Spaces between and

In this paper we consider the spaces that lie between and . We discuss their interpolation properties and the behavior of maximal functions and singular integrals acting on them.

     

Small volume on big -spheres

We consider Riemannian metrics on the -sphere for such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the -dimensional case where Berger has shown that .

     

Principal groupoid -algebras with bounded trace

Suppose is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids and show is integrable if and only if the groupoid -algebra has bounded trace.

     

-algebras and quantization of para-Hermitian spaces

In the present note we describe a family of -algebra structures on the set of square integrable functions on a rank-one para-Hermitian symmetric space .

     

Packing spheres and fractal Strichartz estimates in for

We prove an estimate for the spherical average operator in if . This leads to a lower bound for the Hausdorff dimension of unions of certain collections of spheres and to a Strichartz-type estimate for solutions of the wave equation.

     

-actions with -dimensional fixed point set

We describe the equivariant cobordism classification of smooth actions of the group , considered as the group generated by two commuting involutions, on closed smooth -dimensional manifolds , for which the fixed point set of the action is a connected manifold of dimension and or . For , the classification is known.

     

The -stability of hypersurfaces with zero Gauss-Kronecker curvature

In this paper we give sufficient conditions for a bounded domain in an -minimal hypersurface of the Euclidean space to be -stable. The Gauss-Kronecker curvature of this hypersurface may be zero on a set of capacity zero.

     

Hyperbolic sets exhibiting -persistent homoclinic tangency for higher dimensions

For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits -persistent homoclinic tangency. It provides an open subset of the space of -diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.

     

The -primary classifying space of the fiber of the double suspension

Gray showed that the homotopy fiber of the double suspension has an integral classifying space , which fits in a homotopy fibration . In addition, after localizing at an odd prime , is an -space and if , then is homotopy associative and homotopy commutative, and is an -map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for as well. We go on to give some exponent consequences.

     

On the boundaries of self-similar tiles in

The aim of this note is to study the construction of the boundary of a self-similar tile, which is generated by an iterated function system . We will show that the boundary has complicated structure (no simple points) in general; however, it is a regular fractal set.

     

The torsion of -ramified Iwasawa modules II

In this article we prove the existence of a non-trivial torsion of the -ramified Iwasawa mocule over the -extension of an imaginary quadratic field.

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

Functional relations and special values of Mordell-Tornheim triple zeta and -functions

In this paper, we prove the existence of meromorphic continuation of certain triple zeta-functions of Lerch's type. Based on this result, we prove some functional relations for triple zeta and -functions of the Mordell-Tornheim type. Using these functional relations, we prove new explicit evaluation formulas for special values of these functions. These can be regarded as triple analogues of known results for double zeta and -functions.

     

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.

     

Dynamics of the function and the Green-Tao theorem on arithmetic progressions in the primes

Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . We use ind to denote the least such nonnegative integer . Wushi Goldring [Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.

In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

     

Nonpositive sectional curvature for -complexes

We give a criterion for the nonpositive sectional curvature of -complexes. As a consequence, we show that certain -complexes have locally indicable, coherent and even locally quasiconvex fundamental groups.

     

Spectra of operators with Bishop's property

Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .

     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.

     

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 .

     

A lattice space-time code of rank

For all previous constructions of lattice space-time codes with a positive diversity product, the rank was at most . In this paper, we give an example of a lattice space-time code of rank with a positive diversity product.