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.

This paper contains two parts:—

I. An endeavour to remove the present confusion in polar diagrams.

II. On the curves derived from a given curve by increasing or diminishing the vectorial angle in a constant ratio.

The transpositions that generate a symmetric group can be represented as real reflections: symmetry operations of a regular simplex. Analogous unitary reflections serve to generate other factor groups of the braid group; they are symmetry operations of regular complex polytopes. Certain relationships among these groups have, as geometric counterparts, unexpected plane sections of the polytopes, beginning with the square sections of the regular tetrahedron. In Section 6, 5-dimensional coordinates will be used to exhibit pentagonal sections of the 4-dimensional regular simplex. The most spectacular instance of such “equatorial” sections occurs in the case of the Witting polytope in complex 4-space, so exquisitely drawn by Peter McMullen for the frontispiece of Regular Complex Polytopes [6]. This has a plane section 3{<5}3 which appears thare as Fig. 4.8B on page 48. Shephard [9, p. 92] called it 3(360)3. Its 120 vertices will be seen to be situated “inside” 120 of the 2160 faces 3{3}3 of the Witting polytope. These “faces” are self-inscribed octagons [7, p. 290].

A. G. D. Watson (1939-41), remarking that there are no Ricci principal directions ata world-point of space-time at which the Einstein equations are satisfied, shows how to define at any world-point a set of principal directions intrinsically related to the Riemann tensor Rijkl itself. These directions are unique except when the space-time has any kind of rotational symmetry about the world-point.

A generalised triangle group has a presentation of the form

where R is a cyclically reduced word involving both x and y. When R = xy, these classical triangle groups have representations as discrete groups of isometrics of S2, R2, H2 depending on

In this paper, for other words R, faithful discrete representations of these groups in Isom +H3 = PSL(2, C) are considered with particular emphasis on the case R = [x, y] and also on the relationship between the Euler characteristic χ and finite covolume representations.

Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.