Let

be a quadratic form in n variables (n ≥ 3) with integral coefficients. One of us gave recently (2) a simple proof of a result of Cassels (l), in the following form: if the equation f = 0 is properly soluble in integers, then it has a solution satisfying

where γn–1 denotes Hermite's constant.

Our main object is to give an exposition of a simplified version of Siegel's proof of his mean value formula in the Geometry of Numbers, also proved by A. Weil in a more general context ((1), (2), (3), (4)). In deriving the particular case which is Siegel's formula, however, certain difficulties of convergence arise, which were resolved by Siegel through rather tedious calculations, and by Weil through the use of Fourier analysis in topological groups. A new method of dealing with these difficulties, depending on Lebesgue's bounded convergence theorem and the relatively simple Lemma 8, seems to be our major contribution. In other respects our proof is very similar to Siegel's. At some points we have simplified his treatment; at others, where his work seems rather condensed, we have expanded it. We are indebted to Mahler (9) for some of the simplifications and grateful to him for letting one of us see his unpublished work. By using Lemma 8 and giving a separate proof of the existence of a fundamental domain for the unimodular group, we give a proof which is independent of Minkowski's reduction theory.

In this paper we bring together two aspects of homological algebra, one modern and the other classical. MacLane(6) has suggested the desirability of avoiding superfluous references to the objects of the category in proving fundamental results in homology theory. Accordingly, we consider categories whose maps form a ringoid in the sense of Barratt(1) and impose on the ringoid axioms, expressed in the language of ring theory, but having a close affinity to the axioms of Buchsbaum(2) for an exact category. (See also Cartan and Eilenberg(3); Appendix.) The purpose of these axioms is to ensure that the standard concepts of homology theory are applicable to the ringoid. Consequently, we call such a ringoid a homological ringoid.

In studying the structure of perfect sets, an important method would be to establish homoeomorphisms in general classes of sets. For then the general structural features of sets could be more easily seen by considering the simplest sets of a homoeomorphic class.

In this paper we establish the homoeomorphism in two general classes of plane perfect sets (we consider only bounded sets).

The first two papers of Reifenberg ((4), (5)) under the general heading ‘Parametric Surfaces’ contain a detailed and profound study of the tangential properties of these surfaces. Since their publication one of the fundamental problems in the subject, that of obtaining a convenient representation for the surface, has been solved by Cesari (1). In this paper we obtain direct proofs of Reifenberg's results from Cesari's theorem. Whereas Reifenberg had to contend with both topological and real-variable problems combined the effect of Cesari's theorem is to remove the topological difficulties and to leave a straightforward real variable problem. The definition of approximate tangential plane used here is not the same as either of the two employed by Reifenberg, but the differences between it and one of Reifenberg's definitions ((5), definition 2) are not very important.

In the last two decades a family of closely related functions have been introduced for the solution of quantum-mechanical problems. In dealing with atomic and molecular systems Pauling (12) introduced in 1928 linear combinations of the eigenfunctions of a single atom, which were called hybrids. These have specific symmetry properties which were utilized by Kimball (6) to construct a group-theoretical method for their determination. More recently Lennard-Jones (8) defined linear combinations of molecular eigenfunctions which he called equivalent orbitals. These are formally similar to the hybrids and Lennard-Jones took over the Kimball method for their determination.

In one numerical method for integration of the diffusion equation in one dimension, the time derivative is replaced by a finite difference in a time interval, and the space derivative by the mean of its values at the beginning and end of the interval. This leads to a set of ordinary differential equations, one for each interval, which have to be solved in succession. Each of these equations is second-order with two-point boundary conditions; the process of integration from one end is severely unstable, the more so the smaller the tune interval. This paper is concerned with a practical, direct and stable method for solving them by integration of two first-order equations, one being integrated inwards and the other outwards, one boundary condition being satisfied in each integration. The extension to axially symmetrical diffusion is briefly considered.

For a suitable choice of E*, the solution as t becomes large of the equation

where Y is given independent of t over the space boundaries, tends to the ground state solution of the wave equation

with the same boundary conditions on P as on Y. As a preliminary to using this relation to solve wave equations in more than one variable, the solution of the equation

has been studied. Methods of numerical solution are discussed, and the convergence of these is examined. Some practical experiments using an electronic computing machine are described.

