Table of contents

The low temperature dynamics of the three-dimensional Ising spin-glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t, t_w) shows a crossover at the waiting (or aging) time t_w from algebraic quasiequilibrium decay for times t<<t_w to another, faster algebraic decay for t>>t_w with an exponent similar to one for the remanent magnetization.

A random walk computer model of river network is proposed. It is shown that the length and the width of both network and of individual streams that constitute it exhibit scaling sigma /sub /// approximately L^v// and sigma _{perpendicular to} approximately L^v ( sigma // and sigma _{perpendicular to} are the longitudinal and lateral sizes of the object, L is the overall length of the object). The simulated individual streams display self-similar behaviour at small L (v/sub ///=v_{perpendicular to} =0.80+or-0.03) and self-affine behaviour at large L (v/sub ///=0.99+or-0.03, v_{perpendicular to} =0.50+or-0.03). Similar behaviour is observed for simulated river networks too: v/sub ///=v_{perpendicular to} =0.66+or-0.03 correspond to these in the self-similarity region, while in the self-affinity region v/sub ///=0.74+or-0.03 and v_{perpendicular to} =0.43+or-0.03. Proceeding from the self-affinity of individual rivers and river networks Hack's empirical law L approximately F^beta has been substantiated (L is the length of the main river, F the catchment area), where beta =1/(1+H), H=v_{perpendicular to} /v/sub ///, Hurst's exponent for river networks. The scaling for the water mass distribution over the river network in the self-affine region is also revealed: sigma _m// approximately L^v, sigma _{m perpendicular to} approximately L^v, v_m//=0.72+or-0.03, v_{m perpendicular to} =0.38+or-0.03. It is shown that in this region the water mass M depends upon the network total length and upon the catchment area as a power law: M approximately L^1.67 approximately F^1.43.

The authors construct a q-tensorial structure for q-oscillators. Following an appropriate algebraic formalism they succeeded in constructing classes of tensorial operators and consequently the Casimir element of U_q(u(2)), as a 0-rank tensorial operator. The classic Casimir operator of SU(2) is obtained in the limit q to 1.

A new trigonometric degeneration of the Sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied. A further contraction gives the standard quantum algebra U_q(sl(2)). It is shown that the degenerate Sklyanin algebra contains a subalgebra isomorphic to algebra of functions on the quantum sphere (SU(2)/SO(2))_q¹2/. The diagonalization of general quadratic form in its generators leads in the functional realization to the difference equation for Askey-Wilson polynomials.

By a loop-expansion around Parisi's mean-field theory for an Ising spin-glass it is shown that the overlap of the magnetization patterns belonging to two different temperatures, T and T', vanishes to any order, (s_i)_T(s_i)_T=0, while the correlation overlap (s_is_j)_T(s_is_j)_T calculated to first loop order (and, for technical reasons, for dimensions d>8 only) is found to decay exponentially, with a characteristic length approximately (T-T')^-1.

The author derives an exact scaling form for the low temperature partition function in a model of polymer collapse. This confirms series work and so gives the exponents sigma =1/2, gamma _-=1/4 and chi =3/4 exactly. The model considered is a variant of the self-interacting partially directed self-avoiding walk in two dimensions.

According to classical nucleation theory (CNT), clusters are produced with the help of a reversible gain of single molecules, with no cluster-cluster aggregation. Simulations at T equivalent to T_c seem to disagree. The author shows that, when a growing embryo cluster aggregates with clusters of any size d>or=1, the dominant contribution is of d=1 (single molecules). Production of the biggest clusters obeys, therefore, CNT. Production of other clusters, however, is modified by their slow rate determining aggregation with the biggest. The theory fits excellently the simulations and for the first time combines aggregation with CNT.

A deformation U_q(P(n))=P_q(n) of the extended noncontragredient (strange) superalgebra of P(n), denoted by P(n), is proposed. A realization of P_q(n) in terms of q-bosons and q-fermions is presented. Then a procedure to build up a set of representations of P_q(n) is briefly discussed. As a by-product, a new realization of SU_q(n) is obtained.

The Fisher-Jin crossing criterion derivation of the effective interfacial Hamiltonian for wetting transitions is generalized to consider surfaces of fixed magnetization that may remain bound to the wall in the limit of infinite adsorption. It is shown that an infinite set of such effective Hamiltonians is required to construct the mean-field order-parameter correlation function G(r₁, r₂). Surfaces that remain bound to the wall in the limit of complete wetting are shown to exhibit fluctuations which have a coherent quality. The author emphasizes that the construction of G from an infinite set of effective Hamiltonians is required for full thermodynamic consistency.

Rectification of acoustic waves corresponding to each mode of propagation in crystalline solids (modal acoustic radiation-induced static strains) is predicted directly from the virial theorem for an elastic continuum.

A cellular automaton model for the description of traffic flow is investigated. It generalizes asymmetric exclusion models which have attracted a lot of interest in the past. The authors calculate the so-called fundamental diagram (flow versus density) for parallel dynamics using an improved mean-field approximation which takes into account short-range correlations. For maximum velocity they find that the simplest of these non-trivial approximations gives the exact result. For higher velocities their results are in excellent agreement with numerical data.

