Semigroups and weights for group representations

Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.

     

The Mardesic factorization theorem for extension theory and c-separation

We shall prove a type of Mardesic factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.

A corollary to our result will be that for any weight and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .

We shall show, on the other hand, that there exists a CW-complex which is not finitely homotopy dominated but which has the property that for each weight , there exists a Hausdorff compactum which is universal for the property and weight .

     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

On the density of the set of generators of a polynomial algebra

Let be the polynomial algebra over a field of characteristic . We call a polynomial coordinate (or a generator) if for some polynomials . In this note, we give a simple proof of the following interesting fact: for any polynomial of the form where is a polynomial without constant and linear terms, and for any integer , there is a coordinate polynomial such that the polynomial has no monomials of degree . A similar result is valid for coordinate -tuples of polynomials, for any . This contrasts sharply with the situation in other algebraic systems.

On the other hand, we establish (in the two-variable case) a result related to a different kind of density. Namely, we show that given a non-coordinate two-variable polynomial, any sufficiently small perturbation of its non-zero coefficients gives another non-coordinate polynomial.

     

Elementary proof of Brauer's and Nesbitt's theorem on zeros of characters of finite groups

The following has been proven by Brauer and Nesbitt. Let be a finite group, and let be a prime. Assume is an irreducible complex character of such that the order of a -Sylow subgroup of divides the degree of . Then vanishes on all those elements of whose order is divisible by . The two only known proofs of this theorem use profound methods of representation theory, namely the theory of modular representations or Brauer's characterization of generalized characters. The purpose of this paper is to present a more elementary proof.

     

Integral representation of linear functionals on spaces of unbounded functions

Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.

     

The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

A classification of prime segments in simple artinian rings

Let be a simple artinian ring. A valuation ring of is a Bézout order of so that is simple artinian, a Goldie prime is a prime ideal of so that is Goldie, and a prime segment of is a pair of neighbouring Goldie primes of A prime segment is archimedean if is equal to it is simple if and it is exceptional if In this last case, is a prime ideal of so that is not Goldie. Using the group of divisorial ideals, these results are applied to classify rank one valuation rings according to the structure of their ideal lattices. The exceptional case splits further into infinitely many cases depending on the minimal so that is not divisorial for

     

On the pointwise maximum of convex functions

We study the conjugate of the maximum, , of and when and are proper convex lower semicontinuous functions on a Banach space . We show that on the bidual, , of provided that and satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for and for the ``preconjugate' of .

     

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let be a real Banach space, let be a closed convex subset of , and let , from into , be a pseudo-contractive mapping (i.e. for all and 1)$">. Suppose the space has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path , , defined by the equation is continuous and strongly converges to a fixed point of as , provided that satisfies the weakly inward condition.

     

On the parametrization of self-similar and other fractal sets

We prove for many self-similar, and some more general, sets that if is the Hausdorff dimension of and is Hölder continuous with exponent , then the -dimensional Hausdorff measure of is .

     

A ``universal' dynamical system generated by a continuous map of the interval

In this paper we show that there is a continuous map of the interval such that any -limit set of any continuous map can be transformed by a homeomorphism to an -limit set of . Consequently, any nowhere-dense compact set and any finite union of compact intervals is a homeomorphic copy of an -limit set of .

     

A note on Pontrjagin forms

Let be a principal bundle over a manifold of dimension , and let be its -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class can be realised as the Pontrjagin form of some connection on ?

     

On sequences of -norm

We study maps and give detailed estimates on in terms of and . These estimates are used to prove a lemma by D. Henry for the case . Here is an open subset and and are Banach spaces.

     

-semigroups generated by second order differential operators with general Wentzell boundary conditions

Let us consider the operator where is positive and continuous in and is equipped with the so-called generalized Wentzell boundary condition which is of the form at each boundary point, where This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions.

Under suitable assumptions on , we prove that generates a positive -semigroup on and, hence, many previous (linear or nonlinear) results are extended substantially.

     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

On some properties of the Gamma function

Anderson and Qiu (1997) conjectured that the function is concave for 1$">. In this paper we prove this conjecture. We also study the monotonicity of some functions connected with the psi-function and derive inequalities for and .

     

Projective boundedness and convolution of Fréchet measures

Fréchet measures of order ( -measures) are the measure-theoretic analogues of bounded -linear forms on products of spaces. In an LCA setting, convolution of -measures is always defined, while there exist -measures whose convolution cannot be defined. In a three-dimensional setting, we demonstrate the existence of an -measure which cannot be convolved with arbitrary -measures.