Remarks concerning linear characters of reflection groups

Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a -eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the -eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .

     

A general functional equation and its stability

Suppose that and are vector spaces over or and are scalar such that whenever We prove that if for and

then each is a ``generalized' polynomial map of ``degree' at most

In case and we show that if some is bounded on a set of positive inner Lebesgue measure, then it is a genuine polynomial function.

Our main aim is to establish the stability of (in the sense of Ulam) in case is a Banach space.

We also solve a distributional analogue of and prove a mean value theorem concerning harmonic functions in two real variables.

     

Duals and envelopes of some Hardy-Lorentz spaces

For we describe the dual spaces and Banach envelopes of the spaces for finite values of and for , the closure of the polynomials in . In addition, we determine the -Banach envelopes for the spaces in the cases and .

     

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 D. H. Lehmer

Let be an odd prime number. For we denote the inverse of modulo by with . Given , we prove that in any range of length the probability that has the same parity as tends to as . This result was previously known only to hold true in the full range of length . We will also obtain quantitative results on the pseudorandomness of the sequence for which we estimate the well-distribution and correlation measures as defined by Mauduit and Sárközy (1997).

     

An elementary and constructive solution to Hilbert's 17th Problem for matrices

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let be an symmetric matrix with entries in the polynomial ring . The result is that if is positive semidefinite for all substitutions , then can be expressed as a sum of squares of symmetric matrices with entries in . Moreover, our proof is constructive and gives explicit representations modulo the scalar case.

     

Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence whose moment matrix is positive semidefinite and has finite rank is the sequence of moments of an -atomic nonnegative measure on . We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure having degree at most if the principal submatrix of (indexed by all monomials of degree ) has full rank . This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set , where and . If is positive semidefinite and has a flat extension such that all localizing matrices are positive semidefinite, then has an atomic representing measure supported by . We also review an application of this result to the problem of minimizing a polynomial over the set .

     

Factorization formulae on counting zeros of diagonal equations over finite fields

Let be the number of solutions of the equation over the finite field , and let be the number of solutions of the equation . If , let be the least integer represented by . and play important roles in estimating . Based on a partition of , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.

     

A note on Gabor orthonormal bases

The study of Gabor bases of the form for has interested many mathematicians in recent years. Alex Losevich and Steen Pedersen in 1998, Jeffery C. Lagarias, James A. Reeds and Yang Wang in 2000 independently proved that, for any fixed positive integer , is an orthonormal basis for if and only if is a tiling of . Palle E. T. Jorgensen and Steen Pedersen in 1999 gave an explicit characterization of such for , , . Inspired by their work, this paper addresses Gabor orthonormal bases of the form for and some other related problems, where is as above. For a fixed , the generating function of a Gabor orthonormal basis for corresponding to the above is characterized explicitly provided that , which is new even if ; a Shannon type sampling theorem about such is derived when , ; for an arbitrary positive integer , an explicit expression of the with being an orthonormal basis for is obtained under the condition that .

     

Bounds for the index of the centre in capable groups

A group is called capable if it is isomorphic to for some group . Let be a capable group. I. M. Isaacs (2001) showed that if is finite, then the index of the centre is bounded above by some function of . We show that if , then with some constant and this bound is essentially best possible. We complete a result of Isaacs, showing that if is a cyclic group, then .

     

A sharp result on -covers

Let be a finite system of residue classes which forms an -cover of (i.e., every integer belongs to at least members of ). In this paper we show the following sharp result: For any positive integers and , if there is such that the fractional part of is , then there are at least such subsets of . This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to -covers of the integral ring of any algebraic number field with a power integral basis.

     

The content of a Gaussian polynomial is invertible

Let be an integral domain and let be a nonzero polynomial in . The content of is the ideal generated by the coefficients of . The polynomial is called Gaussian if for all . It is well known that if is an invertible ideal, then is Gaussian. In this note we prove the converse.

     

Necessary conditions for Schatten class localization operators

We study time-frequency localization operators of the form , where is the symbol of the operator and are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order , is that belongs to the modulation space and the window functions to the modulation space . Here we prove a partial converse: if for every pair of window functions with a uniform norm estimate, then the corresponding symbol must belong to the modulation space . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For and , we recapture earlier results, which were obtained by different methods.

     

Hyperelliptic curves over of every -rank without extra automorphisms

We prove that for any pair of integers such that or 0$">, there exists a (hyper)elliptic curve over of genus and -rank whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties over of dimension and -rank such that .

     

Relations approximated by continuous functions

Let be a Tychonoff space, let be the space of all continuous real-valued functions defined on and let be the hyperspace of all nonempty closed subsets of . We prove the following result. Let be a locally connected, countably paracompact, normal -space without isolated points, and let . Then is in the closure of in with the locally finite topology if and only if is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.

     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

Commutator automorphisms of formal power series rings

For a big class of commutative rings , every continuous -automorphism of with the linear part the identity is in the commutator subgroup of . An explicit bound for the number of commutators involved and a -theoretic interpretation of this result are provided.

     

Hopf algebras of dimension

Let be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If is not semisimple and for some odd integer , then or is not unimodular. Using this result, we prove that if for some odd prime , then is semisimple. This completes the classification of Hopf algebras of dimension .

     

Convergence of cascade algorithms associated with nonhomogeneous refinement equations

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form where is the unknown, is a given vector of functions on , is an dilation matrix, and is a finitely supported refinement mask such that each is an (complex) matrix. Let be an initial vector in . The corresponding cascade algorithm is given by In this paper we give a complete characterization for the -convergence of the cascade algorithm in terms of the refinement mask , the nonhomogeneous term , and the initial vector of functions .

     

Weierstrass functions in Zygmund's class

Consider the function

where 1$"> and is an almost periodic function. It is well known that the function lives in the so-called Zygmund class. We prove that is generically nowhere differentiable. This is the case in particular if the elementary condition is satisfied. We also give a sufficient condition on the Fourier coefficients of which ensures that is nowhere differentiable.