Over a field of prime characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative.

Von Neumann's definition of the continuous sum of Hilbert spaces led Segal [3] to define the continuous sum of measures on a measurable space. In this note we employ Segal's definition to investigate the measure structures associated with a self-adjoint transformation of pure point spectrum and a self-adjoint transformation of pure continuous spectrum. While these transformations, as operators on separable Hilbert spaces, are the antithesis of each other we show that in their measure structure one is a particular case of the other.

where be an entire

function represented by a Dirichlet series whose order (R) and proximate order (R) are respectively ρ (0 < ρ < ∞) and ρ(σ). For proximate order (R) and its properties, see the paper of Balaguer [4, p. 28].

Let G be a graph with p points and q lines, and genus γ. The thicknesst(G) has been introduced as the minimum number of planar subgraphs whose union is G. This topological invariant of a graph has been studied by Battle, Harary and Kodama [1], Tutte [7], Beineke, Harary and Moon [3], and Beineke and Harary [2].

It is natural to generalise this concept of the thickness of a graph to the union of graphs with a specified genus. We say that the n-thickness of G is the minimum number of subgraphs of genus at most n whose union is G. Denoting the n-thickness of G by tn, we write in particular t0 the 0-thickness, i.e., the thickness.

Let (L, ≦) be a distributive lattice with first element 0 and last element 1. If a, b in L have complements, then these must be unique, and the De Morgan laws provide complements for a ∧ b and a ∨ b. We show that the converse statement holds under weaker conditions.

Theorem 1. If(L, ≦) is a modular lattice with 0 and 1 and if a, b in L are such that a ≦b and a ≨ b have (not necessarily unique) complements, then a andb have complements.

The Lucas numbers υn and the Fibonacci numbers υn are defined by υ1 = 1, υ2= 3, υn+2 = υn+1 + υn and u1 = u2 = 1, un+2 = un+1 + un for all integers n. The elementary properties of these numbers are easily established; see for example [2].However, despite the ease with which many such properties are proved, there are a number of more difficult questions connected with these numbers, of which some are as yet unanswered. Among these there is the well-known conjecture that un is a perfect square only if n = 0, ± 1, 2 or 12. This conjecture was proved correct in [1]. The object of this paper is to prove similar results for υn, ½un and ½υn, and incidentally to simplify considerably the proof for un. Secondly, we shall use these results to solve certain Diophantine equations.

Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.

Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F.

A graph Gn consists of n distinct vertices x1x2, …, xn some pairs of which are joined by an edge. We stipulate that at most one edge joins any two vertices and that no edge joins a vertex to itself. If xi, and xj are joined by an edge, we denote this by writing xi ∘ xj.

Consider a second graph HN, where n ≦ N. Following Rado [1], we say that a one-to-one mapping/of the vertices of Gn into the vertices of HN defines an embedding if xi ∘ xj implies f(xi) ∘ f(xj), and conversely, for all i, j = 1, 2,…, n. If there exists an embedding of Gn into HN, we denote this by writing Gn <HN. The particular graph HN is said to be n-universal if Gn < HN for every graph Gn with n vertices.

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

In what follows all small Latin letters denote non-negative integers or functions whose values are non-negative integers. Let N = (n1, …, nj) be a j-dimensional vector and let q = q (k; N) = q(k; n1, …, nj) be the number of partitions of N into just k parts, each part being a vector whose components are non-negative integers. We write

for the generating function of q. We have

.

In this note, we first establish an integral transform pair where the kernel of each integral involves the Gaussian hypergeometric function. Special cases of Theorem 1 have been studied by several authors [1, 2, 5, 6]. In Theorem 2 a similar integral transform pair involving a confluent hypergeometric function is given.

In recent years interest in the mixed boundary value problems of mathematical physics has increased appreciably because of various applications. The mixed boundary value problems for simply-connected regions have been investigated widely and it can reasonably be hoped that within a short time the theory will reach a satisfactory stage. It appears, however, that very few problems for multiply-connected domains have been solved. Recently Srivastav [2] has considered the problem of rinding an axisymmetric potential function for a half space with a cylindrical cavity subject to mixed type boundary conditions. In a subsequent paper [1], Srivastav extends the analysis to the asymmetric problem and formulates the problem in terms of dual integral equations involving Bessel functions of the first and second kinds whose solution leads to the solution of the potential problem. The latter paper, however, involves heavy manipulations and complicated contour integrals.

In this note we consider the problem of determining the stress on the boundary y = 0 of the elastic half-plane y ≡ 0 when there are prescribed body forces acting in the interior and the boundary is free from applied stress. Expressions for the components of stress at a general point of the half-plane when the imposed body force is concentrated at a single point have been derived by Melan [1], Sneddon [2] and Green [3], each author making use of a different method.

Let and Hn(x) be the nth Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101]

and the mehler formula [1, p. 377]

This suggests the following problem. Consider the equation

where fa(x) is a polynomial in x of degree n with highest coefficient equal to 1,

A0 = B0 = 1. We shall also assume that ak = 1 and y0y1y2 … yk–1 ≠ 0. We seek all sets of polynomials {fn(x)} which satisfy (1.3), (1.4) and (1.5).