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.

The main object of this note is to show that a proof given by A. J. Macintyre [2] of a result on the overconvergence of partial sums of power series works more easily in the context of Dirichlet series. Applying this observation to the particular Dirichlet series Σane−ns, we can remove certain restrictions which Macintyre finds necessary in the direct treatment of power series.

In this paper two integrals involving E-functions are evaluated in terms of E-functions. The formulae to be established are:

where n is a positive integer,

and

where n is a positive integer,

and

the prime and the asterisk denoting that the factor sin {(s–s)π/2n} and the parameter βq+s–βq+s + 1 are omitted. The definitions and properties of MacRobert's E-function can be found in [1, pp. 348–352] and [3, pp. 203–206].

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

A. Geddes [1, Theorem 3.3] has shown that the partial algebraic system which he has called a power-free group need not be cancellative. In other words, there exist power-free groups containing at least one element a with the property that ab can equal ac when b ≠ c. In the present paper we propose to study the structure of such non-cancellative power-free groups, and we shall in fact obtain a complete solution to this problem.

1.1. Let A = (aμν) be a normal triangular matrix, i.e., one for which aμμ ≠ 0 (μ ≥ 0), aμν = 0 (ν > μ).

where (i) 0≤m<n, (ii) Rμ>0 (μ≥0), (iii) K is a constant, depending on the matrix A and the sequence {Rμ}, but independent of m, n and the finite sequence {sν}.

Let H be any closed bounded convex set in En, and -H be its reflection in the origin. Then the vector sum K = H+ (−H) has the origin as centre and is called the difference set of H. Clearly every closed bounded convex set K with centre at the origin is the difference set of ½K. Excluding this trivial case, we define such a set K to be reducible if it is the difference set of some H which is not homothetic to K.