Mathematical Proceedings of the Cambridge Philosophical Society

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.

The limiting distribution, when n is large, of the greatest or least of a sample of n, must satisfy a functional equation which limits its form to one of two main types. Of these one has, apart from size and position, a single parameter h, while the other is the limit to which it tends when h tends to zero.

The appropriate limiting distribution in any case may be found from the manner in which the probability of exceeding any value x tends to zero as x is increased. For the normal distribution the limiting distribution has h = 0.

From the normal distribution the limiting distribution is approached with extreme slowness; the final series of forms passed through as the ultimate form is approached may be represented by the series of limiting distributions in which h tends to zero in a definite manner as n increases to infinity.

Numerical values are given for the comparison of the actual with the penultimate distributions for samples of 60 to 1000, and of the penultimate with the ultimate distributions for larger samples.

This paper is concerned with methods of evaluating numerical solutions of the non-linear partial differential equation

where

subject to the boundary conditions

A, k, q are known constants.

Equation (1) is of the type which arises in problems of heat flow when there is an internal generation of heat within the medium; if the heat is due to a chemical reaction proceeding at each point at a rate depending upon the local temperature, the rate of heat generation is often defined by an equation such as (2).

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.

It has been pointed out to me that some of the statistical ideas employed in the following investigation have never received a strictly logical definition and analysis. The idea of a frequency curve, for example, evidently implies an infinite hypothetical population distributed in a definite manner; but equally evidently the idea of an infinite hypothetical population requires a more precise logical specification than is contained in that phrase. The same may be said of the intimately connected idea of random sampling. These ideas have grown up in the minds of practical statisticians and lie at the basis especially of recent work; there can be no question of their pragmatic value. It was no part of my original intention to deal with the logical bases of these ideas, but some comments which Dr Burnside has kindly made have convinced me that it may be desirable to set out for criticism the manner in which I believe the logical foundations of these ideas may be established.

The conductivity of thin films of the alkali metals has recently been measured in the H. W. Wills Physical Laboratory, Bristol*. It was found that as the thickness of the film is decreased to that of a few atomic layers the conductivity drops below that of the bulk metal. In the papers quoted the hypothesis was put forward that this effect is due to the shortening of the mean free paths of the conduction electrons of the metal by collisions with the boundaries of the film. The experimental results were compared with a formula derived on the basis of this hypothesis. This formula was, however, obtained subject to a number of simplifying assumptions, and it is the first purpose of this paper to obtain a more accurate formula. I also compare this formula with experiment, and make certain deductions about the surfaces of thin films.

1. Introduction. The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula:

where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X. In particular if, near the boundary, X is isometric to the product Y x R⁺, the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H²(X, R) by an integral formula

where p₁ is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p₁ = (2π)⁻²Tr R². It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general

The paper studies, in a general way, how the random properties of a ‘medium’ influence the percolation of a ‘fluid’ through it. The treatment diifers from conventional diffusion theory, in which it is the random properties of the fluid that matter. Fluid and medium bear general interpretations: for example, solute diffusing through solvent, electrons migrating over an atomic lattice, molecules penetrating a porous solid, disease infecting a community, etc.

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.

The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).

Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.

For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).

Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).

The method used for integrating the equations numerically is outlined (§ 9).

New factors arise in a species by the process of mutation. The frequency of mutation is generally small, but it seems probable that it can sometimes be increased by changes in the environment (1,2). On the whole mutants recessive to the normal type occur more commonly than dominants. The frequency of a given type of mutation varies, but for some factors in Drosophila it must be less than 10⁻⁶, and is much less in some human cases. We shall first consider initial conditions, when only a few of the new type exist as the result of a single mutation; and then the course of events in a population where the new factor is present in such numbers as to be in no danger of extinction by mere bad luck. In the first section the treatment of Fisher (3) is followed.