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.

For dealing with atoms involving many electrons the accurate quantum theory, involving a solution of the wave equation in many-dimensional space, is far too complicated to be practicable. One must therefore resort to approximate methods. The best of these is Hartree's method of the self-consistent field. Even this, however, is hardly practicable when one has to deal with very many electrons, so that one then requires a still simpler and rougher method. Such a method is provided by Thomas' atomic model, in which the electrons are regarded as forming a perfect gas satisfying the Fermi statistics and occupying the region of phase space of lowest energy. This region of phase space is assumed to be saturated, with two electrons with opposite spins in each volume (2πh)3, and the remainder is assumed to be empty. Although this model hitherto has not been justified theoretically, it seems to be a plausible approximation for the interior of a heavy atom and one may expect it to give with some accuracy the distribution of electric charge there.

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.

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.

The probability relations which can occur between two separated physical systems are discussed, on the assumption that their state is known by a representative in common. The two families of observables, relating to the first and to the second system respectively, are linked by at least one match between two definite members, one of either family. The word match is short for stating that the values of the two observables in question determine each other uniquely and therefore (since the actual labelling is irrelevant) can be taken to be equal. In general there is but one match, but there can be more. If, in addition to the first match, there is a second one between canonical conjugates of the first mates, then there are infinitely many matches, every function of the first canonical pair matching with the same function of the second canonical pair. Thus there is a complete one-to-one correspondence between those two branches (of the two families of observables) which relate to the two degrees of freedom in question. If there are no others, the one-to-one correspondence persists as time advances, but the observables of the first system (say) change their mates in the way that the latter, i.e. the observables of the second system, undergo a certain continuous contact-transformation.

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).

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.

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 theoretical calculation of observable atomic constants is often only possible if the effective electric field inside the atom is known. Some fields have been calculated to fit observed data but for many elements no such fields are available. In the following paper a method is given by which approximate fields can easily be determined for heavy atoms from theoretical considerations alone.

In considering the statistics of the ‘no-field’ square Ising lattice in which each unit is capable of two configurations and only nearest neighbours interact, Kramers and Wannier (3) were able to deduce an inversion transformation under which the partition function of the lattice is invariant when the temperature is transformed from a low to a high (‘inverted’) value. The important property of this inversion transformation is that its fixed point gives the transition point of the lattice.

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).

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.

It is shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law. The criterion for marginal stability is obtained for three sets of boundary conditions, and the motion described. If such convection occurs in a stratum through which a bore-hole passes, the usual method of calculation of the heat flow must be modified, but in general the correction will not be large.

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

1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Kármán. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Kármán shows that the equations of motion and continuity are satisfied by taking

Here we shall mention only the results referring to stability. The definitions of the various quantities Tn, Sn, SNn, and the basic hypotheses made concerning their structure will be found in §§ 2·1, 3·1 or 4·1. For convenience we shall introduce some further terminology in this section. The single-server queues {SNn, Tn} arising in connexion with queues in series will be called the component queues, and the queue {Sn, sTn} implicit in the discussion of many-server queues will be called the consolidated queue. We have already in § 2.33 called the single-server queue {Sn, Tn} critical if E(S0-T0) = 0. We shall now call it subcritical if E(S0 − To) > 0 and supercritical if E(S0 − T0) < 0. A system of queues in series is subcritical if each component queue is subcritical, critical if (at least) one component queue is critical and the rest are subcritical, and supercritical if (at least) one component queue is supercritical. A many-server queue will be described in these terms according to the character of its consolidated queue. Finally, a single-server queue {Sn, Tn} will be said to be of type M if it has the property considered in Corollary 1 to Theorem 5: the sequences {Sn} and {Tn} are independent of each other, and one is composed of mutually independent non-constant random variables.

Single-server queues:

(i) Subcritical: stable (Theorem 3).

(ii) Supercritical: unstable (Theorem 2).

(iii) Critical: stable, properly substable, or unstable (examples in §2·33, including one due to Lindley); unstable if type M (Theorem 5, Corollary 1).

Queues in series:

(i) Subcritical: stable (Theorem 7).

(ii) Supercritical: unstable (Theorem 7).

(iii) Critical: stable, properly substable, or unstable, if the component queues are substable (examples in § 3·2); unstable if any component queue is unstable (Theorem 7), and in particular if any critical component queue is of type M (Theorem 7, Corollary).

Many-server queues:

(i) Subcritical: stable or properly substable (Theorem 8, and example in § 4·3).

(ii) Supercritical: unstable (Theorem 8).

(iii) Critical: stable, properly substable, or unstable, if consolidated queue is substable (examples in § 4·3); unstable if consolidated queue unstable (Theorem 8), and in particular if this is of type M (Theorem 8, Corollary).

From Lemma 1 it follows that none of these queues can be properly substable if all the servers are initially unoccupied.

Wright's model of a random process in genetica is modified by supposing that births and deaths occur individually at random so that the generations are no longer simultaneous. An exact solution is then obtained for the distribution of the number of a-genes in a haploid organism when mutation is occurring in both directions. When there is no mutation, the rate of approach to homozygosity is found in more complicated models with two sexes and diploid individuate. This rate is twice that occurring in Wright's models.

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.

A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.

It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.

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.

1. The study of queues has been of interest to mathematicians and engineers for the past forty years, and a very extensive literature on the subject exists. The applications of the theory are many and varied, from Erlang's original work on telephone engineering, to present-day studies in the design of airports. The number of papers which deal with the theoretical side of the subject is, however, small, and it therefore seems desirable to attempt to develop a general theory which will cover the diverse practical requirements, so that the unity in the applications will become apparent. This paper gives such a development in the case where there is a single queue and a single server attending to it; the theory of multiple queues or many servers seems, except under simplifying assumptions which do not always correspond to reality, to be a problem of considerable difficulty. Previous work on the subject has mainly been confined to a special case where the customers arrive at random, as, for example, in a recent paper by Kendall(5);* the present theory makes no such assumption and allows the customers to join the queue in other ways, though the theory simplifies when the more restrictive assumption is made. The centre of interest in the present theory is the waiting times of the customers.