Graph Products,Fourier Analysis and Spectral Techniques

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products.We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size.Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on generalizing some useful results from the case.     

Composition operators on spaces of entire functions

In this paper we study composition operators on spaces of entire functions. We determine which entire functions induce bounded composition operators on the Paley-Wiener space, , and on the spaces. In addition, we characterize compact composition operators on these spaces. We also study the cyclic properties of composition operators acting on .

     

Doubly stochastic measures with hairpin support

Summary Seethoff and Shiflett [5] proved nice uniqueness theorems concerning doubly stochastic measures supported on the union of the graphs of two functions, but the existence theorems were more elusive. In this present paper, using a functional equations approach, not only uniqueness results but also existence theorems are obtained for doubly stochastic measures with support sets of the form gg^-1 where g is an increasing homeomorphism of [0,1] onto itself such that g(x) whenever 0<x<1.     

A boundary uniqueness theorem for regular functions with a finite “Dirichlet-type” integral

One investigated closed uniqueness sets on U for the class of analytic functions in the unit circle U for which The fundamental result of the paper allows the construction of sufficiently small closed uniqueness sets for the above-specified class of functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 180–190, 1983.This, according to the well-known boundary uniqueness theorem, proves that f(z)=0.     

The Banach envelope of Paley-Wiener type spaces

We give an explicit computation of the Banach envelope for the Paley-Wiener type spaces . This answers a question by Joel Shapiro.

     

Graphs with ∏ 1 0 (K)Y-sections

Summary We prove that a Borel subset of the product of two internal setsX andY all of whoseY-sections are ₁ ⁰ (K)( ₁ ⁰ (K)) sets is the intersection (union) of a countable sequence of Borel graphs with internalY-sections. As a consequence we prove some standard results about the domains of graphs in the product of two topological spaces all of whose horizontal section are compact (open) sets. A version of classical Vitali-Lusin theorem for those types of graphs is given as well as a new proof (and an extension) of a classical result of Kunugui.1980 Mathematics Subject Classification (1985) Revision). Primary 03H05. Secondary 04A15, 54H05Dedicated to the meamory of Goran KrkiResearch supported by the Serbian Science Foundation through a grant from the Mathematical Institute in Belgrade, Yugoslavia     

Algorithmic aspects of intersection graphs and representation hypergraphs

Let be a family of sets. The intersection graph of is obtained by representing each set in by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect. Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure. The grandfather of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.The scope of research that has been going on in this general area extends from the mathematical and algorithmic properties of intersection graphs, to their generalizations and graph parameters motivated by them. In addition, many real-world applications involve the solution of problems on such graphs.In this paper a number of topics in algorithmic combinatorics which involve intersection graphs and their representative families of sets are presented. Recent applications to computer science are also discussed. The intention of this presentation is to provide an understanding of the main research directions which have been investigated and to suggest possible new directions of research.     

Solutions globales de certaines équations de Fuchs non linéaires dans les espaces de Gevrey

We consider nonlinear partial differential equations with several Fuchsian variables of type , where is a Fuchsian principal part of weight zero. We prove existence and uniqueness of a global solution to this problem in the space of holomorphic functions with respect to the Fuchsian variable and in Gevrey spaces with respect to the other variable . The method of proof is based on the application of the fixed point theorem in some Banach algebras defined by majorant functions that are suitable to this kind of equation.

     

Carrier and nerve theorems in the extension theory

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space.

Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric -dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than .

     

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.     

Universal spaces for almost -dimensionality

We find universal functions for the class of lower semi-continuous (LSC) functions with at most -dimensional domain. In an earlier paper we proved that a space is almost -dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most -dimensional domain. We conclude that the class of almost -dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdos space.

     

Composition operators on Banach function spaces

We study the boundedness and the compactness of composition operators on some Banach function spaces such as absolutely continuous Banach function spaces on a -finite measure space, Lorentz function spaces on a -finite measure space and rearrangement invariant spaces on a resonant measure space. In addition, we study some properties of the spectra of a composition operator on the general Banach function spaces.

     

Isometric weighted composition operators on weighted Banach spaces of type

We characterize those weighted composition operators on weighted Banach spaces of holomorphic functions of type which are an isometry.

     

On closed convex sets without boundary rays and asymptotes

We study in finite-dimensional spaces the class of closed convex sets without boundary rays and asymptotes, denoted by and introduced by D. Gale and V. Klee. These sets, not necessarily bounded, enjoy many properties satisfied by compacts sets. New properties of this class are given and convergence analysis of this class is investigated. We also introduce the class of closed convex proper functions which have an epigraph in and we give some properties of these functions.     

Itô maps and analysis on path spaces

We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross–Sobolev spaces of differentiable functions and proving their intertwining with solution maps, , of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given , is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.     

Variational Splines and Paley–Wiener Spaces on Combinatorial Graphs

Notions of interpolating variational splines and Paley–Wiener spaces are introduced on a combinatorial graph G. Both of these definitions explore existence of a combinatorial Laplace operator on G. The existence and uniqueness of interpolating variational splines on a graph is shown. As an application of variational splines, the paper presents a reconstruction algorithm of Paley–Wiener functions on graphs from their uniqueness sets.      

Automatic continuity of linear maps on spaces of continuous functions

Let C(S)and C(T) denote the sup-normed Banach spaces of real- or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map AC(T)C(S) is calledseparating if when two functions x and y from C(T) have disjoint cozero sets then so do Ax and Ay. In the spirit of [3] and [4], we show that separating maps are automatically continuous in some important cases (Theorems 2.4 and 2.5). If a separating map is continuous, then it must be a continuous multiple of a composition map (Theorem 2.2). If A is injective, separating and detaching (Def. 2.4) then S and T are homeomorphic (Theorem 2.1).     

Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations

In this article, we describe spaces P such that: if u is a weak (in the sense of Leray [J. Leray, Sur le mouvement d'un fluide visqueux remplissant l'espace, Acta Math. 63 (1934) 193-248]) solution of the Navier-Stokes system for some initial data u₀, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional , where α, β belong to [0,1]. In order to find optimal conditions for the boundedness of F, we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [J. Serrin, The initial value problem for the Navier-Stokes equations, in: R.E. Langer (Ed.), Nonlinear Problems, Univ. of Wisconsin Press, 1963, pp. 69-98], to the recent conditions formulated by Lemarié-Rieusset [P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall, 2003].     

A generalization of the prolate spheroidal wave functions

Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of and and the Jacobi polynomials which are an orthogonal basis of a weighted The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both and a subspace of known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property.

The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.