Viscosity solutions of Hamilton-Jacobi equations
Michael G.
Crandall;
Pierre-Louis
Lions
1-42
Abstract: Problems involving Hamilton-Jacobi equations--which we take to be either of the stationary form $H(x,u,Du) = 0$ or of the evolution form ${u_{t}} + H(x,t,u,Du) = 0$, where $Du$ is the spatial gradient of $ u$--arise in many contexts. Classical analysis of associated problems under boundary and/or initial conditions by the method of characteristics is limited to local considerations owing to the crossing of characteristics. Global analysis of these problems has been hindered by the lack of an appropriate notion of solution for which one has the desired existence and uniqueness properties. In this work a notion of solution is proposed which allows, for example, solutions to be nowhere differentiable but for which strong uniqueness theorems, stability theorems and general existence theorems, as discussed herein, are all valid.
Conformally invariant variational integrals
S.
Granlund;
P.
Lindqvist;
O.
Martio
43-73
Abstract: Let $f:G \to {R^n}$ be quasiregular and $I = \int {F(x,\nabla \,u)\,dm}$ a conformally invariant variational integral. Hölder-continuity, Harnack's inequality and principle are proved for the extremals of $I$. Obstacle problems and their connection to subextremals are studied. If $u$ is an extremal or a subextremal of $ I$, then $u \circ f$ is again an extremal or a subextremal if an appropriate change in $F$ is made.
Trace class self-commutators
C. A.
Berger;
Marion Glazerman
Ben-Jacob
75-91
Abstract: This paper extends earlier results of Berger and Shaw to all $ {W^\ast}$ algebras. The multiplicity of an operator in a ${W^\ast}$ algebra is defined in terms of the trace on the ${W^\ast}$-algebra, and it is shown that if $ T$ is a hyponormal operator in such an algebra, the trace of its self-commutator is bounded by this multiplicity times the area of the spectrum of $T$, divided by $\pi$.
Completeness and basis properties of complex exponentials
Raymond M.
Redheffer;
Robert M.
Young
93-111
Abstract: This paper is concerned with what might be termed the "fine structure" of the completeness and basis properties of complex exponentials. We give new criteria for two sequences to have the same excess in the sense of Paley and Wiener, a result that illuminates and supplements a well-known completeness criterion of Levinson, and new examples and counterexamples pertaining to Riesz bases.
Finitely generic abelian lattice-ordered groups
Dan
Saracino;
Carol
Wood
113-123
Abstract: The authors characterize the finitely generic abelian lattice-ordered groups and make application of this characterization to specific examples.
Almost convergent and weakly almost periodic functions on a semigroup
Heneri A. M.
Dzinotyiweyi
125-132
Abstract: Let $S$ be a topological semigroup, ${\text{US}}(S)$ the set of all bounded uniformly continuous functions on $S,{\text{WAP(}}S)$ the set of all (bounded) weakly almost periodic functions on $S,{E_0}(S): = \{ f \in {\text{UC(}}S):m(\vert f\vert) = 0$ for each left and right invariant mean $m$ on $ {\text{UC(}}S)\}$ and ${W_0}(S): = \{ f \in {\text{WAP}}(S):\:m(\vert f\vert) = 0$ for each left and right invariant mean $ m$ on ${\text{WAP(}}S)\}$. Among other results, for a large class of noncompact locally compact topological semigroups $S$, we show that the quotient space ${E_0}(S)/{W_0}(S)$ contains a linear isometric copy of ${l^\infty }$ and so is nonseparable.
Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas
Alvany
Rocha-Caridi;
Nolan R.
Wallach
133-162
Abstract: In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups. The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen's character sum formula for a quotient of two Verma modules (one of the main results of this article). Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).
Completion of Akahori's construction of the versal family of strongly pseudoconvex CR structures
Kimio
Miyajima
163-172
Abstract: Let $M$ be a compact smooth boundary of a strongly pseudo-convex domain of a complex manifold $ N$ with dim $N \geqslant 4$. We established a sharp a priori estimate for the Laplacian operator associated with Akahori's subcomplex of the ${\bar \partial _b}$-complex to construct the complex analytic versal family (in the sense of Kuranishi) of $CR$ structures of class $ {C^\infty }$ on $ M$.
