Table of contents

The authors consider the O(2) gauge transformation for a two-state vertex model on a lattice, and derive its fundamental algebraic invariants, the minimal set of homogeneous polynomials of the vertex weights which are invariant under O(2) transformations. Explicit expressions of the fundamental invariants are given for symmetric vertex models on lattices with coordination number p=2, 3, 4, 5, 6, generalising p=3 results obtained previously from more elaborate considerations.

The author presents a two-parameter family of Ince equations for which the problem of the periodicity of the general integral is reducible entirely to the solution of a quadratic equation.

The authors present the general expression for the spin-1 Heisenberg chain invariant under the U_q(SO(3)) quantum algebra. Several physical and mathematical implications are discussed.

When a gradient is imposed in the control parameter of a directed process in a (1+1)-dimensional directed process, a front may occur. The position of this front evolves in steps of size one in the direction opposite to the gradient, and in steps of size equal to or larger than one in the other direction. The larger steps are due to overhangs. There is a critical value for the control parameter only if the first moment of the step size distribution in the direction of the gradient is not singular-i.e. it takes on a finite value in the 'thermodynamic' limit. If there is a critical point, this step-size distribution is a power law at this point, with an exponent larger than two, and which the authors conjecture to be equal to 3- beta / nu _{perpendicular to} . beta is the order parameter exponent, and nu _{perpendicular to} is the spatial correlation length exponent. In a finite gradient there is a second exponent governing the upper cut-off in the step-size distribution. This exponent is equal to nu _{perpendicular to} (1+ nu _{perpendicular to} ). These two exponents govern the convergence of the position of the front towards the critical value of the control parameter. The authors exemplify their discussion with directed site percolation, and with two versions of a model originally designed to study catalytic reactions.

The authors determine numerically the pair connectivity for two-dimensional percolation systems subject to different boundary geometries. Their results provide a sensitive and efficient test of the concept of conformal invariance and confirm its validity even for relatively small lattices. The influence of boundaries on electrical conduction in percolative networks is also addressed briefly.

The author formulates a new approach to the Navier-Stokes turbulence. By analogy with Wilson's theory of critical phenomena, he tries to identify the upper critical dimension of the freely decaying Navier-Stokes turbulence. He points out the possibility of many fixed points. He also points out possible future developments.

The authors compute the staggered polarisations of cyclic solid-on-solid models using the Bethe-type eigenvectors. Their derivation assumes some mathematical identities of elliptic functions which they have not yet proved.

The Langevin dynamics of a nonpotential system is considered. As a consequence the system is not invariant under supersymmetric transformations. Using Ward identities the author demonstrates that there occur corrections to the linear fluctuation-dissipation theorem. Supersymmetry is also broken for stochastic systems driven by coloured noise.

The authors consider synaptic neural networks which minimise the output error of the stored patterns when the input patterns are ensembles of their noisy versions with overlap m_t with the clean patterns. When m_t is infinitesimally less than 1, the network automatically attains maximal stability, confirming the usefulness of training noises in enhancing memory associativity. When m_t drops below 1, the field distribution has two bands for large m_t, and one continuous band for small m_t. Errorless retrieval is impossible for training noises of the order N⁰. With the increase in training noise, the retrieval overlap deteriorates, although memory associativity does increase for sufficiently low storage.

The algebra of meromorphic vector fields with multipoles on a Riemann sphere is constructed explicitly. The central extension of this algebra is also given. The properties of meromorphic λ-differentials, as the realisation of the algebra, are investigated.

Starting from a primitive 6-j symbol for SO (2l+1) in which four elementary spinor irreducible representations (irreps) (. . .) appear, and using the properties of the Kronecker products of partially stretched irreps, the authors develop a bootstrap procedure, based on a generalisation of the Biedenharn-Elliott sum rule, for calculating other 6-j and 9-j symbols for SO(2l+1). Several of their formulae can be expressed in terms of SO(3) 3-j symbols and rotation matrices d ^j_mn( pi ). An explanation for this is found by using the complementary groups whose generators involve the spin S and quasispin Q of electrons in an atomic l shell. The relation between the two kinds of coupling, that of the irreps of SO(2l+1) and that of the SO(3) angular momenta, is particularly fruitful if the quasiparticle description is employed for the electrons. A further bootstrap leads to a formula for an SO(2l+1) 6-j symbol whose six irreps are all of the type (11 . . . 10 . . . 0), and this is shown to be consistent with a formula for an Sp(2n) 6-j symbol if one replaces 2n by the negative dimensions -2l-1.

