Let Mbe a σ-field of subsets of a space X. A partition of Xmeans a countable partition Π of Xinto sets belonging to M; the set of partitions is directed by refinement. A. Kolmogoroff in 1930 [1] discussed an integral (Moore-Smith limit as Π gets finer) for set-functions F defined on M. When it exists, IF is σ-additive, and if by chance F is already σ-additive, then IF = F

Journal of the Australian Mathematical Society 4 (1964), 452–453

The second paragraph should be deleted. The alleged commutator identity (3) is false and is certainly not due to Philip Hall. The correct form is

as Dr. N. D. Gupta of Canberra has pointed out to me. According to Professor B. H. Neumann, this identity appeared in his (Professor Neumann's) thesis.

Nevertheless the theorem is valid and the proof given is correct.

A polygon is said to be rational if all its sides and diagonals have rational lengths. I. J. Schoenberg has posed the interesting problem, “Can any polygon be approximated as closely as we like by a rational polygon?” Besicovitch [2] proved that right-angled triangles and parallelograms can be approximated in Schoenberg's sense, the proofs were improved by Daykin [5]. Mordell [7] proved that any quadrilateral can be approximated by a rational quadrilateral. By adapting Mordell's proof, Almering [1] generalised Mordell's result by showing that, if A, B, C are three distinct points with the distances AB, BC, CA all rational, then the set of points P for which PA, PB, PC are rational is everywhere dense in the plane that contains ABC. Daykin [4] extended the results of Besicovitch [3] and Mordell [7] by adding the requirement that the approximating quadrilaterals have rational area.

We consider the motion of a particle of mass m and electrical charge e, moving in a constant magnetic field Bk, where k is a unit vector, and acted upon by a force mf(t). The position vector r(t) of this particle is governed by the differential equation where .

The subject matter of this note is the notion of a dependence structure on an abstract set. There are a number of different approaches to this topic and it is known that many of these lead to precisely the same structure. Axioms are given here to specify the minimal dependent sets for such a structure. They are closely related to conditions introduced by Hassler Whitney in [1] and to a certain “elimination axiom” for the independent sets.

This paper comprises a number of applications of the results of Part I. We use essentially the same notation as in Part I with a few additions necessary for the problems at hand.

In this paper we study certain operators allied to the Poisson operator and the transforms Hα(f) considered by the author in [2]. We define the integrals ψ(α)α(f) and θ(α)a(f) as follows:

In this note we present some results on relationships between certain verbal subgroups of metabelian groups. To state these results explicitly we need some notation. As usual further [x, 0y] = x and [x, ky] = [x, (k—1)y, y] for all positive integers k. The s-th term γs(G) of the lower central series of a group G is the subgroup of G generated by [a1, … as] for all a1, … as, in G. A group G is metabelian if [[a11, a2], [a3, a4]] = e (the identity element) for all a1, a2, a3, a4, in G, and has exponent k if ak = e for all a in G.

We consider in this paper only commutative rings with identity. When R is considered as a subring of S it will always be assumed that R and S have the same identity. If R is a subring of S an element s of S said to be integral over R if s is the root of a monic polynomial with coefficients in R. Following Krull [8], p. 102, we say s is almost integral over R provided all powers of s belong to a finite R-submodule of S. If R1 is the set of elements of S almost integral over R we say R1 is the complete integral closure of R in S.

In Part I of this paper we shall be concerned with the representation as convolutions of continuous linear operators which act on various function- spaces linked with a locally compact group and which commute with left — or right — translations; cf. the results in [12]. For completeness some known results are included whenever they follow from the general procedure. We have tried to follow simple general approaches as much as possible.

Equations of motion of a vibrating string are established in terms of the transverse and longitudinal displacements. These equations contain the terms of lowest order which are neglected in the classical treatment with vanishing amplitude. These extra terms lead to the natural modes being dependent on amplitude. By a simple procedure a solution of these equations is obtained which separates, as in the classical theory. The familiar circular functions are replaced by a Mathiew Function of position and a Jacobi elliptic function of time. Agreement with a previous study is shown.

The study of topological semirings, initiated by Selden [5], arises naturally from the theory of topological semigroups. It is of interest to take a known multiplication and investigate the possible additions. Selden has done this in [5] for several compact semirings.

In a semigroup S the set E of idempotents is partially ordered by the rule that e≦ƒ if and only if eƒ=e=ƒe. We say that S is an ω-semigroup if E={ei: i=0, 1, 2, …}, where

Bisimple ω-semigroups have been classified in [10]. From a group G and an endomorphism α of G a bisimple ω-semigroup S(G, α) can be constructed by a process described below in § 1: moreover, any bisimple ω-semigroup is isomorphic to one of this type.

The definition of a power-free group will be found in [1]. It is a partial algebraic system which, roughly speaking, may be thought of as a group in which (with the exception of the identity) squares and higher powers of an element are not defined.

It has been shown [1, Theorem 3.3] that the usual cancellation laws need not hold in a power-free group. When these laws do hold, the power-free group is called cancellative. In this paper we shall be solely concerned with cancellative power-free groups and the term ‘power-free group’ is to be understood to mean ‘cancellative power-free group’.

Let R be a commutative ring, with an identity element. It is the purpose of this note to establish conditions for an arbitrary but fixed ideal a of R to satisfy the distributive law

for all ideals b and c of R. In particular, in the Noetherian case, this will be related to the decomposition of a into prime ideals. We start with

Proposition 1. For a fixed ideal a in a commutative ring R with an identity element, the following conditions are equivalent.