Table of contents

The q-deformed boson realisation of the quantum group SU(n)_q ((A_n-1)_q) is constructed and certain types of representations of SU(n)_q are obtained in the q-deformed Fock space by this boson realisation. The Jimbo representations of the quantum group SU(2)_q are given as an example in this letter.

For any potential V(x) which holds n bound states, the authors use repeated supersymmetry transformations to construct an n-parameter family of 'strictly isospectral' potentials (identical eigenenergies, reflection and transmission amplitudes for all family members). They investigate how the shape and behaviour of various isospectral potentials changes as the n parameters are varied. Contained in this family are potentials with n widely separated wells, each of which holds a single bound state.

The modified Lax equations resulting from factorisation of scalar Lax operators are shown to commute themselves. The proof is general and bypasses the Hamiltonian formalism of Lax equations. In particular, the modified KP hierarchy and modified generalised Toda lattices are proved to be integrable.

In view of the great interest in band random matrices, the authors investigate the level repulsion for the simplest case of 3 × 3 band matrices. A non-linear repulsion is found.

The solitary-wave solution of the (3+1)-dimensional nonlinear Schrodinger equation is obtained, taking into account the energy exchange between the central and peripheral parts of the pulse inside and outside the induced waveguide in a nonlinear medium.

The tunnelling rate in real time in the semiclassical limit has been calculated for arbitrary energy levels in the cubic potential. For the ground state it agrees well with the result found by the instanton method.

For n-dimensional diffusion in Gaussian random fields, with continuous and singular covariances, the leading long-time behaviour of the averaged population is derived by estimating Brownian motion expectations. It is shown how these results are related to the leading low-energy behaviour of the density of states for a particle in a corresponding random potential and to the strong-coupling limit of the free energy of the Pekar-Frolich polaron.

The number of clusters with no free ends C(n) is enumerated up to 21 bonds on a square lattice for bond dilution. The exponent nu governing the growth of the radius of gyration of the cluster is also calculated. It is found that C(n) approximately n^-0.56 3.15" and nu =0.54+or-0.01.

Gives an example of a short-range Ising model with a unique ground state with two unusual properties: it has some continuous spectrum, and it has fractal symmetry.

Some exact results for the infinitely long-range Ising spin glass are derived at arbitrary temperature and magnetic field below the transition line within Parisi's replica symmetry breaking theory. The distribution of overlaps between pure states, P(q), proves to be non-analytic at q=0 in zero field. The usual relation S=- delta F/ delta T is found to break down, showing that the thermodynamic limit and the temperature derivative do not commute in the ordered phase, at least in non-zero magnetic field. The upper breakpoint of the order parameter function and the Edwards-Anderson order parameter are rigorously derived in the low-temperature limit. The coefficient of the leading low-temperature correction to the latter turns out to be different from that derived from the TAP equations, indicating that Parisi's theory and the TAP equations may not be strictly equivalent.

Considers an Ising model and the related lattice gas on the Kagome lattice with two- and three-site couplings. Using a transformation which effectively decimates the triplet interactions, the authors determine the exact phase diagram of the Ising model as well as the exact boundary of the two-phase region of the lattice gas.

Modelling formal neurons by Ising spins, the authors describe a simple two-neuron interaction which allows optimal storage capacity for hierarchically structured information. This takes care both of the low-activity limit in simple, Hopfield-type, networks and of the correlations which occur inside the classes of information hierarchies.

Solving the lattice RP^n-1 and CP^n-1 models in the large-n limit, the authors show that they undergo a first-order phase transition for dimensions greater than or equal to two. In the latter case, although there is no long-range order, they argue that at least in the RP^n-1 model, the transition corresponds to a condensation of defects.

The algebraic recursive determination of matrix elements of some selected families of functions Q₁(x) between the solutions Psi _j^m(x) of factorisable equations is reinvestigated. The possibilities of the procedure outlined in a previous paper are enlarged by using the connection between factorisation types, i.e. the different possible factorisations of the same differential equation. The computation of matrix elements between the Jacobi eigenfunctions Psi _j^m(x) approximately=(sin(ax/2))^{alpha +1/2} (cos(ax/2))^{beta +1}2/P_v^{( alpha , beta )}(cos ax), where a is a real or pure imaginary constant, is studied in detail. Algebraic recurrence formulae satisfied by matrix elements of Q_t(x)=(cos(ax/2))^p(sin(ax/2))^q(tan(ax/2))^t, Q_t(x)=(sin(ax))^p(tan(ax))^q(cos(ax))^t, Q_t(x)= Psi _j^t(x) and Q_t(x)= Psi _t^m(x) are given, and, for the particular cases Q_t(x)=(tan(ax/2))^t and Q_t(x)=(cos ax)^t, closed-form expressions are obtained. As an illustrative application, it is briefly shown how the expressions can serve to derive analytical approximations of the bound-state energies for the potential V(x)=A exp(-x²)-l(l+1)/x². Some further applications are pointed out.

