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.

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:

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 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.

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.

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 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.

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.