Table of contents

The authors present results on the roughening of growing interfaces obtained from a Kardar-Parisi-Zhang (KPZ) type continuum equation with quenched additive noise, representing frozen in disorder. Close to the pinning transition, for the exponents describing respectively the temporal and the spatial scaling of the surface from numerical integration in 1+1 dimensions they obtain beta =0.61+or-0.06 and alpha =0.71+or-0.08 up to a crossover time. These estimates are in good agreement with the theoretical prediction beta =3/5 and alpha =3/4 they derive from a dimensional analysis of the equation.

The authors develop a Smoluchowski-type mean-field treatment for a recently introduced model of ballistic agglomeration. The predictions of this mean-field theory for the exponent characterizing typical cluster size are in agreement with earlier results for all dimensions. Nevertheless, the predicted monomer decay and particle size distribution are totally at variance with the numerical observations in one dimension. The reason for this discrepancy is found to be the fact that high velocity particles coalesce rapidly independently of their mass, which introduces correlations not taken into account by the mean-field treatment. This is likely to persist in all dimensions, so that the model has no upper critical dimension. The case where the initial velocity distribution function of the particles has a power-law tail is also examined. It is found that, at least in one dimension, the typical cluster size behaves in a way that depends on the specific velocity distribution function, whereas the monomer decay does not.

The authors argue that in order to describe dilute dipolar or RKKY spin-glasses, a broad distribution of couplings (i.e decaying as a power law) is more appropriate than the traditional Gaussian distribution. They present a mean-field 'replica symmetric' theory for this problem. They find that the spin-glass transition is of a different nature than the one occurring in the SK model, and resembles a percolation transition. The deAlmeida-Thouless temperature is lower than the spin-glass transition temperature. The degree of 'replica symmetry breaking', measured by the absolute value of the negative entropy is found to be smaller than in the SK model and nonmonotonic with temperature. Some experimental consequences are briefly addressed.

By a real-space renormalization-group method, the authors have calculated the dynamic hopping conductivity of a one-dimensional hopping system with hierarchically distributed transition rates. Their numerical results show that the conductivity can display rather different types of low- and high-frequency behaviour as the value of the hierarchical parameter R is varied.

A quantitative information theory is developed for both classical and quantum spin systems. The theory is examined by heavy numerical simulations, which show that the information embedded in each weight of the system increases with the size of the system. The difference between the notions of capacity and information is examined carefully and gives an explanation to the surprising result that the capacity per bit can be greater than one.

Cellular automata for modelling quantum systems are presented. The mass at each site is updated according to rules that depend on the masses of neighbouring sites. An essential feature of these local rules is that mass and probability are conserved. In the limit of small spatial and time steps it is shown that the equation defining one class of automata reduces to the Schrodinger equation and the equation defining another class of automata reduces to the Dirac equation. Advantages of these methods are discussed.

A new algebra that is different to the known algebras in both quantum R matrix and quantum enveloping algebras is presented. Its nontrivial Hopf algebraic structures are also discussed.

A generalization of the AGCM method by using Banach * algebraic bundles over locally compact group was presented.

The authors give a positive answer to a question about the existence of any direct link between two apparently unrelated facts for a specific family of solvable Lie algebras: the structure of their modules of functions on one side, and the algebraic form of their Casimir invariants on the other.

The reflection quadratic algebra A(R) associated with the Z₂ model is formulated from the reflection equation. It is proved that A(R) is isomorphic to the matrix-element algebra A(R) of the quantum group related to R, and thus its coalgebra structure is obtained. A new A(R)-comodule structure of A(R) is revealed.

The discrete spectrum of a deformed oscillator is calculated here for the first time according to the noncanonical Heisenberg algebra. The spectrum extends from omega /2(1+ mu ) to omega / mu , where mu is a positive deformation parameter.

The quantum double and the universal R matrix for a new quantum group associated with a nonstandard solution of the Yang-Baxter equation is constructed. This quantum group can be regarded as a generalization of U_q(sl(n)) and the author's construction reduces to that of U_q(sl(n)) for a special choice of parameters.

