In this paper, we consider the effect of selection in a finite population of haploid individuals in which mutation is occurring in both directions. Unlike Wright's models (2) where the members of each generation all die at the same time to be replaced by a new generation, we suppose that at each change in the state of the system only one individual dies and is replaced by a new individual. In a previous paper (1) we have represented the process by a Markov chain, but it is here more convenient to begin by considering a Markov process with a continuous time parameter. This enables us to introduce selection in a plausible manner.

We shall understand the norm, h(A) of a regular matrix A = (amn) to be

The method has a norm, ‖ ‖, ‖ ‖ ≥ 1, given by ‖ ‖ = inf h(A) where the inf is taken over all the matrix methods equivalent to for bounded sequences (b-equivalent). These definitions are due essentially to Brudno(1) though his definition of

In a previous note (Good (7)) it was shown that there is an intimate connexion between Legendre polynomials and ‘trinomial’ random walks, that is walks on a one-dimensional lattice with the same probabilities p–1p0, p1 at each point of taking a step of – 1,0,1. The purpose of the present note is to point out that if the probabilities depend on the current position of the particle, then there is scope for applications of the theory of analytic continued fractions.

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.

A number of writers have noticed Fourier sine or cosine transform relations between certain pairs of series. For example, Duffin(4) has shown that in a variety of circumstances

are a pair of sine transforms, and Guinand(7) that

are cosine transforms, using L2 theory. Other examples are given in (2) and (5); and in (6) power series and general transforms are considered.

In this paper we consider a model of a population in which the individuals are diploid and are of two sexes. The individuals have genotypes aa, Aa and AA and we suppose that mutation is occurring in both directions. By supposing that each step in the process is the result of a single death followed by a birth instead of supposing (as in Wright's models (3)) that all the individuals die together at the same instant, to be replaced by a new generation, the mathematical details are much simplified. In this way it is possible to deal exactly with more complicated situations than that of the population of haploid individuals used by Wright in his study of the effects of mutation in a finite population. Moreover, the present model not only involves zygotes and two sexes but is more realistic for those natural populations in which the generations are overlapping.

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.

1. Introduction. Problems of unrestricted random walks on lattices have been considered by many authors and methods have been discovered for the exact enumeration of the number of walks between two points, for obtaining asymptotic estimates of the distribution functions and for evaluating special parameters such as the probability of ultimate retum to the starting point. By an unrestricted walk we mean one in which the probability of the next step at any stage is not influenced by the previous choice of steps. The corresponding problems for restricted (or correlated) random walks are more difficult and fewer results have been obtained. Restricted walks are, however, of considerable physical interest in connexion with the statistical behaviour of long chain molecules, the theory of cooperative phenomena in crystals and in other applications.

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.

The general radial integral between oscillator-well wave-functions with a Gauss interaction potential is expressed as a generalized hypergeometric function. A relation between contiguous functions is found and used to obtain a recurrence relation between these integrals.

If φ belongs to L2(0, ∞), the Watson transform W φ of φ is defined by

where k(u)/u belongs to L2(0, ∞) and

The operator W is unitary in L2(0, ∞), and its inverse is given by

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.

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

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.

In recent years much attention has been directed towards the asymptotic solution of diffraction problems. In the present work, consideration is given to the relatively simple problem of the diffraction of an E-polarized plane wave by an infinite slit. The solution takes the form of a series in inverse powers of the ratio of slit-width to wavelength of the incident wave, and is based on the solution by successive substitutions of a pair of integral equations. The current densities induced on both halves of the screen are calculated, from which is deduced the electric field in the slit. The far-field is determined from the aperture distribution, and an asymptotic expression is found for the transmission coefficient as a function of the angle of incidence and the ratio of slit-width to wavelength.

While most previous work has been confined to near-normal incidence, the present theory is uniformly valid for all angles of incidence.

It is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.

A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

It is shown that the phenomena of double refraction—optical activity, Faraday rotation, Kerr effect, etc.—may be treated as scattering problems in quantum field theory. An expression for the scattering cross-section of an atom or molecule for photons may be easily obtained and from this expression using the idea of Stokes operators, equating their expectation values to the classical Stokes parameters, the usual quantum mechanical expressions for optical activity, etc. may be obtained. This approach does not require the ideas of refractive index or polarization of the medium, and emphasizes what are the actual observables in these experiments.

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 methods introduced into the theory of atomic spectra by Racah are applied to the calculation of electric quadrupole line strengths for transitions involving d and s electrons. The transitions considered are of the types dn−dn−1s, dn−1s−dn−1s and dn−1s−dn−2s2. Sum rules and rules for almost closed shells are obtained. Detailed numerical results are given for various transitions in and between the configurations d9 and d8s, and d7, d6s and d5s2.

The notions developed in Part I in connexion with the diffraction of an E-polar-ized wave are here applied to the case of H-polarization. Induced current densities, aperture and far fields, and the transmission coefficient are again found in the form of infinite series in inverse powers of the slit-width to wavelength ratio. By means of Babinet's principle, the solution for diffraction by a strip is obtained.

A detailed comparison is made with values of the transmission coefficients for both polarizations found previously and by the present method. It appears that this theory provides accurate information when the slit-width is greater than about a wavelength.