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).

     

Operators on spaces preserving copies of Schreier spaces

It is proved that an operator , compact metrizable, a separable Banach space, for which the -Szlenk index of is greater than or equal to , , is an isomorphism on a subspace of isomorphic to , the Schreier space of order . As a corollary, one obtains that a complemented subspace of with Szlenk index equal to contains a subspace isomorphic to .

     

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.

     

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.

     

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.

     

Stable and finite Morse index solutions on or on bounded domains with small diffusion

In this paper, we study bounded solutions of on (where and sometimes ) and show that, for most 's, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of on with Dirichlet or Neumann boundary conditions for small .

     

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.

     

Maximal holonomy of infra-nilmanifolds with -dimensional quaternionic Heisenberg geometry

Let be the quaternionic Heisenberg group of real dimension and let denote the maximal order of the holonomy groups of all infra-nilmanifolds with -geometry. We prove that . As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a -dimensional quaternionic hyperbolic manifold with cusps is at least

     

Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

The -invariant on certain surfaces with symmetry groups

The global holomorphic -invariant introduced by Tian is closely related to the existence of Kähler-Einstein metrics. We apply the result of Tian, Yau and Zelditch on polarized Kähler metrics to approximate plurisubharmonic functions and compute the -invariant on for .

     

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 .

     

A sharp weak type inequality for martingale transforms and other subordinate martingales

If is a martingale difference sequence, a sequence of numbers in , and a positive integer, then

Here denotes the best constant. If , then as was shown by Burkholder. We show here that for the case 2$">, and that is also the best constant in the analogous inequality for two martingales and indexed by , right continuous with limits from the left, adapted to the same filtration, and such that is nonnegative and nondecreasing in . In Section 7, we prove a similar inequality for harmonic functions.

     

Harmonic maps and 2-cycles, realizing the Thurston norm

Let be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, , of a harmonic map with Morse-type singularities delivers the Thurston norm of its homology class .

In particular, for a map with connected fibers and any well-positioned oriented surface in the homology class of a fiber, we show that the Thurston number satisfies an inequality

Here the variation is can be expressed in terms of the -invariants of the fiber components, and the twist measures the complexity of the intersection of with a particular set of ``bad" fiber components. This complexity is tightly linked with the optimal ``-height" of , being lifted to the -induced cyclic cover .

Based on these invariants, for any Morse map , we introduce the notion of its twist . We prove that, for a harmonic , if and only if .

     

Hardy space of exact forms on

We show that the Hardy space of divergence-free vector fields on has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of . Using the duality result we prove a ``div-curl" type theorem: for in , is equivalent to a -type norm of , where the supremum is taken over all with This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on , study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on .

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

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.

     

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 .

     

Degeneration of linear systems through fat points on surfaces

In this paper we introduce a technique to degenerate surfaces and linear systems through fat points in general position on surfaces. Using this degeneration we show that on generic surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through fat points in general position on a general quartic surface in .

     

Generalized interpolation in with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for functions on the unit disc to problems concerning lifting onto of an operator that is defined on ( is an inner function) and commutes with the (compressed) shift . In particular, he showed that interpolants (i.e., such that ) having norm equal to exist, and that in certain cases such an is unique and can be expressed as a fraction with . In this paper, we study interpolants that are such fractions of functions and are bounded in norm by (assuming that , in which case they always exist). We parameterize the collection of all such pairs and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

     

Semi-linear homology -spheres and their equivariant inertia groups

This paper introduces an abelian group for all semi-linear homology -spheres, which corresponds to a known abelian group for all semi-linear homotopy -spheres, where is a compact Lie group and is a -representation with 0$">. Then using equivariant surgery techniques, we study the relation between both and when is finite. The main result is that under the conditions that -action is semi-free and with 0$">, the homomorphism defined by is an isomorphism if , and a monomorphism if . This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology -spheres.