Character degrees and local subgroups of -separable groups

Let be a finite -solvable group for different primes and . Let and be such that . We prove that every of -degree has -degree if and only if and .

     

On uniqueness of -adic meromorphic functions

Let be a complete ultrametric algebraically closed field of characteristic zero, and let be the field of meromorphic functions in . For all set in and for all we denote by the subset of : zero of order After studying unique range sets for entire functions in in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every , there exist sets of elements in such that, if have the same poles (counting multiplicities), and satisfy , then . We show how to construct such sets.

     

A unified extension of two results of Ky Fan on the sum of matrices

Let be an Hermitian matrix with where are the ordered eigenvalues of . A result of Ky Fan (1949) asserts that if and are Hermitian matrices, then is majorized by . We extend the result in the framework of real semisimple Lie algebras in the following way. Let be a noncompact real semisimple Lie algebra with Cartan decomposition . We show that for any given , , where is the unique element corresponding to , in a fixed closed positive Weyl chamber of a maximal abelian subalgebra of in . Here the ordering is induced by the dual cone of . Fan's result corresponds to the Lie algebra . The compact case is also discussed. As applications, two unexpected singular values inequalities concerning the sum of two real matrices and the sum of two real skew symmetric matrices are obtained.

     

The maximal normal -group

Let be a prime number and let be an abelian -group. Let be the maximal normal -subgroup of and the maximal -subgroup of its centre. Let be the torsion radical of . Then . The result is new for and 3, and the proof is new and valid for all primes .

     

On component groups of and degeneracy maps

For an integer and a prime not dividing , we study the kernel of the degeneracy map , where and are the component groups of and , respectively. This is then used to determine the kernel of the degeneracy map when . We also compute the group structure of in some cases.

     

On an analogue of Selberg's eigenvalue conjecture for

Let be the homogeneous space associated to the group
. Let where and consider the first nontrivial eigenvalue of the Laplacian on . Using geometric considerations, we prove the inequality . Since the continuous spectrum is represented by the band , our bound on can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

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 .

     

The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.

     

Stieltjes moment sequences and positive definite matrix sequences

For a certain constant (a little less than ), every function satisfying , , is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence there is a positive definite matrix sequence which is not of positive type and which satisfies , . For a certain constant (a little greater than ), for every function satisfying , , there is a convolution semigroup of measures on , with moments of all orders, such that , , and for every such convolution semigroup the measure is Stieltjes indeterminate for all .

     

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex in . It is proved that such an expansion is uniformly summable on the simplex for any continuous function if and only if . Moreover, it is shown that means define a positive linear polynomial identity, and the index is sharp in the sense that means are not positive for .

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

A structural result of irreducible inclusions of type

Given an irreducible inclusion of factors with finite index , where is of type , of type , , and are relatively prime positive integers, we will prove that if satisfies a commuting square condition, then its structure can be characterized by using fixed point algebras and crossed products of automorphisms acting on the middle inclusion of factors associated with . Relations between and a certain -kernel on subfactors are also discussed.

     

On the continuity of the evaluation mapping associated with a group and its character group

For an abelian Hausdorff group , let denote the character group endowed with the compact-open topology and let denote the canonical homomorphism. We show that the evaluation mapping from into the torus is continuous if and only if is locally compact and is continuous. If is injective and open, then the evaluation mapping is continuous if and only if is locally compact. Several examples and counterexamples are given.

     

First order differential operators in white noise analysis

Let be the space of test white noise functionals. We first introduce a family of products on including Wiener and Wick products, and then show that with each product , we can associate a first order differential operator, called a first order -differential operator. We next show that a first order -differential operator is indeed a continuous derivation under the product . We finally characterize by means of rotation-invariance and continuous derivation under the product . Here and are the Gross Laplacian and the number operator on , respectively.

     

The equivalence of some Bernoulli convolutions to Lebesgue measure

Since the 1930's many authors have studied the distribution of the random series where the signs are chosen independently with probability and . Solomyak recently proved that for almost every the distribution is absolutely continuous with respect to Lebesgue measure. In this paper we prove that is even equivalent to Lebesgue measure for almost all .

     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

On Eisenstein series and

In this paper, we derive some new identities satisfied by the series using Ramanujan's identities for , and . Our work is motivated by an attempt to develop a theory of elliptic functions to the septic base.