Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

A sharp version of Zhang's theorem on truncating sequences of gradients

Let be a compact and convex set of matrices and let be a sequence in that converges to in the mean, i.e. . I show that there exists a sequence of Lipschitz functions such that and . This refines a result of Kewei Zhang (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 313-326), who showed that one may assume . Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. Rational Mech. Anal. 115 (1991), 329-365) regarding the approximation of valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of can be replaced by quasiconvexity.

     

Capacity convergence results and applications to a Berstein-Markov inequality

Given a sequence of Borel subsets of a given non-pluripolar Borel set in the unit ball in with , we show that the relative capacities converge to if and only if the relative (global) extremal functions () converge pointwise to (). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support in guaranteeing that the pair satisfy a Bernstein-Markov inequality. This implies that the orthonormal polynomials associated to may be used to recover the global extremal function .

     

The ideal structure of some analytic crossed products

We study the ideal structure of a class of some analytic crossed products. For an -discrete, principal, minimal groupoid , we consider the analytic crossed product , where is given by a cocycle . We show that the maximal ideal space of depends on the asymptotic range of , ; that is, is homeomorphic to for finite, and consists of the unique maximal ideal for . We also prove that is semisimple in both cases, and that is invariant under isometric isomorphism.

     

Towards a Halphen theory of linear series on curves

A linear series on a curve is primary when it does not contain the series cut by planes. For such series, we provide a lower bound for the degree , in terms of deg(), g() and of the number . Examples show that the bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered.

     

Summability of Fourier orthogonal series for Jacobi weight on a ball in

Fourier orthogonal series with respect to the weight function
on the unit ball in are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to is uniformly summable on the ball if and only if .

     

On the depth of the tangent cone and the growth of the Hilbert function

For a dimensional Cohen-Macaulay local ring we study the depth of the associated graded ring of with respect to an -primary ideal in terms of the Vallabrega-Valla conditions and the length of , where is a minimal reduction of and . As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to -primary ideals. We also study the growth of the Hilbert function.

     

Discrete threshold growth dynamics are omnivorous for box neighborhoods

In the discrete threshold model for crystal growth in the plane we begin with some set of seed crystals and observe crystal growth over time by generating a sequence of subsets of by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site's prescribed neighborhood. The growth dynamics generated by this model are said to be omnivorous if finite and imply . In this paper we prove that the dynamics are omnivorous when the neighborhood is a box (i.e. when, for some fixed , the neighborhood of is . This result has important implications in the study of the first passage time when is chosen randomly with a sparse Bernoulli density and in the study of the limiting shape to which converges.

     

Dehn surgery on arborescent links

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link is sufficiently complicated, in the sense that it is composed of at least rational tangles with all , and none of its length 2 tangles are of the form , then all complete surgeries on produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let be a tangle with a closed circle, and let . We will show that if and mod , then remains incompressible after all nontrivial surgeries on . Two bridge links are a subclass of arborescent links. For such a link , most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless has a partial fraction decomposition of the form , in which case it does admit non-laminar surgeries.

     

Monomial Bases for

We consider the polynomial algebra as a module over the mod Steenrod algebra, , being an odd prime. We give a minimal set of generators consisting of monomials and characterise all such `monomial bases'.