Depth and cohomological connectivity in modular invariant theory

Let be a finite group acting linearly on a finite-dimensional vector space over a field of characteristic . Assume that  divides the order of so that is a modular representation and let be a Sylow -subgroup for . Define the cohomological connectivity of the symmetric algebra to be the smallest positive integer such that . We show that is a lower bound for the depth of . We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if is -nilpotent and is cyclic, then, for any modular representation, the depth of is .

     

Glauberman-Watanabe corresponding -blocks of finite groups with normal defect groups are Morita equivalent

Let be a finite group and let be a solvable finite group that acts on such that the orders of and are relatively prime. Let be a -block of with normal defect group such that stabilizes and . Then there is a Morita equivalence between the block and its Watanabe correspondent block of given by a bimodule with vertex and trivial source that on the character level induces the Glauberman correspondence (and which is an isotypy by a theorem of Watanabe).

     

Fixed point index in symmetric products

Let be an open subset of a locally compact metric ANR and let be a continuous map. In this paper we study the fixed point index of the map that induces in the -symmetric product of , . This index can detect the existence of periodic orbits of period of , and it can be used to obtain the Euler characteristic of the -symmetric product of a manifold , . We compute for all orientable compact surfaces without boundary.

     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

One-dimensional dynamical systems and Benford's law

Near a stable fixed point at 0 or , many real-valued dynamical systems follow Benford's law: under iteration of a map the proportion of values in with mantissa (base ) less than tends to for all in as , for all integer bases 1$">. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford's law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford's distribution occurs for every , but for essentially nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as , where is with F'(0)$">, also follow Benford's law. Besides generalizing many well-known results for sequences such as or the Fibonacci numbers, these findings supplement recent observations in physical experiments and numerical simulations of dynamical systems.

     

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let be the blow-up of a projective scheme along the ideal sheaf of . It is known that there are embeddings for , where denotes the maximal generating degree of , and that there exists a Cohen-Macaulay ring of the form (which gives an arithmetic Macaulayfication of ) if and only if , for , and is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants and such that is Cohen-Macaulay for all d(I)e + \varepsilon$"> and e_0$">, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form . If has negative -invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if , for 0$">, and is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of for all d(I)e + \varepsilon$"> and e_0$">.

     

On structurally stable diffeomorphisms with codimension one expanding attractors

We show that if a closed -manifold admits a structurally stable diffeomorphism with an orientable expanding attractor of codimension one, then is homotopy equivalent to the -torus and is homeomorphic to for . Moreover, there are no nontrivial basic sets of different from . This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on , .

     

Gröbner bases of associative algebras and the Hochschild cohomology

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let be a finitely generated algebra over a commutative ring with a (possibly infinite) Gröbner base on a free algebra , that is, is the quotient with the ideal of generated by . Given a Gröbner base for an -subbimodule of the free -bimodule generated by a set , we have a morphism of -bimodules from the free -bimodule generated by to sending the generator to the element . We construct a Gröbner base on for the -subbimodule Ker() of , and with this we have the free -bimodule generated by and an exact sequence . Applying this construction inductively to the -bimodule itself, we have a free -bimodule resolution of .

     

Anosov automorphisms on compact nilmanifolds associated with graphs

We associate with each graph a -step simply connected nilpotent Lie group and a lattice in . We determine the group of Lie automorphisms of and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every there exist a -dimensional -step simply connected nilpotent Lie group which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice in such that admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

     

Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and 0$">. Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of .

     

Boundary Hölder and estimates for local solutions of the tangential Cauchy-Riemann equation

We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood of a point when is a generic -concave manifold of real codimension in , where . Our method is to first derive a homotopy formula for in when is the intersection of with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any -closed form on without shrinking. We obtain Hölder and estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage d'un point d'une sous-variété générique -concave de codimension quelconque de . Nous construisons une formule d'homotopie pour le sur , lorsque est l'intersection de et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme -fermée sur . Nous en déduisons des estimations et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur .

     

Extension-orthogonal components of preprojective varieties

Let be a Dynkin quiver, and let be the corresponding preprojective algebra. Let be a set of pairwise different indecomposable irreducible components of varieties of -modules such that generically there are no extensions between and for all . We show that the number of elements in is at most the number of positive roots of . Furthermore, we give a module-theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.

     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.

     

Cremer fixed points and small cycles

Let , , and let denote the sequence of convergents to the regular continued fraction of . Let be a function holomorphic at the origin, with a power series of the form . We assume that for infinitely many we simultaneously have (i) , (ii) the coefficients stay outside two small disks, and (iii) the series is lacunary, with for . We then prove that has infinitely many periodic orbits in every neighborhood of the origin.

     

Weakly compact approximation in Banach spaces

The Banach space has the weakly compact approximation property (W.A.P. for short) if there is a constant so that for any weakly compact set and 0$">there is a weakly compact operator satisfying and . We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space ) have the W.A.P, but that James' tree space fails to have the W.A.P. It is also shown that the dual has the W.A.P. It follows that the Banach algebras and , consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space so that fails to have the W.A.P., but has this approximation property without the uniform bound .

     

Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups

We consider a locally compact group and a limiting measure of a commutative infinitesimal triangular system (c.i.t.s.) of probability measures on . We show, under some restrictions on , or , that belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. on any locally compact group . It is also valid for a limiting measure which has `full' support on a Zariski connected -algebraic group , where is a local field, and any one of the following conditions is satisfied: (1) is a compact extension of a closed solvable normal subgroup, in particular, is amenable, (2) has finite one-moment or (3) has density and in case the characteristic of is positive, the radical of is -defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group , we show that it is always positive for any probability measure on , and it is also multiplicative in case of symmetric commuting measures.

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

Cut numbers of -manifolds

We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

     

A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra

Let and be closed subgroups of the extended Morava stabilizer group and suppose that is normal in . We construct a strongly convergent spectral sequence

where and are the continuous homotopy fixed point spectra of Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in the category of -local -modules.

     

Convergence of double Fourier series and -classes

The double Fourier series of functions of the generalized bounded variation class are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class defined here, have quadrant limits at every point and, for there exist at most countable sets and such that, for and is continuous at . It is shown that the previously studied class contains essentially discontinuous functions unless the sequence satisfies a strong condition.