Table of contents

Large simulations of 2D site percolation on a square lattice provide several estimates of the fractal dimension D_h of the 'hull' or outer boundary of percolation clusters close to criticality. Statistical estimates based on an ensemble of more than 10⁶ clusters with varying sizes yield D_h=1.74+or-0.02. Geometrical measurements of the hull two-point correlation for individual clusters with more than 10⁵ sites give D_h=1.76+or-0.01. The observed scaling behaviour is the same both slightly above and below p_c.

The author studies self-avoiding walks (SAWs) on percolation clusters. A scaling function representation for R, the mean end-to-end distance, is proposed which describes a crossover from ordinary SAWs to SAWs on fractals. He distinguishes between SAWs on a single cluster for which R approximately N^x, where x= nu ₁, and SAWs on all clusters for which R approximately N^z, where z= nu ₂ and N is the number of monomers in the walk. He estimates nu ₁(d=2) approximately=1.285 and the correction-to-scaling exponent Omega (d=2) approximately=1.3, and nu ₁(d=3) approximately=1.38. Two plausible generalisations of the Flory (1953) approximation for nu ₁ are investigated and it is argued that none of them provides a satisfactory approximation for nu ₁ at all dimensions.

The universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two-dimensional systems is shown to be a consequence of conformal invariance. Both periodic and free boundary conditions are considered.

By making use of the method of the phase factor of the standard differential loop, the more general forms of the imbedding SO(4) pseudoparticle solutions with vanishing energy-momentum tensor in SU(N) Yang-Mills theory are obtained.

The authors present a preliminary investigation of the spherical Raman- Nath equation, and discuss the connections between its solution and previously known cases.

A rigorous proof is given of the multiplicative property of the superdeterminant (Berezinian) of a supermatrix. The proof obviates the need to consider the domain of existence of the logarithm function for supermatrices, and devolves on the identity det(1-PQ) det(1-QP)=1, where P and Q are compatible rectangular matrices over the odd part of a Grassmann algebra.

Branching rules for E₈ down arrow SO₁₆ are derived from those of E₈ down arrow SU₂*(E₇ down arrow SU₈) and SO₁₆ down arrow U₁*SU₈ by noting that both group chains share a common U₁*SU₈ subgroup. Additional branching rules are deduced from Kronecker products of E₈ and SO₁₆ irreps. In each case the rules distinguish unambiguously between conjugate irreps of SO₁₆. An alternative labelling scheme for the irreps of SO_2k based on its maximal U₁*SU_k subgroup is outlined.

By adjunction of one or two well chosen functionals taking account simultaneously of the unknown u(x) and of the inhomogeneous term f(x), every Fredholm equation of the second kind can easily be transformed into new nonlinear integral equations, for which the solution of the primary equation remains obviously valid. However, when usual iterative solving methods are tested, the convergence of the various sequences now available are very different and one needs criteria to select the best ones. Fredholm equations of the first kind can also be solved, using the new processes described which are particularly efficient after a preliminary transform pointing out the first iterated operator which is necessarily positive. Optimisation techniques are detailed, in order to work out nonlinear equations of particular interest, i.e. they are very suitable to perform numerically an accurate iterative solution.

The Wronskian formulation of the N-soliton solutions of various nonlinear evolution equations-modified Korteweg-de Vries (with both zero and non-zero asymptotic conditions), sine-Gordon, Korteweg-de Vries, Kadomtsev-Petviashvili and Boussinesq equations-are obtained by inductive use of Backlund transformations in Hirota's (1980) bilinear notation.

The authors obtain a new N-soliton solution to the modified KdV equation with non-zero vacuum parameter b using the inverse scattering method (see Zakharov and Shabat, 1972 and Ablowitz et al., 1973) under a non-vanishing condition (Kawata and Inoue, 1977). As demonstrations, they study the special features of the one-soliton and two-solitons in detail. The amplitude, width, and magnitude of the velocity of these solitons are all dependent on b. They all move in the positive direction-such a characteristic is different from that of the KdV solitons.

Appropriate geometrical machinery for the study of time-dependent Lagrangian dynamics is developed. It is applied to the inverse problem of the calculus of variations, and a set of necessary and sufficient conditions for the existence of a Lagrangian are given, in terms of the existence of a 2-form with suitable properties, which are exactly equivalent to the Helmholtz conditions.

Hypervirial theorems are applied in order to obtain easily perturbation series (PS) for eigenenergies and expected values of power of the radial coordinate of a general D-dimensional isotropic anharmonic oscillator (IAO(D)). In the next step the renormalisation method is introduced to obtain renormalised PS allowing the eigenenergies of IAO(D) to be calculated with a high accuracy in a whole range of quantum numbers n, l and for high anharmonic constants.

