A barrel in a locally convex Hausdorff space E[τ] is a closed absolutely convex absorbent set. A σ-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled (countably barrelled) if every barrel (σ-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (σ-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2).

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

The purpose of this note is to solve a problem of Dr A. M. Sinclair. Denote by Aw(I, T) the algebra with identity generated by a bounded linear operator T in the weak operator topology. We prove the following result.

Let G be a finite group and let S be a subgroup of G with

The linear differential system

where w is a vector with n components and A is an n by n matrix is said to have z = 0 as a regular singular point if there exists a fundamental matrix of the form

such that S is holomorphic at z = 0 and R is a constant matrix ((1), p. 111; (2), p. 73). For such systems A will have at most a pole at z = 0 and we may write

where p is an integer, Ã is holomorphic at z = 0, and Ã(0) ≠ 0. However, the converse is not true. When p ≦ − 1, A is holomorphic at z = 0, and every fundamental matrix is holomorphic at z = 0. If p ≧ 1, the non-negative integer p is called (after Poincaré) the rank of the singularity and there is a significant difference between the cases p = 0 and p ≧ 1. If p = 0 the linear differential system (1) is known to have z = 0 as a regular singular point ((1), p. 111) ; whereas, if p ≧ 1, z = 0 may or may not be a regular singular point.

In his recent book on ordinary differential equations Hille (3) devotes a chapter to complex oscillation theory. Drawing upon his own work in this area and the work of Nehari, Schwarz, Taam, and others, he gives a variety of oscil-lation and nonoscillation theorems for solutions of the differential equation

where z is a complex variable and p is regular in some appropriate domain. There are a number of results for (1.1) with an arbitrary coefficient o and some discussions for special cases of classical interest, such as the Bessel and Mathieu equations. There is a bibliography at the end of the chapter. For other recent work in this area attention is directed to papers by Herold (1, 2) Kim (4, 5) and Lavie (6) where other references are given.

Let R be an associative, commutative ring with identity, and let A be a (unitary) R-module. It is well known that if A is a Noetherian R-module then every submodule of A has a primary decomposition in A. The object of the present paper is to dualise this result; that is, to show that if A is an Artinian R-module then every submodule of A can be expressed as the sum of a finite number of coprimary submodules of A.

Let X be a normed linear space. We regard X as a subspace of its bidual X**. Polars will always be evaluated in the pair (X**, X*). We denote the closed unit ball in X by U, so that U0, U00 are the closed unit balls in X*, X** respectively. The weak topology induced by X on X* (the “weak-star” topology) will be denoted by σ(X), and cl() will denote σ(X)-closure.

Recently several papers on varieties of topological groups have appeared. In this note we investigate the question: if Ω is a class of topological groups, what topological groups are in the variety V(Ω) generated by Ω that is, what topological groups can be “manufactured” from Ω using repeatedly the operations of taking subgroups, quotient groups and arbitrary cartesian products? We seeka general theorem which will be useful for investigating V(Ω) for well-known classesΩ.

Let pn>0 be such that pn diverges, and the radius of convergence of the power series

is 1. Given any series σan with partial sums sn, we shall use the notation

and

Let {Kn} be a sequence of complex numbers, let

and let

Let D be the open unit disc {z: |z| <1}, let be its closure and let .

The primary object of this paper is to prove the two theorems stated below, the first of which generalises a result of Copson (1).

In (5) and (6) we studied certain subgroups of infinite dimensional linear groups over rings. In particular we investigated how the structure of the subgroups was related to the structure of the rings over which the linear groups were defined. It became clear that it might prove useful to study generalised nilpotent properties of rings analogous to Baer nilgroups and Gruenberg groups. We look briefly at some classes of generalised nilpotent rings in this paper and obtain a lattice diagram exhibiting all the strict inclusions between the classes.

In (6) Taylor has introduced the notion of a convolution measure algebra. In the same paper he constructed a canonical embedding of an arbitrary, semisimple commutative convolution measure algebra A into the algebra M(S) of all bounded, regular Borel measures on a compact semigroup S. This embedding has the properties that A is σ(M(S), C(S)-dense in M(S), that if μ is in A and ν is absolutely continuous with respect to μ, then ν is in A and that the set A^ of non-zero complex homomorphisms of A can be identified with the set S^ of continuous semicharacters of S where h ∈ A^ is identified with χ ∈ S^ by the equation

When Etherington (2) introduced linear commutative non-associative algebras in connection with problems in theoretical genetics, he pointed out that various sequences of elements in these algebras represented different mating systems. In all such systems it was however assumed that the generations did not overlap, and this restriction has been kept in later work in this field. In this paper we treat sequences which make it possible to find the probability distribution in successive generations in a discrete time model where the generations may be overlapping. We also consider idempotents in genetic algebras and outline how the method used on the overlapping generation sequence may be applied to other sequences.

In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all the properties of the twenty-seven lines would not become apparent until a better notation than they had given was found. This notation was discovered by Schläfli in 1858 in the double-six theorem (henceforth referred to as given five skew linesa1, …, a5with a single transversal b6such that no four of the ai lie in a regulus, the four ai excluding aj have a second transversal bj and the five lines b1, …, b5thus obtained have a transversal a6—the completing line of the double-six. The other fifteen lines of the cubic surface are then , where ai bj is the plane containing ai and bj.

This paper is a continuation of (4). The main aim of this paper is the introduction of the concept of sex-linked duplication. In addition, we shall give several equivalent definitions for the concept of a genetic algebra and make several remarks on overlapping of generations.

It is well known that sufficient conditions for the existence of a positive vector u which satisfies the matrix equation Au = λu are that A should be non-negative and irreducible. This result, the qualitative part of the Perron-Frobenius theorem, has been proved in a variety of ways, one of the most attractive of which is that given by Alexandroff and Hopf in their treatise “ Topologie ”. The aim of this note is to show how their method can be adapted to deal with the generalised eigenvalue problem defined by Au = λBu where A and B are square matrices.

Earlier this year Professor W. L. Edge drew my attention to the paper (1). Since Black probably wrote his paper about 1890, the integral he considered must have been one of the earliest n-variate integrals to be evaluated. The present paper generalizes Black's result from a column vector as variate to a rectangular matrix—his integral is the case p = m = k = 1 of the integral J below.

In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric function

in the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.

In (2) a finite p-group G is said to be nearly regular if it has the following two properties:

(i) There exists a central subgroup Z of order p and G/Z is regular.

(ii) If x ∈ G and y ∈ γ2(G), then gp{x, y} is regular.