Let G be a group. With each element a in G we associate the mappings ρ(a) and λ(a) of G into itself defined as follows, The product of mappings is defined as usual. Let P(G) and ∧(G) denote respectively the semigroups generated by the set of all ρ's and λ's. These semigroups will be called the commutation semigrowps of G.

The interesting results arising from the study of ‘Four intersecting spheres’ [9] in a solid made the author think of an analogous picture in higher spaces too and thus the present paper arose.

The structure of a bisimple inverse semigroup with an identity has been related by Clifford [2] to that of its right unit subsemigroup. In this paper we give an explicit structure theorem for bisimple inverse semigroups in which the idempotents form a simple descending chain

e0 > e1 > e2.…

We call such a semigroup a bisimple co-semigroup. The structure of a semigroup of this kind is shown to be determined entirely by its group of units and an endomorphism of its group of units.

The statical Reissner-Sagoci problem [1, 2, 3] is that of determining the components of stress and displacement in the interior of the semi-infinite homogeneous isotropic elastic solid z ≧ 0 when a circular area (0 ≦ p ≦ a, z = 0) of the boundary surface is forced to rotate through an angle a about an axis which is normal to the undeformed plane surface of the medium. It is easily shown that, if we use cylindrical coordinates (p, φ, z), the displacement vector has only one non-vanishing component uφ (p, z), and the stress tensor has only two non-vanishing components, σρπ(p, z) and σπz(p, z). The stress-strain relations reduce to the two simple equations

where μ is the shear modulus of the material. From these equations, it follows immediately that the equilibrium equation

is satisfied provided that the function uπ(ρ, z) is a solution of the partial differential equation

The boundary conditions can be written in the form

where, in the case in which we are most interested, f(p) = αρ. We also assume that, as r → ∞, uπ, σρπ and σπz all tend to zero.

In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.

For any nilpotent group B of class c and any given element h of B generating the subgroup H, Wiegold [1] has shown that if, in addition, [B, H] has exponent pr for some prime p and integer r, then B can be embedded in a nilpotent group G such that G also contains psth root for h(s ≧ 1). In fact, Wiegold has gone further and calculated an upper bound for the class of G in terms of the variables c, p, r, s.

Recently some inversion integrals for integral equations involving Legendre, Chebyshev, Gegenbauer and Laguerre polynomials in the kernel have been obtained [1, 2, 3, 5, 6]. In this note, two inversion integrals for integral equations involving Whittaker's function in the kernel are obtained. We shall make use of the following known integral [4, p. 402]

The results of this note are based on the following two integrals, which are derived from (1) by writing u – t = (v – t)x.

for m + 1 > 2v > – 1;

for m + 1 > 2v > – 1.

Let G, H, K be groups such that G is normal in K and G ⊆ H ⊆ K. Let I(H, K) be the set of inner automorphisms of K restricted to H; thus α ∊ I(H, K) if and only if, for some κ∊ K, α(h) = k-1hk for all h ∊ H. Let φ be an isomorphism of H/G onto a subgroup Hφ/G of K/G. An isomorphism Φ of H onto H(φ) is called an extension of ø if

Φ(h)G = φ(hG) for all h∊H.

A series ∑an is said to be summable (C, — 1) to s if it converges to s and nan = o(1) [8]. It is well known that this definition is equivalent to tn→s (n→∞), where tn = sn + nan, sn = ao + … + an. The series is summable | C, — 1 | to s if the sequence t = {tn} is of bounded variation (t ∈ B.V.), i.e. ∑ |; ▲tn |; = ∑ | tn - tn-1 | < ∞, and ∑ ▲tn = lim tn = s. An equivalent condition is ∑ | an |; < ∞, ∑an = s and ∑ | ▲(nan) | < ∞. For, suppose that ∑an = s | C, - 1 |. Since {sn} is the sequence of (C, 1)-means of {tn} and since | C, 0 | ⊂ | C, 1 |, we have ∑ | an | < ∞ and ∑an = s whence ∑ | ▲(nan) | < ∞. Conversely, ∑ | an | < ∞, ∑an = s and ∑ | ▲(nan) | < ∞ imply t ∈ B.V. and ∑▲n = s + lim nan. But lim nan = 0, since ∑ | an | < ∞.

In a recent paper [2] one of the authors has introduced the concept of module type of a ring, for rings with unit. The object of this paper is to generalize this concept to arbitrary rings, without assuming the existence of a unit. This is easily accomplished for rings with one-sided unit, and we shall define the type of such a ring. Theorem 2.5 gives a relation between this type and the module type of [2], and permits the immediate extension of all results in [2] to rings with one-sided unit.

In an unpublished, dissertation Cleaver [1] proved the following

Theorem 1. If L is a lattice in euclidean four-space R4 of determinant d(L) = 1 and with no pair of its points within unit distance apart then any four-sphere of radius 1 contains a point of L.

This paper is concerned with (a) a new simple method of solution of a wide variety of problems of elastic strips by means of Fourier transforms in the complex plane and (b) a direct solution of the elastic annulus. Continuation of functions into adjacent regions of the plane and the solution of differential-difference equations are seen to be unnecessary complications.

We say that a set of closed circular discs of radii r1r2, …, all lying in a Euclidean plane, is saturated if and only if r = inf ri > 0 and any circle of radius r has at least one point in common with a circle of the set. For any set X we use α(X) to denote the area of X. If X denotes the point set union of the discs and X(k) the part of X inside the disc whose centre is the origin and radius k then by the lower density of the covering we mean . The problem is to find the exact lower bound of the lower density for any saturated set of circles. We show that it is φ/(6√3) provided the circles are disjoint. The general case, when they may overlap, remains unsolved.

This paper is concerned with marginal convection in a self-gravitating sphere of uniform incompressible fluid containing a uniform distribution of heat sources. Its purpose is twofold. The first aim is to present the mathematical argument in a form which, the author believes, is more succinct than that which has been given heretofore. The second aim is to determine the effect of the convective motions upon the moments of inertia of the body and, in the light of the results obtained, examine briefly the hypothesis that the moon is in a state of convection.

Let p be a prime, t a positive integer, and P = P(x1, …, xn) a polynomial over the rational field K, in any number n of variables, of degree k = 2, 3, or 4. We shall consider the congruence

Let p be an odd prime and denote by [p], the finite field of residue classes, mod p. In Euclidean n-space, let n denote the lattice of points x = (x1, …, xn) with integral coordinates and C = C(n, p), the set of points of n satisfying

The purpose of this paper is to give a new and improved version of Linnik's large sieve, with some applications. The large sieve has its roots in the Hardy-Littlewood method, and in its most general form it may be considered as an inequality which relates a singular series arising from an integral where S(α) is any exponential sum, to the integral itself.