1. One of the most important techniques in the study of non-linear differential systems is the Lyapunov's second method and its extensions. One of the extensions of the method depends upon the fact that the function satisfying a differential inequality can be majorized by the maximal solution of the corresponding differential equation. This method was used extensively by V. Lakshmikantham and others for obtaining results, in a unified way, on stability and boundedness of systems of differential equations.

The symmetrical Epstein refractive index profile (with free space conditions on each side of the transition) is generalized to a non-symmetrical form including both a barrier and a well by the discovery of a profile that cannot be obtained from the standard Epstein profile merely by adjusting the parameters. The analytical evaluation of the reflexion and transmission properties of the model is more complicated than that which occurs in the study of the Epstein profile.

A convergence criterion is applied to the eigenvalue problems, which occur in lower bound calculations for , when either the energy or the nuclear charge is taken as the eigenvalue. In the former problem the lower bounds do not converge to the exact energy and the continuum causes a larger error for σu (1) than for σg (1). However, when the nuclear charge is taken as the eigenvalue the lower bound procedure converges.

In a previous paper (3) the writer has proved

where is a Jacobi polynomial defined as ((4) p.255)

The order and asymptotic form of the error of a general class of numerical method for solving the initial value problem for systems of ordinary differential equations is considered. Previously only the convergence of the methods, which include Runge-Kutta and linear multistep methods, has been discussed.

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 → ∞.

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 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 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.

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.

An analysis is made of the stability of a layer of an elastico-viscous liquid flowing down an inclined plane in the presence of two-dimensional disturbances. The modified Orr-Sommerfeld equation is solved by a regular perturbation technique for disturbances of large wavelengths. It is shown that in the absence of surface tension, the layer is more unstable as compared with that for an ordinary viscous liquid if Q1 > Q2, Q1 and Q2 being stress relaxation and strain retardation parameters respectively.

Some features of the linear theory of an elastic Cosserat plate are discussed and special attention is given to a bending theory which corresponds to the bending of a transversely isotropic three-dimensional plate. A useful representation for the solution of the system of differential equations characterizing the bending of an elastic Cosserat plate is deduced and the results are applied to the problem of an infinite Cosserat plate with a circular hole, the plate being in a state of uniform bending at infinity.