A cardinal number which is too large to be reached by some process is generally said to be inaccessible by that process. Many kinds of inaccessible cardinals have been discussed and for a general survey the book of H. Bachmann [1, Chapter 7] may be consulted. We consider here two inaccessibility properties. We shall denote the cardinal of a set X by |X|. The first inaccessibility property will be called regularity: the cardinal| X| will be said to be regular if there does not exist a disjoint cover {X1: i ε I} of X such that

(i)|X1|<|X|, for each i in I, and

(ii)|I|<|X|.

1. It is a well known theorem due to Frobenius that the number of solutions of the equation

Xn = 1

in a finite group G, is a multiple of the greatest common divisor (n, g) of n and the order g of G. Frobenius himself proved later that the number of solutions of the equation

Xn = a

where a is a fixed element of G, is a multiple of (n, ga), ga being the order of the centralizer Z(a) of ain G.

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.

Let the closed orientable surface Fh of genus h be a covering of F2 and let and f be the fundamental groups respectively. Then is a subgroup of f of index n = h − 1. A covering is called regular if is normal in f.

Conversely, let be a normal subgroup of f of finite index. Then there is a uniquely determined regular covering Fh such that is the fundamental group of Fh. The covering Fh is an orientable surface. Since the index n of in f is supposed to be finite, Fh is closed, and its genus is given by n = h − 1.

The fundamental group f can be defined by

.

In the study of connected partially ordered spaces a problem of fundamental interest is to determine sufficient conditions to ensure the existence of chains (i.e., simply ordered subsets) which are connected. Recently [5] R. J. Koch proved that, if X is a compact Hausdorff space with continuous partial order (i.e., the partial order has a closed graph), if L(x) = {y: y ≦ x} is connected for each x ∈ X, and if X has a zero (i.e., an element 0 such that 0 ≦ x for all x ∈ X), then each element of X lies in a connected chain containing zero. It is easy to find simple examples which show that this result is false if X is assumed only to be locally compact. However, if it is assumed that the partial order is that of a topological lattice then the existence of such chains can be shown by elementary methods. This solves a problem which was proposed in [3].

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

Over a field of characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. It would be of interest to determine the powers of the radical in the non-semi-simple case [2, p. 61]. In the particular case of p-groups the solution to the problem is known through the work of Jennings [6]. We here consider the special case of group algebras whose radicals have square zero and we relate this condition to the structure of the group itself.

The purpose of this note is to establish the following

Theorem. The centre of a (left) hereditary local ring is either afield or a one-dimensional regular local ring.

Before starting the proof, it is necessary to explain the terminology. A ring R with an identity element is called a left local ring if the elements of R which do not have left inverses form a left ideal I. In these circumstances (see [1, Proposition 2.1, p. 147]), I is necessarily a two-sided ideal and it consists precisely of all the elements of R which do not have right inverses. Furthermore, every element of R which is not in I possesses a two-sided inverse. Thus there is, in fact, no difference between a left local ring and a right local ring and therefore one speaks simply of a local ring. In addition, I contains every proper left ideal and every proper right ideal. We may therefore describe I simply as the maximal ideal of R.

MacRobert [2] in 1937 defined the E-function as

where the symbol denotes that to the expression following it, a similar expression with α and β interchanged is to be added. For (1) he also gave the integral representation

where Re β > 0, | arg z | < π.

Let

It is familiar that

(1) In the addition theorem [3, p. 363]

where

take θ = π and replace v by v−½. Since

we get

In two previous papers [1], [2] the confluent form

of the δ-algorithm [3]

was established, and various properties which the confluent form of the algorithm possesses were discussed. It was shown, among other things, that if in (1)

and the notation

is used, then (1) is satisfied by

and further that under certain conditions, and for a certain n,

identically. It is the purpose of this note to derive another confluent form of the Ɛ-algorithm and to discuss its properties.

Let f(n) = an2+ bn + c be an irreducible quadratic polynomial with integer coefficients, and let D denote the discriminant b2 – 4ac of f(n).We shall assume that (D, k) = 1, and that for all positive integer n, f(n) is positive and coprime with k, where k is a fixed integer greater than 1.

It is well known that when the characteristic p(≠ 0) of a field divides the order of a finite group, the group algebra possesses a non-trivial radical and that, if p does not divide the order of the group, the group algebra is semi-simple. A group algebra has a centre, a basis for which consists of the class-sums. The radical may be contained in this centre; we obtain necessary and sufficient conditions for this to happen.

