We consider non-zero polynomials f(x1, …, xk) in k variables x1, …, xk with coefficients in the finite field GF[q] (q = pn for some prime p and positive integer n). We assume that the polynomials have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. By the degree of a polynomial f(x1, …, xk) will be understood the ordered set (m1, …, mk), where mi is the degree of f(x1 ,…, xk) in x1(i = 1, 2, …, K). The degree (m,…, mk) of a polynomial will be called totally positive if mi>0, i = 1, 2, …, k.

It is well-known that a homomorphism ø(A→B) between groups A and B induces a homomorphism ø*(ZA→ZB) between the corresponding group rings ZA and ZB over the ring of integers Z. The identical congruence O on B and the unit element eB of B can be characterised by the equations x–y = 0 and x–eB = 0 (x,y ∈ B) respectively. Similarly the congruence Γø corresponding to ø and the corresponding normal subgroup of A are

and {x∈A1 = A,(x–eA)ø = 0} respectively.

The object of this note is to generalize the notion of quasi-monotony for sequences of real numbers and to prove corresponding generalizations of certain known theorems. First, we recall the definition of quasi-monotony.

In this note we ask two questions and answer one. The questions can be combined as follows:

Does there exist a polynomial of the form

which starts with prescribed complex coefficients c0, …, cr–1; and satisfies

These differ from the classical problems of Carathéodory in one essential respect: the values of p and its first r–1 derivatives are given at the point z = 1 on the circumference of the unit circle, while in the original problem they were given at z = 0. Carathéodory's own answer was in terms of his “moment curve”, but the forms studied a few years later by Toeplitz yield a more convenient statement of the solution.

In [1], R. L. Goodstein has extended some well-known theorems on functions and equations in a Boolean algebra to the case of a distributive lattice L with 0 and 1. The purpose of this paper is to prove that most of Goodstein's theorems, as well as some additional results, are still valid in the case when L is not required to have least and greatest elements.

The motion of a rigid sphere embedded in an adhering medium and subjected to an external force is analysed exactly in the context of classical elastodynamics. For the limiting case of an incompressible medium it is possible to write down a simple second order differential equation relating displacement of the ball to the external force.

The theory is generalised to the case of a viscoelastic solid and the results obtained are pertinent to recently developed experimental methods for testing the dynamical mechanical properties of very low modulus polymer gels.

Let be any given infinite series with sn as its n-th partial sum.

We write

and

where

I am indebted to Dr. B. Kuttner for having kindly drawn my attention to certain corrections necessary in my paper with the above title, appearing in these Proceedings, Vol. 15 (Series II), Part 1, June 1966, pages 47–55.

The major corrections are as follows. On pages 48-9, Lemma 1 should appear without the factor log (1 – z)−1 of the denominator in the integrand at the end of the lemma; the formula in the proof of Lemma 1 is incorrect, its correct form being