Applications of $q$-Lagrange inversion to basic hypergeometric series
Ira
Gessel;
Dennis
Stanton
173-201
Abstract: A family of $ q$-Lagrange inversion formulas is given. Special cases include quadratic and cubic transformations for basic hypergeometric series. The $ q$-analogs of the so-called "strange evaluations" are also corollaries. Some new RogersRamanujan identities are given. A connection between the work of Rogers and Andrews, and $q$-Lagrange inversion is stated.
Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of ${\bf C}\sp{n}$ $(n>1)$
M. Seetharama
Gowda
203-212
Abstract: Let ${A^{p,\alpha }}(B),{A^{q,\alpha }}(B)$ and ${A^{l,\alpha }}(B)$ be weighted Bergman spaces on the unit ball of ${{\text{C}}^{n}}\,(n > 1)$. We prove: Theorem 1. If $ 1/l = 1/p + 1/q$ then ${A^{p,\alpha }}(B) \cdot {A^{q,\alpha }}(B)$ is of first category in ${A^{l,\alpha }}(B)$. Theorem 2. Theorem 1 holds for Hardy spaces in place of weighted Bergman spaces. We also show that Theorems 1 and 2 hold for the polydisc ${U^n}$ in place of $B$.
On neighbourly triangulations
K. S.
Sarkaria
213-239
Abstract: A simplicial complex is called $d$-neighbourly if any $d + 1$ vertices determine a $d$-simplex. We give methods for constructing $1$-neighbourly triangulations of $ 3$- and $4$-manifolds; further we discuss some relationships between $d$-neighbourly triangulations, chromatic numbers and the problem of finding upper and lower bounds on the number of simplices and locating the zeros of the characteristic polynomial of a triangulation. A triangulation of an orientable manifold is called order-orientable if there exists some ordering of the vertices which orients the manifold. We give necessary conditions for their existence; also we construct such triangulations on $3$-dimensional handlebodies and discuss the problem of recognising finite monotone subsets of an affine space by using these ideas.
Discrete series characters and Fourier inversion on semisimple real Lie groups
Rebecca A.
Herb
241-262
Abstract: Let $G$ be a semisimple real Lie group. Explicit formulas for discrete series characters on noncompact Cartan subgroups are given. These formulas are used to give a simple formula for the Fourier transform of orbital integrals of regular semisimple orbits.
On derivations of certain algebras related to irreducible triangular algebras
Baruch
Solel
263-273
Abstract: This paper deals with derivations on algebras that are generated by a maximal abelian selfadjoint algebra of operators $\mathcal{A}$ on a Hilbert space and a group of unitary operators acting on it. A necessary and sufficient condition for such a derivation to be implemented by an operator affiliated with $ \mathcal{A}$ is given. The results are related to the study of derivations on a certain class of irreducible triangular algebras.
Systems of fixed point sets
A. D.
Elmendorf
275-284
Abstract: Let $G$ be a compact Lie group. A canonical method is given for constructing a $ G$-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as $ G$-Eilenberg-Mac Lane spaces and Postnikov towers.
Axioms for Stiefel-Whitney homology classes of some singular spaces
Darko
Veljan
285-305
Abstract: A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds mod$\, 2$ and homology manifolds mod$\, 2$. As a consequence, it is shown that on homology manifolds mod$ \, 2$ the generalized Stiefel conjecture holds.
Semistability at the end of a group extension
Michael L.
Mihalik
307-321
Abstract: A $1$-ended $ {\text{CW}}$-complex, $ Q$, is semistable at $ \infty$ if all proper maps $r: [0,\infty) \to Q$ are properly homotopic. If ${X_1}$ and ${X_2}$ are finite $ {\text{CW}}$-complexes with isomorphic fundamental groups, then the universal cover of ${X_1}$ is semistable at $\infty$ if and only if the universal cover of $ {X_2}$ is semistable at $ \infty$. Hence, the notion of a finitely presented group being semistable at $ \infty$ is well defined. We prove Main Theorem. Let $1 \to H \to G \to K \to 1$ be a short exact sequence of finitely generated infinite groups. If $ G$ is finitely presented, then $G$ is semistable at $\infty$. Theorem. If $ A$ and $ B$ are locally compact, connected noncompact $CW$-complexes, then $A \times B$ is semistable at $ \infty$. Theorem. $\langle\;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle $ is semistable at $ \infty$. The proofs are geometrical in nature and the main tool is covering space theory.
