Most work in genetic algebras has been concerned with inheritance which is symmetric with respect to sex, in that the characters studied are determined by genes located at autosomal loci, and it is assumed that the segregation pattern is the same in males and females. When asymmetric situations are studied, the development of the theory is complicated by the higher dimensions of the algebras, and by a feature to which Etherington (3, p. 40) drew attention, namely the fact that the passage from the gametic to the zygotic algebra no longer quite corresponds to the process of duplication, as it does in the symmetric case. Etherington gave some results for the gametic and zygotic algebras of a single sex linked diallelic locus, and its properties were discussed further by Gonshor (4, p. 44). In a second paper (5, p. 334) Gonshor studied sex linkage in the case of multiple alleles, choosing a canonical basis which exhibited very clearly the multiplication table and ideal structure of the algebra. His treatment from the statement of the multiplication table in terms of the natural basis to its expression in terms of a canonical basis, is repeated in the displayed relations (4)–(8) below, for completeness and to establish the present notation.

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

If we consider a semigroup, its algebraic structure may be such that it is isomorphic to a subsemigroup of a group, or is algebraically embeddable in a group. This problem was investigated in 1931 by Ore who obtained in (4) a set of necessary conditions for this embedding. A necessary condition is that the semigroup should be cancellative: for any a, x, y in the semigroup either xa = ya or ax = ay implies that x = y. Malcev in (3) showed that this was not sufficient. It is enough to note that his example was a non-commutative semigroup: a commutative cancellative semigroup is embeddable algebraically in a group.

In this paper we introduce a generalised Hankel operator and generalised Erdélyi-Kober operators and deduce some relations between them. The operators are then applied to obtain solutions to some dual integral equations which have applications in diffraction theory.

A smooth map of a differentiable n-manifold into Euclidean (n+k)-space is called an immersion if its Jacobian has rank n at each point of M. If f is also 1-1, it is called an embedding.

Some Hilbert spaces of continuous functions satisfying a mean value property are studied in which the generalised eigenfunctions of any selfadjoint operator again satisfy the same mean value property. Applications are made to nullspaces of some differential operators. The classes of functions involved in these applications are less general than those studied by K. Maurin (6); however, the Hilbert space norms may be arbitrary, while Maurin only considered L2-norms.

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequality

then it is convergent. This I was able to prove by considering the corresponding difference equation

Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by letting

where α(x, y) is a non-zero element of K, subject to the condition that

which is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and if G has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.

We shall use results of Palmer (10, 11) and of Edwards and Ionescu Tulcea (6) to show that a commutative V*-algebra (with identity) of operators on a weakly complete Banach space is isomorphic to such an algebra on a Hilbert space, the isomorphism extending to the weak closures of the algebras. This result leads to an extension of Stone's theorem on unitary groups (a similar extension is proved by different methods in (2, p. 350) and of Nagy's theorems on semigroups of normal operators. The same technique yields an easy proof of Dunford's theorem on the existence of a σ-complete extension of a bounded Boolean algebra of projections on a weakly complete Banach space. We are indebted to H. R. Dowson for suggesting this topic and for help and guidance in pursuing it.

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

Throughout this note A denotes a ring with identity, and “ module ” means “ left unitary module ”. In (2), C. Yohe studied elemental annihilator rings (e.a.r. for brevity). An e.a.r. is defined as a ring in which every ideal is the annihilator of an element of the ring. For example, a semi-simple, Artinian ring is an e.a.r. A is a l.e.a.r. (left elemental annihilator ring) if every left ideal is the left annihilator of an element of the ring. A r.e.a.r. (right elemental annihilator ring) is denned similarly.