Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group and a vector space decomposition such that and , where denotes the integral closure. Moreover, is Cohen-Macaulay if and only if is Cohen-Macaulay. Furthermore, is polynomial if and only if is polynomial, and if and only if

where and .

     

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

     

Open colorings, the continuum and the second uncountable cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or ``open coloring axioms.' In particular it will be shown that the conjunction of two well-known axioms, and , implies that the size of the continuum is .

     

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

     

Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

We show that integrals of the form

and

satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all and in the second case when is even, these integrals are reducible to zeta values. In the case of odd , we combine the known results for Euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight .

The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section.

     

The Aronsson equation for absolute minimizers of -functionals associated with vector fields satisfying Hörmander's condition

Given a Carnot-Carathéodory metric space generated by vector fields satisfying Hörmander's condition, we prove in Theorem A that any absolute minimizer to is a viscosity solution to the Aronsson equation

under suitable conditions on . In particular, any AMLE is a viscosity solution to the subelliptic -Laplacian equation

If the Carnot-Carathéodory space is a Carnot group and is independent of the -variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation

under suitable conditions on . As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic -Laplacian equation is established on any Carnot group .

     

Classification of abelian hereditary directed categories satisfying Serre duality

In an ongoing project to classify all hereditary abelian categories, we provide a classification of -finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence.

In order to prove the classification, we will study the shapes of Auslander-Reiten components extensively and use appropriate generalizations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.

     

On the maximal difference between an element and its inverse in residue rings

We investigate the distribution of where

Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on . We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study than a more traditional way using exponential sums.

     

-estimates for the linearized Monge-Ampère equation

Let be a strictly convex domain and let be a convex function such that    det in . The linearized Monge-Ampère equation is

where det is the matrix of cofactors of . We prove that there exist and depending only on , and such that

for all solutions to the equation .

     

The first cohomology group of the generalized Morava stabilizer algebra

There exists a -local spectrum with = . Its Adams-Novikov -term is isomorphic to

where

In this paper we determine the groups

for all 0$">. Its rank ranges from to depending on the value of .

     

Telescoping, rational-valued series, and zeta functions

We give an effective procedure for determining whether or not a series telescopes when is a rational function with complex coefficients. We give new examples of series , where is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set is linearly independent, where is the Riemann zeta function.

Some series of the form , where is a quotient of symmetric polynomials, are shown to be telescoping, as is . Quantum versions of these examples are also given.

     

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

We address the problem of approximating numerically the solutions of stochastic evolution equations on Hilbert spaces , with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of , where is a linear operator. First, we introduce estimates on the behavior of . Then we characterize the error induced by the substitution of with the solution of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of to , where denotes the explicit Euler method. Finally, we consider an extrapolation method based on the Euler scheme. An application to the quantum harmonic oscillator system is included.

     

Integral motives and special values of zeta functions

For each field , we define a category of rationally decomposed mixed motives with -coefficients. When is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain groups.

     

Homological algebra for the representation Green functor for abelian groups

In this paper we compute some derived functors of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.

When the group is a cyclic -group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor .

When the group is , we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired functors.

     

Existence and Lipschitz regularity for minima

We prove the existence, uniqueness and Lipschitz regularity of the minima of the integral functional

on ( ) for a class of integrands that are convex in and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumptions on .

     

Noetherian hereditary abelian categories satisfying Serre duality

In this paper we classify -finite noetherian hereditary abelian categories over an algebraically closed field satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories.

As a side result we show that when our hereditary abelian categories have no non-zero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

     

Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .

     

A hierarchy of thin sets related to the boundedness of trigonometric series

We study the family  of the sets on which some series of the form is uniformly bounded. We show that the families  of all sets admitting the boundary  form a hierarchy which is incontinuous with respect to the operations of intersection and union.

     

Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop' conjecture.

     

Gorenstein derived functors

Over any associative ring it is standard to derive using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product using Gorenstein flat modules.