Table of contents

Using thermodynamic arguments we find that the probability that there are no eigenvalues in the interval (-s, infinity ) in the double scaling limit of Hermitian matrix models is O(exp(-s^{2 gamma +2})) as s to + infinity . Here gamma =m-1/2, m=1, 2, ... determines the mth multi critical point of the level density: sigma (x) alpha b(1-(x/b)²)^gamma , x in (-b, b) and b² alpha N. Furthermore, the size of the transition zone, where the eigenvalue density becomes vanishingly small at the tail of the spectrum, is approximately=N_{( gamma -1}/2( gamma +1)) in agreement with earlier work based upon the string equation.

We simulate 50 off-lattice diffusion limited aggregation (DLA) clusters of one million particles each. The probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically. For increasing cluster size N, the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power-law in N. From this specific viewpoint, asymptotically large DLA clusters are locally isotropic.

It is shown that a kind of solution of the n-simplex equation can be obtained from representations of the braid group. The symmetries in its solution space are also discussed.

For pt.III, see ibid, vol.26, p.6667 (1993). This paper is a continuation of a series of papers (1993 J. Phys. A: Math. Gen. 26, 6635, 6649, 6667), and makes use of the same assumptions. The paper presents a model of the first stage of the spatial scale splitting of hydrodynamic vortex structures. It is shown that this model is one of a first-order non-equilibrium phase transition, which leads to the forming of a heterogeneous low-conducting current state of the exploding conductor. As this takes place, its dynamics are determined by three interacting order parameters. As a result the exploding conductor is broken down into metallic particles with the size of the conductor diameter. It is shown that the transition from a high- to low-conducting state by the splitting of the spatial scale is energetically more profitable than the transition by excitation of current vortex structures. Parameters of the model, which describe the formation stage of hydrodynamic and current vortex structures, are identified by experimental results for the electrical explosion of conductors. The problems which emerged while bringing together the models describing these stages are also discussed.

Interactions between fluid membranes are studied by renormalization group techniques. To make the zero-flow condition at the origin compatible with the renormalization, there should be a nonrenormalizable region near the origin. The renormalization procedure involving such a region picks up an inhomogeneous contribution which will also be recast during the renormalization. This paper gives an example to show how the morphology of the fixed-point potential undergoes a series of transitions due to this inhomogeneous contribution.

A simple, local cluster interaction is presented, which has as (only) ground states perfectly quasicrystalline tilings from a single local isomorphism class. Since these tilings do not allow for any perfect matching rules, it is thereby shown that the class of structures which are the ground state of some finite range interaction is considerably larger than previously anticipated. Cluster interactions having a quasicrystalline ground state turn out to be simple and robust, and therefore provide an attractive explanation for the existence of quasicrystals. A simplified version of our cluster interaction is found to have super-tile random tiling ground states. Due to the large size of the super-tiles, these random tilings still look perfect on a local scale.

We introduce two kinds of discrete-time non-equilibrium lattice models (stochastic cellular automata), which we call the creation process eta _t and the branching process eta . In special cases the former process can be identified with the site or bond directed percolation models. When the system is defined on a d-dimensional finite lattice with size L, these processes are determined by 2(L^d)*2(L^d) transition matrices, M_L and M_L, respectively. It is proved that subject to certain relations between the parameters of these models, M_L and the transpose of M_L are conjugate and thus the characteristic polynomials become equal to each other, det (M_L- lambda E)=det(M_L- lambda _E), for arbitrary L>or=n, where E is the identity matrix. Since dynamical critical exponents as well as critical values will be determined by the asymptotic behaviour in the limit L to infinity of the large eigenvalues of the transition matrix, our result implies that, if continuous phase transitions and critical phenomena are observed, these two processes belong to the same universality class. In proving the equality, we use the relation which is called the coalescing duality in probability theory.