The homomorphisms from the free group F₂ with two generators to the complex unimodular group Sl(2,C) are classified. The action induced by the automorphism group Phi ₂ of F₂ on pairs of elements from Sl(2,C) is expressed in a geometry inspired by Fricke and Klein.

All quantum group structures are found on the supergroup GL(1 mod 1). These structures are described by two 2-parameter families, GL_q, _Q(1 mod 1) and GL_h1,_h2(1 mod 1). Each family possesses a central multiplicative quantum superdeterminant, which allows one to define the quantum supergroups SL_q,Q(1 mod 1) and SL_h1,h2(11).

Quantum deformations of SU(3) is discussed. Special attention is given to SO_q(3) and SO(3) subalgebras of SU_q(3) and SU(3). Various chains of SU_q(3) and SU(3) are realized by using the deforming functionals. Both boson and q-boson realizations are also discussed.

The author constructs Lagrangian and Hamiltonian functions for stationary flows of some class of Hamiltonian nonlinear dynamical systems. The method is illustrated by the example of stationary flows of Sawada-Kotera hierarchy.

A 1-parameter group of nonlocal symmetries of nonhomogeneous, nonlinear Schrodinger system with the coupling function linear in x is presented. On using this symmetry the author is able to insert the spectral parameter into the nonparametric Lax pair in all cases which are known to be integrable. A working algorithm to isolate integrable geometries, proposed by Levi, Sym and Tu (1990), is extended.

For x- and t_n-finite-dimensional Hamiltonian systems obtained from the decompositions of zero-curvature equations, it is shown that their Lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems. As a consequence, the zero-curvature representation for soliton hierarchy with source is presented.

Two classes of generalized Coulomb potentials related to the SO(2,1) dynamical algebra are rigorously solved by path integration in terms of parabolic coordinates.

The q-deformed harmonic oscillator within the framework of the recently introduced Schwenk-Wess q-Heisenberg algebra is considered. It is shown, that for 'physical' values q approximately 1, the gap between the energy levels decreases with growing energy. Comparing with the other (real) q-deformations of the harmonic oscillator, where the gap instead increases, indicates that the formation of the macroscopic energy gap in the Schwenk-Wess q-quantum mechanics may be avoided.

The authors consider the electrostatic potential due to a uniform distribution of charge on a Cantor set. Mellin transforms are used to find expansions for the potential at a distance r from the end of the charge distribution. It is found that the potential is a power law multiplied by a function periodic in ln r, together with a power series in r. A recursion relation for the moments of the distribution is used to reveal a similar structure in these moments. Finally Fourier transform techniques are used to find explicit representations for the distribution and its moments.

The author presents a new algorithm for simulating random walks which is simple, versatile and efficient. It uses recursive function calls and can be used to obtain unbiased samples with any given length distribution. This makes it particularly useful in disordered geometries where the effective connectivity constant is not known a priori. When applying it to self-avoiding random walks on two-dimensional media with quenched randomness, he finds evidence for at least two new renormalization group fixed points.

The globular state of a randomly branched disordered heteropolymer-the simplest model of RNA with given secondary structure-is investigated using the replica approach in the mean-field approximation. It is shown that for dimensions of space less than four there is replica-symmetry breaking with power behaviour of the Parisi order parameter. The scale of fluctuations of links decreases when heterogeneity of interaction in the molecule increases.

The well known Potts Hamiltonian formalism for the general study of site-bond correlated percolation in lattice spin models is introduced for the first time in the context of microscopic theories of microemulsions. The approach is sufficiently general to be used in connection with any lattice model of microemulsions. Within a mean-field approximation, the authors obtain equations for the percolation thresholds, average cluster size and cluster size distribution for each of the molecular species of the mixture, which require only the knowledge of the structure functions of the model and an adopted bond activation probability. The theory is applied to a spin-1 model for microemulsions.

The authors study the Ising spin glass with a +or-J distribution in six dimensions by Monte Carlo simulations. Their results for the transition temperature agree very well with the value from series expansions. The exponents are consistent with mean-field values modified by logarithmic corrections as expected.