A variety of methods are described for expanding particular generating functions in terms of their S-function content. Plethysms are used to establish conjugacy relations between S-function series and the task of evaluating certain products of classical S-function series introduced by Littlewood (1950) is then completed. Slinky diagrams are introduced to represent non-standard S-functions and their modification to give standard S-functions. The role of slinkies in determining the S-function content of other generating functions is then explained and exemplified.

The authors study rigorously the resistance and fluctuation of resistance of a large deterministic fractal lattice in the limit of an infinite number of resistors. They give estimates on corrections to the effective medium approximation of the total resistance. They prove scaling laws for the relative fluctuation, and prove that the normalised relative fluctuation converges in distribution to the standard normal variable. This is a kind of non-linear law of large numbers.

The problem of deriving explicit coordinates for quasicrystal approximants is solved in all cases where the quasicrystal has an inflation symmetry. In the higher-dimensional space Rⁿ, from which the quasiperiodic pattern is obtained by the cut method, the inflation symmetry is represented by a hyperbolic modular matrix (with integer entries) leaving the 'physical' space invariant. But this matrix also generates, by iteration, a sequence of (rational) approximant spaces which converges to the irrational space. A simple algorithm is described, providing the approximant periodic lattice and the set of vertices within a unit cell.

The authors show that for a wide class of dynamical systems (described by Hamiltonians of the form usually considered in time-dependent perturbation theory) the divergence of the Magnus expansion in the Schrodinger picture for large time intervals is due to pole singularities inherent to the Baker-Campbell-Hausdorff formula.

It has been overlooked that-with the exception of especially favourable circumstances-the answer seems to be negative. For single-frequency systems, however, an averaging procedure is shown to allow for an experimental investigation of the Hannay angles.

The quantum group SU(2)_q is discussed by a method analogous to that used by Schwinger to develop the quantum theory of angular momentum. Such theory of the q-analogue of the quantum harmonic oscillator, as is required for this purpose, is developed.

The author reports a study of the infrared properties of the electron propagator in a framework which uses a gauge-invariant electromagnetic potential. The quantisation of the electromagnetic potential is performed in the Zwanziger formulation (1979) of quantum electrodynamics to avoid the usual infrared divergence. The Kallen-Lehmann representation of the infrared electron propagator in the photon vacuum is shown to be well defined and convergent in this framework. Furthermore, he shows that the probability density of an electron wavepacket propagating in the vacuum decreases like (time)^-3, a well known property which is recovered here in quantum electrodynamics.

The authors have examined the cost distribution of locally optimal solutions in certain combinatorial optimisation problems. This distribution is found to be peaked about a value characteristics of the algorithm involved, with a width that decreases with the system size N. It is shown that the distribution of costs epsilon is of the form exp(Ng( epsilon )) and that g( epsilon ) is self-averaging. Consequently, estimation of the optimal cost is best achieved by curve fitting to g( epsilon ). Possible forms for g( epsilon ) are proposed.

In cellular automaton fluids, where binary collision laws admit reflections, the Stosszahl ansatz and the Boltzmann equation are no longer valid at low densities. This is proven for a class of Lorentz gas models, defined on cubic lattices, for which the diffusion coefficient is calculated from a self-consistent ring kinetic equation or effective medium theory. The calculated results are in excellent agreement with results from molecular dynamics simulations over the whole density interval.

The author employs an 'orbifold procedure' to construct new exactly solvable lattice statistical mechanical models. Starting with the su(3) models at level k=3j, with j an integer, he obtains new models with the same bulk free energy. The bulk free energy is known for the su(n) models in a two-parameter subspace of the full parameter space, so the new models are also exactly solvable in this subspace. He expresses the toroidal partition function of the original model as a sum over partition functions of the orbifold model with twisted boundary conditions. Each model has a critical point, described by a conformal field theory with central charge c=2(1-12/(k+3)(k+2)).

