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.

     

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.

     

-identities for the Grassmann algebra: The conjecture of Henke and Regev

Let be an algebraically closed field of characteristic 0, and let be the infinite dimensional Grassmann (or exterior) algebra over . Denote by the vector space of the multilinear polynomials of degree in , ..., in the free associative algebra . The symmetric group acts on the left-hand side on , thus turning it into an -module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The -modules and are canonically isomorphic. Letting be the alternating group in , one may study and its isomorphic copy in with the corresponding action of . Henke and Regev described the -codimensions of the Grassmann algebra , and conjectured a finite generating set of the -identities for . Here we answer their conjecture in the affirmative.

     

Dieudonné rings associated with

We use Dieudonné theory for periodically graded Hopf rings to determine the Dieudonné ring structure of the -graded Morava -theory , with an odd prime, when applied to the -spectrum (and to ). We also expand these results in order to accomodate the case of the full Morava -theory .

     

Spectral radius algebras and contractions

We consider the spectral radius algebras associated to contractions. If is such an operator we show that the spectral radius algebra always properly contains the commutant of .

     

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree

Let , , be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform of degree . It is proved that if is not in the Maaß space, then there exist infinitely many primes for which the sequence , , has infinitely many sign changes.

     

Rademacher multiplicator spaces equal to

Let be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space of measurable functions such that for every a.e. converging series , where are the Rademacher functions. We characterize the situation when . We also discuss the behaviour of partial sums and tails of Rademacher series in function spaces.

     

Topology of three-manifolds with positive -scalar curvature

Consider an -dimensional smooth Riemannian manifold with a given smooth measure on it. We call such a triple a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjecture , where and is the scalar curvature of . In this note, we study the topological obstruction for the -stable minimal submanifold with positive -scalar curvature in dimension three under the setting of manifolds with density.

     

The effective Chebotarev density theorem and modular forms modulo

Suppose that (resp. ) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers of a number field . For any ideal , we bound the first prime for which (resp. ) is zero ( ). Applications include the solution to a question of Ono (2001) concerning partitions.

     

A note on -bases of a regular affine domain extension

Let be a tower of commutative rings where is a regular affine domain over an algebraically closed field of prime characteristic and is a regular domain. Suppose has a -basis over and . For a subset of whose elements satisfy a certain condition on linear independence, let be a set of maximal ideals of such that is a -basis of over . We shall characterize this set in a geometrical aspect.

     

Polynomials with roots in for all

Let be a monic polynomial in with no rational roots but with roots in for all , or equivalently, with roots mod for all . It is known that cannot be irreducible but can be a product of two or more irreducible polynomials, and that if is a product of irreducible polynomials, then its Galois group must be a union of conjugates of proper subgroups. We prove that for any , every finite solvable group that is a union of conjugates of proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with ) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of .

     

Upper bound for isometric embeddings

The isometric embeddings (, ) over a field are considered, and an upper bound for the minimal is proved. In the commutative case ( ) the bound was obtained by Delbaen, Jarchow and Pełczyński (1998) in a different way.

     

The stability of exceptional bundles on complete intersection -folds

A very long-standing problem in Algebraic Geometry is to determine the stability of exceptional vector bundles on smooth projective varieties. In this paper we address this problem and we prove that any exceptional vector bundle on a smooth complete intersection -fold of type with and is stable.

     

Representation of measures with polynomial denseness in , , and its application to determinate moment problems

It has been proved that algebraic polynomials are dense in the space , , iff the measure is representable as with a finite non-negative Borel measure and an upper semi-continuous function such that is a dense subset of the space    as equipped with the seminorm . The similar representation ( ) with the same and ( , and is also a dense

subset of ) corresponds to all those measures (supported by ) that are uniquely determined by their moments on ( ). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach has also been examined.

     

The limiting distribution of the coefficients of the -Catalan numbers

We show that the limiting distributions of the coefficients of the -Catalan numbers and the generalized -Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small , we conjecture that for sufficiently large , the coefficients are unimodal and even log-concave except for a few terms of the head and tail.

     

bounds for oscillatory hyper-Hilbert transform along curves

We study the oscillatory hyper-Hilbert transform

(1)

along the curve , where are some real positive numbers. We prove that if , then is bounded on whenever . Furthermore, we also prove that is bounded on when . Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an boundedness result for some strongly parabolic singular integrals with rough kernels.

     

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.

     

Maps on the -dimensional subspaces of a Hilbert space preserving principal angles

In a former paper we studied transformations on the set of all -dimensional subspaces of a Hilbert space which preserve the principal angles. In the case when , we could determine the general form of all such maps. The aim of this paper is to complete our result by considering the problem in the remaining case .

     

A central set of dimension

The central set of a domain is the set of centers of maximal discs in . Fremlin proved that the central set of a planar domain has zero area and asked whether it can have Hausdorff dimension strictly larger than . We construct a planar domain with central set of Hausdorff dimension .

     

Near-symmetry in and refined Jones factorization

We use variants of the Hardy-Littlewood maximal and the Cruz-Uribe-Neugebauer minimal operators to give direct characterizations of and that clarify their near symmetry and yield elementary proofs of various known results, including Cruz-Uribe and Neugebauer's refinement of the Jones factorization theorem.