The new rotor coherent-state construction of the SU(3) algebra is used to derive simple expressions for SU(3)SO(3) Wigner coefficients.

The authors extend the Weyl graphical method to the two-body operator matrix elements in the many-particle system. The evaluation is accomplished by a simple and efficient recursion scheme.

A method based on the Darboux transformation in the form of a pole-expansion is presented for finding soliton solutions of the derivative nonlinear Schrodinger equation. Related matters concerning Alfven waves in plasmas are also discussed. A generalised Zakharov-Shabat system of equations and its reduced form are deduced simply from the Darboux transformations. The expected asymptotic behaviour of the multisoliton solution is derived from its expression in terms of determinants. An explicit expression for the N-soliton solution is given by means of algebraic techniques from the generalised Zakharov-Shabat equations.

Continued-fraction solutions to the matrix Riccati equation are discussed which are constructed by using the concept of form invariance. It is demonstrated that this technique is related to the AKNS method of deriving integrable nonlinear lattice systems. This gives an explanation why continued-fraction solutions related to the Toda lattice were obtained in a previous work. Continued fractions corresponding to Kac-Van Moerbeke, discrete nonlinear Schrodinger and discrete modified KdV lattice equations are constructed. A method for linearising the Kac-Van Moerbeke lattice equations is rederived and particular solutions are generated. The authors approach demonstrates the crucial role played by the boundary condition at the finite end of the lattice for the existence of this method. These results are extended to the other two lattice systems above in the semi-infinite case and corresponding particular solutions generated in terms of Bessel functions.

It is shown that the stability of ideal quantum gases can be measured by means of the Riemann scalar curvature R of the parameter space. The components of the metric tensor were assumed to be the second moments of energy and the number of particle fluctuations. As a result, R is a function of the second and third moments of those quantities. For bosons R is positive and increases monotonically from zero at the classical limit to positive infinity in the condensation region. A system is less stable if R is bigger and vice versa. For fermions R is negative and this means that Fermi gases are more stable than the ideal Bose and ideal classical systems.

A geometrical approach to statistical thermodynamics is proposed. It is shown that any r-parameter generalised Gibbs distribution leads to a Riemannian metric of parameter space. The components of the metric tensor are represented by second moments of stochastic variables. The scalar curvature R, as a geometrical invariant, is a function of the second and third moments, so is strictly connected with fluctuations of the system. In the case of a real gas, R is positive and tends to infinity as the system approaches the critical point. In the case of an ideal gas, R=0. The obtained results, and the results of the authors previous work, suggests that for a wide class of models R tends to + infinity near the critical point. They treat R as a measure of the stability of the system. They propose some sort of statistical principle: only such models may be accepted for which R tends to infinity if the system is approaching the critical point. It is shown that, if this criterion is adopted for a class of models for which the scaling hypothesis holds, then they obtain the new inequalities for the critical indices. These inequalities are in good agreement with model calculations and experiment.

The authors have analysed a multiphase evolution of population growth. Individual birth and immigration are assumed to be the consequence of the evolution of an individual through a sequence of phases whose duration form a family of independent non-negative random variables. The population model is then adapted to describe the evolution of photons in the cavity and, in particular, it is shown that a multiphase immigration model corresponds to the photons resulting from a stream obtained by amplitude mixing of coherent and chaotic beams. The model is also shown to bring out the characteristics of the multiplicity distribution of particles produced in high-energy collisions.

The first-quantised 'spinor chain' path-integral description of spin-1/2 particles due to Jacobson (1984) is a novel non-Grassmannian approach to the problem of finding a path integral which yields the Dirac or Weyl equation. Unlike conventional path integrals, the spinor chain description does not explicitly use a path integral of the form integral exp iS, instead summing over all contributing spinor chains (paths) with equal weight, and hence the question of the existence of a canonical pseudoclassical action is left open. The authors examine the possibility of recasting the spinor chain system in the form of an action, and show that this involves subtleties concerning the weighting of unruly paths. They find the true action corresponding to the spinor chain which when canonically quantised yields the Weyl equation. Their action is extendable to the problem of the massive Dirac equation in a manner unlike the spinor chain. They discuss the classical solutions and compare these new actions with a similar non-Grassmannian action due to Barut and co-workers (1989).

