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.

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 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 + ∞.

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]).

We begin this paper by considering a Boolean algebra as a lattice which is relatively pseudo-complemented (i.e., residuated with respect to intersection) and give, in this case, certain properties of the equivalences of types A, B and F(as introduced by Molinaro [1]). We then show how these results carry over to the case of Boolean matrices, which form a Boolean algebra residuated also with respect to matrix multiplication. Other properties of matrix residuals are established and we conclude with three algebraic characterisations of invertible Boolean matrices.

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.

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.

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shape

with y > 0. We write (for s > 1)

The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

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.