In problems in the mathematical theory of elasticity related to the symmetric deformation of an infinite elastic solid with an external crack we encounter the problem of determining an axisymmetric function φ(ρ, z) which is harmonic in the half-space z>0 and satisfies the mixed boundary conditions

on the plane boundary z = 0, where it is assumed that f(ρ) is continuously differentiable in [1, ∞). Further φ→0 as √(ρ2+z2)→∞.

We consider some new types of realization problem for obstructions in the Browder-Livesay groups by homotopy equivalences of closed manifold pairs. We give several examples of calculations. We also consider relations with classical surgery problems.

In text books of Plane Coordinate Geometry, two methods are usually given for investigating the condition that the general equation of the second degree:

may represent a pair of real or imaginary straight lines.

Let AOB, DOE be any two intersecting chords of a circle. Required to inscribe a circle in one of the compartments, as BOE.

Draw OF bisecting the angle BOE and let it cut the circle in F. Draw FM perpendicular to AB.

The Fibonacci groups are a special case of the following class of groups first studied by G. A. Miller (7). Given a natural number n, let θ be the automorphism of the free group F = 〈x1, …, xn |〉 of rank n which permutes the subscripts of the generators in accordance with the cycle (1, 2, …, n). Given a word w in F, let R be the smallest normal subgroup of F which contains w and is closed under θ. Then define Gn(w) = F/R and write An(w) for the derived factor group of Gn(w). Putting, for r ≦ 2, k ≦ 1,

with subscripts reduced modulo n, we obtain the groups F(r, n, k) studied in (1) and (2), while the F(r, n, 1) are the ordinary Fibonacci groups F(r, n) of (3), (5) and (6). To conform with earlier notation, we write A(r, n, k) and A(r, n) for the derived factor groups of F(r, n, k), and F(r, n) respectively.

Clay (3), Johnson (5) and Krimmel (6) have each considered the near-rings with identity on dihedral groups. Krimmel actually generalised the class of dihedral groups and investigated the class of finite non-abelian groups with a cyclic normal subgroup of prime index: we shall call this class . Krimmel considered the near-rings with identity that might be denned on members of and he determined the subclass of groups in which support near-rings of this kind. He also managed to calculate the number of non-isomorphic near-rings involved for certain cases. His methods were essentially combinatorial, and his results were expressed in terms of various integers which characterised the individual members of . Certain features of this work led us to investigate the structure of near-rings on members of from a more algebraic point of view and thereby to complete and extend Krimmel's programme. Part of the work in this paper formed the basis of the second author's thesis (7). We should like to thank Dr. J. Krimmel for permission to include some of his results, and Dr. J. Meldrum who detected an error in our original formulation of Theorem 7.1.

The differential equation of Mathieu, or the equation of the elliptic cylinder functions

is known by the theory of linear differential equations to have a general solution of the type

φ and ψ being periodic functions of z, with period 2π.

The object of the paper was to suggest for the teaching of electrostatics a leading idea, which should readily co-ordinate all the facts, introduce no misleading inferences, and guide the course of learners in the direction of the most recent investigations—in all which respects the notion of attraction and repulsion is at least a partial failure. The leading idea or fact referred to is, that almost all electrostatic distributions, however complex, can be analysed into one or more repetitions of a certain simple system, which is called in the paper “an electrostatic system,” and which may be described as follows:—Two equally and oppositely electrified conducting surfaces, facing each other, separated by any dielectric, and insulated from each other. A complete study of one system of this kind, and of the very simple ways in which the establishment of one such system often necessitates the establishment of others, is therefore the fundamental study of electrostatics.

There are a number of classes of distributive lattices whose members can be characterised as the coproduct A * L of suitable distributive lattices A and L. For example, Post algebras [1], pseudo-Post algebras [4], Post Lalgebras ([6], [8[9]) and the lattices [D]n of [4]. Moreover, the α-completeness and α-representability of some(though not all) of these algebras have been investigated in [7], [2], [6], and [10].

Let the set b1b2, b3, …., bk; of k non-negative integers be denoted by Bk. Let ξ;Bk denote the set ξb1 ξb2, ξb3, …. ξbk; ξ being any integer > 1. Without loss of generality, we can suppose that b ≦bi+1.

We establish the existence of positive radially symmetric solutions of Δu + f(r, u, u′) = 0 in the domain R1 <r<R0 with a variety of Dirichlet and Neumann boundary conditions. The function f is allowed to be singular when either u = 0 or u′ = 0. Our analysis is based on Leray-Schauder degree theory.