If the set K of r+1 distinct integers k0, k1 …, kr has the property that the (r+1)r differences ki–kj (0≦i, j≦r, i≠j) are distinct modulo r2+r+1, K is called a perfect difference set modr2+r+1. The existence of perfect difference sets seems intuitively improbable, at any rate for large r, but in 1938 J. Singer [1] proved that, whenever r is a prime power, say r = pn, a perfect difference set mod p2n+pn+1 exists. Since the appearance of Singer's paper several authors have succeeded in showing that for many kinds of number r perfect difference sets mod r2+r+1 do not exist; but it remains an open question whether perfect difference sets exist only when r is a prime power (for a comprehensive survey see [2]).

1. A number of inclusion theorems have been given in connection with methods of summation which include the Riesz method (R, λ, κ). Lorentz [4, Theorem 10] gives necessary and sufficient conditions for a sequence to sequence regular matrix A = (an, v) to be such that A ⊃ (R, λ, 1)†. He imposes restrictions on the sequence { λn}, so that A does not include all Riesz methods of order 1. In Theorem 1 below, we generalize the Lorentz theorem by giving a condition without restriction on λn, If the matrix A is a series to sequence or series to function regular matrix, there do not appear to be any results concerning the general inclusion

A ⊃ (R, λ, κ).

However, when A is the Riemann method (ℜ, λ, μ), Russell [7], generalizing earlier results, has given sufficient conditions for (ℜ, λ, μ) ⊃ (R, λ, κ). Our Theorem 2 gives necessary and sufficient conditions for A ⊃ (R, λ, 1), where A satisfies the condition an, v → 1 (n →co, ν fixed). Thus Theorem 2 applies to any series to sequence regular matrix A. In Theorem 3 we give a further representation for matrices A which include (R, λ, 1), and finally make some remarks on the problem of characterizing matrices which include Riesz methods of any positive order κ.

1. We use Cassels's notation and define h (m, n), Q (m, n), Zh (s), Zh (1) – ZQ (1) and G (x, y) as in [1]. Rankin [5] proved that the Epstein zeta-function Zh (s) satisfies, for s ≧ 1·035, the

THEOREM. For s > 0, Zh (s) — ZQ (s) ≧ 0 with equality if and only ifh is equivalent to Q. Rankin then asked whether the theorem is true for all s > 1. Cassels [1] answered this question in the affirmative and proved further that the theorem is true for all s > 0.

The nature of the eigenvalues of a square quaternion matrix had been considered by Lee [1] and Brenner [2]. In this paper the author gives another elementary proof of the theorems on the eigenvalues of a square quaternion matrix by considering the equation Gy = μȳ, where G is an n x n complex matrix, y is a non-zero vector in Cn, μ is a complex number, and ȳ is the conjugate of y. The author wishes to thank Professor Y. C. Wong for his supervision during the preparation of this paper.

The nth order polylogarithm Lin(z) is defined for |z| ≦ 1 by

([4, p. 169], cf. [2, §1. 11 (14) and § 1. 11. 1]). The definition can be extended to all values of zin the z-plane cut along the real axis from 1 to ∝ by the formula

[2, §1. 11(3)]. Then Lin(z) is regular in the cut plane, and there is a differential recurrence relation [4, p. 169]

It is convenient to extend the sequence Lin(z) backwards in the manner suggested by (2) and define

Then Li1(z)= – log(l–z), and Lin(z) is a rational function of z for n= 0, – 1, – 2,…. Formula (2) now holds for all integers n.

In [2], Tosiro Tsuzzuku gave a proof of the following:

THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.

For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

In 1958 Ragab [3] deduced the sums of certain infinite series involving a product of two E-functions in terms of E-functions. MacRobert [2] gave a very simple alternative method for proving the results of Ragab. Later, Ragab [4, 5] in 1962 used a method similar to the one given by MacRobert to deduce a number of summations involving products of E-functions. In this paper, some more general summations of E-functions, which contain Ragab's results as special cases, are given. It may be mentioned that all the series summed run from n = – ∞ to + ∞co instead of n = 0 to + ∞.

Throughout this paper, E will denote a finite-dimensional vector space over an ordered field . The real number field will be denoted by ℜ and its rational subfield by . Many of the basic notions in the theory of convexity (convex set, extreme point, hyperplane, etc.) can be defined in the general case just as they are when , but their behaviour may be different from that in the real case. By way of example, we consider the following theorem (due essentially to Minkowski), which is of fundamental importance both for geometric investigations and for the applications of convexity in analysis:

(1) Suppose . If K is a convex subset of E which is linearly closed and linearly bounded, then. K = con ex K; that is, K is the convex hull of its set of extreme points.