The authors present a straightforward method for finding implicit solutions for nonlinear evolution and wave equations. The method is illustrated by finding implicit single solitary wave solutions for the Harry Dym, Korteweg-de Vries, modified Korteweg-de Vries, Boussinesq and the generalised Korteweg-de Vries equations.

The generalization of the Baker-Hausdorff lemma and its anticommutator analogue is formulated. Additionally, the anticommutator analogues of several well known operator identities involving repeated commutators are derived. It is pointed out that these are more convenient for application whenever the operators in question satisfy simpler repeated anticommutator relations (and, in particular, when they anticommute) than the repeated commutator relations. The diagonalization of two spin-1 Hamiltonians, in which the anticommutator analogue of the Baker-Hausdorff lemma is used to good advantage, is presented.

The authors show that, while the classical equations of motion of Belobrov, Zaslavski and Tartakovski (1976) are rigorously obtained from the thermodynamic limit of the Heisenberg equations of motion of the many-atom version of the spin-boson model in suitable states, those of Milonni, Ackerhalt and Galbraith (1983) are derivable from a quantum Hamiltonian only if s)1/2, where s is the spin quantum number. Both equations are known to display chaotic behaviour for large coupling constants.

The authors present a discussion of the occurrence of supercritical effects induced by an attractive δ potential. A detailed analysis of the one-dimensional Dirac equation in the presence of such a potential shows that the ground state never dives into the Dirac sea and therefore the supercritical effects are absent. In the three-dimensional spherically symmetric δ-potential case, in contrast, they find supercritical effects. This situation represents an interesting problem in relativistic quantum mechanics because the δ potential induces discontinuities in the wavefunction, in contrast to the well known Schrodinger case where only the first derivative of the wavefunction becomes discontinuous.

The problem of a SAW starting next to an excluded line, say the negative x axis, is studied. In two dimensions, using arguments based on conformal invariance and scaling, it is shown that the moments of the displacement along the x axis after N steps, x_N, grow like (x^2k+1_N) approximately N^(2k+1)v. The argument can, however, be simplified and extended also to the case of three dimensions.

The authors show that the shell model, currently used in the lattice dynamics study of ionic crystals, is a singular system. The model is treated as a constrained Hamiltonian system. The use of the Dirac generalized Hamiltonian formalism for constrained systems permits them to solve the difficulties which appear in quantum mechanics and statistical mechanics formulations. The constraints, first-class Hamiltonian and Dirac bracket are obtained. The general expression for the partition function of the quantum statistical mechanics is defined. Finally some considerations about the perturbative and non-perturbative methods are performed.

The authors present a simple neural network model in which the outputs of groups of neurons are chemically modulated. This feature takes into account the existence of neuromodulators which are known to have a strong effect on the behaviour of real biological networks. The effective (non-symmetric) connection matrix thereby becomes a function of the modulator concentrations. By deriving evolution equations for the pattern overlaps they show that, during recall, the system can select distinct subclasses from the stored patterns by varying these concentrations. This model may increase the understanding of the function of neuromodulators.

Expressions for the interior and exterior potentials of a homogeneous ellipsoid are given in terms of ellipsoidal coordinates and Lame polynomials. It is shown that Wang's (1988,9) corresponding solutions for spheroidal bodies may be considered as special cases of these general triaxial expressions. Wang's results are also readily obtained by employing the relationship between the Lame and the Legendre polynomials in the case where the ellipsoid is degenerated into an oblate or prolate spheroid.

The classical Boussinesq system and its associated higher-order flows in the bilinear form are considered. An infinite family of homogeneous rational solutions of this hierarchy is verified by the use of the Wronskian technique developed by Freeman and Nimmo (1983).

The Stormer theory (1955) of charged particles in a magnetic dipole field has been generalized by superposing an axially symmetric electric field. A formula describing the borders between 'allowed' and 'forbidden' regions for a particle with a given azimuthal component of generalized momentum is derived. The theory has been tested by some numerically calculated proton orbits in the fields of a parallel rotator.

On the basis of complex variable theory, the author presents an alternative demonstration of the equivalence of m quasiparticles to one true particle in the Laughlin state.

The self-avoiding walk problem on the L lattice is treated with a position-space renormalization group approach using a 'centre-rule' weight function.