Regularity and Algebras of Analytic Functions in Infinite Dimensions

A Banach space is known to be Arens regular if every continuous linear mapping from to is weakly compact. Let be an open subset of , and let denote the algebra of analytic functions on which are bounded on bounded subsets of lying at a positive distance from the boundary of We endow with the usual Fréchet topology. denotes the set of continuous homomorphisms . We study the relation between the Arens regularity of the space and the structure of .

     

Distinguished representations and quadratic base change for

Let be a quadratic extension of number fields. Suppose that every real place of splits in and let be the unitary group in 3 variables. Suppose that is an automorphic cuspidal representation of . We prove that there is a form in the space of such that the integral of over is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.

     

Uniform harmonic approximation of bounded functions

Let be an open set in and be a relatively closed subset of . We characterize those pairs which have the following property: every function which is bounded and continuous on and harmonic on can be uniformly approximated by functions harmonic on . Several related results concerning both harmonic and superharmonic approximation are also established.

     

Topological centers of certain dual algebras

Let be a Banach algebra with a bounded approximate identity. Let and be, respectively, the topological centers of the algebras and . In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras and , we study the sets , , the relations between them and with several other subspaces of or .

     

Duality and Polynomial Testing of Tree Homomorphisms

Let be a fixed digraph. We consider the -colouring problem, i.e., the problem of deciding which digraphs admit a homomorphism to . We are interested in a characterization in terms of the absence in of certain tree-like obstructions. Specifically, we say that has tree duality if, for all digraphs , is not homomorphic to if and only if there is an oriented tree which is homomorphic to but not to . We prove that if has tree duality then the -colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the -property studied by Gutjahr, Welzl, and Woeginger.

We then focus on the case when itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads for which the -colouring problem is -complete. We contrast these with several families of oriented triads which have tree duality, or bounded treewidth duality, and hence polynomial -colouring problems. If , then no oriented triad with an -complete -colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad . We prove that none of the oriented triads with -complete -colouring problems given in the companion paper has tree duality.

     

On the ordering of -modal cycles

The forcing relation on -modal cycles is studied. If is an -modal cycle then the -modal cycles with block structure that force form a -horseshoe above . If -modal forces , and does not have a block structure over , then forces a -horseshoe of simple extensions of .

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

A Concordance Extension Theorem

Let be a manifold approximate fibration between closed manifolds, where , and let be the mapping cylinder of . In this paper it is shown that if is any concordance on , then there exists a concordance such that and . As an application, if and are closed manifolds where is a locally flat submanifold of and and , then a concordance extends to a concordance on such that . This uses the fact that under these hypotheses there exists a manifold approximate fibration , where is a closed -manifold, such that the mapping cylinder is homeomorphic to a closed neighborhood of in by a homeomorphism which is the identity on .

     

On quadratic forms of height two and a theorem of Wadsworth

Let and be anisotropic quadratic forms over a field of characteristic not . Their function fields and are said to be equivalent (over ) if and are isotropic. We consider the case where and is divisible by an -fold Pfister form. We determine those forms for which becomes isotropic over if , and provide partial results for . These results imply that if and are equivalent and , then is similar to over . This together with already known results yields that if is of height and degree or , and if , then and are equivalent iff and are isomorphic over .

     

Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions

A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in , the Jack parameter shifted by . More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families.

The coefficients of a second series, essentially the logarithm of the first, specialize at values and of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in , and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.

     

Powers in Finitely Generated Groups

In this paper we study the set of -powers in certain finitely generated groups . We show that, if is soluble or linear, and contains a finite index subgroup, then is nilpotent-by-finite. We also show that, if is linear and has finite index (i.e. may be covered by finitely many translations of ), then is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.

     

Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to

     

On extension of cocycles to normalizer elements, outer conjugacy, and related problems

Let be an ergodic automorphism of a Lebesgue space and a cocycle of with values in an Abelian locally compact group . An automorphism from the normalizer of the full group is said to be compatible with if there is a measurable function such that at a.e. . The topology on the set of all automorphisms compatible with is introduced in such a way that becomes a Polish group. A complete system of invariants for the -outer conjugacy (i.e. the conjugacy in the quotient group is found. Structure of the cocycles compatible with every element of is described.

     

Iterated Spectra of Numbers---Elementary, Dynamical, and Algebraic Approaches

sets and central sets are subsets of which arise out of applications of topological dynamics to number theory and are known to have rich combinatorial structure. Spectra of numbers are often studied sets of the form . Iterated spectra are similarly defined with coming from another spectrum. Using elementary, dynamical, and algebraic approaches we show that iterated spectra have significantly richer combinatorial structure than was previously known. For example we show that if and , then is an set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition.

     

Linear additive functionals of superdiffusions and related nonlinear P.D.E.

Let be a second order elliptic differential operator in a bounded smooth domain in and let . We get necessary and sufficient conditions on measures under which there exists a positive solution of the boundary value problem

The conditions are stated both analytically (in terms of capacities related to the Green's and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called -superdiffusions).

We also investigate a closely related subject --- linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain in , we establish a 1-1 correspondence between a class of such functionals and a class of -excessive functions (which we describe in terms of their Martin integral representation). The Laplace transform of satisfies an integral equation which can be considered as a substitute for (*).

     

The Bergman kernel function of some Reinhardt domains

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points . Let be the Reinhardt domain

where , ; and let be the Bergman kernel function of . Then there exist two positive constants and and a function such that

holds for every . Here

and is the defining function for . The constants and depend only on and , not on .

     

Nonnegative Radix Representations for the Orthant

Let be a nonnegative real matrix which is expanding, i.e. with all eigenvalues , and suppose that is an integer. Let consist of exactly nonnegative vectors in . We classify all pairs such that every in the orthant has at least one radix expansion in base using digits in . The matrix must be a diagonal matrix times a permutation matrix. In addition must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set can be diagonally scaled to lie in . The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.

     

Finite-dimensional lattice-subspaces of

Let be linearly independent positive functions in , let be the vector subspace generated by the and let denote the curve of determined by the function , where . We establish that is a vector lattice under the induced ordering from if and only if there exists a convex polygon of with vertices containing the curve and having its vertices in the closure of the range of . We also present an algorithm which determines whether or not is a vector lattice and in case is a vector lattice it constructs a positive basis of . The results are also shown to be valid for general normed vector lattices.

     

Functorial structure of units in a tensor product

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let be an algebraically closed field. Let and be reduced rings containing , having connected spectra. Let be a unit. Then for some units and .

Here is a deeper consequence, stated for simplicity in the affine case only. Let be a field, and let be a homomorphism of finitely generated -algebras such that is dominant. Assume that every irreducible component of or is geometrically integral and has a rational point. Let be a faithfully flat homomorphism of reduced -algebras. For a -algebra, define to be . Then satisfies the following sheaf property: the sequence

is exact. This and another result are used to prove (5.2) of [7].

     

Gross spaces

A Gross space is a vector space of infinite dimension over some field , which is endowed with a symmetric bilinear form and has the property that every infinite dimensional subspace satisfies dim dim. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals , a Gross space in dimension exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.