-Betti numbers

A standing conjecture in -cohomology says that every finite -complex is of -determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing, e.g., all extensions of residually finite groups with amenable quotients, all residually amenable groups, and free products of these. If, in addition, is -acyclic, we also show that the -determinant is a homotopy invariant -- giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for -Betti numbers.

     

On modules of finite upper rank

For a group and a prime , the upper -rank of is the supremum of the sectional -ranks of all finite quotients of . It is unknown whether, for a finitely generated group , these numbers can be finite but unbounded as ranges over all primes. The conjecture that this cannot happen if is soluble is reduced to an analogous `relative' conjecture about the upper -ranks of a `quasi-finitely-generated' module for a soluble minimax group . The main result establishes a special case of this relative conjecture, namely when the module is finitely generated and the minimax group is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)     

Local derivations on -algebras are derivations

Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.

     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

Berezin transform on real bounded symmetric domains

Let be a bounded symmetric domain in a complex vector space with a real form and be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.

     

Groups with two extreme character degrees and their normal subgroups

We study the finite groups for which the set of irreducible complex character degrees consists of the two most extreme possible values, that is, and . We are easily reduced to finite -groups, for which we derive the following group theoretical characterization: they are the -groups such that is a square and whose only normal subgroups are those containing or contained in . By analogy, we also deal with -groups such that is not a square, and we prove that if and only if a similar property holds: for any , either or . The proof of these results requires a detailed analysis of the structure of the -groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than , then the index of the centre is small, and in some cases we may even bound the order of .

     

Sharp Sobolev inequalities with lower order remainder terms

Given a smooth compact Riemannian -manifold , this paper deals with the sharp Sobolev inequality corresponding to the embedding of in where the remainder term is replaced by a lower order term.

     

Good ideals in Gorenstein local rings

Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.

     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.

     

A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of

For a domain contained in a hemisphere of the -dimensional sphere we prove the optimal result for the ratio of its first two Dirichlet eigenvalues where , the symmetric rearrangement of in , is a geodesic ball in having the same -volume as . We also show that for geodesic balls of geodesic radius less than or equal to is an increasing function of which runs between the value for (this is the Euclidean value) and for . Here denotes the th positive zero of the Bessel function . This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of and having a fixed value of the one with the maximal value of is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for . Various other results for and of geodesic balls in are proved in the course of our work.

     

-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations

We develop a general -framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach. We apply our -framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an error estimate for an upwind-central finite difference scheme.

     

On the invariant faces associated with a cone-preserving map

For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.

     

On the number of -equivalent non-isomorphic models

We prove that if is consistent then is consistent with the following statement: There is for every a model of cardinality which is -equivalent to exactly non-isomorphic models of cardinality . In order to get this result we introduce ladder systems and colourings different from the ``standard' counterparts, and prove the following purely combinatorial result: For each prime number and positive integer it is consistent with that there is a ``good' ladder system having exactly pairwise nonequivalent colourings.

     

Noncrossed products over

Noncrossed product division algebras are constructed over rational function fields over number fields by lifting from arithmetic completions . The existence of noncrossed products over -adic rational function fields is proved as a corollary.

     

On the telescopic homotopy theory of spaces

In telescopic homotopy theory, a space or spectrum is approximated by a tower of localizations , , taking account of -periodic homotopy groups for progressively higher . For each , we construct a telescopic Kuhn functor carrying a space to a spectrum with the same -periodic homotopy groups, and we construct a new functor left adjoint to . Using these functors, we show that the th stable monocular homotopy category (comprising the th fibers of stable telescopic towers) embeds as a retract of the th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving ``infinite -suspension spaces.' We deduce that Ravenel's stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel's th stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson -homology but nontrivial -periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is -equivalent to a suspension spectrum. As an unstable chromatic application, we determine the -localizations and -localizations of infinite loop spaces in terms of -localizations of spectra under suitable conditions. We also determine the -localizations and -localizations of arbitrary Postnikov -spaces.

     

Arithmetic discriminants and morphisms of curves

This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pull-backs of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition and differ when is an algebraic point on a curve and is a nonconstant morphism of curves. In particular, it is demonstrated that , with at most finitely many exceptions, whenever the degrees of and are sufficiently small, relative to the difference between the genera and . The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2.

     

Transfers of Chern classes in BP-cohomology and Chow rings

The -module structure of for extraspecial -groups is studied using transfer and Chern classes. These give rise to -torsion elements in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained by Totaro.

     

Properties of Anick's spaces

We prove three useful properties of Anick's space  . First, at odd primes a map from into a homotopy commutative, homotopy associative -space can be extended to a unique -map from into . Second, at primes larger than , is itself homotopy commutative and homotopy associative. And third, the first two properties combine to show that the order of the identity map on  is .

     

Two-weight norm inequalities for Cesàro means of Laguerre expansions

Two-weight norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted' cases, by including all values of for all positive orders of the Cesàro summation and all values of the Laguerre parameter -1$">. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.