Table of contents

The authors present a construction of the exceptional Lie algebra G₂ and of corresponding quantum algebra U_q(G₂) using quasi-parafermionic creation and annihilation operators and their quantum analogue. As a by-product, a new realization of U_q(A₂) is found.

The structure and Verma module of the quantum matrix element algebra A(n)_q of the quantum group GL(n)_q are studied. The method used here is similar to that for studying the structure and Verma module of semisimple Lie algebras. The concept of the Cartan subalgebra, raising and lowering operators and their pairs is defined. The q-boson and the corresponding Heisenberg-Weyl relation realizations are generally studied and the cyclic representations of A(n)_q are obtained. The explicit examples A(2)_q and A(3)_q are discussed in detail.

Discusses the quantum group GL_q(n,H) obtained by a hyperbolic complexification of the quantum group GL_q(n), and prove that GL_q(n,H) is isomorphic to the direct product GL_q(n)*GL_q(n). In particular, SU_q(2,H) and SU_q(1,1,H) are isomorphic to SL_q(2).

A certain inverse problem concerning the specification of a class of infinite Jacobi matrices in terms of their eigenvalues and eigenfunctions is formulated and solved.

For pt.I see ibid., vol.24, p.L215, (1991). It has been shown that the differential operator for the extremal trajectory of a stochastic process can be written as a square of operators, i.e. the Langevin systematic operator times its adjoint. Here the author shows that one can go further and also write directly from the Langevin equation, first integrals (conservation principles) of the extremal path differential equation. Linearity in the Langevin operator and Gaussianity for the fluctuation are assumed.

An exact solution to the bound state problem for the N-dimensional generalized Dirac-Coulomb equation, whose potential contains both the Lorentz-vector and Lorentz-scalar terms of the Coulomb form, is obtained.

There exists an approach for finding of exact solutions of the two-dimensional sine-Gordon equation. Using this approach three classes of solutions have been found. One of the classes consists of running wave solutions which are a generalization of the solutions of the one-dimensional sine-Gordon equation. This class of solutions is studied here.

The existence of analytic normal forms for an area-preserving map with an elliptic fixed point is considered. If the linear frequency is diophantine, the complexified map can be analytically conjugated with an integrable map in a disc of the complex radial coordinate r excluding a sequence of sets, whose measure decreases exponentially fast as the origin is approached; the analyticity in the angle theta is just a strip. The non-analyticity domains correspond to the regions where the topology of the orbits changes since nonlinear resonances are present and the modulus of the residue of the leading poles, provided by perturbation theory, is the square of the width of the corresponding chains of islands. This suggests a relation between clusters of singularities of the normalizing transformation and changes in the topology of the orbits.

A certain power series, arising in the Gaussian model for fluids, has been conjectured to reduce to a polynomial when the dimension parameter is a negative even integer. This conjecture is confirmed here, using a graph-theoretical interpretation of the coefficients.

The problem of tunnelling in 'topological' (massless) quantum mechanics with an anharmonic potential is considered. A simple method to calculate the imaginary part of action for the family of potentials, U(x,y)=(xy)ⁿ, is proposed. The tunnelling action is determined by the pole contribution only (on the complex-time plane), in a contrast to the ordinary (massive) quantum mechanics, in which the tunnelling action is always determined by the branch point contribution.

Using a simple method, it is proved that the quantum propagator for some polynomially perturbed harmonic oscillators is close to the propagator for the unperturbed oscillators for a small coupling constant and arbitrarily large times.

Integrating out the gauge field in the Chern-Simons interaction of anyons, the authors recover in the non-relativistic limit the winding number in the action for particles with fractional statistics.

The authors present an analytic argument for the critical exponents ( nu _{perpendicular to} and nu /sub ///) of the fully directed self-avoiding walks on a family of Sierpinski carpets. In contrast to the cases of random walks or isotropic self-avoiding walks on a fractal lattice, they find that both exponents do not depend on the fractal dimension of the underlying carpet but take a trivial value of unity. Only the correction-to-scaling exponents vary with the fractal property of the underlying lattice. Numerical simulations confirm the prediction.

The authors study the statistics of the ideal chain (or equally weighted trajectories) for the first time on a two-dimensional critical percolation cluster. They discuss the asymptotic behaviour of the mean end-to-end distance and the number of chains for the long chain limit by exact enumeration. The results strongly suggest that this problem does not belong to the same universality class as the random walk (or kinetically weighted trajectories) on the same fractal cluster.

The influence of long-range self-avoiding forces on the statistical mechanics of a D-dimensional tethered membrane is studied in the limit of large embedding space dimension d to infinity . The author finds three distinct isotropic (crumpled) phases and several flat phases, and thermally driven crumpling transitions for D>2. The effect of attraction and of short-range repulsion for D>2 is also discussed.

