Table of contents

For the principal chiral model, two hierarchies of symmetries and the related Virasoro and Kac-Moody algebras are derived by using a recursion operator.

An integrable system in two discrete spatial variables and continuous time is presented. It contains the Volterra model as a particular case. Both the system and its solutions are characterised into the framework of asymptotic modules. Rational and soliton-like solutions are exhibited.

Using the IWOP technique (integration within an ordered product), the authors find canonical coherent-state representations of two kinds of squeeze operators. These representations manifestly show that they are quantum maps imaged by certain symplectic transformations in coordinate-momentum phase space. In terms of this formalism the coherent-state propagator of parametric amplifiers is easily obtained.

The authors present a new method for growing and analysing diffusion-limited aggregates (DLA). The method is based on the exact enumeration approach which enables one to calculate exactly the probability density of a random walker starting from an outer circle (at r=r₁). The method yields the exact growth probabilities, p_i, of the perimeter sites, i, for a given configuration as a function of time. The authors study the histogram, n(p), i.e. the number of perimeter sites having growth probability p, as a function of time, for several different boundary conditions. Their results suggest that the fluctuations in the survival times of the particle are very small compared with the large fluctuations in the growth probabilities. They find that at times of the order of r₁² all growth probabilities are essentially converged. Very long survival particles have only a negligible effect on the histogram n(p) and thus on the DLA structure.

The thermodynamical description of fractals that has recently attracted much interest both experimentally and theoretically in the study of dynamical systems is, in some ways, limited, being essentially an additive theory. The author presents a subadditive thermodynamic formalism for which he derives a variational principle and shows how it may be used to study the dynamics of non-conformal transformations. In particular the author discusses an analogue of Bowen's formula for the dimension of a mixing repeller.

The 6-j symbols of a group are independent of the subgroup chain chosen to define the basis states. The authors present an improved algorithm for calculating the primitive 6-j symbols for a compact group with a faithful irrep by recursive building up using only the Kronecker product rules and the general properties of 6-j. Previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6-j. The authors show that the primitive 6-j may all be easily solved in terms of a subclass, the core 6-j. They discuss how the core 6-j can usually be solved, proving that the method is complete for SO₃. The authors conjecture that the algorithm is complete for all groups.

The mixed Yukawa couplings of the 27 and 27* superfields to other superfields are studied in a model stemming from a superstring model compactified on a generic Calabi-Yau manifold. The coupling to the standard E₆-algebra-valued flux-loop operator, as well as an analogous E₆-invariant operator, is argued to be related to certain Yukawa couplings. A vacuum expectation value of the latter operator is shown to give masses to all E₆-singlet scalar superfields, except for those stemming from the supergravity multiplet.

A non-Markovian stochastic process modelled by a linear first-order differential equation involving quadratic Ornstein-Uhlenbeck noise is investigated. The generator of an evolution operator of the process is constructed and linear propagators of a one-dimensional probability distribution are built. The initial correlation functions are presented and evolution equations for the moments of the process are derived. Some approximative methods are verified.

Period doubling in three symmetrically coupled two-dimensional area-preserving maps is numerically studied. It is found that there is a two-dimensional manifold on which all period-2ⁿ (n=0, 1, 2, . . .) orbits of a period-doubling bifurcation sequence lie. On this manifold, the universality classes of period doubling are just the three classes for two symmetrically coupled two-dimensional area-preserving maps, each characterised by its own Feigenbaum constants, reported by Mao and Helleman (Phys. Rev. A. vol.35, p.1847, 1987). It is also reported that, for three non-symmetrically coupled two-dimensional area-preserving maps, three maps which are fixed under the renormalisation operator have been found. The relevant eigenvalues of perturbation around the fixed maps are again the same as those found for two non-symmetrically coupled two-dimensional area-preserving maps, by Mao and Greene (Phys. Rev. A vol.35, p.3911, 1987). The above-mentioned numerical and renormalisation results for the six-dimensional maps agree with each other.

