Table of contents

Concerns a generalisation of the fundamental theorem on super-multiplicative functions for which the super-multiplicative inequality is replaced by a_n+m>or=a_na_f(m) with lim_{m to infinity} m^-1f(m)=1.

The Kadomstev-Petiviashvile and sine-Gordon equations in two spatial dimensions are shown to be reducible to Painleve transcendental equations of the first and third kind respectively in terms of suitable Lie-invariant variables.

Using perturbation theory, asymptotic expansions are derived for the eigenenergies and eigenfunctions of the wave equation for the interaction lambda x²/(1+gx²) in the range of small values of g and large values of lambda . The first few energy eigenvalues are calculated and found to be comparable with the non-perturbative results obtained by Mitra (1978).

A discrete Hamiltonian to describe the vulcanisation which occurs when linear polymer chains are mixed with cross-linking units is proposed. Here the vulcanisation of the chains can occur via clusters of cross-linking units. The Hamiltonian is a simple combination of the n-vector model in the limit when n goes to zero, and the m-states Potts model when m goes to unity. The partition function is discussed. The Migdal renormalisation group shows that the chain behaviour is always controlled by the self-avoiding walk (SAW) fixed point. The vulcanisation is described by percolation exponents except in the vicinity of a higher-order critical point where it crosses over the SAW exponents.

The generating function C(x) for the number of self-avoiding walks on the face-centred cubic lattice is extended by two terms to order 14. The series coefficients are analysed for a singularity of the form A₁t^{- gamma} +A₂t^{- gamma +1}+Bt^{- gamma + Delta 1} with t=1- mu x, where mu is the connective constant. Two cases of interest are studied, (a) gamma =1¹/₆, B=0 is conjectured in earlier work on series expansions and (b) gamma =1.1615, Delta ₁=0.465 as predicted by renormalisation group (RG) calculations. It is found that the series coefficients are better fitted to the RG predictions (b).

It is conjectured that the singularities in the free energy for the q-state Potts model on the square lattice lie on two self-dual circles in the complex x plane where x=exp(-J/kT) is the usual low-temperature variable. These circles are natural generalisations of the known circles for the Ising model (q=2). The conjecture leads to the prediction that there is a single antiferromagnetic critical point at x=1+(q/(q-1))¹2/.

A modified antiferroelectric model (MF-model) is introduced. It is shown that the general solution of this model, whose basic ingredients are two energies epsilon ₁ and epsilon ₂, includes as special cases the F-model ( epsilon ₁= epsilon ₂= epsilon ) and the Ising model ( epsilon ₁= epsilon , epsilon ₂= infinity ). The MF-model has a polymer equivalent in which a polymer bond can assume one of three available states, one trans state (energy zero) and two gauche states with energies epsilon ₁ and epsilon ₂.

The authors have re-examined the high-temperature susceptibility series of the spin-¹/₂ Ising model on the three-dimensional tetrahedron lattice. They conclude that the series data provide strong evidence for the result gamma =1.250 and are inconsistent with the recent renormalisation group prediction gamma =1.2402+or-0.0009.

Assuming that ultra-dense matter behaves as a fluid of quarks rather than hadrons, the authors investigate the possible superfluid order parameters which may arise.

A new series of unitary representations of the general covariant group in N-dimensional real vector space (group Diff R^N) is constructed. The matrix elements of the finite transformations in the space of the principal series of unitary representations of the special linear group SL(N,R)(N>3) and in the space of all series of unitary representations of SL(2,R), SL(3,R) are found.

A method for obtaining a simple and compact expression for the generating function of the Weyl invariants is developed. The approach starts from the generating function of the representation basis in Bargmann space and uses the properties of this space in connection with Gaunt's integral. From the generating function of the Weyl invariants it is possible to derive the 3-j symbols of SU(N). When applied to SU(2), the method merely leads to the well-known Schwinger generating function of the 3-j symbols. The generating function of the SU(3) representation basis is built, and the SU(3) representation matrix elements are calculated; the equivalence between a part of these matrix elements and the Beg and Ruegg harmonic functions is proved. A new parametrisation for SU(N) is proposed, the invariant measure of which is explicitly determined. As an illustration, the approach is applied to the determination of the generation function of the Weyl invariants in some particular representations of SU(3) and SU(N).

The methodology of earlier papers by Butler and Wybourne (1976) is used to obtain algebraic formulae for 6j symbols of the double dihedral and cyclic groups and the 3 jm factors for all possible imbeddings: D_m contains/implies D_n and D_m contains/implies C_n. The usual 3 jm symbols of angular momentum theory, that is for SO₃ contains/implies SO₂, do not have a phase choice which allows their factorisation into SO₃ contains/implies D_infinity and D_infinity contains/implies SO₂ 3jm factors. The authors derive the change of phase necessary for factorisation, thus obtaining a relation between SO₃ contains/implies D_infinity factors and the SO₃ contains/implies SO₂ 3jm symbols of standard angular momentum theory. The use of maximal imbeddings has removed the multiplicity problems encountered by other methods.

