The purpose of this article is to give an introduction to the study of a class of stochastic partial differential equations and to give a brief review of some of the recent developments in this field. This study has evolved naturally out of the theory of stochastic differential equations initiated in a pioneering paper of K. Itô [13]. In order to set this review in its appropriate setting we begin by considering a simple scalar stochastic differential equation.

These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form

(1)

in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.

The aim of this paper is to give a number of characterizations of the rings of the title. In particular, these turn out to be precisely those exchange rings whose idempotents are all central. They are also those rings in which every element is the sum of a unit and a central idempotent.

This note is concerned with some new integral inequalities which are extensions of the results in [2]. The method by which these results are obtained is due to D. C. Benson [1]. Throughout the present note we shall assume 1<p<∞ and f(x) a non-negative measurable function.

The coarseness, c(G), of a graph G is the maximum number of edge disjoint nonplanar subgraphs contained in G For the n-dimensional cube Qn we obtain the inequalities

If f is a suitable meromorphic function then, by a classical technique in the calculus of residues, one can evaluate in closed form series of the form

1.1

The notion of functionality for an internal horn functor H in a concrete category K was introduced in Banaschewski and Nelson [1], formalizing the condition that the structure on the H(A, B) is "pointwise" structure on sets of functions.

In this note, we have shown that the set of conditions given by Gel'fand and Vilenkin for the positive definiteness of a functional L(ϕ) [1, Theorem 7, p. 285] though sufficient is not necessary, thereby providing an answer to an open problem posed by them [1, Theorem 7, p. 285].

We define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that

(1)

(2)

In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined by

where the summation is over a square-reduced residue system (mod r).

B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and

(3)

As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function

Equivalence problems in Riesz-summability lead to the problem of finding the zeros of special power series in the unit circle, i.e.

Let P stand for a polynomial set (p.s.), i.e., a sequence {P0(x), P1(x), P2(x),...} such that for each n P0(x) is a polynomial in x of exact degree n and P0(x)≠0. We refer to Pn(x) as the nth component of P.

L'objet de cette note est de montrer que le dérivé d'un cuboïde eulérien n'est jamais parfait.

(1) Appelons cuboïde entier un cuboïde dont les arêtes et les diagonales des faces sont des entiers; appelons cuboïde parfait un cuboïde entier dont la diagonale intérieure est un entier.

Let A1, A2 be commutative semi-simple Banach algebras and M(A1), M(A2) their multiplier algebras. Birtel in [2] has proved that every isomorphism of A1 onto A2 induces an isomorphism of M(A1) onto M(A2). In this note, we extend this result to the noncommutative case. We also show that if A is a dual A*-algebra which is a dense two-sided ideal of a B*-algebra B, then M(A) is isomorphic to M(B). Thus the converse of the previous result cannot hold. All algebras under consideration are over the complex field.