A new ring-shaped potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, is investigated. The Schrodinger equation is solved in spherical, circular cylindrical, prolate and oblate spheroidal coordinates. As in the case of the Hartmann potential, the 'accidental' degeneracies occurring in the spectrum are shown to be due to an su(2) dynamical invariance algebra. This establishes a close connection between both ring-shaped potentials.

The authors considers the derivation of some statistical results for the 'inverted' harmonic oscillator (one with negative kinetic and potential energies), which are equivalent to ones that exist for the more familiar simple harmonic oscillator. It is an object that is frequently employed in modelling amplifiers in quantum optics, but also arises in statistical and quantum mechanics and is of interest in its own right. The author hopes that this account may help to elucidate its role as a potential amplifier.

The critical behaviour of fully directed Levy flight on a square lattice is studied using the Monte Carlo method. The obtained critical exponents nu , nu /sub /// and nu _{perpendicular to} are independent of the parameter u. This seems to be interesting compared with the Levy flight previously discussed by Halley and Nakanishi (1985), for which nu depends on u in addition to d. It also indicates that the introduction of the direction plays a dominant role in directed Levy flight just as in directed SAW, with nu , nu /sub /// and nu _{perpendicular to} independent of d.

The spectral dimension d of a network governs the massless singularity of a free field and the asymptotic behaviour of the diffusion on the network. Approximate values of d for two types of three-dimensional generalisations of the dual Sierpinski carpet are obtained using the POP approximation method. For one of them which is generated by a cube of side length three, with one block at the centre taken away, the value obtained is d(POP)=2 log 26/log(884/93) approximately=2.89. For the other one with seven cubes taken away, d(POP)=2 log 20/log (40/3) approximately=2.31. The algorithm of the POP method is explained. The results for 2D symmetric dual Sierpinski-type carpets are also reported.

The authors present three techniques for determining rigorous bounds for site percolation critical probabilities of two-dimensional lattices. A technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that p_c(D)>or=0.5020 for the dice lattice D. Combining this method with Kesten's duality result simplifies Toth's derivation of the lower bound p_c(S)>or=0.5034 for the square lattice S. The authors also present a technique for deriving upper bounds for bipartite graphs. A technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice H: p_c(H)<or=0.8079 and p_c(H)<or=p_c(S). The grouping technique is applied to the dice lattice to find the upper bound p_c(D)<or=0.7937.

A detailed analysis is given of high-temperature series expansions for the susceptibility of an O(2)-symmetric spin model with discretely valued spin-spin interaction on SC, BCC and FCC lattices. This analysis indicates that the model is in the same universality class as the regular 3D O(2) spin model. New results are given for the critical amplitudes. The authors also list and discuss series expansions for the free energy and specific heat.

A count of the number of metastable states is employed to obtain indications on the retrieval and spin-glass properties of asymmetrically diluted neural networks. It is found that the main effect is on the retrieval states. Their position, distribution and number depend essentially on the normalised storage parameter alpha , the ratio of the number of memories to the mean connectivity. The effect of asymmetrical dilution on metastable states uncorrelated with the memories depends on the dilution mode; the number of such states, however, still grows exponentially with system size, even for completely asymmetrical networks. To the extent that asymmetry destabilises this spin-glass phase it must be doing it by modifying the dynamics and not by eliminating metastable states. It is also shown that there are no individual retrieval states with significant basins of attraction, for the symmetric as well as the asymmetric neural network.

The Parisi-Sourlas mechanism is demonstrated directly for fields over pseudo-Euclidean space, without the use of Wick rotations, superspace or Berezin integration. An irreducible field supermultiplet for the Lie superalgebra iosp(m,n/2) is constructed using a modified version of the method of produced representations, and a (1,1)-dimensional reduction obtained through examination of the metric.

Using an extension of a factorisation given by Ding (ibid. vol.20, p.6293, 1987) using radial ladder operators, a one-parameter family of potentials with spectra coinciding with that of the hydrogen atom is generated.

Using the thermofield dynamics formalism the authors study the effect of temperature on supersymmetry within the context of supersymmetric quantum mechanics. The model considered involves an interaction not of polynomial type and it is shown that the finite-temperature effect causes spontaneous breaking of supersymmetry.