The principle of basis set representation in terms of coordinate interchange matrices, of which the Pauli spin matrices are an example in two dimensions, are extended to three and four dimensions. The four-dimensional basis set of coordinate interchange matrices satisfies the usual conditions of completeness, but the three-dimensional basis set cannot be complete under any circumstances and an 'anticomplete' property is assigned to it. The coefficients of the basis set, when used to represent an arbitrary matrix, form a Hadamard transform of the cyclically interchanged arbitrary matrix.

Inhomogeneous differential approximants (J/L;M)_f(x), (J/L;M,N)_f(x,y) etc. are defined for functions of one or more variables given as power series expansions, and some of their properties are exposed. The approximants are easily computable, and numerical studies are reported (for single-variable series) which demonstrate their utility in circumstances where the customary direct or logarithmic derivative Pade approximants (which are limiting cases) are inadequate.

In terms of a previously suggested Berger et al. (1977) iterative method for finding the lowest-lying eigenstate of hermitian matrixes the problem of pseudo-convergence, i.e. the tendency to converge to an undesired eigenvalue, is discussed. A simple extension of this algorithm is presented which provides a means of identifying and circumventing pseudo-convergence in many cases. This extended scheme may also be used simultaneously to obtain excited states in a numerically economical manner.

A systematic method for the generation of multiple time scales expansion of the oscillator (d²x/dt²)+ omega ²x= epsilon Sigma _n=1^N Sigma _m=1^mg_nmxⁿ(dx/dt)^m to any order is presented. The excessive freedom which is inherent in the process is conveniently controlled, thus allowing one to generate easily different expansions to the same problem. This option was used to study the extent by which different uses of this freedom can affect the accuracy of the expansion, concluding that the effect may be significant. The new method was applied to the Duffing and the van der Pol oscillators. The complicated algebraic computations involved were accomplished by a computer.

For any central potential, it is possible to construct a constant of the motion which generalises the Runge-Lenz vector. This is a vector pointing from the centre of force to the nearest point of the orbit where r is maximum (or minimum). In general, the direction of this 'constant' vector thus changes abruptly whenever r is minimum (or maximum).

A differential equation in momentum space is derived for the case of an attractive Coulomb potential. The bound-state energies and the momentum eigenfunctions are shown to arise from this differential equation in a simple fashion. The zero-energy partial-wave momentum eigenfunctions are also derived. The s-wave bound states and their momentum eigenfunctions in a three-dimensional linear potential and an attractive 1/(r+ beta ) potential are derived by similar techniques. The connection between the quantisation formulae in these potentials and the classical action integral in momentum space is explored.

It is demonstrated that the density matrix method may be applied to problems in time-dependent perturbation theory. The advantages of the method are that it is compact, systematic and the effects of radiation damping can be readily treated on a phenomenological basis.

The spectral intensity distribution of resonance fluorescence radiation emitted by N two-level atoms is calculated for the case of a weak electromagnetic field. Using these results the spectrum is discussed in terms of the number of atoms, their localisation and the orientation of the transition dipole moments. It is shown that the spectrum can be expressed as a sum of Lorentzian lines only in some specific cases. The general condition describing these cases is given.

The author investigates the graded Lie algebra of generators of symmetries assuming that the even subspace of this algebra is spanned by the generators of the de Sitter group and of internal symmetries. He finds that the generators of the de Sitter group form the Lie algebra of O(3,2) and not of O(4,1) and that, as in the case of conformal symmetry, there is a complete fusion between geometric and internal symmetries. In particular the internal symmetry group SO(N) is generated by Fermi charges.

A new approach to the problem of the motion of a self-interacting massive charged particle in general relativity is presented. a charged Robinson-Trautman solution is used as a general relativistic model of such a particle. Such a solution is shown to generate a unique world line in its own H space, which is interpreted as the world line of the particle. Using the R-T dynamical relations, the equation of motion of the particle is derived, which, in the limiting case of zero curvature, is shown to be the same as the classical Lorentz-Dirac equation of motion.

For pt.II see ibid., vol.12, no.7, p.106 (1979). The author considers a line-element which, in linearised gravity, is associated with the gravitational field of an arbitrary accelerating, rotating, axially symmetric source. If the body rotates about its symmetry axis and moves along its symmetry axis when viewed in the background flat space-time and if its linearised field (Riemann tensor) is free of wire singularities then the body must move with uniform acceleration and rotation and its mass must be constant.

For pt.III see ibid., vol.12, no.10, p.1771 (1979). The author continues previous work on equations of motion in linearised gravity in which a new approach was adopted to study the motion of the sources of some Robinson-Trautman fields. He now considers the introduction of an external field to drive the source. This is demonstrated for the Levi-Civita fields of both a charged and uncharged uniformly accelerating mass.

The nu vacuum discussed recently by Fulling (1977) in his general study of alternative vacuum states in space-times with horizons is investigated specifically for the scalar and Dirac fields in Kerr space-time. An explicit evaluation of energy flux shows that, except for the effects due to super-radiance of scalar waves, this vacuum corresponds to a black hole in thermal equilibrium with its environment.

