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

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.

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.

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.

If the elements of a symmetric matrix lie in the real field it is well known that the roots of its characteristic equation are real. This implies that the discriminant of that equation (i.e. the product of the squared differences of the roots) is a polynomial in the elements which is non-negative and the same must be true for the leading coefficients of all the other Sturm functions associated with the characteristic equation. One would expect that it should be possible to express them as a sum of squares. Conversely, such an expression would establish the reality of the roots.

The inequality

which is valid for positive, non-null f, g in the spaces Lp(0, ∞), Lq(0, ∞), where p > 1, (1/p)+(1/q) = 1, is a well-known generalisation of the classical inequality of Hilbert (see for instance Chapter 9 of Hardy, Littlewood, and Polya (1)).

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

Let A be a commutative, semi-simple, convolution measure algebra in the sense of Taylor (6), and let S denote its structure semigroup. In (2) we initiated a study of some of the relationships between the topological structure of A^ (the spectrum of A), the algebraic properties of S, and the way that A lies in M(S). In particular, we asked when it is true that A is invariant in M(S) or an ideal of M(S) and also whether it is possible to characterise those measures on S which are elements of A. It appeared from (2) that if A is invariant in M(S) then S must be a union of groups and that A^ must be a space which is in some sense “ very disconnected ”. In (3) we showed that if A^ is discrete then A is “ approximately ” an ideal of M(S). (What is meant by “ approximately ” is explained in (3); it is the best one can expect since algebras which are approximately equal have identical structure semigroups and spectra.) In this paper we round off some of the results of (2) and (3). We show that if A is invariant in M(S) then A^ is totally disconnected, and that if A^ is totally disconnected then S is an inverse semigroup (union of groups). From these two crucial facts it is fairly straight-forward to obtain a complete characterisation of algebras A (and their structure semigroups) for which (i) A^ is totally disconnected, (ii) A is invariant in M(S), or (iii) A is an ideal of M(S).

G denotes a locally compact abelian group and M(G) the convolution algebra of regular bounded Borel measures on G. An ideal I of M(G) closed in the usual (total variation) norm topology is called an L-ideal if μ ∈ I, ν≪ μ (ν absolutely continuous with respect to μ) implies that ν ∈ I. Here we are concerned with the L-idealsL1(G), , and M0(G) where, as usual, L1(G) denotes the set of measures absolutely continuous with respect to Haar measure, denotes the radical of L1(G) in M(G) and M0(G) denotes the set of measures whose Fourier-Stieltjes transforms vanish at infinity.

In this paper we investigate some triple equations involving the inverse of the finite Mellin transform MR which is defined by the equation

This transformation is one of four which were first introduced by D. Naylor in his paper (1) and some of its properties have been listed by the author in paper (2), where its relationship to the Mellin transform is discussed in detail.

The recent papers (6), (7) of J. T. Marti have revived interest in the concept of extended bases, introduced in (1) by M. G. Arsove and R. E. Edwards. In the present note, two results are established which involve this idea. The first of these, which is given in a more general setting, restricts the behaviour of the coefficients for an extended basis in a certain type of locally convex space. The second result extends the well-known weak basis theorem (1, Theorem 11).

An irreducible curve in S4, projective 4-space, may arise as the complete intersection of three given irreducible threefolds. At a simple point P on such a curve there is an osculating solid, and we would like to have its equation. This solid, necessarily containing the tangent line to the curve at P, belongs to the net spanned by the tangent solids at P to the threefolds. We seek the appropriate linear combination of the known equations for these tangent solids.