Let us consider an experiment, the possible outcomes of which are random events. Let A1, A2, …, An be some of these outcomes. Define the random variable ηn as the number of the events occurring simultaneously among the A1, A2, …, An in an experiment. Denote by Pk = (ηn = k) (k = 0,1, 2, …, n), the distribution of the random variable ηn and by , the binomial moments of ηn. Here is the symbol of the probability and & that of the expectation.

Bartlett and Rajalakshman (3) have derived two types of large sample goodness of fit tests for the hypothesis that a multivariate (or vector) time series {X(t)}, t = 0, ± 1, ± 2, …, is generated by a linear autoregressive process. This may be defined as the stationary solution of an equation of the form

where the Ai are square matrices such that the roots of the determinantal equation ‖ zp + A1zp−1 + … + Ap ‖ = 0 have moduli less than unity, and {U(t)} is a sequence of independent random vector variables with a common distribution.

The paper deals with weak ergodicity, i.e. the tendency for a chain to ‘forget’ the distant past. This may occur in non-homogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases.

Investigation of the general conditions of weak ergodicity leads to the definition of a special subclass of regular matrices. Conditions for chains involving regular matrices are also given (§§2,3).

Section 4 discusses similarities in asymptotic behaviour between chains whose transition matrices differ only by small amounts.

Dirac's wave equation of an electron enables one to solve for the wave function of an electron moving in an electromagnetic field. The wave function ψ has 4 components ψ1, ψ2, ψ3, ψ4, and the electromagnetic field is described by a four-vector Aμ, consisting of a scalar potential φ and a vector potential A.

In the usual problem, we solve for the wave function ψ when the potential Aμ is given. By doing so, we obtain the wave function of an electron which moves in a given electromagnetic field. It is of some interest to consider the reverse question: given the wave function ψ, what can we say about the electromagnetic potential Aμ, which is connected with ψ by Dirac's equation? Is Aμ uniquely determined, and if not, what is the extent to which it is arbitrary ?

The general equation for the adiabatic second-order derivative of the energy En of an eigenstate with respect to parameters λ and λ′ occurring in the Hamiltonian ℋ is

The applications of this equation to molecules (λ, λ′ = nuclear position coordinates) and to enclosed assemblies of interacting particles (λ = λ′ = volume) are discussed, and the classical analogue of the equation for a micro-canonical ensemble is derived.

In a previous paper by the author (3) it was shown how the theory of distributions of L. Schwartz enables a mathematically consistent formalism to be given for a system composed of point charges interacting through their classical electromagnetic field. In the present paper a definition of the energy and momentum of the field lying in a space-like surface is given, and it is shown that from this four-vector it is possible to derive the usual equation of conservation of total energy and momentum.

Within the limitations of the small-deflexion plate theory, complex variable methods are used in this paper to obtain an exact solution for the problem of a thin circular plate supported at several interior or boundary points, and subjected to a certain normal loading spread over the area of an eccentric circle, the boundary of the plate being free. The load considered includes as a special case a linearly varying load over the circle and, as the radius of the loaded circle tends to zero, this load can be made to tend to a couple nucleus at its centre. As limiting cases the procedure adopted provides us with solutions appropriate to a circular plate, an infinitely large plate and a half-plane having free boundaries and acted upon by any normal system of concentrated forces and concentrated couples in equilibrium. Formulae for the moments, shears and deflexions relating to special examples are worked out in detail.

Within the restrictions of the classical plate theory, complex variable methods are used in this paper to develop an exact expression for the transverse displacement of an infinitely large isotropic plate having a free outer boundary and elastically restrained at an inner circular boundary, the plate being subjected to a general type of loading distributed over the area of a circle. The limiting case of a half-plane clamped along the straight edge and acted upon normally by the same loading is also considered.

It is pointed out that the free particle, the harmonic oscillator and certain other systems are such that the Feynman summation over histories is independent of the class of histories summed over. These examples are therefore useless as tests of methods of defining and carrying out the summation.