The initial-Neumann problem for the heat equation in Lipschitz cylinders
Russell M.
Brown
1-52
Abstract: We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in ${L^p}$, $1 < p < 2+\varepsilon$, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space ${H^1}$. Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and half of a time derivative in ${L^p}$, $1 < p < 2+\varepsilon$, with a corresponding Hardy space result when $p = 1$. Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in $ {L^q}$,
Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains
Jeff E.
Lewis
53-76
Abstract: The symbolic calculus of pseudodifferential operators of Mellin type is applied to study layer potentials on a plane domain ${\Omega ^ + }$ whose boundary ${\partial\Omega ^ + }$ is a curvilinear polygon. A "singularity type" is a zero of the determinant of the matrix of symbols of the Mellin operators and can be used to calculate the "bad values" of $p$ for which the system is not Fredholm on ${L^p}(\partial {\Omega ^ + })$. Using the method of layer potentials we study the singularity types of the system of elastostatics $\displaystyle L{\mathbf{u}} = \mu \Delta {\mathbf{u}} + (\lambda + \mu )\nabla \operatorname{div} {\mathbf{u}} = 0.$ in a plane domain ${\Omega ^ + }$ whose boundary ${\partial\Omega ^ + }$ is a curvilinear polygon. Here $ \mu > 0$ and $-\mu \le \lambda \le +\infty$. When $\lambda = +\infty$, the system is the Stokes system of hydrostatics. For the traction double layer potential, we show that all singularity types in the strip $0 < \operatorname{Re} z < 1$ lie in the interval $\left( {\frac{1} {2},1} \right)$ so that the system of integral equations is a Fredholm operator of index 0 on ${L^p}(\partial {\Omega ^ + })$ for all $p$, $ 2 \le p < \infty$. The explicit dependence of the singularity types on $ \lambda$ and the interior angles $\theta$ of $ {\partial\Omega ^ + }$ is calculated; the singularity type of each corner is independent of $\lambda$ iff the corner is nonconvex.
A continuous localization and completion
Norio
Iwase
77-90
Abstract: The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localization and completion of an ${A_n}$-space is given to be an $ {A_n}$-homomorphism between ${A_n}$-spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent $ G$-CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent $\mathbf{D}$-CW-complex for a polyhedral category $\mathbf{D}$.
$L\sp p$ inequalities for entire functions of exponential type
Qazi I.
Rahman;
G.
Schmeisser
91-103
Abstract: Let $f$ be an entire function of exponential type $\tau$ belonging to ${L^p}$ on the real line. It has been known since a long time that
Classification of crossed-product $C\sp *$-algebras associated with characters on free groups
Hong Sheng
Yin
105-143
Abstract: We study the classification problem of crossed-product $ {C^ * }$-algebras of the form $C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf{Z}}$, where $G$ is a discrete group, $\chi$ is a one-dimensional character of $ G$, and ${\alpha_\chi}$ is the unique $ *$-automorphism of $C_r^ * (G)$ such that if $U$ is the left regular representation of $ G$, then $ {\alpha_{\chi}(U_{g})=\chi(g)U_{g}}$, $g \in G$. When $ {G = F_{n}}$, the free group on $n$ generators, we have a complete classification of these crossed products up to $*$-isomorphism which generalizes the classification of irrational and rational rotation $ {C^ * }$-algebras. We show that these crossed products are determined by two $ K$-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural $ \operatorname{Aut} ({F_n})$-action, and that these two orbit invariants completely classify the characters up to automorphisms of $ {F_n}$. The classification of crossed products follows from these results. We also consider the same problem for $G$ some other groups.
Local rigidity of symmetric spaces
V.
Schroeder;
W.
Ziller
145-160
Abstract: We show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds, then the new metric is isometric to the old one. We also prove an analytic continuation property for symmetric spaces of rank $\ge 3$.
Weight strings in nonstandard representations of Kac-Moody algebras
Meighan I.
Dillon
161-169
Abstract: We consider the weights which occur in arbitrary irreducible highest weight representations of Kac-Moody algebras and determine conditions under which certain weights may or may not occur.
Minimal identities of symmetric matrices
Wen Xin
Ma;
Michel L.
