In (2) a simple proof was given of the following inequality of Atkinson, Watterson and Moran (l), a slight generalization of a result of Scheuer and Mandel ((4), see also (3)) which is of importance in genetics. If aij, pi, qj (i, j = 1, 2, 3,…) are non-negative numbers with

and if

then

and if then This result is shown in (2) to be a special case of a large class of inequalities involving sets of non-negative numbers depending on several indices, and ‘partial averages’ over subsets of those indices.

Consider the eigenvalue problem

where q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λr (r = 0, 1, 2, …) and λn ∽ n2π2 (b − a)−2 as n → ∞.

This paper is a contribution to the study of set functions which, although additive, fail to be measures by not being countably additive. It is motivated by the suggestion (to be found, for instance, in (2), section 2.3) that such functions might be used instead of measures in the theory of probability, and by speculation about the effect which the adoption of such a suggestion would have on the theory of stochastic processes.

In the theory of finite state, discrete time, non-homogeneous Markov chains, different notions of ergodicity have been introduced in the literature. These notions are concerned with the long-run behaviour of chains and with their tendency to get some stability properties after a sufficiently long period of time. The aim of this paper is the study of non-homogeneous Markov chains with a denumerable number of states. It will be shown that some theorems which are valid in the finite case are also valid for chains with a denumerable number of states as well. Moreover, a new notion of stability is introduced and it is shown to be satisfied for some chains. Although the paper is self-contained some familiarity with the theory of finite state non-homogeneous Markov chains is desired. Without any attempt of completeness we list for the interested reader the papers: (1–3, 8, 9).

In this paper we present results for the critical percolation probabilities of a number of two- and three-dimensional lattices. These results are based upon Monte Carlo studies of the way in which cluster-size distributions vary as the number of occupied sites in a lattice is progressively increased; the principle of the method has been described in some detail in an earlier publication (Dean (1)) in which the results of studies carried out on the ACE computer were reported. The use of a KDF 9 computer with a 32K word high-speed store has now enabled us to obtain results of a very high accuracy indeed; in those cases where our values for critical probabilities can be checked against exactly known values, they differ by no more than 0·1%.

The notion of a general dynamical system was introduced by Barbashin(1). In this paper we consider a general dynamical system in a metric space following Zubov(6) where Lyapunov's method has been extended to investigate the stability properties using a single Lyapunov functional.

An addition theorem, expressing the spherical harmonics in terms of the spherical harmonics and , is derived.

1. It was known to Euler that p(n), the number of unrestricted partitions of n into non-increasing integral parts, is generated by

with the usual convention that p(0) = 1.

The generalization of the Dirac and quaternion algebras to Riemannian spaces is outlined. The components of various elements of the algebras are interpreted as physical quantities (tensors) and their symmetries and algebraic properties are linked with the properties of the algebra. The generalization of the quaternion algebra is of particular interest in that it resolves the anomalies that arise in the usual identification of quaternions with rank two spinors. Algebraic expressions for the electromagnetic energy momentum tensor, the Ricci tensor and Einstein tensor are obtained in both E-number and quaternion form. Extension of the principles to EF-numbers yields a proof of the symmetry properties of Bel's tensor and a simple expression for its divergence.

The paper examines the structure obtained by abstracting from the conventional (manifold) representation of relativistic space-time the concept of an event-set equipped with two partial orderings, whose counterparts are the notions ‘causally precedes’ and ‘chronologically precedes in the history of some observer’.

The integral equation for the isotropic scattering of radiation in a medium bounded laterally by planes ±y0 is solved as a series in powers of exp (−y0). This series is certainly convergent for y0 large enough, and rapidly convergent for most practical purposes.

A generalization of the Neville–Aitken method is described which allows the construction of interpolating functions, other than polynomials, by means of simple recurrence relations. In particular, simple constructions are given for rational functions and trigonometric series which interpolate prescribed function values at non-equispaced positions of the independent variable.

Restrictions imposed by requiring the interpolating functions to be invariant under linear transformations of the coordinates are discussed, and application of the technique to the problem of inverse interpolation is also considered.

It is proved that for a second-order, homogeneous, random process on a globally symmetric space a filter, that is a closed linear operator which is invariant under a group of isometries of the space, may be fully described through a response function, that is that it has a direct integral decomposition into components which are scalar multiples of the identity.

Recently Hoyle and Narlikar(2) have put forward a conformal theory of gravitation in which they obtain an explicit solution of the field equations using certain approximations and assumptions of symmetry. To obtain the solution, the metric is assumed to have a special form. In the present paper a more general form of the metric is considered, and it is shown that many of the features of the special solution are shared by the solution in the more general case. In particular, the implications of the special solution as concerns the sign of the gravitational constant remain valid in the more general case under certain assumptions.

The influence of a uniform external magnetic field on the propagation of harmonic waves in a perfectly conducting elastic plate extending to infinity in vacuum, with its plane boundaries free of mechanical tractions, has been studied. With the plane of the field and the direction of wave propagation being normal to the plate, it is found that the effects of electromagnetic radiation into the adjoining free space cannot be neglected in the case of waves polarized in the plane. When the field is aligned with the direction of propagation, the maximum influence appears to be on long flexural waves, whereas if the field is normal to the plate then the effect is very much pronounced on long plate waves, and the flexural waves propagate almost with Alfvén velocity.

A simple sufficient condition is given for the existence of a lacuna in two-dimensional wave propagation governed by an equation of the second order. This work was inspired by Petrowsky's very general work (Petrowsky (3)) but for waves in two space dimensions we do not need the sophisticated machinery developed by him.

For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts {rθ}, namely:

the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for which

the step problem: for the set

rearranged in ascending order, to determine the steps into which the interval [0,1] is thereby partitioned. There are subsidiary problems of the numbers of gaps or steps of the various lengths, and also of the permutation of 1,…, N when (2) is arranged in order.

The laminar boundary layer on a magnetized plate, when the magnetic field oscillates in magnitude about a constant non-zero mean, is analysed. For low-frequency fluctuations the solution is obtained by a series expansion in terms of a frequency parameter, while for high frequencies the flow pattern is of the ‘skin-wave’ type unaffected by the mean flow. In the low-frequency range, the phase lead and the amplitude of the skin-friction oscillations increase at first and then decrease to their respective ‘skin-wave’ values. On the other hand the phase angle of the surface current decreases from 90° to 45° and its amplitude increases with frequency.

Introduction. The distribution of stress produced in the interior of an infinite elastic solid when a constant pressure is applied over the entire surface of a penny-shaped crack has been solved by Sneddon(6), (7). The problem considered here is the closely allied one of a plane longitudinal wave, harmonic in time, moving in the positive direction of the z-axis and impinging on the surface of a penny-shaped crack. The analysis follows the methods adopted for dealing with an axisymmetrical vibrating punch acting on a semi-infinite elastic solid, Robertson (5).