The free product A* B of groups A and B can be described in two ways.

We can construct the set of reduced words in A and B. Define a binary operation on by concatenating two words and performing as many reductions as possible. Prove that is a group; the difficult step is the proof of associativity. Define A * B = .

Let Λ be the group of all real non-singular 2 × 2 matrices and let Ω be the group of all real 2 × 2 matrices with determinant 1 in which a matrix is identified with its negative. Then Ω is isomorphic to the group of all real linear fractional transformations of determinant 1. In a previous paper ((3)) the authors determined all faithful representations of the modular group (more generally of the Hecke groups) by a discrete subgroup of Ω, in which the representations were partitioned into conjugacy classes over Λ. In this paper we consider the question for the more general situation of the free product of any two cyclic groups of finite order. Our results parallel the results of (3) quite closely, but some significant differences in the details of the proofs arise. In particular Theorem 3 of section 3 below, which is purely group-theoretic, is of independent interest and should prove useful in other investigations.

Let Q be the field of rational numbers, and let A be an Abelian curve (an Abelian variety of dimension one) defined over Q. Following Weil, the conductor of A is

where p runs over all primes, and the exponent is explained below; the primes dividing N are exactly those for which A has degenerate reduction modulo p.

Conrad ((2)), has shown that any lattice group which obeys (C.F.) each strictly positive element exceeds at most a finite number of pairwise orthogonal elements may be constructed, from a family of simply ordered groups, by carrying out, alternately, the operations of forming finite direct sums and lexico extensions, at most a countable number of times. The main result of this paper, Theorem 3.1, gives necessary and sufficient conditions for a multilattice group, which obeys (ℋ*), to be isomorphic to a multilattice group which is constructed from a family of almost ordered groups, by carrying out, alternately, the operations of forming arbitrary direct sums and lexico extensions, any number of times; we call such a group a lexico sum of the almost ordered groups.

Introduction. In his paper ((1)), Baumslag has shown that the wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group, the prime p being the same for both groups. Liebeck ((3)) has obtained the exact nilpotency class of A wr B when A and B are Abelian. Let A be an Abelian p -group of exponent pn and let B be a direct product of cyclic groups, whose orders are pβ1, …, pβn, with β1 ≤ β2 ≤ … βn. Then A wr B has nilpotency class .

In two recent papers of Conner and Floyd ((2)) and ((3)), the additive structure of the SU-cobordism (or bordism) ring was completely determined. The object of this note is to point out that their results can also be used to determine the somewhat complicated multiplicative structure.

Introduction. Let X be a submanifold of Y, in either the topological, smooth, or piecewise linear ( = PL) categories. A normal cell bundle on X in Y is a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such that E is a neighbourhood of X in Y and p: E → X is a retraction. The triple (Y, X, ξ) is a tubular neighbourhood, or briefly, a tube. For convenience we may refer to a tube by its cell bundle.

Introduction. This paper is a continuation of (1), where definitions may be found. All manifolds, maps, bundles, etc., are to be piecewise linear. Recall that cells are always closed.

In this paper, we give an extension (Theorem 1) of the following well-known result of Banach ((l)): If X, Y are sets and Θ: X → Y, ψ: Y → X are injective mappings, then there exist partitions X = X1 ∪ X2, Y = Y1 ∪ Y2 such that Θ(X1) = Y1 and ψ(Y2) = X2. Here, as is usual, we say that X = X1 ∪ X2 is a partition of the set X = X1 ∩ X2 = π. Our theorem is applied in sections 2 and 3 to problems concerned with the existence of common representatives for two families of sets.

Introduction. Let sn denote the infimum of the set of real numbers X, where x belongs to X if, and only if, x is the Besicovitch dimension of the residual set of a packing of n-spheres into the unit n-cube In, of Rn. In recent work ((1)) I have shown that s2 is greater than one, and, quite naturally, I have since been asked whether or not the proof can be generalized to prove the analogous result, sn greater than n − 1, in Rn. Whilst it is clear, in theory, that this could be done, in practice the details might become rather complicated. However, such a generalization is unnecessary, for the result sn > n − 1 is a trivial consequence of combining the result s2 > 1 with the following theorem.

A theory of quadratic forms is developed over R(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.

In a recent paper, Smith ((l)), considering a statement or conjecture of Isbell ((2)), proved that the Hausdorff uniformities arising from two different uniformities on the same set must induce different topologies on the ‘hyperspace’ of subsets, except possibly in one case which remained open, namely, when the two given uniformities induce the same proximity and neither is precompact. I now give an example to show that different uniformities can (in this case) induce the same topology on the hyperspace (in the example one of the uniformities is metric). This disproves Isbell's conjecture.

Spectral theorems and a functional calculus for families of permutable self-adjoint operators in a separable Hilbert space are developed in a form convenient for physical applications. A recent paper of Jauch and Misra is discussed.

Introduction. In this paper we consider the two cross-product combinations of Bessel functions

where δ = (k − 1) z and (') denotes differentiation with respect to the argument. Here Jν and Yν designate respectively the Bessel functions of the first and second kind of order ν.

Certain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.

In the usual notation let

where .

Also, for the generalized hypergeometric function pFq(x) let us employ a contracted notation and write

Throughout the present paper i will run from 1 to p, I from 1 to P, and so on. Thus ((a) )m is to be interpreted as

,

and similar interpretations hold for ((A))m, etc.

The problem of diffraction of a plane wave by a semi-infinite grating of iso-tropic scatterers leads to the consideration of a non-linear integral equation. This bears a resemblance to Chandrasekhar's integral equation which arises in the study of radiative transfer through a semi-infinite atmosphere. It is shown that methods which have been used with success to solve Chandrasekhar's equation are equally useful here. The solution to the non-linear equation satisfies a more simple functional equation which may be solved by factoring (in the Wiener-Hopf sense) a given function. Subject to certain additional conditions which are dictated by physical considerations, a solution is obtained which is the unique admissible solution of the non-linear integral equation. The factors and solution are found explicitly for the case which corresponds to closely spaced scatterers.

Two natural definitions for the distribution function of the height of an ‘arbitrary local maximum’ of a stationary process are given and shown to be equivalent. It is further shown that the distribution function so defined has the correct frequency interpretation, for an ergodic process. Explicit results are obtained in the normal case.

This paper contains the derivation of a metric for flat space-time associated with the supertranslations of the Bondi-Metzner group, and a description of the geometry of the null hypersurfaces on which the metric is based.

The effect of restricting the displacements of a quantum mechanical harmonic oscillator to semi-infinite and finite regions is studied. It is shown how the eigenvalue problem may be solved, both graphically, and from certain series expansions.