Table of contents

The branching rule is obtained for the decomposition of the osp(1/2N,r) oscillator-like unitary irreducible representations into sp(2N,r) representations. Raising operators for osp(1/2N,r)sp(2N,r) are determined and used to construct a basis in the irreducible representation carrier space. The matrix elements of the odd generators between two sp(2N,r) lowest-weight states are calculated. The general results so obtained are illustrated with the osp(1/4, r) example.

Feynman-Vernon theory is used to calculate the energy levels of a quantum harmonic oscillator interacting linearly with a heat bath.

The energy spectrum of the quantum elliptic billiard is obtained by solving the Schrodinger equation in elliptic coordinates. The corresponding eigenstates also diagonalise an operator B which commutes with H. A numerical search of the exact eigenvalues of B and H permits one to follow each state as a function of the deformation parameter mu . Geometrical arguments, also valid for the simpler problem of the rectangular box, allow one to understand the results obtained.

Some useful algebraic relations between the anomaly of D_mu J^mu and anomalous commutators are derived using only the canonical commutators and the equation of motion. Their applications are discussed.

Fractals constructed by recursive processes are introduced to model growth phenomena. These fractals are simultaneously directed and self-similar in analogy with patterns growing under diffusion-limited conditions. The multifractal nature of the harmonic measure associated with Laplacian interfaces is qualitatively interpreted using the models. Calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion-limited aggregates.

Renormalisation of the recently proposed field-theoretic model for the 'true' self-avoiding random walk is analysed. Apart from a special case, which corresponds to the problem of a random walk in a random environment, the model is shown to be non-renormalisable in the case of short-range interactions, contrary to earlier conjectures. The long-range version of this field-theoretic model is shown to be renormalisable in certain regions of the parameter space, and renormalisation group analysis of the model is carried out to two-loop order in the region corresponding to the 'true' self-avoiding random walk with long-range repulsion. In a different region of the parameter space the anomalous dimension of the diffusion coefficient is determined exactly in perturbation theory.

A one-parameter family of transformations defined on the set of all one-dimensional cellular automata is studied. The class of a given cellular automaton is unchanged under the set of these transformations. For class 3 cellular automata, transformed statistical quantities satisfy simple scaling properties.

Exact results are obtained for the local magnetisation and correlation functions of the two-dimensional Ising model with a line defect which produces large-scale inhomogeneity in its vicinity. It is shown that near the critical point the correlation functions exhibit a non-scaling behaviour.

The author has constructed a Stephen-type mean-field theory for the elastic granular disc model in two dimensions and calculated the mean-field value of two crossover exponents phi ₁ and phi ₂, which describe the ways in which the two-point elastic susceptibility and the angle-angle correlation function scale with the distance, respectively. The author has found that phi ₁=2 and phi ₂=1. The possible relation of phi ₁ and phi ₂ to the bulk modulus exponent f is discussed.

The dynamics of neural networks with constant stabilities is studied analytically for states in the vicinity of a stored pattern. Only two parameters of the synaptic matrix determine the dynamics of the neurons in the cases considered. The resulting equations are able to predict the basin of attraction near the critical storage capacity. The authors compare their analytical results with simulations of two network models.

Using an explicitly finite method which only needs the discrete levels of Schrodinger equations, the non-zero quantum contribution to the soliton mass in the SUSY sine-Gordon model is found.

A new modified induced module construction is presented for all finite-dimensional irreducible typical and atypical modules of a type-I basic classical Lie superalgebra. The method is illustrated with some low-dimensional representations of gl(m mod n) and all representations of gl(2 mod 1).

Simple methods are described for constructing the Weyl orbits of exceptional Lie algebras and for identifying the orbit of an arbitrary weight. The methods make use of convenient bases in weight space. They are applied to G₂, F₄, E₆ and E₇. A complete table of all weights of all orbits of F₄ is given. The weights of the 37 orbits of smallest weight length of E₆ and E₇ are given in tabular form. The depth structures of the orbits are discussed.