It is shown that in non-hyperbolic Hamiltonian systems where correlations decay as a power law, transient chaos and chaotic scattering demonstrate different power laws. Numerical evidence of this effect for the stadium billiard is presented.

For the r-adic one-dimensional maps, the author explicitly constructs the decaying eigenstates and adjoint eigenstates associated with the Ruelle resonances. It is shown that the eigenfunctions of the corresponding Frobenius-Perron operator are the well known Bernoulli polynomials. The adjoint eigendistributions are obtained as derivatives of the Dirac distributions at the end points of the unit interval. The resulting expansion of the initial density in terms of the decaying eigenstates is given by the Euler summation formula.

The authors performed extensive Monte Carlo simulations for different types of walks (random walks, ideal chains and self-avoiding walks) on the Penrose quasilattice. The critical exponent nu -for each process-is found to be the same as for periodic two-dimensional lattices, thus universality seems to hold also for the Penrose tiling.

The dynamics of a random sequential adsorption process on a quasi-one-dimensional lattice is solved exactly. It is shown that the filling process exhibits some features typical to two-dimensional systems. In particular the jamming density is rho _r=¹/₂(1-(1/2e))=0.4080301397 . . ., considerably lower than the one-dimensional jamming density.

Develops a scaling theory for the decay of the spin-spin correlation function at two-dimensional (d=2) wetting transitions in systems with short-ranged forces. For the critical wetting transition effective Hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous. In contrast, for the complete wetting transition, scaling and the Ornstein-Zernike theory are both valid. The author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d=2.

Compares the generalization probability of a perceptron using the Delta -rule with other learning algorithms. The numerical results indicate that the Delta -rule gives results which are superior to Hebb's rule, and comparable with the maximal stability perceptron rule (minover), both for randomly generated linear separable example sets and for optimal queries.

The problem of escape of a classical particle from a multidimensional potential well due to the influence of random forces is studied. A new solution to the problem is obtained which shows that with fairly large friction anisotropy the qualitative picture of the process principally differs from that underlying the conventional multidimensional Kramers-Langer theory. According to the new process picture particles escape the well before reaching the saddle point; the limiting stage of the process is the particle attainment of the transition region (i.e. well dynamics), not during passage through the barrier regions as the conventional theory suggests. The new expression for the rate constant predicts a slower process rate in comparison with the Kramers-Langer theory. With decreasing friction anisotropy the new expression coincides with the conventional one.

A fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered. It represents the simplest generalization of the fractional diffusion equation valid in Euclidean systems. The solution of the fractional equation in one dimension is discussed, and compared with exact results for the fractional Brownian motion and the one-dimensional version of the 'standard' diffusion equation on fractals. In higher dimensions, it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals, as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality.

A fractional equation for diffusion in isotropic and homogeneous fractal structures is discussed. It generalizes the fractional diffusion equation valid for d-dimensional Euclidean systems. The asymptotic behaviour of the probability density function is obtained exactly. Analytical expressions are derived for the scattering and relaxation functions, which can be studied by X-ray and neutron scattering experiments on fractals.

The authors investigate numerically the distribution of currents in a two-dimensional diode network at the directed percolation threshold. They obtain the multifractal spectrum of this distribution as well as other scaling properties such as the fractal dimension of the backbone, and the scaling of the conductivity. The analysis of the multifractal spectrum provides information about the singly connected bonds in the lattice.

The authors consider two lattice animal models for the collapse of dilute branched polymers in a good solvent. In both cases, the collapse is driven by a near-neighbour contact fugacity, the two models differing in the way molecular weights are assessed, either by the site content or the bond content of the animal. They describe some rigorous results, including bounds for the temperature dependence of their reduced limiting free energies on a d-dimensional hypercubic lattice and compare these results with numerical estimates derived from exact enumeration data. From the specific heat, they estimate the collapse temperature of both models on a variety of lattices. In addition, they estimate the cross-over exponent phi and find that for both models phi =0.60+or-0.03 (d=2) and phi =0.82+or-0.02 (d=3).

The authors use the method of 'damage spreading' to measure the time taken for the equilibrium damage (the magnetization) to disappear in finite systems (10<or=L<or=103) at the critical temperature of the infinite system. This time is found to scale as L^z with an exponent z of 2.16+or-0.02. This value is compatible with the expected value of the dynamic exponent for the 2D Ising model, but they argue that this method is unable to measure fluctuations properly which they show are responsible for the higher values recently reported in the literature. They suggest that the slowest mode of relaxation has an exponent of z>or=2.27.