On lexicographically shellable posets
Anders
Björner;
Michelle
Wachs
323-341
Abstract: Lexicographically shellable partially ordered sets are studied. A new recursive formulation of $ {\text{CL}}$-shellability is introduced and exploited. It is shown that face lattices of convex polytopes, totally semimodular posets, posets of injective and normal words and lattices of bilinear forms are $ {\text{CL}}$-shellable. Finally, it is shown that several common operations on graded posets preserve shellability and $ {\text{CL}}$-shellability.
Real-analytic submanifolds which are local uniqueness sets for holomorphic functions of ${\bf C}\sp{3}$
Gary A.
Harris
343-351
Abstract: The following problem is considered. Given a real-analytic two-dimensional submanifold, $M$, of complex Euclidean three-space, are ambient holomorphic functions determined by their values on $M?$ For a large class of submanifolds a necessary and sufficient condition is found for $M$ to be a local uniqueness set for holomorphic functions on complex three-space. Finally, the general problem is shown to be related to two-dimensional Nevanlinna theory.
Conservation laws of free boundary problems and the classification of conservation laws for water waves
Peter J.
Olver
353-380
Abstract: The two-dimensional free boundary problem for incompressible irrotational water waves without surface tension is proved to have exactly eight nontrivial conservation laws. Included is a discussion of what constitutes a conservation law for a general free boundary problem, and a characterization of conservation laws for two-dimensional free boundary problems involving a harmonic potential proved using elementary methods from complex analysis.
Spectra of invariant uniform and transform algebras
I.
Glicksberg
381-396
Abstract: For $G$ a locally compact abelian group, any closed invariant proper subalgebra of ${C_0}(G)$ has analytic discs in its spectrum. Related results are given for $A(G)$ and $B(G)$.
Twisting cochains and duality between minimal algebras and minimal Lie algebras
Richard M.
Hain
397-411
Abstract: An algebraic duality theory is developed between $1$-connected minimal cochain algebras of finite type and connected minimal chain Lie algebras of finite type by means of twisting cochains. The duality theory gives a concrete relationship between Sullivan's minimal models, Chen's power series connections and the various Lie algebra models of a $1$-connected topological space defined by Quillen, Allday, Baues-Lemaire and Neisendorfer. It can be used to compute the Lie algebra model of a space from the algebra model of the space and vice versa.
Decay of Walsh series and dyadic differentiation
William R.
Wade
413-420
Abstract: Let ${W_2}\,n\,[f]$ denote the ${2^n}{\text{th}}$ partial sums of the Walsh-Fourier series of an integrable function $f$. Let $ {\rho _n}(x)$ represent the ratio $ {W_2}n[f,x]/{2^n}$, for $x \in [0,1]$, and let $T(f)$ represent the function ${(\Sigma \rho _n^2)^{1/2}}$. We prove that $T(f)$ belongs to $ {L^p}[0,1]$ for all $0 < p < \infty$. We observe, using inequalities of Paley and Sunouchi, that the operator $f \to T(f)$ arises naturally in connection with dyadic differentiation. Namely, if $f$ is strongly dyadically differentiable (with derivative $\dot Df$) and has average zero on the interval [0, 1], then the ${L^p}$ norms of $f$ and $T(\dot Df)$ are equivalent when $1 < p < \infty$. We improve inequalities implicit in Sunouchi's work for the case $p = 1$ and indicate how they can be used to estimate the ${L^1}$ norm of $ T(\dot Df)$ and the dyadic $ {H^1}$ norm of $ f$ by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if $f$ is strongly dyadically differentiable in dyadic ${H^1}$, then $\int_0^1 {\Sigma _{N = 1}^\infty \vert{W_N}[f,x] - {\sigma _N}[f,x]/N\,dx < \infty}$.
Relative genus theory and the class group of $l$-extensions
Gary
Cornell
421-429
Abstract: The structure of the relative genus field is used to study the class group of relative $l$-extensions. Application to class field towers of cyclic $l$-extensions of the rationals are given.