Table of contents

A new generalization of the t - J model with a nearest-neighbour hopping is formulated and solved exactly by the Bethe ansatz method in the thermodynamic limit. The model describes the dynamics of fermions with different spins and with isotropic and anisotropic interactions.

We show that any solution of the KP hierarchy leads to a solution of Frenkel's hierarchy. This allows us to make an explicit relation between his hierarchy and the one introduced by Haine and Iliev in 1997.

The critical behaviour associated with a transverse magnetic field applied at the edge of a semi-infinite chain is calculated using field theory techniques. Contrary to a recent claim, we find that the long-time behaviour is given by a renormalization group fixed point corresponding to an infinite field which polarizes the spin at the edge. The zero-temperature entropy and position-dependent magnetization are calculated.

We study numerically and analytically the average length of reduced (primitive) words in the so-called locally free and braid groups. We consider the situations when the letters in the initial words are drawn either without or with correlations. In the latter case, we show that the average length of the reduced word can be increased or lowered depending on the type of correlation. The ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth model.

The kinetics of annihilating random walks in one dimension, with the half-line x>0 initially filled, is investigated. The survival probability of the nth particle from the interface exhibits power-law decay, , with for n = 1 and all odd values of n; for all n even, a faster decay with is observed. From consideration of the eventual survival probability in a finite cluster of particles, the rigorous bound is derived, while a heuristic argument gives . Numerically, this latter value appears to be a lower bound for . The average position of the first particle moves to the right approximately as , with a relatively sharp and asymmetric probability distribution.

Using Monte Carlo techniques and a star-triangle transformation, Ising models with random, `strong' and `weak', nearest-neighbour ferromagnetic couplings on a square lattice with a (1,1) surface are studied near the phase transition. Both surface and bulk critical properties are investigated. In particular, the critical exponents of the surface magnetization, , of the correlation length, , and of the critical surface correlations, , are analysed.

A one-dimensional model having a unique ground state and countable number of extreme limit Gibbs states is constructed.

We prove a recent conjecture on the duality relation for correlation functions of the Potts model for boundary spins of a planar lattice. Specifically, we deduce the explicit expression for the duality of the n-site correlation functions, and establish sum rule identities in the form of the Möbius inversion of a partially ordered set. The strategy of the proof is by first formulating the problem for the more general chiral Potts model. The extension of our consideration to the many-component Potts models is also given.

We present here a graphical approach to the Feenberg multiple-scattering expansion and discuss examples of Hamiltonians with topological and substitutional disorder where the path-contribution technique allows us to understand the effects of such disorder and generate physically relevant approximations.

The Green function for the Schrödinger equation with an isotropic, three-dimensional harmonic-oscillator potential is given in closed form. A similar closed form is obtained when the Schrödinger equation also contains a magnetic interaction and the magnetic field is such that the precession and oscillation frequencies are equal. The latter Green function is used to obtain energy and Sturmian eigenvalues that occur in the theory of atom-atom collisions.

The quantum double for the quantized BRST superalgebra is studied. The corresponding -matrix is explicitly constucted. The Hopf algebras of the double form an analytical variety with coordinates described by the canonical deformation parameters. This provides the possibility to construct the nontrivial quantization of the proper time supergroup cotangent bundle. The group-like classical limit for this quantization corresponds to the generic super Lie bialgebra of the double.

This paper deals with the derivation of non-polynomial solutions to the q-difference form of the Harper equation. Only quasiclassical approximations proceeding this time to first and second orders are discussed. The exact non-polynomial zero-energy solution to the above q-difference equation has also been presented.

The ideas of geometrical algebra are used to investigate the physical content of the Kustaanheimo-Stiefel regularization procedure, which transforms the potential into the harmonic oscillator potential.

A geometric formulation of a generalization of Nambu mechanics is proposed. This formulation is carried out, wherever possible, in analogy with that of Hamiltonian systems. In this formulation, a strictly non-degenerate constant 3-form is attached to a 3n-dimensional phase space. Time evolution is governed by two Nambu functions. A Poisson bracket of 2-forms is introduced, which provides a Lie-algebra structure on the space of 2-forms. This formalism is shown to provide a suitable framework for the description of non-integrable fluid flow such as Arter flow, Chandrashekhar flow and of coupled rigid bodies.

We define the invariant spectra of rotation angles and twist angles (angular dynamical spectra) and study the properties of their moments (angular moments) in a model two-dimensional map. The angular moments give the main frequencies of the orbits. A main frequency is defined both for a regular and chaotic orbit. For KAM curves around a centre this frequency corresponds to the rotation number. Inside islands of stability, we obtain both the main frequency (rotation number) and the `epicyclic frequency'. A fast detection of thin chaotic layers is obtained on the basis of the behaviour of the frequency curve. We explore the resonant structure near a last KAM boundary. The secondary islands of various multiplicities, forming Farey sequences, and the noble tori between them are located. We find a criterion to determine whether various resonant chaotic zones communicate, or are separated by invariant tori.

Quantum Lorentz groups H admitting quantum Minkowski space V are selected. The natural structure of a quantum space is introduced, defining a quantum group structure on G only for triangular H (q = 1). We show that it defines a braided quantum group structure on G for |q| = 1.