For pt.I see ibid. vol.24, p.4873-87 (1991). The space groups of the periodic approximants (PAs) to the dodecagonal quasilattice in two dimensions are investigated. The Bravais lattices of the PAs are tetragonal (p4mm), rectangular (pmm), hexagonal (p6mm) and rhombic (cmm). The author presents several series of PAs, each of which is derived from a prototype PA by a successive application of the deflation-and-rescaling (DAR), and the space group is common among the members. Each series is composed of two subseries; the consecutive members in each subseries are related by the 'proper' DAR and the two subseries are related to each other by the 'improper' DAR.

The author shows that the central charge c of the Virasoro algebra is determined by the spectrum of the Hamiltonian L₀+NL₀ corresponding to a partition function which is invariant under the subgroup of modular transformations generated by S. Using this result he discusses in detail a new possibility of determining c for a given system at criticality which turns out to give excellent estimates even if the lattices accessible to numerical calculation are very small. This enables him to predict the central charge of some spin systems. Furthermore his approach to the determination of c leads to new universal functions interpolating between criticality and off-criticality.

The authors study two lattice models: (a) c-animals with an interaction energy alpha between nearest-neighbour pairs of vertices; (b) c-animals as in (a) but with an additional interaction energy omega between the vertices of the animal and an adsorption surface. By assuming that the partition functions satisfy A_n(c, alpha ) approximately n^{- theta c( alpha )} lambda _c( alpha )ⁿ and A_n(c, alpha , omega ) approximately n^{- theta c( alpha , omega )} lambda _c( alpha , omega )ⁿ as n to infinity with c fixed, they show that theta _c( alpha )= theta ₀( alpha )-c, - infinity < alpha < infinity and theta _c( alpha , omega )= theta ₀( alpha , omega )-c, - infinity < alpha , omega < infinity , where theta ₀( alpha ) and theta ₀( alpha , omega ) are the corresponding exponents for trees.

Mazenko's theory of phase ordering dynamics is generalized to an n-component nonconserved vector order parameter. The scaling functions for the equal-time and two-time correlation functions are calculated, as well as the exponent characterizing the decay of autocorrelations. The equal-time correlation function is numerically quite close to that recently calculated by Bray and Puri (1991), and by Toyoki (1991), and exhibits the same power-law tail in its Fourier transform, the time-dependent structure factor.

The authors show that a Potts glass neural network can be used for invariant recognition. Each neuron is modelled as a q state Potts variable and p Potts configurations are stored in the network. The learning rule is a generalized Hebb rule, so that all pq! permutations of the patterns are stabilized simultaneously. They analyse the model for an extensive number of patterns and present results of numerical simulations to confirm the analytical results and to illustrate the relaxation to equilibrium. The model is applied to the recognition of isomorphic graphs. Finally they show how to generalize the model, such that its stable states are patterns, were the permutational symmetry is broken in a hierarchical manner.

The presence of many attractors in neural networks give rise to interesting competitive phenomena. The authors consider dilute recurrent (or attractor) neural networks with sign-constrained weights and storing uncorrelated patterns with maximal stability. The dynamics of these networks is governed by the competitive effects of retrieval, nonretrieval and uniform (i.e. ferromagnetic) attractors, which result in basin encroaching, shrinking, splitting and wedging. They have found the parameter regions in which each of these attractors exist. The basins of attraction of the uniform attractors enlarge at the expense of the other attractors even when the weight signs are slightly imbalanced, but can be compensated by the introduction of a dynamical threshold.

Finite temperature effects in the chaotic maser model are studied in the framework of a mean field variational approach. Analytic expressions for the relevant thermodynamic properties are given. The chaotic dynamics of the system is extended so as to include finite-temperature effects. Temperature may dramatically influence the behaviour of classical trajectories. A simple physical interpretation is provided.

Matrices representing the Clebsch-Gordan series for the irreducible representations of a finite group form a complete set of commuting operators whose eigenvalues are the group characters and whose eigenvectors are the columns in the character table. These operators are dual to another complete set of commuting operators which represent the class multiplication structure constants. The duality between these two complete sets of commuting operators is made explicit.

A generalized deformed oscillator, with eigenvalues equal to the eigenvalues of the Schrodinger equation with the modified Poschl-Teller potential, is constructed. For special values of the potential depth the deformed oscillator algebra has a finite-dimensional irreducible representation. The polynomial representation and the associate deformed operations of integration and differentiation are studied. The results of this study are general and they can be applied directly to the case of the q-deformed oscillator, with q being a root of unity.

The author investigates a class of linear multiplicative stochastic differential equations and demonstrates the existence of a striking noise-induced transition in the structure of the resulting asymptotic stationary probability distribution for the dependent variable. The transition amounts to a change from a bounded distribution to an unbounded one with only a finite number of convergent moments. It occurs when the range of fluctuation of one of the variables driven by the underlying stochastic process increases sufficiently to permit changes of sign for the variable. It seems likely that the phenomenon is a general one and occurs in a wider class of models than that discussed in this paper. He obtains explicit results for simple cases which he confirms by appropriate numerical simulations. This gives him the opportunity of assessing the applicability of perturbation theory which is one of the few calculational methods employed on these models up until now.

