In his paper on sets of primes with intermediate density Golomb1 proved the following theorem: Let 2 < p1 < P2 < … be any sequence of primes for whichfor every i and j. Denote by A (x) the number of P's not exceeding x. Then.

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.

It is known that the theory of Cauchy's problem for differential equations with two independent variables is réducible to the corresponding problem for systems of quasi-linear equations. The reduction is carried further, by means of the theory of characteristics, to the case of systems of equations of the special form first considered by H. Lewy [1]. The simplest case is that of the pair of equationswhere the aii depend on z1 and z2. The problem to be considered is that of finding functions z1(x, y), z2(x, y) which satisfy (1) and which take prescribed values on x + y = 0.

Let G be a group of order pn where p is some prime. Denote by nr (G) the number of subgroups of G of order pr. If H is the elementary abelian group of order pn, then.

This paper is chiefly concerned with inequalities for the numbers of subgroups of a finite p-group. The following are typical results. Let G be a p-group of order pa, and let nG(pk) denote the number of subgroups of G of order pk.

Papers I, II of this projected series lay the algebraic foundations of the theory of the Clifford groups; I deals with the case p > 2, II with the case p = 2. The present introduction refers to both papers. Our theory has applications in group theory, geometry and number theory.

The present paper deals with the Clifford groups in the case p = 2. For the most part, it runs parallel to the previous paper I ([1]) on the case p > 2, and a number of proofs are therefore either given in outline or omitted. A general introduction to both papers is given in I, § 1.

A similarity method is used to develop a solution of the wave equation within a sector with mixed boundary conditions. In this manner the field which results from the diffraction of an incident pulse of step function time dependence is found.

The Siegel conjecture on the rational approximation to algebraic numbers was proved a few years ago by K. F. Roth [1] with the following theorem: Let α be any algebraic number, not rational. If has an infinity of solutions in integers h and q (q > 0) tehn k ≤ 2.

In the controversy which has been continuing over the “clock paradox”, it has been claimed by supporters ([1]) of the orthodox interpretation that the difference in behaviour of clocks carried by observers in motion relative to each other can be traced to differencesin the motion relative to the universe. Whilst A remains unaccelerated relative to the universe, B for a portion of his journey does not. Professor Dingle, in reply ([2]), remarked that he would “not dispute a possible slight effect” but could not understand why, if the acceleration of B were so important, it did not appear explicitly in the answer.

If A and B are two complex matrices and if U is a complex unitary matrix such that UAUCT = B (where UCT denotes the conjugate transpose of U), then A and B are said to be unitarily similar. Necessary and sufficient conditions that two matrices be unitarily similar have been dealt with in [5] (from the point of view of group representation theory) and in [2] (from the point of view of developing a canonical form under unitary similarity).