1. The properties of the circulant determinant or the circulant matrix are familiar. The circulant matrix C of order 4 x 4, with elements in the complex field, will serve for illustration.

The four matrix coefficients of c0, c1 c2, c3 form a reducible matrix representation of the cyclic group ℐ4, so that C is a group matrix for this. Let ω be a primitive 4th root of 1. Then Ω as below, its columns being normalized latent vectors of C,

is unitary and symmetric, and reduces Cto diagonal form thus,

where the μr, the latent roots of C, are given by

All of the above extends naturally to the n x n case.

In his book on Fourier Integrals, Titchmarsh [l] gave the solution of the dual integral equations

for the case α > 0, by some difficult analysis involving the theory of Mellin transforms. Sneddon [2] has recently shown that, in the cases v = 0, α = ±½, the problem can be reduced to an Abel integral equation by making the substitution

or

It is the purpose of this note to show that the general case can be dealt with just as simply by putting

The analysis is formal: no attempt is made to supply details of rigour.

In this note we consider measures on a left coset space G/H, where G is a locally compact group and H is a closed subgroup. We assume the natural topology in G/H and we denote the generic element of this space by xH (x∈G). Every element t∈G defines a homeomorphism of G/H given by t(xH) = (tx)H. A. Weil showed that a Baire measure on G/H invariant under all these homeomorphisms can exist only if

Δ(ξ) = δ(ξ) for each ξ ∈ H,

where Δ(x), δ(ξ) denote the modular functions in G, H [6, pp. 42–45]. We shall devote our investigations to inherited measures on G/H (cf. [3] and the definition below) invariant under homeomorphisms belonging to a normal and closed subgroup T ⊂ G.

In a previous note [3], Mennicke and I showed that the relations

(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=E

define a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group

[3, 7]+: B7=(AB)2=(A-1B)3=E

and to any factor group of this group which in turn has Klein's simple group of order 168, defined by

(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,

as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1].

Coxeter [1] has studied groups defined by the relations

and gives lists of finite groups known to be completely defined by such sets of relations. In a later paper [2] he shows that G3, n, p is finite if n, p are both even and satisfy

and expresses the conjecture that the restriction to even values may be removed. The only case satisfying this inequality and not already known to be finite is G3, 7, 16. In this note we show that G3, 7, 16 is indeed finite, being of order 21504 = 210.3.7, by showing that its subgroup

of index 2 is finite and of order 10752. Thus we add one entry to each of the lists of finite groups in Coxeter [1].

The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equation

and its Hankel transform of order v is defined by the equation

The object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

Let Rn denote real Euclidean space of n dimensions. If

define , and (as usual) so that, by the inequality of arithmetic and geometric means,

Let Λ0 be the integer lattice, consisting of those points in Rn whose co-ordinates are integers. A non-singular n × n matrix M will be called a Minkowski matrix, if, for any point a ∈ Rn, there exists a point x ∈ Λ0 such that

It was shown by Minkowski that, when n = 2, every non-singular matrix is a Minkowski matrix, and that, for general n, every rational non-singular matrix is a Minkowski matrix. Minkowski is also said to have conjectured that every non-singular matrix is a Minkowski matrix, whatever the value of n. For n = 3, this was proved by Remak [5], and a much simpler proof was given later by Davenport [2]. For n = 4, it was proved by Dyson [3], who used a method similar to that of Remak and Davenport, but required the methods of algebraic topology to deal with some of the complications which arise in the higher dimension. Since this method depends also on the reduction of quadratic forms, it is quite likely that it might fail for higher values of n even if the topological difficulties could be overcome. Therefore, an alternative proof for n = 3, due to Birch and Swinnerton-Dyer [1], is of some interest, though in this dimension it is more complicated than Davenport's proof. A good deal of their analysis applies to general n, and they showed that for all n there is a neighbourhood of the unit matrix I that consists entirely of Minkowski matrices.

The torsion of beams of L-cross-section was studied for the first time, from a mathematical standpoint, by Kotter [1]. He solved the problem in the case of an L-section both arms of which are infinite. Some time later, Trefftz [2], in his work on the torsion of beams of polygonal cross-section, applied his method also to an infinite L-section. In 1934, Seth [3] solved the case of a beam of an L-section with only one infinite arm. In 1949, Arutyanyan [4] solved the torsion problem of an L-section that has both arms finite, but of equal length, reducing the problem to that of solving an infinite system of equations.