The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.

Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kind

and the triple series equations of the second kind

where f, f1, g, g1h and h1 are all known functions,

is the Jacobi polynomial (3).

Let {pn} be a positive sequence. The Nörlund transformation (N, pn) maps the sequence {Sn} into the sequence {tn} by means of the equation

where

Circularly and transversely polarised (henceforth called circular and transverse) waves have been shown to occur as solutions of non-linear equations governing a wide range of physical phenomena, including finite elasticity (1), magnetohydrodynamics (2), and gyromagnetism (3), but only when the material properties of the medium are isotropic with respect to the direction of wave propagation. This paper is an attempt to unify and generalise these results.

Over a field of characteristic p>0 the group algebra of a finite group has a unique maximal nilpotent ideal, the Jacobson radical of the algebra. The powers of the radical form a decreasing and ultimately vanishing series of ideals and it would be of interest to determine the least vanishing power. Apart from the work of Jennings (3) on p-groups little is known in general (cf. (5)) about this particular power of the radical (cf. Remarks of Brauer in (4), p. 144. Problem 15). In this paper we give non-trivial lower bounds for the index of the least vanishing power of the radical when the group is p-soluble. Of the lower bounds we give we show that that lower bound, which is dependent solely on the order of the group, is the best possible such bound and that this bound is invalid if the assumption of p-solubility is omitted.

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

We shall call a finite semigroup S arithmetical if there exists a positive integer N and a monomorphism μ of S into the multiplicative semigroup RN of the ring of residue classes of the integers modulo N. In 1965 P. C. Baayen and D. Kruyswijk [1] posed the problem' Is every finite commutative semigroup arithmetical? ' The purpose of this paper is to answer this question.