Generating functions for the number of states with given total angular momentum and spin, for a system of identical fermions or bosons, are presented, along with generating functions for the number of states of a system of identical particles with any specified permutational symmetry and total spin. The spin of each particle could be any integral or half-integral number. A specific example for a system of particles with a pseudospin s=3/2 is presented. The extension of the generating functions for a system of particles which carry more than two angular momenta is made.

The authors describe and compare the dissipative effects on the scaling laws of the Obukhov-Corrsin universal equilibrium theory as intermittency corrections formulated according to the log-normal model of Obukhov and Kolmogorov (1962) and the beta model of Frisch et all (1978) under conditions of Prandtl numbers away from unity.

N-phase solutions of the Kadomtsev-Petviashvili (KP) equation (1970), that are periodic in space variables x and y, were obtained and effectively investigated using the Schottky uniformisation, of which a short description is given. Many wave patterns are represented graphically as contour plots and as isometric projections for different parameter values of two-, three- and four-phase solutions of the KP equation.

The author studies the way in which electromagnetic waves at arbitrary angles of incidence are localised in the direction normal to the strata of a randomly stratified medium. The author points out differences from the case of normal incidence, and, in particular, differences between the two polarisation modes which amount to a type of birefringence.

The equations for the laser with a saturable absorber in the mean-field limit and exact resonance are considered. For critical values of the number of active atoms in the laser and the absorber the equations exhibit a codimension-two degeneracy of the Takens-Bogdanov type, where a stationary bifurcation and a Hopf bifurcation with infinite period coalesce. The envelopes of the electric field and the atomic polarisations are treated as complex variables. This allows one to take differently polarised laser fields into account. The authors compute the relevant normal form coefficients and divide the space of the physical parameters into a number of regions that give rise to qualitatively different bifurcation scenarios. These include hysteretic and continuous stability exchanges between circularly polarised and modulated linearly polarised laser fields.

The authors exploit some exact soluble model potentials V^N( alpha , beta )=V(A,B)+ lambda x⁶ in order to extrapolate (as lambda to 0) reliable eigenvalues of the one-dimensional Schrodinger equations with either anharmonic or symmetric double well potentials of the form V(A,B)=1/2Ax²+Bx² (B>0). Their procedure, which corresponds to low-order Rayleigh-Schrodinger perturbation theory, is found to be competitive with both high-order Pade summation of conventional RSPT and large-scale variational calculations using harmonic oscillators basis functions.

The authors study the question of quantum integrability for two-dimensional Hamiltonian systems with special attention on the ℏ² correction terms in the potential. A class of Hamiltonians of type H=1/2(p_x² + p_y²)+ v₄y⁴ + y²(f₂(x) + α ₆) + f₀(x) with a second invariant of type I = p_x⁴ + A(x,y)p_x² + B(x,y)p_xp_y + C(x,y) p_y² + D(x,y) is considered. The general solution for f₂ involves elliptic integrals. For quantum integrability the potential must be modified with ℏ²-dependent terms. They construct a point transformation which, after coupling constant metamorphosis, takes the Hamiltonian to a new quantum integrable Hamiltonian for which no correction terms are necessary. The new system does not in general have a flat space kinetic part.

The symmetry algebra of the m-dimensional quantum anisotropic harmonic oscillator Hamiltonian H with commensurate frequencies is shown to be u(m). Each eigenspace of H carries a single irreducible representation of u(m). However, distinct eigenspaces can yield equivalent u(m) representations. The dynamical algebra is the non-compact symplectic algebra sp(m,r). For m+3, the anisotropic Hamiltonian is relevant to superdeformed nuclei.

A rich set of integrable two-dimensional quantum field theories are obtained from integrable lattice vertex models with q states per bound (q>or=2) in the scaling limit by a generalisation of the light-cone lattice approach. Chiral fermion models with any simple Lie group of symmetry arise in this way (for finite q) as well as bosonic models like the principal chiral model (for q= infinity ). The Hamiltonian, momentum and colour-conserved currents are constructed on the lattice and the bare equations of motion are derived. The renormalised mass spectrum is given explicitly for the set of models considered here. All these integrable vertex models yield conformal invariant theories if one takes the scalding limit in an appropriate different way. It is argued that the values one obtains for the central charges are the same as those provided by the Sugawara construction (in the continuum) for all simple Lie algebras.