We show that the dynamical order parameters can be re-expressed in terms of the distribution of the staggered auto-correlation and response functions. We calculate these distributions for the out of equilibrium dynamics of the Sherrington-Kirkpatrick model at long times. The results suggest that the landscape this model visits at different long times in an out-of-equilibrium relaxation process is, in a sense, self-similar. Furthermore, there is a similarity between the landscape seen out of equilibrium at long times and the equilibrium landscape. The calculation is greatly simplified by making use of the superspace notation in the dynamical approach. This notation also highlights the rather mysterious formal connection between the dynamical and replica approaches. We also perform numerical simulations which show good agreement with the analytical results for the out of equilibrium dynamics.

We study analytically and numerically a model of random sequential adsorption (RSA) of segments on a line, subject to some constraints suggested by two types of physical situation: (i) deposition of dimers on a lattice where the sites have a spatial extension and (ii) deposition of extended particles which must overlap one (or several) adsorbing sites on the substrate. Both systems involve discrete and continuous degrees of freedom and, in one dimension, are equivalent to our model which depends on one length parameter. When this length parameter is varied, the model interpolates between a variety of known situations: monomers on a lattice, the 'car parking' problem and dimers on a lattice. An analysis of the long time behaviour of the coverage as a function of the length parameter exhibits a 1/t² approach to the jamming limit at the transition point between the fast exponential kinetics, characteristic of the lattice model, and anomalous to the 1/t law of the continuous model.

Simulations of an elastic string in 1+1 dimensions near and at the depinning threshold f_c are presented. The spatial dependence of the roughness is characterized by two exponents. One, eta approximately=0.9, describes the height-height correlations. The other, chi approximately=1.15, measures how the transverse width, W, of the string increases with increasing string length. Since chi >1 any physical string will eventually break as its length is increased. The growth of the width of the string with time is controlled by an exponent beta approximately=0.9. As the applied driving force approaches f_c from above the width increases according to W approximately (f-f_c)^{- sigma} with sigma approximately=. The dependence of f_c on the strength of the random potential is found to be in agreement with the collective pinning theory by Larkin and Ovchinnikov (1979).

We consider learning and generalization in the multi-interacting feed-forward network model recently proposed by H O Carmesin (1994). With an a priori definition of the net architecture, based on symmetries presented by the function to be learnt, we define a generalized Hebb rule, extend the maximum stability learning algorithm to multi-interactions, and obtain training and generalization curves. For rules where different orders of synapses are not correlated the results obtained for the simple perceptron concerning the Hebb rule and through replica calculations in the space of couplings may be straightforwardly adapted to multi-interactions through a simple renormalization of the total number of independent couplings. Analytical and numerical simulation results are compared and show excellent agreement.

We introduce a path-integral-like partition function for chaotic mappings. This path integral is based on arbitrary non-Markovian stochastic processes generated by the symbolic dynamics of the map rather than the Wiener process. Our approach can be regarded as an extension of the thermodynamic formalism to infinitely many inverse temperatures. The concept of Renyi entropies is generalized to entropy functionals. A generalized transfer operator is introduced, which allows us to calculate the entropy functionals with high numerical precision. Several examples are worked out in detail.

We consider the multiplicity problem of the branching rule GL(2k+1,C) down arrow SO(2k+1,C). Finite-dimensional irreducible representations of GL(2k+1,C) are realized as right translations on subspaces of the holomorphic Hilbert (Bargmann) spaces of q*(2k+1) complex variables. Maps are exhibited which carry an irreducible representation of SO(2k+1,C) into these subspaces. An algebra of commuting operators is constructed. Eigenvalues and eigenvectors of certain of these operators can then be used to resolve the multiplicity in the branching rule.

We give a practical method for the construction of finite-dimensional representations of U_q(G), where G is a non-twisted affine Kac-Moody algebra with no derivation and zero central charge. It is well known that the finite-dimensional representations of a quantum group U_q(G) at generic q are simply deformations of the corresponding irreps of the classical Lie algebra G. This is, however, no longer true when one goes over to an affine Kac-Moody algebra G and its q-deformation U_q(G). In most cases it is necessary to take the direct sum of several irreducible U_q(G)-modules to form an irreducible U_q(G)-module which then becomes reducible at q=1. We illustrate our technique by working out explicit examples for G=C₂ and G=G₂. These finite-dimensional modules may, for example, determine the multiplet structure of quantum solitons in affine Toda theory.

