomial algebra generated by one hermitian element. Although the notion of standardness implies that of selfadjointness, the converse is not true in general. Therefore, we consider under what conditions a $ ^{\ast}$-representation is standard.
Geometric condition for universal interpolation in $\hat{\mathcal{E}}'$
William A.
Squires
401-413
Abstract: It is known that if $ h$ is an entire function of exponential type and $Z(h) = {\{ {z_k}\} _{k = 1}}$ with $\vert h^{\prime}({z_k})\vert \geqslant \varepsilon \exp (- c\vert{z_k}\vert)$ for constants $\epsilon$, $C$ independent of $k$, then $ \{ {z_k}\} _{k = 1}^\infty$ is a universal interpolation sequence. That is, given any sequence of complex numbers $\{ {a_k}\} _{k = 1}^\infty$ such that $ \vert{a_k}\vert \leqslant A\,\exp (B\vert{z_k}\vert)$ for constants $A,B$ independent of $ K$ then there exists $ g$ of exponential type such that $g({z_k}) = {a_k}$. This note is concerned with finding geometric conditions which make $ \{ {z_k}\} _{k = 1}^\infty$ a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for $ \{ {z_k}\} _{k = 1}^\infty$ to be a universal interpolation sequence is that $ \int_0^{\vert{z_k}\vert} {n({z_k},t)\,dt/t \leqslant C\vert{z_k}\vert + D,k = 1} , 2,\ldots$, where $ n({z_k},t)$ is the number of points of $ \{ {z_k}\} _{k = 1}^\infty$ in the disc of radius $t$ about ${z_k}$, excluding ${z_k}$, and $C,D$ are constants independent of $ k$. Results for the space $ \hat{\mathcal{E}}^\prime= \{ f\;{\text{entire}}\vert\vert f(z)\vert \leqslant A\;\exp [B\vert\operatorname{Im} z\vert + B\log (1 + \vert z\vert^{2})]\}$ are given but the theory is not as complete as for the above example.
Local spectra of seminormal operators
Kevin F.
Clancey;
Bhushan L.
Wadhwa
415-428
Abstract: The local spectral theory of seminormal operators is studied by examining the connection between two naturally occurring contractive operator functions. These results are used to control the local spectra of cohyponormal operators. An invariant subspace result for seminormal operators whose real part has thin spectra is provided.
Erratum to: ``The derived functors of the primitives for ${\rm BP}\sb\ast (\Omega S\sp{2n+1})$''
Martin
Bendersky
429