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.

     

The Lipschitz continuity of the distance function to the cut locus

Let be a closed submanifold of a complete smooth Riemannian manifold and the total space of the unit normal bundle of . For each , let denote the distance from to the cut point of on the geodesic with the velocity vector The continuity of the function on is well known. In this paper we prove that is locally Lipschitz on which is bounded; in particular, if and are compact, then is globally Lipschitz on . Therefore, the canonical interior metric may be introduced on each connected component of the cut locus of and this metric space becomes a locally compact and complete length space.

     

Maximal degree subsheaves of torsion free sheaves on singular projective curves

Fix integers with k>0$"> and . Let be an integral projective curve with and a rank torsion free sheaf on which is a flat limit of a family of locally free sheaves on . Here we prove the existence of a rank subsheaf of such that . We show that for every there is an integral projective curve not Gorenstein, and a rank 2 torsion free sheaf on with no rank 1 subsheaf with . We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.

     

Computing the homology of Koszul complexes

Let be a commutative ring and an ideal in which is locally generated by a regular sequence of length . Then, each f. g. projective -module has an -projective resolution of length . In this paper, we compute the homology of the -th Koszul complex associated with the homomorphism for all , if . This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if , we compute the homology of the complex where and denote the functors occurring in the Dold-Kan correspondence.

     

A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I

Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of  explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.

     

Siegel discs, Herman rings and the Arnold family

We show that the rotation number of an analytically linearizable element of the Arnold family , , , satisfies the Brjuno condition. Conversely, for every Brjuno rotation number there exists an analytically linearizable element of the Arnold family. Along the way we prove the necessity of the Brjuno condition for linearizability of and , , at 0. We also investigate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of and are quasidiscs with a critical point on the boundary if the rotation number is of constant type.

     

Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2

Let be an algebraically closed field of characteristic zero. Let be the ring of (-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension which is the tensor product of two regular commutative affine domains of Krull dimension . Simple holonomic -modules are described. Let a -algebra be a regular affine commutative domain of Krull dimension and be the ring of differential operators with coefficients from . We classify (up to irreducible elements of a certain Euclidean domain) simple -modules (the field is not necessarily algebraically closed).

     

Livsic theorems for connected Lie groups

Let be a hyperbolic diffeomorphism on a basic set and let be a connected Lie group. Let be Hölder. Assuming that satisfies a natural partial hyperbolicity assumption, we show that if is a measurable solution to a.e., then must in fact be Hölder. Under an additional centre bunching condition on , we show that if assigns `weight' equal to the identity to each periodic orbit of , then for some Hölder . These results extend well-known theorems due to Livsic when is compact or abelian.

     

Invariant ideals of abelian group algebras and representations of groups of Lie type

This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group is the additive group of a finite-dimensional vector space over a locally finite field of prime characteristic , and the automorphism group is a simple infinite absolutely irreducible subgroup of . Thus is isomorphic to an infinite simple periodic group of Lie type, and is realized in via a twisted tensor product of infinitesimally irreducible representations. If is a Sylow -subgroup of and if is the unique line in stabilized by , then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus on . At present, we are able to handle the case where is a rational representation with character field equal to .

     

A measurable cardinal with a closed unbounded set of inaccessibles from

We prove that is sufficient to construct a model in which is measurable and is a closed and unbounded subset of containing only inaccessible cardinals of . Gitik proved that is necessary.

We also calculate the consistency strength of the existence of such a set together with the assumption that is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set generate the closed unbounded ultrafilter of while remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.

     

Sur le rang du

On the rank of the -class group of . Let be a square-free positive integer and be a prime such that . We set , where or . In this paper, we determine the rank of the -class group of .

RÉSUMÉ. Soit , un corps biquadratique où ou bien un premier et étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du -groupe de classes de .

     

Homotopy commutativity of -spaces with finitely generated cohomology

We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .

     

On the relation between upper central quotients and lower central series of a group

Let be a group with a normal subgroup contained in the upper central subgroup . In this article we study the influence of the quotient group on the lower central subgroup . In particular, for any finite group we give bounds on the order and exponent of . For equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as . Our proofs involve: (i) the Baer invariants of , (ii) the Schur multiplier of relative to a normal subgroup , and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.

     

A growth dichotomy for o-minimal expansions of ordered groups

Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .

     

Random variable dilation equation and multidimensional prescale functions

A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation

We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.

     

Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball

The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on for the equation

where is the classical Laplacian operator, and denotes the open ball of center and radius in When is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.

     

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.

     

The Serre spectral sequence of a multiplicative fibration

In a fibration we show that finiteness conditions on force the homology Serre spectral sequence with -coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, is ``almost' isomorphic to . One consequence is the conclusion that is elliptic if and only if and are.

     

Periodic solutions of conservation laws constructed through Glimm scheme

We present a periodic version of the Glimm scheme applicable to special classes of systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in , for Bakhvalov's class, and in , in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in with , for some constant which is for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, , are uniformly bounded in , for all 0$">, where is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the -systems with , , , which, for , model isentropic gas dynamics in Lagrangian coordinates.