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 equation

need not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where

The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).

In the present paper we determine the Laplace transforms of the modified Bessel function of the second kind Kn(t±mx), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part.

We prove that if t is a compact linear operator that is not quasi-nilpotent and is appropriately normalised, then the closed semi-algebra A(t) generated by t is locally compact. The theory of locally compact semi-algebras (2) is therefore applicable to A(t), and we show that it can be used to obtain spectral properties of t.

Definition. The isogonals of the medians of a triangle are called the symmedians

If the internal medians be taken, their isogonals are called the internal symmedians or the insymmedians; if the external medians be taken, their isogonals are called the external symmedians, or the exsymmedians

The purpose of this paper is to provide a vector version of the characterisations of the multipliers for L1(G) (see (9), p. 2). This problem was considered by Akinyele (1). However, the Banach algebras involved in that paper are commutative semi-simple Banach algebras and the proof of the main theorem seems to be incomplete. Indeed we give at the end of this paper an example which shows that statements (i) and (ii) of Theorem 3.2 of Akinyele ((1), p. 490) are not equivalent in general.

M. L. Cartwright has given ((2), 180–181) the following theorem, together with a neat proof of it.

Theorem C.Suppose that f(z) is regular and

in the half-strip S

of the complex plane.

Suppose also that for some constant α in α<a<β

as y→∞. Then for every δ>0.

uniformly asy→∞ for α+δ≦x≦β–δ.

§ 1. 1 have ventured to bring the formulae of this paper before the Society, as I have been unable to find reference to them in any text-book or any original contribution to mathematical literature which I have come across. I confine my attention completely to the central surface, as the corresponding formulae for the paraboloids are very readily deduced by a similar process.

The hypergeometric function1F(a, b; c; z) is analytic in the domain |arg(−z)| < π, and, when |z| < 1, may be represented by the series

When |z| = 1 in the domain |arg(−z)| <π, this series converges2 to F(a; b; c; z) if R(a+b−c) < 0 (integral values of a, b and c are excluded in the present paper).

The construction of all irreducible modules of the symmetric groups over an arbitrary field which reduce to Specht modules in the case of fields of characteristic zero is given by G. D. James. Halicioğlu and Morris describe a possible extension of James' work for Weyl groups in general, where Young tableux are interpreted in terms of root systems. In this paper we show how to construct submodules of Specht modules for Weyl groups.

If X is a Tychonoff topological space, and if βX is the Stone-Cech compactification of X, then βX\X will denote the complement of X in βX. If A is a subset of X, then cl [A: X] will denote the closure of A in X, and int [A: X] will denote the interior of A in X. In Isbell ((3), p. 119) a property of βX\X is called a property which X has at infinity, and it is the aim of this paper to give necessary and sufficient conditions for X to be finite at infinity. Since βX is T1 we can say that if X is finite at infinity, then βX\X is closed in βX. So we lose nothing by restricting our attention to locally compact, Tychonoff spaces, and for the remainder of the paper X will denote such a space.

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.

1. Theorem I. If , where a and n are positive integers, and if f(n) is a finite single-valued function of n for values of n蠇a, then

In a previous paper we studied the asymptotic distribution of the multiparameter eigenvalues of uniformly right definite multiparameter Sturm–Liouville eigenvalue problems. In this paper we extend the analysis to deal with multiparameter Sturm–Liouville problems satisfying uniform left definiteness, and non-uniform left and right definiteness.