The authors study a number of non-uniform specified topologies and show rigorously that for certain topologies with cut edges, the critical exponent gamma _t is in agreement with a conjecture given by Gaunt et al. (1984) and that the exponent nu _t= nu , the exponent for self-avoiding walks. The authors also find that the scaling relations gamma _t- gamma _t¹ and gamma _t- gamma _t¹¹ are the same as for self-avoiding walks, previously conjectured only for uniform networks. By assigning an interaction energy to a nearest neighbour contact, they prove that the collapse transition for these topologies is the same as that for self-avoiding walks.

The authors investigate the problem of loading a single-layered perceptron of Ising (+or-1) weights, focusing on the cases of linear and binary output neurons. Previous studies of this problem have been made using the canonical ensemble, which in many cases require the breaking of replica symmetry. They consider here an alternative approach, based on the microcanonical ensemble, and show that many results that were obtained previously using replica symmetry breaking within the canonical ensemble can be easily obtained using the replica-symmetric assumption within the microcanonical ensemble. Since the nature of the replica symmetry breaking in many of the models with discrete weights is similar, they believe that the results can immediately be extended to other cases such as learning a rule from examples, and utilizing discrete weights of larger synaptic depth.

A conserved quantity K of a Hamiltonian H implies a conserved quantity J of the S-matrix only if the forward and the backward asymptotic limits of K are equal. A conserved quantity J of S implies a conserved quantity of H only if J is also a conserved quantity of the free Hamiltonian H₀. Therefore the integrability or chaoticity of S does not necessarily decide the integrability of H.

The matrix elements of the quantum Hamiltonian, which corresponds to the classical KAM system, H=H₀+ epsilon V (where H₀ is integrable) in an energy ordered eigenbasis of H₀ are considered. Their statistical properties are comprised in the definition of the sparsed banded random matrix ensemble (SBRME). In analysing the spectral statistics of this ensemble the authors find the power law level repulsion and the level spacing distribution which is well described by the Brody distribution. The numerical study of SBRME shows two interesting scaling laws: (i) the connection between the scaling variable x= alpha b^3/2 ( alpha =mean diagonal increment, b=bandwidth) and the level repulsion parameter beta as deduced from the Brody distribution, and (ii) the connection between the same scaling variable x and an expression which contains two types of localization length (entropy and geometric). The universal aspects and the importance of SBRME for the Hamiltonian systems in the transition region between integrability and chaos are discussed.

New realizations of quantum algebras su_q(2) and su_q(1,1) are constructed in terms of the q-bosonic and the q-fermionic oscillators. They satisfy the defining commutation relations of quantum algebras without giving any new structure on the q-oscillators. A new relation between the q-oscillators defined for q and those for q^-1 is found. It converts the defining relations of the q-oscillators for q^-1 to those for q. Using this relation, it can be shown that these realizations are invariant under q to or from u^-1.

All finite-dimensional indecomposable solvable Lie algebras L(n,f), having the Heisenberg algebra H(n) as the nilradical, are constructed. The number of non-nilpotent elements f that can be added to H(n) satisfies f<or=n+1. The Casimir and generalized Casimir operators of the algebras L(n, f) are obtained.

Some methods have been developed to calculate the su_q(2) Clebsch-Gordan coefficients (CGC). Here the authors develop a method based on the calculation of Clebsch-Gordan generating functions through the use of 'quantum algebraic' coherent states. Calculating the su_q(2) CGC by means of this generating function is an easy and straightforward task.

Quantum groups are of current interest because of their applications in many fields of physics. In the present paper the authors discuss a q-analogue to the Hamiltonian of the nuclear shell model. If q is written as q=exp(i tau ) where tau is a real number in the interval 0<or= tau <or=2 pi , then for tau =0 they recover the ordinary nuclear shell theory Hamiltonian where the U(3) symmetry is broken by the presence of the spin-orbit coupling term as well as the one depending on L². On the other hand if tau is in the interval 0.5<or= tau <or=2, the levels of corresponding to a given number of quanta N almost collapse to a single one, thus recovering the U(3) symmetry. In the conclusion the authors compare this result with other procedures to re-establish the U(3) symmetry.