We point out that the dissipative force-free Duffing oscillator and Holmes-Rand nonlinear oscillator admit infinite dimensional symmetry algebras for certain parametric choices for which the systems become integrable. From the nature of the symmetry vector fields one can also write down the integrals of motion for the above systems. In addition we point out that the other dissipative systems discussed in the literature also admit infinite dimensional Lie algebras.

With each second-order differential equation Z in the evolution space J¹(M_n+1) we associate, using a new differential operator A_z, four families of vector fields and 1-forms on J¹(M_n+1) providing a natural set-up for the study of symmetries, first integrals and the inverse problem for Z. We analyse the relations between the four families pointing out the symmetric structure of this set-up. When a Lagrangian for Z exists, the bijection between dynamical and dual symmetries is included in the whole context, suggesting the corresponding bijection between dual-adjoint and adjoint symmetries. As an application, we show how some results of the inverse problem can be framed naturally in this geometrical context.

The concept of positons is introduced for the Toda lattice equation. It is shown that these multiparametric oscillating and slowly decaying solutions, when inserted as potentials in the finite-difference Schrodinger equation of the corresponding Lax pair, lead to a trivial S-matrix. The resulting eigenvalues are embedded in the continuous spectrum of this infinite Jacobi matrix. The singularities connected with the one-positon solution are discussed and compared with those of the positons of continuous integrable models. The special features of the soliton-positon interaction are analysed.

A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sate approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, under which the Moyal bracket collapses to the Poisson bracket, is particularly simple.

We construct generalized Painleve expansions with logarithmic terms for a general class of ('non-integrable') scalar equations, and describe their structure in detail. These expansions were introduced without logarithms by Weiss-Tabor-Carnevale (WTC-1983). The construction of the formal solutions is shown to involve semi-invariants of binary forms, and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series. The structure of the series depends strongly on whether 1 is or is not a resonance. The convergence of these series is obtained as a consequence of the general results of Littman and Kichenassamy (1993). The results are illustrated on a family of fifth-order models for water-waves, and other examples. We also give necessary and sufficient conditions for -1 to be a resonance.

Starting from a proposal of Volovik (1978) we investigate surface point, line, and wall defects with the aid of relative homotopy groups. The exact sequence of homotopy groups is used in interpreting the relations between surface and bulk singularities. In particular, we consider the possibilities for surface singularities to move into the bulk, for bulk singularities to leave the medium through the surface, and for singular loops to be broken apart by the surface. The theory is extended to the case where the surface induces a thermodynamic phase differing from that in the bulk. An example is given of a biaxial nematic surface with the Klein bottle as the order-parameter space. Due to the fundamental theorem for homotopy groups of fibre bundles, the classification scheme of surface defects is identical for all pairs of bulk and surface order-parameter spaces which are related by an inverse-bundle projection.

We use a model of constant negative curvature Riemann surfaces to study the effect of coupling together several Hall systems on the Hall conductances of the individual systems. We find that on average, the coupling does not effect the conductances. The method we use is based on the fact that the average conductances are given by Chern numbers, and hence are stable when one varies parameters of the Hamiltonian, as long as eigenvalues do not cross. The parameters we change are the surface parameters, and we find the averaged conductances of the surface by 'twisting' and 'pinching' it. This method is applicable to other systems as well.

