Table of contents

It is shown that a set of six series representations, and correspondingly a set of six ₃F₂(1)'s is necessary and sufficient to account for the 72 symmetries of the 3j-coefficient.

A three-parameter solution of Ernst's equation, unfortunately not asymptotically flat, is presented. For special values of the parameters, the solution reduces to the Curzon solution.

The concept of thermal equilibrium in a Robertson-Walker universe is discussed. A finite-temperature Green function is constructed and used to calculate (T₀₀)_{beta 0}. A recognised static form results.

A renormalisation group approach is applied to the square site percolating system using a scaling transformation of a finite lattice in real space. The scale of length b for renormalisation has been changed for b=2, 3 and 4. For each scale of length, the location of the fixed point and the correlation length critical exponent are calculated and listed in tabular form.

Analytic results for the absorption spectrum in cooperative resonance fluorescence are presented. An initial broad-band absorption spectrum on the cooperative branch develops a sharp spike as the instability point is approached. The usual one-atom absorption spectrum is regained on the non-cooperative branch. The photon statistics of the scattered light are studied through a calculation of the second-order correlation function g⁽²⁾(0). On the cooperative branch the value of g⁽²⁾(0) is unity reflecting the coherent scattering from the cooperative atomic state. On the non-cooperative branch the value of g⁽²⁾(0) jumps to two, reflecting the incoherent scattering from independently scattering atoms.

A method is presented for estimating definite and indefinite integrals over a density function, such as a local density of states, defined by a three-term recurrence relation. This may be generated, for example, by the 'recursion method' applied to some Hamiltonian, and properties of the approximation are given, and the results derived, in that context.

The subject under consideration is a system of n non-interacting particles with spin, being described by a tensor product of n unitary irreducible massive representations of the Poincare group. A unitary transformation that effects a separation of centre-of-mass variables and internal variables is constructed. By means of this transformation, for any of the n particles there are constructed two position operators both acting on the internal variables only. The first of these operators (called the internal position) commutes with all spatial translations and with the centre-of-mass position, the second one (called the impact position) commutes moreover with all time translations. It is argued that the impact position may be interpreted as the internal position taken at the instant when the n particles are most closely together. By considering spatial separation properties, it is shown that both the internal position and the impact position of any particle depend essentially on the location of all particles of the system.

The asymptotic behaviour of the general solution of Schrodinger's equation for an electron in a logarithmic potential is derived from stability theorems, and the energy spectrum is investigated in both the attractive and repulsive cases. The general solution is represented by a uniformly convergent perturbation expansion which may be used to calculate electron interferences.

For pt.I see ibid., vol.11, p.679 (1978). The authors derive the asymptotic behaviour of the general solution of Dirac's equation for an electron in a logarithmic potential from stability theorems and investigate the energy spectrum, thereby recognising significant differences in comparison with corresponding properties of the Schrodinger equation. A uniformly convergent perturbation expansion of the general solution is presented, in analogy to the treatment of the Schrodinger equation, which may be used to compute biprisma interferences of electron waves.

It is shown that the sums involving squares of particular 3-j symbols, can be expressed in terms of gamma functions and higher transcendentals for complex k_i and for any natural number n. Simplified formulae are derived for integral and half-integral k_i values.

The authors present some interesting, conformally-flat metrics for a pseudo-Riemannian space-time with a divergence-free curvature tensor. The effective (gravitational) potentials for these metrics show both repulsive and periodic properties.

The critical temperature T_c(b) of a d-dimensional system with finite thickness b in one of its dimensions approaches its bulk limit as b to infinity according to T_c( infinity )-T_c(b) varies as b^{- lambda} . To first order in epsilon =4-d the shift exponent lambda is found to be given by lambda =2-((n+2)/(n+8)) epsilon +O( epsilon ²), and thus satisfies the scaling relation lambda = nu _d^-1, where nu _d is the correlation length exponent of the bulk system and n is the order parameter dimensionality.

The authors are concerned with the number of figure eights, dumbbells and theta graphs weakly embeddable in a two-dimensional lattice. They show rigorously that, for the square lattice, the dominant limiting behaviour of the numbers of dumbbells and theta graphs is the same as the limiting behaviour of the number of self-avoiding walks and that the total number of figure eights is less than or equal to the number of rooted polygons. Numerical data are presented to suggest that this bound may be the best possible. Estimates of critical exponents for dumbbells and theta graphs are also obtained.

Following Griffiths the author presents some further methods in the Landau model, for higher order critical points. Techniques due to Arnol'd (1974) allow one to exhibit the codimension and incidence scheme of critical points in the theory for two and three order parameters. Unlike the procedures of Griffiths who analyses the global behaviour of the free energy, the results here are obtained by pasting local solutions together. That the combination of such strata is a complete description is an assumption valid in some cases.

The value of the anomalous magnetic moment of an electron as a function depending on the parameters of the external electromagnetic field is obtained. Physical analysis of the obtained results is presented.

Green tensor expansions for the vector harmonic wave equation, in spheroidal coordinate systems, are derived. The tensor expansions are suitable for application to problems in which the fields are circularly (rotationally) symmetric. Two expansions of the Green tensor are given; one appropriate to infinite free space and the other appropriate to the exterior of a spheroid having a vector Dirichlet boundary condition prescribed on its surface.

The intensity-dependent frequency shift in Thomson scattering, having been a controversial effect in the literature and being as yet unobserved, is calculated within classical electrodynamics for the interaction of a strong linearly polarised plane wave laser pulse with a high temperature thermal plasma. For the highest reported intensities an approximate redshift of Delta omega / omega _i=10^-2 is found. The simplifying assumptions involved in the derivation of the appropriate cross section are discussed. A few points relating to the possible detection of the frequency shift with available experimental facilities are touched upon.

The general scattering problem of vector electromagnetic waves from a stochastic medium containing particles with random size, orientation and position distributions is formulated and solved in terms of a generalised dyadic average scattering amplitude. The independence approximation used in the solution is verified by considering the results of an exact solution to the scattering from two spheres as a function of their displacement. The specific case of a rain-filled medium and propagation in the centimetre and millimetre wave regions is considered. The general theory is applied in order to derive a more generalised form of the Van de Hulst propagation equation for the coherent field in such a medium. Conditions are then formulated for multiple scattering to be negligible and numerical examples presented covering the frequency range up to 100 GHz which show that indeed the effects of multiple scattering are negligible for the raindrop medium.

Complex coordinates are used under the assumptions of geometrical optics to study the relations between phase and power density distributions on plane apertures of reflector antenna systems. No symmetry assumptions are made and the results form a generalisation of existing work in the field. The synthesis of a dual reflector system to produce given aperture phase and power distributions when illuminated by a point source of prescribed power density is shown to depend on the solution of a particular partial differential equation of the Monge-Ampere type. Finally it is shown that even in cases of symmetry the use of complex coordinates simplifies the design equations.

Theory given previously, in a series of papers (1963-1966), relating surface phenomena to the electric double layer by a continuum theory of electrostatic forces in a fluid medium, has been found to be affected by a systematic error due to the attachment of the wrong sign to the mean curvature of the surface. In the present note, one of the main results of that theory, namely, the formula given for surface tension, is corrected and a discussion given of the corrected result.