Analytical formulae are presented of the first- to fifth-order renormalised perturbation theory (RPT), allowing highly accurate calculation of energy levels of a strongly anharmonic oscillator (AO) and an oscillator with a double-well potential (DWP).

The arrival time at a point in the case of the one-dimensional motion of a classical particle is only predictable if initially the position and velocity are both known precisely. It is shown that such an arrival time can be defined in a probabilistic sense when only the initial means and standard deviations of position and velocity are known. The arrival time so defined depends on the subjective concept of confidence limit. It is further shown that arrival time in the latter sense goes over to quantum mechanics. A lower bound on the transit time is derived for this situation by use of the Mandelstam-Tamm inequality.

The authors investigate the quantum mechanical analogue of the classical integrable system named in the title. The Goryachev-Chaplygin (GC) gyrostat is a generalisation of the GC top, where the Coriolis interaction is taken into account. The problem is formulated in terms of the Euclid E(3) group. The integrals of motion are derived. The separation of variables is based on the connection between the degenerated representation of the E(3) group and the special representation of the SO(3,2) group. Some numerical results on the spectra of energy and separation constant are presented. The strong field limit for the integrals of motion is considered in detail, as well as a correlation diagram connecting the states in the limits of weak and strong fields. It appears that the dependence of energy and separation constant on the strength of the Coriolis interaction has the zone behaviour.

It is shown that the spectral density of the Ford-Kac-Mazur (1965) noise for the quantum Langevin equation can be derived from the general requirement of the validity of the fluctuation-dissipation theorem.

The asymmetric clock model is studied on a Cayley tree and commensurate and incommensurate phases are found. The transition lines are obtained from stability conditions and characteristic points in the phase diagram are analysed by numerical iteration. A critical endpoint is found for the case of three states per site and a Lifshitz point for the case of four states per site.

The dimer problem is solved exactly for a hexagonal lattice with general boundary using a known generating function from the theory of partitions. It is shown that the leading term in the entropy depends on the shape of the boundary.

Continuum models of percolation and self-avoiding walks are introduced with the property that their series expansions are sums over linear graphs with intrinsic combinatorial weights and explicit dimension dependence.

The bond percolation critical probability of a planar graph with square and triangular faces, obtained by inserting a diagonal in every other face of the square lattice, is the root of 1-p-6p²+6p³-p⁵=0 in (0, 1), which is approximately 0.404 518. The proof uses the star-triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical.

The (1+1)-dimensional Ashkin-Teller model is studied by the renormalisation group method and by the finite lattice extrapolation technique. The possible phases and their boundaries in an extended space of couplings are determined from the degeneracy structure of the low-lying levels in the thermodynamic limit. The critical behaviour is investigated for ferromagnetic couplings using increasing block sizes in the renormalisation transformation. The line of fixed points and its bifurcation at the Potts fixed point is studied.

The existence of a shock wave propagating in relativistic magnetofluids is assumed and the shock strength is determined. The jump discontinuities of the flow and field parameters across the shock wave are explicitly expressed in terms of parameters defined on the shock surface itself and the flow variables just on the upstream side of the shock. The flow gradients at the rear of the shock have been determined in terms of the flow parameters just ahead of the shock and their interior derivatives along the shock surface itself. The expressions for vorticity and current density generated by a hydromagnetic shock propagating in relativistic magnetofluids have been obtained. A few results of astrophysical interest have also been derived.

The transverse dynamic response of the dilute classical Heisenberg antiferromagnetic chain is calculated near the percolation threshold as k to 0 and k to pi . In the critical limit the universal dynamic structure function, which is calculated in closed form, exhibits dynamic scaling with exponents z=1, eta _T=1. This function displays the crossover from spin wave (k- pi ) xi _p>>1 to hydrodynamic (k- pi ) xi _p<<1 response. The relevance of the results to the experimental situation is discussed.

It is proved that Coniglio's (1981, 1982) lemma holds for any pair of points in any graph in the low-density series expansion of the cut-edge weighted pair connectedness.

For pt.I see ibid., vol.16, no.16, p.3805 (1983). The existence is established of homogeneous time-dependent invariants of arbitrary degree in the coordinate and momentum for the harmonic oscillator with variable mass and frequency. In the constant mass, constant frequency case the cubic and quartic invariants are calculated and some comments are made on the connection with the linear and quadratic invariants.