We obtain the numerical solution of the Smoluchowski kinetic equation for model kernels U(M₁, M₂) varies as (M₁+M₂)^lambda and U(M₁, M₂) varies as (M₁M₂)^lambda 2/, 0< lambda <or=2. We show that the behaviour of the solution for the kernel U varies as (M₁+M₂)^lambda at 0< lambda <or=1 and U varies as (M₁M₂)^lambda/2 at 0< lambda <or=2 becomes self-similar after some time. The shape of the scaling function is analysed; in particular, a simple approximate expression for it at U varies as (M₁+M₂)^lambda is found. An interesting result is obtained for U varies as (M₁M₂)^lambda 2/, 0< lambda <1: the asymptotic behaviour of the scaling function proved to be non-power. We develop the procedure for determining t_cr, the critical indices and the exponent of the power-law asymptotics of the Smoluchowski equation solution. The concrete values of these quantities for the model kernels are obtained. The stability conditions of the algorithm for numerical solving are analysed. The possibilities afforded by numerical solution for investigation of the Smoluchowski equation are discussed.

It is widely known, in classical theory, that nonlinear Hamiltonian systems depending on several parameters often exhibit bifurcation in their trajectories against the changes in the parameters. A Hamiltonian pitchfork bifurcation in the oscillator's periodic trajectories is a typical example. In this article, to seek a quantum counterpart of that bifurcation, the Maslov quantization is applied to a perturbed 1:1 resonant oscillator in the Birkhoff-Gustavson normal form with two parameters. By using a geometric method the degeneracy of energy levels is found to be taken as a quantum counterpart to that bifurcation. The bifurcation set for the Hamiltonian pitchfork bifurcation in classical theory is viewed as the classical limit of a 'bifurcation set' for the degeneracy of energy levels in quantum theory.

The problem of thin flame front propagation with curvature-dependent speed in a weak turbulent flow has been considered, and its connection with classical problems in the physics of disordered systems such as polymers in a random medium, growing interfaces and n-body interaction problems has been discussed. By a path-integral approach, an explicit formula for the randomly moving flame front in terms of the fluctuating velocity field has been derived. The steepest-descent approximation has been used to find the random configuration of the flame surface in the limit of small Markstein diffusivity. New expressions for the turbulent burning velocity (the overall propagation rate of a flame surface subject to a random velocity field) involving the random velocity field along the Lagrangian trajectories have been derived.

This paper presents a solution of 2D electromagnetic wave propagation problems in complicated terrestrial domains. Scalar 2D parabolic approximations are derived from Maxwell's equations for both vertical and horizontal polarization. The parabolic equations are then solved using a new technique involving Pade rational function approximations of the macroscopic operator. This method allows for larger-than-normal range stepping, speeding up computational time significantly. The Pade approximation and the numerical implementation are fully discussed. The discontinuity at the Earth's surface is handled directly by using classical continuity conditions and deriving exact interface conditions for linking the fields in the atmosphere to those in the terrain. The interface conditions are then implemented using the concept of virtual points. Preliminary benchmark tests show the interface treatment to work well. Finally, several example runs are presented illustrating results.

Analytical solutions of the Schrodinger equation for two-dimensional hydrogen in a homogeneous magnetic field (perpendicular to the plane in which the electron is located) are found for a denumerably infinite set of field strengths. The number of solutions for the Nth excited state is N.

We use a gauge-invariant 'reference section' and define the geometric phase for all quantum evolutions in a closed form. This geometric phase is obtained by integrating the inner product of the 'reference section' and its path derivative along the evolution curve which is valid for non-cyclic, non-unitary and non-Schrodinger evolutions of quantum systems. Two non-trivial examples are studied to realize our new expression.

We obtain a general Euclidean connection for the su(1,1), su_q(1,1) algebras. Our Euclidean connection allows an algebraic derivation for the S-matrix. These algebraic S-matrices reduce to the known ones in suitable circumstances. We also obtain a map between the su(1,1) and su_q(1,1) representations.

Angular momentum projection integrals arising from the coherent state representation of bosonic spectrum generating algebras are evaluated using analytical and algebraic methods. For the vibron model of diatomic molecules, a closed expression is derived by direct integration. For the general case, the integral is evaluated as a 1/N expansion by exploiting the symmetries of the underlying boson system and making use of computer algebra.