Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

     

On almost representations of groups

We say that a group belongs to the class if every nonunit quotient group of has an element of order two.

Let be a Hilbert space and let be its group of unitary operators. Suppose that groups and belong to the class and the order of is more than two. Then the free product has the following property. For any there exists a mapping satisfying the following conditions :

1)

2) for any representation the relation

holds.

     

Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems where , are real spectral parameters. It will be shown that if the functions and are `generic' then for all integers , there are smooth 2-dimensional manifolds , , of `semi-trivial' solutions of the system which bifurcate from the eigenvalues , , of , , respectively. Furthermore, there are smooth curves , , along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, , which `links' the curves , . Nodal properties of solutions on and global properties of are also discussed.

     

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .

     

A one-point attractor theory for the Navier-Stokes equation on thin domains with no-slip boundary conditions

In an earlier paper related to recent results of Raugel and Sell for periodic boundary conditions, we considered the incompressible Navier-Stokes equations on 3-dimensional thin domains with zero (``no-slip') boundary conditions and established global regularity results. We extend those results here by developing an attractor theory. We first show that under similar thinness restrictions trajectories of solutions approach each other in -norm exponentially. Next, for constant-in-time forcing data we suppose that in as and show that if and solve the equations with forcing data and , respectively, then as For similar thinness restrictions we show that the steady-flow equations with forcing data have a unique solution . Under both thinness assumptions we then have that all solutions converge to in as ; thus we have a one-point attractor for strong solutions. In fact, we have a one-point attractor for the Leray solutions as well. Moreover, under significantly more relaxed thinness assumptions we are able to show that Leray solutions nonetheless eventually become regular.

     

Von Neumann algebras and linear independence of translates

For and , define and if , define . It has been conjectured that if , then is linearly independent over ; one motivation for this problem comes from Gabor analysis. We shall prove that is linearly independent if and is contained in a discrete subgroup of , and as a byproduct we shall obtain some results on the group von Neumann algebra generated by the operators . Also, we shall prove these results for the obvious generalization to .

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

On a problem of Dynkin

Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

     

An extremal problem for trigonometric polynomials

Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Derivations and the integral closure of ideals

Let be a complete local domain containing the rationals. Then there exists an integer such that for any ideal , if , , then there exists a derivation of with .

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem on ; on , where is a bounded region in , is an indefinite weight function and may be positive, negative or zero.

     

An application of the regularized Siegel-Weil formula on unitary groups to a theta lifting problem

Let and be the pair of unitary groups over a global field and an irreducible cuspidal representation of which satisfies a certain -function condition. By using a regularized Siegel-Weil formula, we can show that the global theta lifting of in is non-trivial if every local factor of has a local theta lifting (Howe lifting) in .

     

type free resolutions of monomial ideals

Let be a monomial ideal of . Bayer-Peeva-Sturmfels studied a subcomplex of the Taylor resolution, defined by a simplicial complex . They proved that if is generic (i.e., no variable appears with the same non-zero exponent in two distinct monomials which are minimal generators), then is the minimal free resolution of , where is the Scarf complex of . In this paper, we prove the following: for a generic (in the above sense) monomial ideal and each integer , there is an embedded prime of . Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, is shellable). We also study a non-generic monomial ideal whose minimal free resolution is for some . In particular, we prove that if all associated primes of have the same height, then is Cohen-Macaulay and is pure and strongly connected.

     

Free boundary value problems for analytic functions in the closed unit disk

We shall prove (a slightly more general version of) the following theorem: let be analytic in the closed unit disk with , and let be a finite Blaschke product. Then there exists a function satisfying: i) analytic in the closed unit disk , ii) , iii) in , such that

satisfies

This completes a recent result of Kühnau for , , where this boundary value problem has a geometrical interpretation, namely that preserves hyperbolic arc length on for suitable
. For these important choices of we also prove that the corresponding functions are uniquely determined by , and that is univalent in . Our work is related to Beurling's and Avhadiev's on conformal mappings solving free boundary value conditions in the unit disk.

     

Common fixed points of commuting holomorphic maps in the unit ball of

Let be the unit ball of (). We prove that if are holomorphic self-maps of such that , then and have a common fixed point (possibly at the boundary, in the sense of -limits). Furthermore, if and have no fixed points in , then they have the same Wolff point, unless the restrictions of and to the one-dimensional complex affine subset of determined by the Wolff points of and are commuting hyperbolic automorphisms of that subset.

     

Sur les -corps

Here we give new examples of fields in characteristic whose -invariant and -invariant are different: or . These fields are also -fields.

RSUM. Nous donnons ici de nouveaux exemples de corps en caractéristique dont le -invariant et le -invariant diffèrent. Plus précisément: et ou . Ces corps sont aussi des -corps.