Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection of completely regular Hausdorff spaces and , the Stone-Cech compactification functor yields a surjection . For an -fold covering map , we show that the fibres of , while never containing more than points, may degenerate to sets of cardinality properly dividing . In the special case of the universal bundle of a -group , we show more precisely that every possible type of -orbit occurs among the fibres of . To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

     

Sufficient conditions for zero-one laws

We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size is bounded above by a polynomial in , then has a monadic second order - law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most is bounded above by a polynomial in , then this class has a first order - law. These results cover all known natural examples of classes of structures that have been proved to have a logical - law by Compton's method of analyzing generating functions.

     

Uniform and Lipschitz homotopy classes of maps

If is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line into an element of where is the space of uniformly continuous functions from to and is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to is surjective. If is the -dimensional torus, it is bijective, while if is a compact orientable surface of genus 1$">, it is not injective.

In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds to compact smooth manifolds With each such Lipschitz homotopy class we associate an element of where is the dimension of is the space of bounded continuous functions from the positive real axis to and is the set of all such that

     

On the Glauberman and Watanabe correspondences for blocks of finite -solvable groups

If is a finite -solvable group for some prime , a solvable subgroup of the automorphism group of of order prime to such that stabilises a -block of and acts trivially on a defect group of , then there is a Morita equivalence between the block and its Watanabe correspondent of , given by a bimodule with vertex and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).

     

Sums of squares in real analytic rings

Let be an analytic ring. We show: (1) has finite Pythagoras number if and only if its real dimension is , and (2) if every positive semidefinite element of is a sum of squares, then is real and has real dimension .

     

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for the identity map id: is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a -concave symmetric Banach sequence space the identity map is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in . Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator on with values in a -concave symmetric Banach sequence space is a multiplier from into . Furthermore, we prove an asymptotic formula for the -th approximation number of the identity map , where denotes the linear span of the first standard unit vectors in , and apply it to Lorentz and Orlicz sequence spaces.

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

Topological dynamics on moduli spaces II

Let be an orientable genus 0$"> surface with boundary . Let be the mapping class group of fixing . The group acts on the space of -gauge equivalence classes of flat -connections on with fixed holonomy on . We study the topological dynamics of the -action and give conditions for the individual -orbits to be dense in .

     

Invariant ideals and polynomial forms

Let denote the group algebra of an infinite locally finite group . In recent years, the lattice of ideals of has been extensively studied under the assumption that is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra stable under the action of an appropriate automorphism group of . Specifically, in this paper, we let be a quasi-simple group of Lie type defined over an infinite locally finite field , and we let be a finite-dimensional vector space over a field of the same characteristic . If acts nontrivially on by way of the homomorphism , and if has no proper -stable subgroups, then we show that the augmentation ideal is the unique proper -stable ideal of when . The proof of this result requires, among other things, that we study characteristic division rings , certain multiplicative subgroups of , and the action of on the group algebra , where is the additive group . In particular, properties of the quasi-simple group come into play only in the final section of this paper.

     

Trees and branches in Banach spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree of a certain type on a space is presumed to have a branch with some property. It is shown that then can be embedded into a space with an FDD so that all normalized sequences in which are almost a skipped blocking of have that property. As an application of our work we prove that if is a separable reflexive Banach space and for some and every weakly null tree on the sphere of has a branch -equivalent to the unit vector basis of , then for all 0$">, there exists a subspace of having finite codimension which embeds into the sum of finite dimensional spaces.

     

Ribbon tilings and multidimensional height functions

We fix and say a square in the two-dimensional grid indexed by has color if . A ribbon tile of order is a connected polyomino containing exactly one square of each color. We show that the set of order- ribbon tilings of a simply connected region is in one-to-one correspondence with a set of height functions from the vertices of to satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph.

Using these facts, we describe a linear (in the area of ) algorithm for determining whether can be tiled with ribbon tiles of order and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order- ribbon tilings of can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order- ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

     

Detection of renewal system factors via the Conley index

Let be an isolating neighborhood for a map . If we can decompose into the disjoint union of compact sets and , then we can relate the dynamics on the maximal invariant set to the shift on two symbols by noting which component of each iterate of a point lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to . In essence, we measure the difference between the Conley index of and the sum of the indices of and .

     

Spin structures and codimension two embeddings of -manifolds up to regular homotopy

We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into -space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of -manifolds , the following phenomenon occurs: there exists a codimension two immersion of the -sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of . This involves introducing and studying an equivalence relation on the set of spin structures on . Their associated -invariants also play an important role.

     

Self-tilting complexes yield unstable modules

Let be a group and a commutative ring. Let be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of -modules. The subgroup of consisting of self-equivalences fixing the trivial -module acts on the cohomology ring . The action is functorial with respect to . The self-equivalences which are 'splendid' in a sense defined by J. Rickard act naturally with respect to transfer and restriction to centralizers of -subgroups in case is a field of characteristic . In the present paper we prove that this action of self-equivalences on commutes with the action of the Steenrod algebra, and study the behaviour of the action of splendid self-equivalences with respect to Lannes' -functor.

     

Thick points for intersections of planar sample paths

Let denote the number of visits to of the simple planar random walk , up to step . Let be another simple planar random walk independent of . We show that for any , there are points for which . This is the discrete counterpart of our main result, that for any , the Hausdorff dimension of the set of thick intersection points for which , is almost surely . Here is the projected intersection local time measure of the disc of radius centered at for two independent planar Brownian motions run until time . The proofs rely on a ``multi-scale refinement' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius centered at by for general sets .