Table of contents

A connection between non-linear systems of differential equations containing limit cycles, transformation groups, and perturbation techniques is discussed.

It is shown that U( xi )=u₀(cosh² a xi +(ab)^-2 sinh² a xi )^-1 is the exact stationary wave solution to the Whitham equation in a two-layer fluid of finite depth and represents the natural connection between the Benjamin-Ono deep water (a to 0) and Kortweg-de Vries shallow water (b to infinity ) theories.

Eigenvalues of the three-dimensional Schrodinger equation with a radial Gaussian potential are obtained using the Liouville-Green uniform asymptotic method. The results are compared with those obtained by direct integration of the Schrodinger equation.

An appropriate uncertainty measure for a compact Lie group is the invariant dispersion ( Delta F)²=(g^rs(F_r-(F_r))(F_s-(F_s))). It is proven that it is minimized for maximum weight vectors (of greatest length in the weight space), and those unitarily related to them.

The gravitational contribution to the anomaly in the axial current divergence, i.e. RR*/384 pi ², is derived via dimensional continuation.

With the aid of a functional equation the long-wavelength limits of partial structure factors of multicomponent systems are expressed in terms of compressibility and dilatation factors. This generalises similar formulae for a binary system discovered by Bhatia and Thornton (1970).

Renormalisation group methods are used to analyse a phenomenological model of fluids with a coupling to a non-ordering parameter y, which is asymmetric with respect to the order parameter. The correlation function of y is studied above and below T_c. The critical isochore is found to have a singular term of the order mod T-T_c mod ^{3 beta + delta} , in addition to a shift in the critical field and the critical order parameter.

The wavevector q associated with the helical order varies along the paramagnetic-helical critical line as mod q mod approximately mod p mod ^{beta k}, as a Lifshitz point (T=T_L, p=0) is approached. Renormalisation group techniques in d=4+¹/₂m- epsilon ( epsilon >0) dimensions are used to calculate the critical exponent beta _k, associated with an m-fold Lifshitz point, to second order in epsilon .

A lower bound transformation of the type introduced by Kadanoff is applied to the discrete s-state Potts in d=3 dimensions. For all values of s=2,3,4,5,6 the transformation predicts a continuous transition.

It is suggested that the correct processing of data in a recent experiment of Kimble, Dagenais and Mande (1977) implies that the short-time value of the correlation function of photons emitted by a single atom in resonance fluorescence is zero. This result is found by firstly correcting an apparent error in the treatment of the background radiation and secondly by taking account of Poisson fluctuations in the number of radiating atoms.

The frequency up-conversion process with stochastic coupling function is considered. It is shown that the total photon number increases with time. The antibunching effect (g⁽²⁾<1) is predicted regardless of the mean photon numbers available in the process.

It is shown that a steady state non-linear electron plasma wave can have saw-toothed shaped structure.

Recoupling coefficients and isoscalar factors of a chain of compact topological groups (finite or continuous) are discussed in terms of character formulae. Special emphasis is put on the chain SU₂ contains/implies G where G is isomorphic with the spinor group of a point symmetry group.

An explicit formula for general harmonic oscillator brackets is derived. The formula is particularly useful and convenient in numerical computations.

The critical exponent of the intensity moment is computed from the Thorn-Arnol'd classification of caustics as catastrophes. The caustics are studied on the torus whose coordinates are 16 N random phases theta ₁... theta _N for members of the ensemble describing the phase screen or inhomogeneous medium responsible for the disorder of the wave. The results indicate that when N to infinity (Gaussian random medium) the exponents depend only on whether the waves propagate in two or three space dimensions.

The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.

The author has previously presented a new three-parameter family of exact asymptotically flat stationary axisymmetric vacuum solutions of Einstein's equations which contains the solutions of Kerr and Tomimatsu-Sato (TS) as special cases. In this paper, he considers two special cases of the previous family which must be constructed by a limiting process. These are interpreted as a 'rotating Curzon metric' and a 'generalised extreme Kerr metric'. In addition, approximate forms for the original metrics are given for the case of slow rotation and small deformation.

A homothetic conformal transformation of the metric tensor is both necessary and sufficient to ensure that a source-free non-null electromagnetic field in four dimensions undergoes a duality rotation leaving its energy-momentum tensor invariant. For null electromagnetic fields the condition is sufficient but not necessary. The result suggests that vacuum Einstein-Maxwell space-times exist which admit groups of homothetic motions. If the order of such a group is greater than unity it must be associated with isometry transformations of the metric tensor other than those obtained as a special case of the conformal group.

A model of discrete space-time is presented which is, in a sense, both Lorentz invariant and has no restriction on the relative velocity between particles (except v<c). The space-time has an inbuilt indeterminacy.

The three-state Potts model on a triangular lattice with nearest neighbour interaction K is studied. Using a simple position-space renormalisation group transformation it is found that, in addition to the usual continuous transition to the ferromagnetic state which occurs for positive K, a continuous transition to an antiferromagnetic state occurs for negative K. The transformation introduces a three-spin interaction. In the space of these two interactions, the lines of ferromagnetic and antiferromagnetic transitions meet at a bicritical point. These results differ from those predicted by Landau theory.

S_sigma (k, beta ), of a given form, over a d-dimensional lattice are analysed for small beta and k= mod k mod . In particular, for non-integral sigma , S_sigma (k, beta ) is shown to have a singular part and to exhibit cross-over phenomena. Explicit results for the cross-over function are obtained.

A discussion of supersymmetric spin systems is presented extending the results obtained in a recent paper (see ibid., vol.9, p.1497 (1976)). A two-dimensional model is given and gauge invariance is defined; the latter is shown to necessitate the introduction of further spin operators.

The formula for the dielectric constant of a liquid, derived by Onsager, is modified to take account of the effect, on molecular orientations in water, of hydrogen bonding. This is achieved by using the lattice model of Bell. By a suitable choice of parameters, isobars are obtained for the dielectric constant as a function of temperature. These are in fair agreement with experiment and illustrate well the important role of hydrogen bonding.

Mode coupling techniques are used to derive, from the exact equation of motion, an expression for the diffusion tensor, and hence the friction tensor, of a polymer chain in dilute solution. This theory allows one to analyse the approximations needed to obtain an expression for the friction tensor similar to the usual hydrodynamic results, and provides a framework in which non-hydrodynamic contributions can be discussed.

The effects of an external alternating electric field on radiation by a charged particle traversing an infinite isotropic dielectric with a uniform velocity nu ₀ exceeding the phase velocity of light c/n, where n is the refractive index of the dielectric, are considered. In the first case the field E=E₀ sin omega ₀t is applied parallel to the direction of motion of the particle and in the second case it is applied perpendicular to it. Exact formulae for the energy loss due to Cerenkov and Doppler radiations are obtained and the loss is evaluated for different applied alternating fields. In both cases, it is found that the Cerenkov radiation is reduced considerably. For the same field frequency and field strength, the parallel field reduces Cerenkov radiation more effectively than the perpendicular one. With the application of an external alternating electric field, Doppler radiation starts appearing, reaches a maximum and decreases as the field intensity increases.