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.

     

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.

     

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.

     

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.

     

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 sectional genus of quasi-polarized 3-folds

Let be a smooth projective variety over and a nef-big (resp. ample) divisor on . Then is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that , where is the sectional genus of and is the irregularity of . In general it is unknown whether this conjecture is true or not, even in the case of . For example, this conjecture is true if and . But it is unknown if and . In this paper, we prove if and . Furthermore we classify polarized manifolds with , , and .

     

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'.

     

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.

     

On the coefficients of Jacobi sums in prime cyclotomic fields

Let and be prime numbers, and let be a primitive root mod . For , denote by the Jacobi sum . We study the integers such that and . We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of , in particular to the study of Vandiver's conjecture. For , let be the number of distinct roots of in . We show that . We give closed formulas for the numbers and in terms of quadratic and cubic power residue symbols mod .

     

Exact Hausdorff measure and intervals of maximum density for Cantor sets

Consider a linear Cantor set , which is the attractor of a linear iterated function system (i.f.s.) , , on the line satisfying the open set condition (where the open set is an interval). It is known that has Hausdorff dimension given by the equation , and that is finite and positive, where denotes Hausdorff measure of dimension . We give an algorithm for computing exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When (or more generally, if and are commensurable), the algorithm also gives an interval that maximizes the density . The Hausdorff measure is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters , it is possible to choose the translation parameters in such a way that , so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.

     

Cohomology of uniformly powerful -groups

In this paper we will study the cohomology of a family of -groups associated to -Lie algebras. More precisely, we study a category of -groups which will be equivalent to the category of -bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group in this category, its -cohomology is that of an elementary abelian -group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of in the case that is associated to a Lie algebra . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups which are defined to be the kernel of the mod reduction

     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures

We consider the set of diffeomorphisms of the 2-torus , provided the conditions that the tangent bundle splits into the directed sum of -invariant subbundles , and there is such that and . Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the set has no SBR measures. This is related to a result of Hu-Young.

     

Galois coverings of selfinjective algebras by repetitive algebras

In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form where is the repetitive algebra of an artin algebra and is an admissible group of automorphisms of . If is of finite global dimension, then the stable module category of finitely generated -modules is equivalent to the derived category of bounded complexes of finitely generated -modules. For a selfinjective artin algebra , an ideal and , we establish a criterion for to admit a Galois covering with an infinite cyclic Galois group . As an application we prove that all selfinjective artin algebras whose Auslander-Reiten quiver has a non-periodic generalized standard translation subquiver closed under successors in are socle equivalent to the algebras , where is a representation-infinite tilted algebra and is an infinite cyclic group of automorphisms of .

     

Nullification and cellularization of classifying spaces of finite groups

In this note we discuss the effect of the -nullification and the -cellularization over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe by means of a covering fibration, and we classify all finite groups for which is -cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of and

     

Classes of singular integrals along curves and surfaces

This paper is concerned with singular convolution operators in , , with convolution kernels supported on radial surfaces . We show that if , then boundedness holds if and only if . This statement can be reduced to a similar statement about the multiplier in . We also construct smooth for which the corresponding operators are bounded for but unbounded for , for given . Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.

     

Golubev series for solutions of elliptic equations

Let be an elliptic system with real analytic coefficients on an open set and let be a fundamental solution of Given a locally connected closed set we fix some massive measure on . Here, a non-negative measure is called massive, if the conditions and imply that We prove that, if is a solution of the equation in then for each relatively compact open subset of and every there exist a solution of the equation in and a sequence () in satisfying such that for This complements an earlier result of the second author on representation of solutions outside a compact subset of