Table of contents

The authors investigate the ability of the lattice Boltzmann equation to reproduce the basic physics of fully turbulent two-dimensional flows and present a qualitative estimate of its computational efficiency with respect to other conventional techniques.

For the spin-1 system in a slowly varying magnetic field, the authors investigate its classical analogue to calculate the Hannay angle and explicitly check that this is related to Berry's phase according to the well known semiclassical formula.

The author studies the bosonisation of the Abelian Thirring model in the presence of vortices in the framework of path integrals.

The authors construct a family of isotropic spin-s quantum chains consisting of sums of operators satisfying a Temperley-Lieb algebra. Exact values for the infinite lattice limit of the ground-state energy per site and for the (non-zero) gap to the lowest energy excited state follow from the Temperley-Lieb equivalence with a Bethe ansatz soluble XXZ model. The family of spin chains includes the biquadratic spin-1 model.

The exponents for the resistance of a random walk path and the 'random walk on a random walk' problem are related to a number of other exponents for random walks. Some rigorous inequalities for these exponents are then established.

Biased diffusion on hierarchical comb structures is studied within an exact renormalisation group scheme. The scaling exponents of the moments of the first-passage time for random walks are obtained. It is found that the scaling properties of the diffusion depend only on the direction of bias. In a particular case, the presence of bias may give rise to a new multifractality.

Using Seeley's heat kernel expansion, the authors compute the asymptotic density of states of the Dirac operator coupled to a magnetic field on a two-dimensional manifold with boundary ('fermionic billiard'). Local boundary conditions compatible with vector current conservation depend on a free parameter alpha . It is shown that the perimeter correction identically vanishes for alpha =0. In that case, the next-order constant term is found to be proportional to the Euler characteristic of the manifold. These results are independent of the external magnetic field and of the shape of the billiard, provided the boundary is sufficiently smooth. For the flat circular billiard, the constant term is found to be -¹/₂, in agreement with a numerical result of Berry and Mondragon.

BRST cohomology calculus in the space of superstring differential forms is treated in detail. The Hodge-star duality transformation is introduced and the explicit expressions of cohomology operators are derived for superforms of arbitrary order.

The author writes down the regular representation of the Temperley-Lieb algebra T_k(q) in the basis of reduced words on the k Temperley-Lieb generators. When k=2n-1, representations of these generators may be used to construct the transfer matrices for statistical mechanical models on an n-site wide lattice. He shows that the generically irreducible representation of T_2n-1(q) responsible for the unique free energy in such models may be restricted to the regular representation of T_n-1(q). He gives equivalent forms for the regular representation, derived from lattice models, which manifest its indecomposable structure when k and q are such that T_k(q) is semisimple. He hence generalises to obtain the structure of T_k(q) when not semisimple. A transfer matrix built with generators in the regular representation gives the long distance properties of all possible such models. He shows how to find the semisimple quotient algebra which gives these long distance properties when T_k(q) itself is not semisimple. He hence classifies the long distance properties of these statistical mechanical models.

Analysis is performed of general features of schemes for calculating the topological charges of heterophase gauge field configurations and order parameter configurations. The feasibility is substantiated of using the results and methods of the theory of topologically non-trivial order parameter configurations for studying complicated heterophase gauge field configurations and vice versa. Structures are revealed of topological charges for string-like configurations of the order parameter which contain ball defects in their cores, as well as for string-like configurations of gauge fields, containing ball monopoles in their cores. Within the framework of the topological approach, string-like defects are studied in superfluid ³He and nematic liquid crystals, as well as string-like configurations of gauge fields in the gauge theory with SU(3) symmetry.

In the Lagrange formulation of a classical constrained dynamics the properties of non-point transformations (i.e. those depending not only on coordinates but also on their time derivatives) which result in physically equivalent theories are studied as well as their analogues in the corresponding Hamilton dynamics.

The paper is devoted to the results of experiments studying aerosol behaviour in magnetic and electric fields of various configurations. A complete analogy with the behaviour of ferromagnetic particles in uniform magnetic and electric fields is presented. Possible mechanisms of movement of the ferromagnetic particles in the constant magnetic field and the validity of a model with a magnetic charge are discussed.

The author considers the Poincare (or multipolar) gauge with finite and infinite reference point in connection with singularities of the Aharonov-Bohm solenoid and Dirac magnetic monopole type. Families of paths on which the Poincare gauge potentials are defined may give rise to 'shadow' surface or regions on which the Poincare gauge potentials are singular. These singularities may be avoided by changing the family of paths to another family (based on the same reference point) and this is equivalent to changing the gauge. He considers the Dirac magnetic monopole using the Poincare gauge with a family of parallel straight paths from reference points situated at (0,0,0,+or- infinity ), producing the 'overlapping' potentials for the monopole. A method is given for calculating the Poincare gauge potentials on the shadow surface arising from a singularity, and this is illustrated by considering the solenoid problem in which the solenoid is given a finite radius ( in ), and it is shown that the shadow surface in this case contains singularities of the Dirac delta function type.

In a previous paper the authors have employed a group theoretic method to find the evolution operator for a quantum system having a SU(2) Hamiltonian. In this paper, they consider a more complicated system whose Hamiltonian consists of SU(2) and h(4) group generators. A transformation method will be introduced, in conjunction with the above group theoretic method, to tackle this quantum problem. The result thus obtained will be applied to the problem of a mass-varying harmonic oscillator under an external force. Their result shows that the equation of motion of this oscillator is identical to a damped harmonic oscillator under an external force. In addition, it is shown that an initial coherent state will evolve as a squeezed state under the above Hamiltonian.

For a simple non-relativistic fermion model the author shows that the Schwinger anomaly can be viewed as an effect of the infinite depth of the Dirac sea.

The renormalisation group (RG) method is applied to the investigation of an exactly solvable phase transition model in which only the interaction between fluctuations with equal and antiparallel momenta is taken into account. The RG equation for the model is derived, its exact solution and critical asymptotics are obtained. It is shown that direct calculation of the partition function and solution of the RG equation for the model lead to identical results.

The authors demonstrate that there exists a definite mapping between hopping on hierarchical structures, with an arbitrary number of sons, and random walking on one new infinitely large family of deterministic loopless fractals. In addition, by explicit calculation of the corresponding spectral dimensions of the introduced fractals, they corroborate the validity of the Einstein diffusion relation for inhomogeneous media.