The main features of a hyperbolic Kac-Moody algebra (denoted by A₁⁽¹⁾), which appears in the dimensional reduction of N=1 supergravity from four to one dimensions, are presented. A vertex construction is exhibited and the structure of the fundamental representations is discussed. The vertex operator realization is presented in full generality, i.e. for any indefinite Kac-Moody algebras.

Some features of the probability E(n, R) of a region R in certain log-potential systems containing precisely n particles are noted. First, it is shown that a quantity analogous to E(n, R) for a new solvable two-component log-gas can be expressed in terms of the Toeplitz determinant discretization of a Fredholm determinant which occurs in the calculation of E(n, R) for Hermitian random matrices. Second, the first two terms of the asymptotic large-R expansion of E(n, R) for complex random matrices, when R is a disk, are derived by using an electrostatic/thermodynamic argument based on an analogy with the two-dimensional one-component plasma. Finally, by using the Fisher-Hartwig 'conjecture' from the theory of Toeplitz determinants, the asymptotics of E(0,R) for a class of one-dimensional lattice systems is shown to obey a sum rule which has been conjectured to be applicable to all fluid systems with exclusively mobile species.

A number of highly accurate analytic approximations to solitary waves on lattices are detailed, and the results of numerical tests presented. Two types of approximation are considered. The first set of approximations are generalizations of the continuum approximation, based around various Pade expansions of the operator. These are convergent to the exact solution in the limit of wavespeed c to 1, or a dense lattice. The author also considers ways of constructing approximations to fast solitary waves from a continuum-type theory. The two types are closely related. In both cases significant improvements are made on existing approximations. A symplectic Hamiltonian integration scheme is used to perform computer simulations. A new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike, to compare the various methods considered.

The Schrodinger equation i Psi =(A+Bt) Psi with constant Hermitian matrices A and B is studied. In the form where B is diagonalized, this equation is a generalization of the Landau-Zener problem to an arbitrary number of crossing energy levels. An approach-the independent crossing approximation-leading to a partial understanding of the general case in terms of the two-level problem is introduced. It is found that certain S-matrix elements (and thus the corresponding transition probabilities) for the general problem are exactly given by formulae of unexpected simplicity, suggesting that some kind of general analytic solution might exist. The full S-matrix is calculated for a solvable special case of the problem. The solution of another special case, previously discovered for systems with three states, is generalized to any number of states. The asymptotic behaviour of the system is discussed in general and given explicitly to lowest order in 1/t.

Some time ago, in a remarkable paper Hofstadter (1976) showed that the energy spectrum of a spinless tight-binding electrons, moving in a constant magnetic field B, has a very complicated recursive structure. In this article, it is shown that, these energy levels satisfy a simple sum rule. Then, as an application of this sum rule, the author shows why a proposition on the absolute minimum of the energy of this system, which was made by Hasegawa et al (1989), holds in general cases.

The strong coupling expansion (SCE) for the generating functional of the Green functions in the ( phi ^* phi )_d² model is investigated by the method of Schwinger equations. No lattices are used. The problem of boundary conditions is discussed. The exact (in all orders of SCE) relation between the two-particle function and the propagator has been obtained. From this relation the equation for the propagator is obtained. In the lower order this equation coincides with the known gap equation. A modification of the second Legendre transformation in the SCE region is presented.

A new integrable model of a strong correlated electronic system is formulated as a model of hole superconductivity. The model is solved by using the Bethe ansatz. The critical exponents describing the decrease of correlation functions on long distances are derived. The behaviour of these correlations indicates that Cooper pairs of holes are formed in the repulsive region of the model. This conclusion is also confirmed by the calculation of the conductivity and the effective transport mass. In the attractive region, the model is a highly conducting system in which the current carriers with small effective mass are 'light fermions'.

It is pointed out that the equilibrium solutions for pure coagulation and pure fragmentation which Dubovskii et al (ibid., vol.25, p.4737, 1992) purport to obtain are in error, and that such solutions do not in fact exist.