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.

     

Properties of extremal functions for some nonlinear functionals on Dirichlet spaces

Let be the set of holomorphic functions on the unit disc with and Dirichlet integral not exceeding one, and let be the set of complex-valued harmonic functions on the unit disc with and Dirichlet integral not exceeding one. For a (semi)continuous function , define the nonlinear functional on or by . We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

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 .

     

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 .

     

Representations of monoids arising from finite groups of Lie type

A class of finite monoids constructed from a group of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of . The sandwich matrix of is decomposed into a product of matrices corresponding to maximal parabolic subgroups of .

     

Extremal functions for Moser's inequality

Let be a bounded smooth domain in , and a function with compact support in . Moser's inequality states that there is a constant , depending only on the dimension , such that

where is the Lebesgue measure of , and the surface area of the unit ball in . We prove in this paper that there are extremal functions for this inequality. In other words, we show that the

is attained. Earlier results include Carleson-Chang (1986, is a ball in any dimension) and Flucher (1992, is any domain in 2-dimensions).

     

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.

     

Lévy group action and invariant measures on

For let be defined by . We investigate permutations of , which satisfy as with for (i.e. is in the Lévy group , or for in the subspace of Cesàro-summable sequences. Our main interest are -invariant means on or equivalently -invariant probability measures on . We show that the adjoint of maps measures supported in onto a weak*-dense subset of the space of -invariant measures. We investigate the dynamical system and show that the support set of invariant measures on is the closure of the set of almost periodic points and the set of non-topologically transitive points in . Finally we consider measures which are invariant under .

     

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.

     

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 .

     

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.

     

New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces

Let be a real Banach space and a bounded, open and convex subset of The solvability of the fixed point problem in is considered, where is a possibly discontinuous -dissipative operator and is completely continuous. It is assumed that is uniformly convex, and A result of Browder, concerning single-valued operators that are either uniformly continuous or continuous with uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let The effect of a weak boundary condition of the type on the range of operators is studied for -accretive and maximal monotone operators Here, with sufficiently large norm and Various new eigenvalue results are given involving the solvability of with respect to Several results do not require the continuity of the operator Four open problems are also given, the solution of which would improve upon certain results of the paper.

     

Divisors on Generic Complete Intersections in Projective Space

Let be a generic complete intersection of hypersurfaces of degree in -dimensional projective space. We study the question when a divisor on is nonrational or of general type, and give an alternative proof of a result of Ein. We also give some improvement of Ein's result in the case .

     

Regularity theory and traces of -harmonic functions

In this paper we discuss two different topics concerning -
harmonic functions. These are weak solutions of the partial differential equation

where for some fixed , the function is bounded and for a.e. . First, we present a new approach to the regularity of -harmonic functions for . Secondly, we establish results on the existence of nontangential limits for -harmonic functions in the Sobolev space , for some , where is the unit ball in . Here is allowed to be different from .

     

Tensor products over abelian Walgebras

Tensor products of Calgebras over an abelian Walgebra are studied. The minimal Cnorm on is shown to be just the quotient of the minimal Cnorm on if or is exact.

     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure 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.

     

On the group of homotopy equivalences of a manifold

We consider the group of homotopy equivalences of a simply connected manifold which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group for special manifolds , for example if is a connected sum of products of spheres. In particular the group is determined completely. Also the connection of with the group of isotopy classes of diffeomorphisms of is studied.