A one-dimensional mixture system of impenetrable, hard-core particles is analysed numerically. The particles have masses either m₁ or m₂, with sigma identical to m₁/m₂=1 and 1.2. Several realisations of stochastic boundary conditions simulating the contact with thermal walls are considered. The authors report on the relaxation from initial states with velocities +or-1 and on the properties of the final stationary non-equilibrium state. In particular, they describe the breakdown of Fourier's law when sigma =1 and the good ergodic properties of the case sigma >1, and compare with some existing theories for sigma =1.

The authors enumerated the number of self-avoiding polygons on the honeycomb lattice to 82 steps. Analysis of the resulting series gives for the connective constant mu =1.8477594+or-6*10^-6, in agreement with Nienhuis' value (1984) of mu =(2+2)¹2/=1.84775906. . .. The critical exponent alpha =0.4999+or-10^-4 (unbiased), or alpha =0.49998+or-2*10^$0-⁵ (biased). A non-physical singularity on the negative real axis is accurately located, and some curious results concerning the confluent singularity are obtained.

The authors prove that the Hausdorff dimension of the two-dimensional Edwards' random walk (1965) is two with probability one. This result is in contradiction with the widely accepted conjecture that the Hausdorff dimension of a random walk is the inverse of the critical exponent v; actually, v is 3/4 for the two-dimensional Edwards' random walk.

Simple models for crack growth which are closely related to the diffusion-limited aggregation (DLA) model have been explored using computer simulations. In these models the bond-breaking probabilities for bonds at the surface of a growing crack in a two-dimensional (triangular) network of bonds and nodes are proportional to ( delta _i)^eta and delta _i is the bond strain. The authors' results indicate that the cracks generated by these models have a fractal structure and that their effective dimensionalities depend on both the bond-breaking probability exponent ( eta ) and the boundary conditions (which bonds are considered to be surface bonds that can be broken) at the crack surface. Very similar results were obtained using shear and dilational strain.

The authors have investigated several models of monodisperse reaction-limited cluster-cluster aggregation in two and three dimensions and have obtained the first systematic theoretical estimations of the overall anisotropy of cluster-cluster aggregates, and the way in which small clusters approach scaling. Their results are supported by simulation and confirm that in the monodisperse case the approach to scaling is very rapid and the clusters are significantly aspherical in aspect ratio.

Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.

The operator content of the cyclic solid-on-solid models is derived under general boundary conditions by mapping the row-to-row transfer matrix onto the six-vertex model and XXZ Heisenberg chain. For the shift boundary conditions, the dimensions and spins of the primary operators are derived analytically from finite-size corrections to the scaling spectra. The full operator content agrees with results for a quantum chain model having the same local symmetry.

The authors investigate the structure of extended conformal algebras generated by local chiral fields of spin 1,2,. . .,n using the Gelfand-Dickey algebra of formal pseudodifferential operators for the gl(n) algebra. The commutation relations of Zamolodchikov's spin-3 operator algebra (1985) are derived in the presence of an additional U(1) conserved current.

For pt.I see ibid., vol.21, p.4245-50 (1988). The internal potential for a homogeneous spheroid in a rectangular coordinate system may be obtained with the substitution of the root of the equation for confocal spheroids being zero in the expression of the external potential. In a spheroidal coordinate system, the integration of the internal potential would be composed of two parts: inner and outer volumes with respect to the interior point of a spheroid, which generates the explicit forms of the internal potential and the expressions on spheroidal surface, where the external and internal potentials should merge, for both prolate and oblate spheroids. The reductions for the potential at the centre of the spheroid to a sphere and the numerical results compared between the rectangular and spheroidal coordinate systems have certainly confirmed the correctness of the formulae of the internal potentials for both spheroids.

The relation between mass, M, and size, R, for self-avoiding random surfaces enclosing a hard sphere of radius r is examined. It is suggested that M approximately R²F((R - r)/r), with F proportional to the surface roughness. These functions are investigated using experimental data and theoretical arguments.