An algorithm for the calculation of q-dependent spin characters of the symmetric group is given with an explicit example of S₄. A method for constructing the q-analogue of the vertex operators is developed. A 1:1 correspondence between the space V of twisted q-vertex operators and the ring of q-deformed symmetric functions Lambda (X)_ZQ(q,t) is established and a mapping from V to Lambda (X)_ZQ(q,t) is defined. A number of relevant theorems are given.

The determination of conjugacy classes within the group of all automorphisms of an affine Kac-Moody algebra is discussed in detail, and the limited role of Cartan preserving automorphisms is emphasized. A comprehensive method of dealing with all the automorphisms of an untwisted affine Kac-Moody algebra is developed. This is based on a matrix formulation of the untwisted affine Kac-Moody algebras. Four different types of automorphism (called 'type 1a', 'type 1b', 'type 2a' and 'type 2b') are identified and analysed within this matrix formulation. The development of the detailed properties of automorphisms in the matrix formulation includes formulae for the products and inverses of automorphisms, together with the conjugacy and involutive conditions.

The matrix formulation of automorphisms of an affine Kac-Moody algebra that was developed in a previous paper is here applied to the determination of all the conjugacy classes of involutive automorphisms of the affine Kac-Moody algebra A₁⁽¹⁾, where it is compared with the application of more traditional structural techniques.

Structure of the reflection coefficient of the Fokker-Planck equation is investigated. Asymptotic expressions of the reflection coefficient for small wavenumber (low frequency) and large wavenumber (high frequency) are presented. As an application of this analysis, a method for calculating the eigenvalues of the Fokker-Planck equation is derived.

An expression for the elastic energy of the ferroelectric, chiral smectic C* liquid crystal phase is presented. It is shown that due to the chirality of the system, compared with the elastic energy of the smectic C phase, the elastic energy of the smectic C* phase contains three additional terms, one of which is a surface term. Of the two remaining terms one is responsible for the helicoidal ordering of the c-director, while the other term will impose a tendency on the smectic layers to be nonuniform in space. The physical interpretation of the presence in the elastic energy of such a term is discussed.

It is shown that the Korteweg-de Vries-Burgers equation possesses the Painleve property conditionally. Using an algorithmic approach a travelling wave solution is reproduced.

The operator ordering problem is investigated in the framework of geometrical quantization in the Schrodinger coordinate representation. It is shown that, for quantization of polarization-preserving classical observables pf(q), correct Hermitian quantum operators are derived; for quantization of polarization-unpreserving classical observables p²f(q), the BKS kernel method actually defines an ordering rule which is not Weyl's rule, but is the GCT-invariant rule proposed by DeWitt (1957).

The authors give a general analysis of negative-result experiments, and argue that they do not require the hypothesis of wavefunction collapse, as suggested by other workers. They point out that the regeneration phenomenon involving neutral kaons provides an experimentally tested example of this type of measurement. They also show that the phenomenon of interrupted fluorescence in the V-configuration of atomic physics may be explained in a straightforward way without the use of collapse.

A new and powerful mathematical technique is described for evaluating the binding energies of excitons within the framework of the variational method. The technique is applied to infinite wells and the binding energies of 1s- and 2s-type excitons calculated as a function of well width.

A higher-order Schrodinger equation containing parameters, which is used to describe pulse propagation in optical fibres, is shown to admit an infinite-dimensional prolongation structure for exactly four combinations of the parameters, besides the classical NLS equation. For each of these cases, the structure of the resulting prolongation algebra is determined explicitly. For the first three cases the prolongation algebra is essentially a sub-algebra of A₁⁽¹⁾, the fourth case turns out to be sub-algebra of the twisted Kac-Moody algebra A₂⁽²⁾. Using vector-field representations, related systems of differential equations for the (pseudo-) potential functions are given for each of the cases. The cases found here correspond exactly to the derived NLS equations I and II, the Hirota equation and the equation recently considered by Sasa and Satsuma (1991). The result of this paper strongly indicates that the considered higher-order NLS equation is completely integrable for precisely these four cases.

The authors comment on the Lie point symmetries for the Khokhlov-Zabolotskaya equation as calculated by Roy Chowdhury and Nasker (1986), and demonstrate that their result for the coefficients of the vector field is correct but incomplete.

The authors show that a previously presented solution for the system of coupled nonlinear differential equations leads to inconsistent results except in the case where the system is decoupled.