Racine
171-192
Abstract: Let ${H_n}(F)$ denote the subspace of symmetric matrices of ${M_n}(F)$, the full matrix algebra with coefficients in a field $F$. The subspace ${H_n}(F)\subset {M_n}(F)$ does not have any polynomial identity of degree less than $2n$. Let $\displaystyle T_k^i({x_1}, \ldots ,{x_k}) = \sum\limits_{\begin{array}{*{20}{c}... ... {{{( - 1)}^\sigma }{x_{\sigma (1)}}} {x_{\sigma (2)}} \cdots {x_{\sigma (k)}},$ , and $e(n) = n$ if $n$ is even, $n + 1$ if $n$ is odd. For all $ n \geq 1,T_{2n}^i$ is an identity of ${H_n}(F)$. If the characteristic of $ F$ does not divide $ e(n)!$ and if $ n \ne 3$, then any homogeneous polynomial identity of ${H_n}(F)$ of degree $2n$ is a consequence of $T_{2n}^i$. The case $n = 3$ is also dealt with. The proofs are algebraic, but an equivalent formulation of the first result in graph-theoretical terms is given.
The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry
J. C.
Lagarias
193-225
Abstract: This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar's linear programming algorithm. A nonlinear change of variables, projective Legendre transform coordinates, is introduced to study these trajectories. The projective Legendre transform mapping has a coordinate-free geometric interpretation in terms of the notion of "centering by a projective transformation." Let ${\mathsf{H}}$ be a set of linear programming constraints $ \{ \langle {{\mathbf{a}}_j},{\mathbf{x}}\rangle \geq {b_j}:1 \leq j \leq m\}$ on $ {{\mathbf{R}}^n}$ such that its polytope of feasible solutions ${P_{\mathsf{H}}}$ is bounded and contains ${\mathbf{0}}$ in its interior. The projective Legendre transform mapping ${\psi _{\mathsf{H}}}$ is given by $\displaystyle {\psi _{\mathsf{H}}}({\mathbf{x}}) = \frac{{{\phi _{\mathsf{H}}}(... ...{{{\mathbf{a}}_j}}} {{\langle {{\mathbf{a}}_j},{\mathbf{x}}\rangle - {b_j}}}.}$ Here $ {\phi _{\mathsf{H}}}(x)$ is the Legendre transform coordinate mapping introduced in part II. ${\psi _{\mathsf{H}}}({\mathbf{x}})$ is a one-to-one and onto mapping of the interior of the feasible solution polytope $ \operatorname{Int} ({P_{\mathsf{H}}})$ to the interior of its polar polytope $ \operatorname{Int} (P_{\mathsf{H}}^\circ )$. The set of projective scaling trajectories with objective function $\langle {\mathbf{c}},{\mathbf{x}}\rangle - {c_0}$ are mapped under ${\psi _{\mathsf{H}}}$ to the set of straight line segments in $\operatorname{Int} (P_{\mathsf{H}} ^\circ )$ passing through the boundary point $- {\mathbf{c}}/{c_0}$ of $P_{\mathsf{H}} ^\circ$. As a consequence the projective scaling trajectories (for all objective functions) can be interpreted as the complete set of "geodesics" (actually distinguished chords) of a projectively invariant metric geometry on $\operatorname{Int} ({P_{\mathsf{H}}})$, which is isometric to Hilbert geometry on the interior of the polar polytope $ P_{\mathsf{H}}^\circ$.
A regularity theory for variational problems with higher order derivatives
F. H.
Clarke;
R. B.
Vinter
227-251
Abstract: We consider problems in the calculus of variations in one independent variable and where the Lagrangian involves derivatives up to order $N$, $N \ge 1$. Existence theory supplies mild hypotheses under which there are minimizers for such problems, but they need to be strengthened for standard necessary conditions to apply. For problems with $N > 1$, this paper initiates investigation of regularity properties, and associated necessary conditions, which obtain strictly under the hypotheses of existence theory. It is shown that the $N$th derivative of a minimizer is locally essentially bounded off a closed set of zero measure, the set of "points of bad behaviour". Additional hypotheses are shown to exclude occurrence of points of bad behaviour. Finally a counter example suggests respects in which problems with $N > 1$ exhibit pathologies not present in the $N = 1$ case.
Elliptic problems involving an indefinite weight
M.
Faierman
253-279
Abstract: We consider a selfadjoint elliptic eigenvalue problem, which is derived formally from a variational problem, of the form $Lu = \lambda \omega (x)u$ in $ \Omega$, ${B_j}u = 0$ on $\Gamma$, $j = 1, \ldots ,m$, where $ L$ is a linear elliptic operator of order $2m$ defined in a bounded open set $ \Omega \subset {{\mathbf{R}}^n}\quad (n \geq 2)$ with boundary $\Gamma$, the ${B_j}$ are linear differential operators defined on $\Gamma$, and $\omega$ is a real-valued function assuming both positive and negative values. For our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem. We also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces.
Using subnormality to show the simple connectivity at infinity of a finitely presented group
Joseph S.