The authors provide a general scheme for the treatment of the thermodynamics of mean-field site-random quantum spin systems, including systems where bond randomness is expressed as an underlying site randomness. They use the method to find the phase structure of a mean-field Heisenberg model in a random field, and of a mean-field Heisenberg spin glass model.

The authors study the time evolution of the damage and the macroscopic distance between two configurations of three-dimensional +or-J Ising spin glasses in an external magnetic field. Similarly to the zero-field case, a phase transition and three temperature regimes are found for the damage. The multifractal properties of the distribution of probability for a site to be damaged n times are also investigated at the onset of the chaotic phase.

Numerical evidence is presented to support the conjecture that the gap-exponent relation remains valid for multi-defect Ising systems where exact solutions are unavailable. Predictions for exponents corresponding to arbitrary lattice anisotropy are given.

The authors study the diffusion of a particle in a d-dimensional lattice where disorder arises from a random distribution of waiting times associated with each site of the lattice. Using scaling arguments they derive, in addition to the leading asymptotic behaviour, the correction-to-scaling terms for the mean square displacement. They also perform detailed Monte Carlo simulations for one, two and three dimensions which give results in substantial agreement with the scaling argument predictions.

There exist some criticisms of the Bethe ansatz formulation predicting a discontinuity for the sine-Gordon soliton mass at finite temperatures. The basic ideas and some key steps of the Bethe ansatz thermodynamics are examined and clarified along with associated criticisms. The validity of the Bethe ansatz theory is demonstrated and an experimental implication is discussed.

The authors investigate the optimal storage capacity of attractor neural networks with sign-constrained weights, which are prescribed a priori. The storage capacity is calculated by considering the fractional volume of weights which can store a set of random patterns as attractors, for a given stability parameter. It is found that this volume is independent of the particular distribution of signs (gauge invariance) and that the storage capacity of such constrained networks is exactly one half that of the unconstrained network with the corresponding value of the stability parameter.

The authors present results from a computer simulation study of a radical initiated kinetic growth model in which the radicals (initiators) are created with a constant rate r_c. Simulations of the growth process were carried out on simple cubic lattices as large as 100*100*100 for two values of the initiator creation rate. Finite-size scaling analyses of the bulk properties yield critical exponents which are the same, to within error bars, as those found in similar kinetic gelation models and in percolation studies. The critical amplitude ratio C^-/C⁺ is consistent with the values obtained in the kinetic gelation studies with fixed initiator concentrations but is clearly different from percolation values. The cluster size distribution shows the same monotonic behaviour as seen in percolation and is markedly different from the oscillatory behaviour seen in an investigation of a similar gelation model. The scaling properties of the cluster size distribution, however, cannot be described using simple droplet theory. Instead, they use a generalised scaling form which produces a good fit to the cluster data.

A hierarchical neural network model capable of storing and retrieving an exponential number of states is introduced. Formulated on a spin glass analogy, the network spins (neurons) are organised into a multitier cluster hierarchy such that, for an N-spin system, the number of stored states grows exponentially with N. Relaxation occurs at zero temperature by what is essentially a tunnelling process and can be implemented using either a bottom-up or top-down updating procedure. As a result of the encoding prescription, the stored states are highly correlated and can be embedded within an ultrametric topology. The information capacity is determined, as well as the model's ability to content-address its stored memory patterns. Numerical simulations illustrating the operation and effectiveness of various hierarchical systems are also presented.

A certain series of Bessel functions-recently discussed by Lee (1988)-is an asymptotic expansion of an integral of a Bessel function. Here the asymptotic properties of the series are investigated in more detail, and it is shown that the series is not only asymptotic, but also convergent under suitable restrictions. For large positive real arguments finite numbers of terms of the series give good approximations to the integral, but the infinite sum is different from the integral.

It is pointed out that the contents of the recent article in this journal by De-gang Zhang and Jie Liu (1988) regarding the geometrical equivalence of a deformed continuous spin equation to the Hirota equation are merely a special case of a more generalised set-up already published in the literature some time.

Two transformation formulae of the solutions of the two-dimensional Toda lattice are presented. The first formula, which can be obtained by the Zakharov-Shabat dressing method, is proved by applying the Hirota bilinear transformation method (1988) and gives the basis for the second new formula.