Si M est une variété riemannienne de dimension n, n ≧ 2, et de métrique g on s'intéresse au problème: trouver toutes les fonctions régulières F : M → IR qui sont valeurs propres ainsi que leur carré de l'opérateur de Laplace-Beltrami, Δ, associé à g.

We consider the group Ta, its group of characters Za, and an arbitrary order P on Za. For x ∊ Za, let sgnpx be 1, - 1 , or 0 according as x € P\{0}, x € (-P)\{0}, or X = 0. For f in Lp(Ta), 1 < p < ∞, it is known that there is a function in Lp(Ta) such that for all X in Za. Summability methods for are also available. In this paper, we obtain summability methods for that apply for in L1(Ta), and we show how various properties of can be derived from our construction.

This paper studies a pointwise definition of sets of uniqueness on compact, 0-dimensional, metric groups. It is shown that this definition is equivalent for closed sets to one based on supports of pseudo functions. An analog of Rajchman's theorem is given leading to examples of sets of uniqueness.

We present a short, computational proof of the parametric Gauss-Green theorem for a broad class of closed chains. The proof involves only measure theory and the basic theory of differential forms: in particular, no constructions from topology are used. For completeness, the standard properties of winding numbers are also established by methods from analysis.

A condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

Let us suppose one of the following conditions: (a) p ≧ 2 and F is a closed subspace of a projective limit (b) p = 1 and F is a complemented subspace of an echelon Köthe space of order 1, Λ(X,β,μ,gk); and (c) 1 > p > 2 and F is a quotient of a countable product of Lp(μn) spaces. Then, F is Montel if and only if no infinite dimensional subspace of F is normable.

We examine the convergence and analytic properties of a continued fraction of Ramanujan and its connection to the orthogonal polynomials of Meixner-Pollaczek.

Recently we have introduced an enriched cohomology theory for categories that are tripleable (algebraic) over a category of modules. The cohomology admits a circle product, related to the obstruction problem for algebraic deformations, making the total complex a graded ring. We here offer similar constructions in two other situations - coalgebraic and bialgebraic categories. Examples include categories of bialgebras, sheaves of modules, and sheaves of algebras over a sheaf of rings.

An easy proof will be given to show that for finite dimensional Hopf-algebras with nilpotent augmentation ideal over the field of p elements, the upper and lower Loewy-series coincide. In particular, this holds for the restricted universal envelope of nilpotent Lie-p-algebras with nilpotent p-map.

The definition of strong Cesaro summability with respect to a modulus is extended to a definition of strong A -summability with respect to a modulus when A is a nonnegative regular matrix summability method. It is shown that if a sequence is strongly A-summable with respect to an arbitrary modulus then it is A-statistically convergent and that Astatistical convergence and strong A-summability with respect to a modulus are equivalent on the bounded sequences.

The aim of this paper is two-fold: First we prove the Radotype inequality Here denote the weighted geometric means of

with where the pi are positive weights. Thereafter we investigate under which conditions the sequence

is convergent as n → ∞

Uncountable collections of continua of dimension m embeddable in En are investigated, where the difference between m and n is not restricted to one. Collections of isometric copies of continua equivalent to Menger universal continua and collections of continua analogous to G. S. Young's Tn-sets are the main considerations.

In this note, we develop the notion of a Zp-tower in a Demuskin group, and apply the results of Koch and Wingberg on the uniqueness of so-called Demuškin formations to give a classification of such towers in the case p ≠ 2.

In this paper we establish a relationship between the rational solutions (x(t), y(t)), over C(t), of the diophantine equation:

and the solutions which parametrize the elliptic curve E, y2 = 4x3 — g2x — g3 admitting complex multiplication by λ. We first characterize the form of all rational solutions of diophantine equation (1). The rational solutions are derivable from the subsititutions

in which μ = 0,ω1,ω2,ω1 + ω2 = ω3. Using techniques established in elliptic function theory, we prove that the complex multiplier λ, associated with a unique rational solution (x(t), y(t)), must be of a certain form. Next we construct all rational solutions of diophantine equation (1) by using the addition theorems valid for the Weierstrass function, Specific examples are finally worked out for the cases

An R-module is a simple divisible module if it is a nonzero divisible module that has no proper non-zero divisible submodules. We study simple divisible modules and their endomorphism rings, give some examples and determine all simple divisible modules over some classes of rings.

The purpose of this note is to prove that a regular accessible category has a full regular embedding into a set-valued functor category.

It is shown that for totally real number fields the Main Conjecture in Iwasawa-Theory for p = 2 proposed by Fédérer implies the 2-primary part of the Birch-Tate-Conjecture in analogy with the case p odd.

Balinski has shown that the graph of a d-polytope is d-connected. In this note we give a new proof of Balinski's theorem.

It is shown that the concepts of normal radical as introduced by Jaegermann [2] and of invariant radical as introduced by Anderson, Divinsky and Sulinski [1] are equivalent.