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.

     

Boundary and Lens Rigidity of Lorentzian Surfaces

Let be a Lorentzian metric on the plane that agrees with the standard metric outside a compact set and so that there are no conjugate points along any time-like geodesic of . Then and are isometric. Further, if and are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of maximize the distances between their points and and are ``boundary isometric', then there is a conformal diffeomorphism between and and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

     

Packing dimension and Cartesian products

We show that for any analytic set in , its packing dimension can be represented as , where the supremum is over all compact sets in , and denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if . In contrast, we show that the dual quantity , is at least the ``lower packing dimension' of , but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)

     

Cesàro Summability of Two-dimensional Walsh-Fourier Series

We introduce p-quasi-local operators and the two-dimensional
dyadic Hardy spaces defined by the dyadic squares. It is proved that, if a sublinear operator is p-quasi-local and bounded from to , then it is also bounded from to . As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from to and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from to for every .

     

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.

     

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 .

     

Relatively free invariant algebras of finite reflection groups

Let be a finite subgroup of is a field of characteristic and acting by linear substitution on a relatively free algebra of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if is a pseudo-reflection group and contains the polynomial

     

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 .

     

Real connective K-theory and the quaternion group

Let be the real connective K-theory spectrum. We compute and for groups whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients of the fixed points of the Tate spectrum for and . The results provide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.

     

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 .

     

Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities

We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation , as or where is a smooth function defined on a open interval The hypotheses we assume on the nonlinearity allow us to cover the case (or ) and having superlinear growth at infinity, as well as the case (or ) and having a singularity in (respectively in ). Applications are given also to situations like (including the so-called ``jumping nonlinearities'). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.

     

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 .

     

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.

     

An Application of Convex Integration to Contact Geometry

We prove that every closed, orientable -manifold admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov's convex integration technique and the -principle. Similar methods can be used to show that admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If is a closed -manifold with contact form whose contact distribution admits everywhere linearly independent sections, then admits linearly independent contact forms with linearly independent Reeb vector fields.

     

C*-Algebras with the Approximate Positive Factorization Property

We say that a unital -algebra has the approximate positive factorization property (APFP) if every element of is a norm limit of products of positive elements of . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial , and stable rank 1. In the unital case, the group separates the tracial states. The APFP passes to matrix algebras, and if is an ideal in such that and have the APFP, then so does . We also give some new examples of -algebras with the APFP, including type factors and infinite-dimensional simple unital direct limits of homogeneous -algebras with slow dimension growth, real rank zero, and trivial group. Simple direct limits of homogeneous -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a -algebra that may be of interest in the future.

     

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.