The problem considered here is that of a torsional impulsive body force within a semi-infinite elastic solid. The surface of the solid is assumed to be either stress-free or rigidly clamped. The basic solution is found to be essentially the same as that for the surface loading of a half space previously considered by Eason [1]. The displacement is determined in terms of elementary functions for one particular type of body force, although using the results in [1] the solution can be obtained for other types of force. Some numerical results for the surface displacement of the stress-free solid are presented in graphical form.

Let K be any convex body in En, and K any given class of convex sets in En. Then we shall say that K is approximable by the class K if there exists a sequence of sets {Ki}, such that, as i→∞, Ki→K in a suitable metric (for example, the Hausdorff metric), where each set Ki is a vector sum of (a finite number of) sets of the class K An approximation problem is to determine necessary and sufficient conditions for K to be approximable by a given class K.

A slow steady motion of incompressible viscous liquid, bounded by an infinite rigid plane, which is generated when a rigid sphere of radius a moves steadily without rotation in a direction parallel to, and at a distance d from, the plane is considered. Use is made of bispherical coordinates, which were employed some years ago by G. B. Jeffery [1] and Stimson and Jeffery [2] in solving the axi-symmetrical problems in which the sphere is fixed and rotates about a diameter perpendicular to the plane, or when two spheres move without rotation along their line of centres in infinite liquid. The coordinate system has been used recently by Dean and O'Neill [3] in solving the problem in which the sphere is fixed and rotates about a diameter parallel to the plane.

Wave propagation in an elastic medium rotating with a constant angular velocity Ω about an axis is studied by the method of Lighthill [1]. The waves are created by a concentrated periodic force of fixed frequency ω at the origin. The changes in the wave pattern and the decay of amplitude are discussed when 2Ω. increases from zero to a value greater than ω.

Professor C. L. Siegel has pointed out that the statement following equation (9) on page 98 of [1] is false, but can be made correct by adding to the conditions (7) of [1] the further condition:

When Souslin and Lusin initiated and developed the theories of the Souslin operation, of projective sets and of analytic sets, they attached great importance to the constructive nature of their definitions (see [1], [2] and [3]). When Choquet (see [4], [5] and [6]) made his very successful extension of these theories to an arbitrary Hausdorff space Ω, he defined an analytic set in Ω to be a continuous image in Ω of a Kσδ-set in an unspecified compact Hausdorff space X. Thus, a priori, the construction of the analytic sets in Ω requires the preliminary construction of all compact Hausdorff spaces X.

In the theory of the interaction between simple electrically neutral systems with dipole moments, the interaction energy between two such systems when they are identical, one in an excited state and the other in the ground state, is of current interest. It is well-known that, within the Coulomb force approximation for the electron-electron interaction, the energy varies as

where q(r) is the electric dipole moment of the system r = 1, 2, and R is the vector displacement of system 2 from system 1. This is the so called resonance attraction between the systems. On the other hand it has been known since 1948 (see [1]) that for two systems both in their ground states the potential of interaction falls off at large separation faster than the London formula for the energy, namely

predicts. In equation (2) α(r) is the polarization of the system r, in terms of the dipole moments (here induced)

where E is the energy separation between the two states considered, i.e., the ground state and the excited state reached from the ground state by electric dipole transitions. In fact the asymptotic form of the potential energy at separation was given by Casimir and Polder as

Let K be a finite algebraic extension of the rational number field Q, and let R denote the ring of algebraic integers in K. The algebraic integers in a finite extension field of K form a ring which may be considered as a module over R. The structure of these modules has been entirely determined in Fröhlich [2], where, in particular, necessary and sufficient conditions have been established deciding when such a module will be a free R-module.

In the present note we show that the elements of GF(q2) (q = 2n) can be represented in “polar form” in such a way that GF(q2) acts like an “Argand diagram” over its “real subfield” GF(q). From this polar representation it is easy to develop a trigonometry of the plane GF(q2), including definitions of circles and orthogonality. As an application of these ideas we show, in §4, that the circles and lines orthogonal to a given circle yield a new model satisfying Graves' axioms for finite homogeneous hyperbolic planes.

In 1956 Cassels proved the following result, which generalized a theorem of Marshall Hall on continued fractions. Let λ1 …, λr be any real numbers. Then there exists a real number α such that

for all integers u > 0 and for q = 1,…,r, where C = C(r) > 0. Thus all the numbers α+ λ1, …, α+ λr are badly approximable by rational numbers, which is equivalent to saying that the partial quotients in their continued fractions are bounded. In a previous paper I extended Cassels's result to simultaneous approximation. In the simplest case—that of simultaneous approximation to pairs of numbers—I proved that for any real λ1, …, λr and μ1, …, μr there exist α, β such that

for all integers u > 0 and for q=1,…, r, where again C = C(r) > 0. Both the construction of Cassels and my extension of it to more dimensions allow one to introduce an infinity of arbitrary choices, and consequently the set of α for (1) and the set of α, β for (2) may be made to have the cardinal of the continuum.