Positive definite spherical functions on Olshanskii domains

Let be a simply connected complex Lie group with Lie algebra , a real form of , and the analytic subgroup of corresponding to . The symmetric space together with a -invariant partial order is referred to as an Olshanskii space. In a previous paper we constructed a family of integral spherical functions on the positive domain of . In this paper we determine all of those spherical functions on which are positive definite in a certain sense.

     

The -Lagrangian of a convex function

At a given point , a convex function is differentiable in a certain subspace (the subspace along which has 0-breadth). This property opens the way to defining a suitably restricted second derivative of at . We do this via an intermediate function, convex on . We call this function the -Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.

     

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra and Dyer-Lashof algebra have both striking similarities and differences arising from their common origins in ``lower-indexed' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra , whose module actions are equivalent to, but quite different from, those of and . The exact relationships emerge as ``sheared algebra bijections', which also illuminate the role of the cohomology of . As a bialgebra, has a particularly attractive and potentially useful structure, providing a bridge between those of and , and suggesting possible applications to the Miller spectral sequence and the structure of Dickson algebras.

     

Ultrafilters on -their ideals and their cardinal characteristics

For a free ultrafilter on we study several cardinal characteristics which describe part of the combinatorial structure of . We provide various consistency results; e.g. we show how to force simultaneously many characters and many -characters. We also investigate two ideals on the Baire space naturally related to and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.

     

Cohomology of uniformly powerful -groups

In this paper we will study the cohomology of a family of -groups associated to -Lie algebras. More precisely, we study a category of -groups which will be equivalent to the category of -bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group in this category, its -cohomology is that of an elementary abelian -group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of in the case that is associated to a Lie algebra . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups which are defined to be the kernel of the mod reduction

     

A new result on the Pompeiu problem

A nonempty bounded open set () is said to have the Pompeiu property if and only if the only continuous function on for which the integral of over is zero for all rigid motions of is . We consider a nonempty bounded open set with Lipschitz boundary and we assume that the complement of is connected. We show that the failure of the Pompeiu property for implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in , , has the Pompeiu property. So far the result was proved only for solid tori in . We also examine the case of planar domains. Finally we extend the example of solid tori to domains in bounded by hypersurfaces of revolution.

     

Characterizations of spectra with -injective cohomology which satisfy the Brown-Gitler property

We work in the stable homotopy category of -complete connective spectra having mod homology of finite type. means cohomology with coefficients, and is a left module over the Steenrod algebra .

A spectrum is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum satisfies the Brown-Gitler property if the natural map is onto, for all spacelike .

It is known that there exist spectra satisfying the Brown-Gitler property, and with isomorphic to the injective envelope of in the category of unstable -modules.

Call a spectrum standard if it is a wedge of spectra of the form , where is a stable wedge summand of the classifying space of some elementary abelian -group. Such spectra have -injective cohomology, and all -injectives appear in this way.

Working directly with the two properties of stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if is a spectrum with -injective cohomology, the following conditions are equivalent:

(A) there exist a spectrum whose cohomology is a reduced -injective and a map that is epic in cohomology, (B) there exist a spacelike spectrum and a map that is epic in cohomology, (C) is monic in cohomology, (D) satisfies the Brown-Gitler property, (E) is spacelike, (F) is standard. ( is reduced if it has no nontrivial submodule which is a suspension.)

As an application, we prove that the Snaith summands of are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture.

Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an -connected space admits a map to an -fold suspension that is monic in mod homology, then is onto in mod homology.

     

A generalized Brauer construction and linear source modules

For a complete discrete valuation ring with residue field , a subgroup of a finite group and a homomorphism , we define a functor from the category of -modules to the category of -modules and investigate its behaviour with respect to linear source modules.     

coefficients

Let be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain (). We consider the differential operator

where the coefficients are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

for a.e. , every , some constant . Moreover, we assume that the coefficients belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields . We prove the following local -estimate:

for every , . We also prove the local Hölder continuity for solutions to for any with large enough. Finally, we prove -estimates for higher order derivatives of , whenever and the coefficients are more regular.

     

Conditions for the Existence of SBR Measures for ``Almost Anosov' Diffeomorphisms

A diffeomorphism of a compact manifold is called ``almost Anosov' if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure that has absolutely continuous conditional measures on unstable manifolds. The measure is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, tends to either an SBR measure or for almost every with respect to Lebesgue measure. ( is the Dirac measure at .) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of at .

     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

Banach spaces with the Daugavet property

A Banach space is said to have the Daugavet property if every operator of rank satisfies . We show that then every weakly compact operator satisfies this equation as well and that contains a copy of . However, need not contain a copy of . We also study pairs of spaces and operators satisfying , where is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with is as small as possible and give characterisations in terms of a smoothness condition.

     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

Trees and valuation rings

A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra , and one main result here is a positive answer to this question where can be chosen as a quaternion division algebra over a commutative field.

     

Structural interactions of conjugacy closed loops

We study conjugacy closed loops by means of their multiplication groups. Let be a conjugacy closed loop, its nucleus, the associator subloop, and and the left and right multiplication groups, respectively. Put . We prove that the cosets of agree with orbits of , that and that one can define an abelian group on . We also explain why the study of finite conjugacy closed loops can be restricted to the case of nilpotent. Group is shown to be a subgroup of a power of (which is abelian), and we prove that can be embedded into . Finally, we describe all conjugacy closed loops of order .

     

The distributivity numbers of /fin and its square

We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length , the distributivity number of /fin is , whereas the distributivity number of r.o./fin) is . This answers a problem of Balcar, Pelant and Simon, and others.

     

Exact Hausdorff measure and intervals of maximum density for Cantor sets

Consider a linear Cantor set , which is the attractor of a linear iterated function system (i.f.s.) , , on the line satisfying the open set condition (where the open set is an interval). It is known that has Hausdorff dimension given by the equation , and that is finite and positive, where denotes Hausdorff measure of dimension . We give an algorithm for computing exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When (or more generally, if and are commensurable), the algorithm also gives an interval that maximizes the density . The Hausdorff measure is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters , it is possible to choose the translation parameters in such a way that , so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.

     

Products and duality in Waldhausen categories

The natural transformation from -theory to the Tate cohomology of acting on -theory commutes with external products. Corollary: The Tate cohomology of acting on the -theory of any ring with involution is a generalized Eilenberg-Mac Lane spectrum, and it is 4-periodic.