Profio
281-292
Abstract: A CW-complex $ X$ is simply connected at infinity if for each compact $C$ in $X$ there exists a compact $D$ in $X$ such that loops in $X - D$ are homotopically trivial in $X - C$. Let $G$ be a finitely presented group and $ X$ a finite CW-complex with fundamental group $G$. $G$ is said to be simply connected at infinity if the universal cover of $X$ is simply connected at infinity. B. Jackson and C. M. Houghton have independently shown that if $ G$ and a normal subgroup $ H$ are infinite finitely presented groups with $G/H$ infinite and either $H$ or $G/H$ $1$-ended, then $G$ is simply connected at infinity. In the case where $H$ is $1$-ended, we exhibit a class of groups showing that the "finitely presented" hypothesis on $H$ cannot be reduced to "finitely generated." We address the question: if $N$ is normal in $H$ and $H$ is normal in $G$ and these are infinite groups with $ N$ and $G$ finitely presented and either $ N$ or $G/H$ is $1$-ended, is $G$ simply connected at infinity? In the case that $ N$ is $1$-ended, the answer is shown to be yes. In the case that $G/H$ is $1$-ended, we exhibit a class of such groups that are not simply connected at infinity.
Nonmetrizable topological dynamics and Ramsey theory
Vitaly
Bergelson;
Neil
Hindman
293-320
Abstract: Applying ideas from topological dynamics in compact metric spaces to the Stone-Cěch compactification of a discrete semigroup, several new proofs of old results and some new results in Ramsey Theory are obtained. In particular, two ultrafilter proofs of van der Waerden's Theorem are given. An ultrafilter approach to "central" sets (sets which are combinatorially rich) is developed. This enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central. Also, a fortuitous answer to a question of Ellis is obtained.
The Schr\"odinger equation with a quasi-periodic potential
Steve
Surace
321-370
Abstract: We consider the Schràdinger equation $\displaystyle - \frac{{{d^2}}} {{d{x^2}}}\psi + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))\psi = E\psi$ where $\varepsilon$ is small and $\sigma$ satisfies the Diophantine inequality $\displaystyle \vert p + q\alpha \vert \geq C/{q^2}{\text{for}}p{\text{,}}q \in {\mathbf{Z}},q \ne 0.$ . We look for solutions of the form $\displaystyle \psi (x) = {e^{iKx}}q(x) = {e^{iKx}}\sum {{\psi _{mn}}{e^{inx}}} {e^{im(\alpha x + \vartheta )}}$ . If we try to solve for $\psi = {\psi _{mn}}$ we are led to the Schràdinger equation on the lattice ${{\mathbf{Z}}^2}$ $\displaystyle H(K)\psi = (\varepsilon \Delta + V(K))\psi = E\psi$ where $ \Delta$ is the discrete Laplacian (without diagonal terms) and $V(K)$ is some potential on ${{\mathbf{Z}}^2}$ . We have two main results: (1) For $ \varepsilon$ sufficiently small, $H(K)$ has pure point spectrum for almost every $ K$. (2) For $\varepsilon$ sufficiently small, the operator $\displaystyle - {d^2}/d{x^2} + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))$ has no point spectrum. To prove our results, we must get decay estimates on the Green's function ${(E - H)^{ - 1}}$. The decay of the eigenfunction follows from this. In general, we must keep track of small divisors which can make the Green's function large. This is accomplished by a KAM (Kolmogorov, Arnold, Moser) type of multiscale perturbation analysis.
Analysis of a class of probability preserving measure algebras on compact intervals
William C.
Connett;
Alan L.
Schwartz
371-393
Abstract: The measure algebras of the title are those which are also hypergroups with some regularity conditions. Examples include the convolutions associated with Jacobi polynomial series and Fourier Bessel series. It is shown here that there is a one-to-one correspondence between these hypergroups and a class of Sturm-Liouville problems which have the characters of the hypergroup as eigenfunctions. The interplay between these two characterizations allows a detailed analysis which includes a Hilb-type formula for the characters and asymptotic estimates for the Plancherel measure and the eigenvalues of the associated Sturm-Liouville problem.
A topological characterization of ${\bf R}$-trees
John C.
Mayer;
Lex G.
Oversteegen
395-415
Abstract: $\mathbf{R}$-trees arise naturally in the study of groups of isometries of hyperbolic space. An $\mathbf{R}$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals $ \mathbf{R}$. Actions on $\mathbf{R}$-trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of $\mathbf{R}$-trees. Our main theorem is the following: Let $(X,p)$ be a metric space. Then $ X$ is uniquely arcwise connected and locally arcwise connected if, and only if, $X$ admits a compatible metric $ d$ such that $ (X,d)$ is an $\mathbf{R}$-tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.