The class of commutative rings known as Baer rings was first discussed by J. Kist [4], where many interesting properties of these rings were established. Not necessarily commutative Baer rings had previously been studied by I. Kaplansky [3], and by R. Baer himself [1]. In this note we show that commutative Baer rings, which generalize Boolean rings and p-rings, satisfy the Birkhoff conditions for a variety. Next we give a set of equations characterising this variety involving + and * as binary operations, – and as unary operations, and 0 as nullary operation. Finally we describe Baer-subdirectly irreducible commutative Baer rings and state the appropriate representation theorem.

Let {Pn} be a given sequence of constants, real or complx, such that then , defines the sequence {tn} of (N, pn) means of Σnan. The series Σnan is said to be summable I|N, pn|, if {tn} ∈ BV, −|−tn−1| ∞.

Let A and B denote subsets of an abelian group (G, +). The pair (A, B) is called a complementing pair for A + B provided each element of the sum is uniquely represented. A complementing pair (A, B) for C is denoted (A, B) ~ C.

Suppose that Ф is a topological space equipped with a Hausdorff topology and that T is a continuous function mapping Ф into Ф. We discuss the existence and uniqueness of fixed points of T and the convergence of the Picard sequence of iterates, from the viewpoint of the existence of a homotopy with special properties.

In 1957 Kurzweil [1] proved some theorems concerning a generalized type of differential equations by defining a Riemann-type integral, but he did not study its properties beyond the needs of that research. This was done by R. Henstock [2, 3], who named it a Riemann-complete integral. He showed that the Riemann-complete integral includes the Lebesgue integral and that the Levi monotone convergence theorem holds. The purpose of the present paper is to give a necessary and sufficient condition for a function to be Riemann-complete integrable and to establish a termwise integration theorem for a uniformly convergent sequence of Riemann-complete integrable functions.

Let H be a group, let {G1: i ∈ I} be a set of groups and, for each i, let θi be a a monomorphism: H → G1, with Hθ1 = H1. We call such a system of groups and monomorphisms an amalgam and denote it by [G1; H; θi; Hi]. By an embedding of the amalgam into a group G is meant a set of monomorphisms ϕi: G1 → G such that θiϕi= θjϕjfor all i, j and G1ϕi = Hθϕk, for all i, j, k.

In this paper solutions of the generalized metaharmonic equation in several independent variables where λ > 0 are uniquely decomposed into the sum of a solution regular in the entire space and one satisfying a generalized Sommerfeld radiation condition. Due to the singular nature of the partial differential equation under investigation it is shown that the radiation condition in general must hold uniformly in a domain lying in the space of several complex variables. This result indicates that function theoretic methods are not only the correct and natural avenue of approach in the study of singular ordinary differential equations, but are basic in the investigation of singular partial differential equations as well.

The study of complemented Banach*-algebras taken up in [1] was confined mainly to B*-algebras. In the present paper we extend this study to (right) complemented Banach*-algebras in which x*x = 0 implies x = 0. We show that if A is such an algebra then every closed two-sided ideal of A is a *-ideal. Using this fact we obtain a structure theorem for A which states that if A is semi-simple then A can be expressed as a topological direct sum of minimal closed two sided ideals each of which is a complemented Banach*-algebra. It follows that A is an A*-algebra and is a dense subalgebra of a dual B*-algebra U, which is determined uniquely up to *-isomorphism.

In this note we define a rather pathological connectedness property of Hausdorff spaces which is stronger than ordinary connectedness. We obtain a few basic properties of such spaces and derive a method for constructing them. It turns out that countable Hausdorff spaces having the connectedness property are as easily constructed as uncountable ones. Hence we have still another method for constructing countable connected Hausdorff spaces.

Let f(x) be continuous in (-π, π) and periodic with period 2π and let , The aim of this paper is to prove the following theorem.

We suppose throughout that G is a finite group with a faithful matrix representation X over the complex field. We suppose that X affords a character π of degree r whose values are rational (hence rational integers). If the matrices in some representation of G affording a character π0 are all permutation matrices, then π0 is called a permutation character. Permutation characters have non-negative integral values. In the general case, we consider what properties of permutation characters are true of π, and in particular, under what circumstances π is a permutation character. Note that assuming X to b faithful is equivalent to considering the image group X(G) instead of G.

In a previous paper of this series – hereinafter to be referred to as [1] – the author introduced a new method for a large class of boundary value problems connected with the flexure analysis of elastic plates of arbitrary shape where the concept of ‘Lines of Equal Deflection’, i.e. lines which are obtained by intersecting the bent plate by planes parallel to the original plane of the plate, was introduced. The present paper extends this analysis to the buckling analysis of thin elastic plates with various forms of boundary conditions. It is shown that the proposed method appears to be a powerful tool for the investigation of those problems of elastic stability which could not be solved by conventional methods because of the difficulty of the mathematical treatment.

The purpose of this note is to derive an alternative expression to that given in Lemma 1 below (due to Hsu, and to von Bahr) for the absolute moments of a random variable, in terms of the characteristic function.

Finite matrices with entries pij Fij (x1,…, xk), where {pij} is stochastic and Fij(.) is a k-variate probability distribution are discussed. It is shown that the matrix of k-variate Laplace-Stieltjes transforms of the Pij Fij(x1, …, xk) has a Perron-Frobenius eigenvalue which is a convex function in k variables in a suitably defined region. The values of the partial derivatives near the origin of this maximal eigenvalue are exhibited. They are quantities of interest in a variety of applications in Probability theory.

The cancellation law is a necessary condition for a semigroup to be embedded in a group. In general, this condition is not sufficient; necessary and sufficient conditions are rather complicated (see [1]). It is, therefore, of interest to find large classes of semigroups for which the cancellation law is sufficient to ensure embeddability in a group.

In this note we give a basis for the radical of the group algebra of a p-nilpotent group over a field of characteristic p in terms of the ordinary representation theory of the group. We use our result to calculate the exponent of the radical for such a group.

I. Šafarevič [2] proves that for a pro-finite p-group G, the deficiency of G (in the topological sense) equals the negative of d(m(G)). This means that if G is a finite p-group, where d(m(G)) = n, then there exists a group K with deficiency −n, such that G is the maximal p factor of K.