The -local -homology of

In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the -homology of a space. As an application, we compute the -homology of in a manner analogous to Mahowald and Milgram's computation of the -homology .

     

The distribution functions of and

Let be the sum of the positive divisors of . We show that the natural density of the set of integers satisfying is given by , where denotes Euler's constant. The same result holds when is replaced by , where is Euler's totient function.

     

Whittaker models for , a -adic field

Let be a non-archimedean local field with odd residual characteristic. Let be the group We construct explicit Whittaker models for any supercuspidal representation of with positive level.

     

planes in

We establish a homeomorphism between the moduli space of ordered -tuples of 2-dimensional linear subspaces (mod ) and the quotient by simultaneous conjugation of a certain open subset . For , this leads to an explicit computation of the moduli space of central 2-arrangements in mod and its subspace of those classes that contain a complex hyperplane arrangement.

     

A -sampling theorem and product formula for continuous -Jacobi functions

In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.

     

Exact multiplicity result for the perturbed scalar curvature problem in

Let denote the closure of in the norm Let and define the constants and Let We consider the following problem for

We show an exact multiplicity result for for all small .

     

Pseudofree -actions on surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order on a surface, and prove the existence of such an action which cannot be realized as a smooth action on the standard smooth surface.

     

On the index and spectrum of differential operators on

If is an system of differential operators on having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that is Fredholm from to for all or no value We prove that both the index (when defined) and the spectrum of are independent of

     

On weighted cohomology

Consider an orientable manifold with countably many complete components of bounded dimension. Suppose that its rational homology is infinitely generated in some degree. Then there is no choice of weight function for which the natural map from weighted cohomology to de Rham cohomology is surjective in that degree.

     

The atomic decomposition in

In this paper we present an atomic decomposition of integrable functions. As an application we compute the distance of in to the Hardy space .

     

On -congruences

The phenomenon of -congruences was recently studied by Ahlgren and Ono (2005) and by Elkies, Ono and Yang (2005). We provide a necessary and sufficient condition which improves their general results.

     

A condition under which follows from

We will give some sufficient conditions for a -hyponormal operator, , to be normal, and a sufficient condition for a triplet of operators , , with , self-adjoint and unitary such that necessarily satisfies .

     

spaces associated with the sections

spaces associated with the sections are introduced. It is shown that some properties which hold for the classical space related to the balls (or cubes) remain valid for the space related to the sections.

     

A note on -intersection bodies

The concept of an intersection body is central for the dual Brunn-Minkowski theory and has also played an important role in the solution of the Busemann-Petty problem. A more general concept of -intersection bodies is related to the generalization of the Busemann-Petty problem. In this note, we compare classes of -intersection bodies for different and examine the conjecture that these classes increase with . In particular, we construct a -intersection body that is not a -intersection body.

     

Inner sequence based invariant subspaces in

A closed subspace is said to be invariant if it is invariant under the Toeplitz operators and . Invariant subspaces of are well-known to be very complicated. So discovering some good examples of invariant subspaces will be beneficial to the general study. This paper studies a type of invariant subspace constructed through a sequence of inner functions. It will be shown that this type of invariant subspace has direct connections with the Jordan operator. Related calculations also give rise to a simple upper bound for , where are zeros of a Blaschke product.

     

Form estimates for the -Laplacean

We consider the problem of establishing conditions on that ensure that the form associated with the -Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that - unlike the constant case - this is not possible if has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in with and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous Our basic criteria involve restrictions on and its gradient.

     

The geography of symplectic -manifolds with an arbitrary fundamental group

In this article, for each finitely presented group , we construct a family of minimal symplectic -manifolds with which cover most lattice points with large in the region . Furthermore, we show that all these -manifolds admit infinitely many distinct smooth structures.

     

Subsequences of the Haar basis consisting of full levels in for

We find a condition to be satisfied by strictly increasing sequences of natural numbers guaranteeing that corresponding subsequences of the Haar basis (consisting of full levels) are equivalent or span isomorphic spaces. This applies in particular to Hardy spaces for . We also construct a continuum of greedy nonequivalent bases in .

     

Harmonic homeomorphisms of the closed disc to itself need be in , , but not

Harmonic maps from the closed disc onto bounded convex sets of the plane obey .

     

A remark on the semi-classical measure from with a degenerate potential

This note is motivated by Evans (2004) and Anantharaman (2004). We study the semiclassical measure arising from the operator when the potential has degenerate minimum points. We will use the technique of integration by parts and some identities of Evans to derive information on the support of the measure.