The mean-square fluctuation in the number of colloidal particles (radius approximately 45 nm) in a small volume element ( approximately 8 mu m³) of an aqueous dispersion was measured by photon-correlation laser light scattering. Both randomly distributed, non-interacting particles and those showing a 'liquid-like' spatial arrangement owing to long-range repulsive Coulombic interactions were studied. The magnitude of the reduced fluctuations in the latter case agreed with that predicted from the structure of the dispersion, which was determined independently from the angular dependence of the average scattered light intensity. This provides the first direct experimental verification of the fundamental Ornstein-Zernike relationship between numbers fluctuations and the pair distribution function g(r) in a system of interacting particles. Possible extensions of the experiment, including the measurement of four-particle correlations, are discussed briefly.

The spin-¹/₂ Ising model high-temperature series are re-examined by an analysis that parallels the method used to analyse the continuum phi ⁴ spin model. Results of the Ising model in three dimensions do not agree with results in the literature based on more conventional methods of analysis. The authors cannot decide using only the series terms presently available, which of the methods is to be preferred. They conclude that previous error assessments are unduly optimistic and that the three-dimensional, spin-¹/₂, Ising model may satisfy hyperscaling in agreement with the continuum results. An attempt is made to estimate the number of additional high-temperature series terms necessary to resolve the hyperscaling question.

It is shown that, by taking the zero-temperature limit of the annealed dilute bond Ising model, approximate expressions for quantities of interest in the percolation problem can be obtained. These are compared with computer simulations for a square lattice and shown to be reasonable over the entire concentration range whilst giving incorrect critical exponents.

The statistical analysis previously used for the temperature behaviour of clusters for the Ising model is applied to Monte Carlo samples of percolation clusters. Three cases are considered: (a) positive correlation (T=2T_c ferromagnetic): (b) random (T= infinity ); (c) negative correlation (T=2T_c antiferromagnetic). It is found that the exponents which characterise the decay of the cluster-size distributions do not depend on correlation. These distributions can be fitted over their whole range by assuming that percolation critical exponents are independent of correlation, but the scaling functions which then result do depend on correlation. Statistical parameters which are related to the compactness or ramification of clusters change smoothly with correlation. However, some features of negative correlation are significantly different in behaviour.

The mapping of the critical points in the q-state model for q<or=4 onto the Baxter line in the eight-vertex model makes it possible, by a comparison of the exactly known critical exponent y_T^8v with the approximately known values for y_T^P, to conjecture the relation (y_T^8v-2)(y_T^P-3)=3. This relation confirms weak universality.

A lattice fluid model on a two-dimensional quadratic lattice is observed in which molecules are capable of preferential bonding between second neighbours. A Hamiltonian is introduced which has interactions which differentiate between parallel and antiparallel spins and parallel and orthogonal spins. A number of interesting special cases are considered including a four-state dilute Potts model and a five-state Potts model. The phase transitions are investigated using Landau symmetry theory. The fluid transition is studied using a mean-field approximation. Within the limitations of this method the system is predicted to have water-like properties.

The many-parameter Gaussian theory of a polymer chain is in the asymptotic region N to infinity (N is the number of monomer units) equivalent to a one-parameter scaled Gaussian theory. Such theories fulfil the necessary condition of minimum free energy only in a polymer chain without remote interactions along the chain. Therefore they cannot be used as a reliable basis in SCF calculations in polymers.

Free electromagnetic vector potentials in Coulomb and Gupta-Bleuler gauges are shown to be unitary equivalent in both Minkowski and Euclidean regions. For covariant gauges, the Euclidean electromagnetic potential is Markovian but non-reflective, whereas for the Coulomb gauge it is reflective but only satisfies the Markov property with respect to special half-spaces. The Feynman-Kac-Nelson formula can be established for the case of the Coulomb gauge.

For pt.II see ibid., vol.12, p.1105 (1979). The dynamical process by which the steady propagation of a coherent light pulse of long width takes place in a dielectric medium is studied. In the absence of direct interaction between atomic dipoles, the nonlinear polariton is unstable against a small perturbation and develops self-modulation of its envelope. Nonlinear Schrodinger equations describing this self-modulation are derived for the two cases where the carrier wave frequency lies outside and inside the polariton gap. It is shown that an arbitrary incoming pulse of long width outside the polariton gap evolves as composite pulse of multiple peak structure, which is regarded as a bound state of the steady pulses obtained in a previous paper. The evolution process of pulse inside the polariton gap and the effect of direct interaction between atoms are also discussed.

Photon-photon and photon-phonon anticorrelation effects in Raman scattering are studied theoretically. The Heisenberg equations of motion describing Raman scattering are solved in the short-time approximation. The laser, Stokes, anti-Stokes and phonon modes are quantised. The joint normally ordered characteristic functions are evaluated for modes exhibiting anticorrelation effects. The problem of existence of a Glauber-Sudarshan quasi-distribution function related to anticorrelation effects is considered. The magnitude of the anticorrelation effects depends on the initial statistical properties of the photon and phonon modes.