Table of contents

Errors in the recent paper 'Relativity of rotation' by Browne, (see ibid., vol.10, p.727 (1977)), are indicated.

The action corresponding to a two-dimensional spinor non-linear realisation of supersymmetry is consistently coupled to supergravity. The resulting formal model has local supersymmetry invariance, which permits gauging away of the fermion.

An action is presented for an extended two-dimensional system which possesses, in addition to reparametrisation invariance, local supersymmetry. It is therefore a candidate to generate the theory of an intrinsically spinning membrane.

Spontaneous and stimulated emission from atoms prepared in the super-radiant state are studied, in the non-resonant case. An exact equation of motion for R₃ and approximate solutions are obtained.

The self- and collective motions of one-component plasmas are investigated with the aid of a kinetic theory of classical fluids developed by Sjorgren and Sjolander (1977). Calculations show a strong coupling between self- and collective motions and a change in the behaviour of the plasma dispersion relation with increasing density, in agreement with computer simulations.

The response of a stratified compressible fluid (an 'atmosphere') to an oscillatory point source has in the past been discussed in an astrophysical context, with a view to increasing understanding of the complex problem of wave generation by turbulence. Much use has been made in such analyses of a technique developed by Lighthill in 1960; namely a method of determining the asymptotic functional form of the solution of the governing differential equation. The wave equation for acoustic-gravity waves is solved exactly by three-fold Fourier integral techniques in a manner similar to that of Rao (1973). Use is made of Lighthill's radiation condition to pick the correct physical solution.

The problem is considered of assigning a probability distribution on the basis of a limited number of moments. The Jaynes-Shannon maximum-entropy method is adapted to numerical computations by the construction of a suitable type of histogram. A brief account is given of the numerical methods involved and some numerical examples and results are presented, including an application to ion implantation.

A double perturbation theory for generator coordinate integral equations is developed. For the treatment of collective oscillations the well known harmonic approximation turns out to be a suitable unperturbed problem. A test calculation is performed on Lipkin's exactly solvable many body system.

A straightforward uniform estimate has been made of the difference between the Jost functions corresponding to scattering of one particle by two close potentials within the region Imk>or=0. The Jost function for an approximating piecewise potential was found by developing a very simple technique of fitting. The exact solution of the original problem can be represented as a uniform limit of the n-fold product of 2*2 matrices at n to infinity .

A new three-parameter family of exact solutions of the stationary axisymmetric vacuum Einstein equations, which represent rotating bounded sources, are presented. This family contains the solutions of Kerr and Tomimatsu-Sato as special cases, and may be regarded as a generalisation of the latter to arbitrary continuous delta parameter. The final form of the metric depends on two ordinary differential equations of the second order. When delta is not an integer, these equations define unfamiliar transcendental functions for which rapidly converging series expansions of several types are available. When delta is an integer, the solutions are polynomial or rational functions of spheroidal coordinates and define the discrete Tomimatsu-Sato series, for which those authors give the cases delta =1, 2, 3, 4.

The transport coefficients for a relativistic degenerate electron plasma embedded in a strong magnetic field are calculated in a relaxation time approximation for the case of a collision dominated plasma. The heat conduction gives rise to two thermal conduction coefficients (parallel and transverse) plus a diffusion term. The two usual viscosity coefficients (bulk and shear) split into five viscosity coefficients (bulk, parallel, transverse, cross shear, plus bulk/shear). Transverse and longitudinal electric conductivities are also calculated. The model depends on a relaxation time which has to be evaluated according to the specific physical situation under consideration. The techniques used throughout the paper are those of the covariant Wigner distributions studied elsewhere.

The standard form of the scalar-tensor theory gives eleven equations for eleven unknowns, the metric tensor g_ij and the scalar field psi . Here we eliminate the scalar field to produce a theory that has just ten equations for ten unknowns g_ij. The resulting expression for the action of fields and matter is contained completely in a single expression.

Power series in number density are used to study the distribution of cluster sizes in a continuum analogue of bond percolation on a lattice. The clusters are formed by overlapping of geometrical regions that are randomly distributed in space. The regions are circles and oriented squares in two dimensions, and spheres and oriented cubes in three dimensions. The power series are based on a graphical expansion, using topological weights from percolation theory and probabilities obtained from cluster integrals. Analysis of the mean cluster size series provides estimates of the critical percolation density, and the associated critical exponent.

The crossover behaviour between Gaussian and Heisenberg critical behaviour is studied in an isotropic n-vector model, in which the effect of a lattice cutoff, a= Lambda ^-1, is imitated by means of (nabla² phi )² and (nabla nabla² phi )² terms in the effective Hamiltonian. A field theoretic method is used to construct, to first order in in =4-d, a crossover scaling function for the susceptibility containing the corrections to scaling which involve Lambda directly. The leading corrections vanish, near the Heisenberg fixed point, as ((T-T_c)^v/ Lambda )² where v is the corresponding correlation length exponent, and near the Gaussian fixed point as ((T-T_c)¹2// Lambda )².

Critical properties of Dyson's hierarchical Ising model in one dimension with 'potential' falling off like r^{-(1+ sigma )} are examined in the range 0< sigma <1 where a phase transition is known to occur. A new exact renormalisation group recursion relation is derived for a 'dual' spin probability density function. Together with a scaling-type assumption the authors obtain eta =2- sigma and delta =(1+ sigma )/(1- sigma ) for all 0< sigma <1. Independent evidence, however, suggests that delta =3 for 0< sigma <or=1/2, along with other classical critical exponents in this region. In the non-classical region 1/2< sigma <1 accurate numerical values are obtained for the critical exponents nu and gamma and various Delta sigma = sigma -1/2 expansions to third order in Delta sigma .

The role of the phase autocorrelation function in the generation of speckle, focusing and other scintillation effects produced when radiation is scattered by a deep random phase screen is investigated. New analytical formulae are obtained which reveal the mathematical origin of the observed physical phenomena, and these are supported by a range of numerically computed results.

Poisson's equation in spherical symmetry is solved numerically with the assumptions of zero field and zero electron energy at the cathode surface. The computed current is expressed in terms of an analytic solution obtained for the extreme relativistic case. Internal and external anode configurations are treated, with ratio of electrode radii extending to 10000 and potential difference up to 15 MV. Application of this analysis to the achievement of controlled fusion through pellet compression by charged particles is considered.

It is shown that the free energy per vertex for a ferroelectric ice model on a triangular lattice can be expressed in terms of elementary functions. An exact upper bound for the residual entropy of ice on a Kagome lattice is obtained.