The authors discuss random walks and self-avoiding walks on a self-affine fractal which is a Cartesian product of two fractals. Mean-square distances on the two sub-fractals have different scalings. Flory-type formulae are derived.

The authors investigate the ground state of an XY model with a transverse field, characterized by an exchange matrix, whose highest eigenvalue, lambda _max, is supposed to be positive and finite. They prove rigorously that for a spin S the ground state changes at the transverse field value K_c=S lambda _max.

A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz (1965) is discussed. A simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analogue of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.

The author considers probabilities p_M(t) for quantum mechanical systems to be found in a subspace M of the Hilbert space of states and derive an upper bound on the change in time of P_M(t) which contains only the Hamiltonian and the projector P_M onto M but not the density matrix.

The author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour. Using scaling arguments a relevance/irrelevance-type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined. The author has performed exact calculations for the surface magnetization of aperiodic quantum Ising chains and for a (1+1)-dimensional directed polymer in a longitudinal aperiodic environment. The results for both systems are in accord with scaling considerations. For a marginal aperiodicity the critical exponents are found to be non-universal.

Using the replica method the authors investigate learning from correlated data in a perceptron. Both types of association-'spatial'-within each example, and 'semantic'-among the different ones-are considered for both Boolean and continuous weights and output functions. 'Spatial' correlations of data may significantly improve the learning properties, whereas the 'semantic' ones do not practically influence them.

The authors examine an Ising spin system in which both spins and the interactions between them may evolve in time, although on disparate timescales, such that the couplings change adiabatically. In thermal equilibrium they find a novel application of the replica method, but for finite replica number, representing the ratio of the temperatures of the spin and interaction systems. Regimes where the motion of the couplings has non-trivial effects are found in addition to those where solely the stochasticity of these interaction weights in significant, and this issue is closely related to the orders of the transitions between the various phases observed. Simulation results lend support to the analysis.

The authors investigate the effects of the synaptic noise in neural networks at finite temperatures. The analytic results are obtained through the use of the path-integral formulation which facilitates performing the quenched average over the random patterns and random noises. We consider the noise effects in the diluted Hopfield model, the fully-connected Hopfield model and the dynamic model, laying emphasis on the interplay with the temperature. In the phase diagrams drawn as functions of the temperature, the storage, and the noise strength, interesting features including re-entrance, first-order transitions as well as second-order transitions are found.

A probabilistic automata network SIS (susceptible-infective-susceptible) model for the spread of an infectious disease in a population of moving individuals is studied. The local rule consists of two subrules. The first one, applied synchronously, models infection and recovery. It is a probabilistic cellular automaton rule. The second, applied sequentially, describes the motion of the individuals. The model contains three parameters, the probabilities p_i to get infected and p_r to recover, and the average number of tentative moves per individual m. Depending upon the values of these parameters, in the infinite-time limit, the system is either in the disease-free state or in the endemic state. It goes from one state to the other through a transcritical bifurcation similar to a second-order phase transition characterized by a non-negative order parameter, whose role is played, in this model, by the stationary density of infected individuals. The (p_i,p_r) phase diagram and the critical behaviour of the stationary density of infectives in the neighbourhood of the phase transition, are studied as a function of m. According to whether the individuals perform short- or long-range moves, it is found that the parameters characterizing the transition have a qualitatively different behaviour as m varies. When m is very large, the correlations created by the application of the subrule modelling infection and recovery are destroyed, and, as expected, the behaviour of the system is then correctly predicted by a mean-field-type approximation which assumes a homogeneous mixing of the individuals. When m is not large, this assumption is no longer correct.

By enumerating irreducible bridges exactly up to 40 steps, and by obtaining a lower bound to the number of bridges with less than 125 steps, a lower bound for the connective constant for square lattice self-avoiding walks of 2.62 is obtained.

The author studies finite-size effects in lattice models for dynamical phase transitions. For the Ziff-Gulari-Barshad (ZGB) model on the Cayley tree, it is shown that the order of the transition from the reactive phase to the monomer-saturated phase depends upon the adsorption rule for dimers at the boundary. He also examines the ZGB model on finite-sized square lattices connected by a single bond. The finite-size effects associated with the first-order and the second-order transitions are markedly different. This phenomenon can be exploited to determine the order of phase transitions numerically, with modest computing requirements. His findings emphasize the importance of finite-size effects and geometrical constraints in dynamical lattice models. They also help explain the puzzling difference between the behaviour of the ZGB model on percolation clusters and on the Sierpinski gasket.

Exact results are obtained for a square lattice fifteen-vertex model and related O(n) model recently proposed by Nienhuis (1990). The partition function per site and finite-size corrections are obtained, along with the critical exponents via the Temperley-Lieb equivalent six-vertex model. The exact exponents nu =⁴/₇ and gamma =⁶/₇ follow in the limit n=0 for even system sizes. Odd system sizes are also discussed. The fifteen-vertex model is also the vertex formulation of the anisotropic, or q-deformed, spin-1 biquadratic model.

Gardner's analysis (1989) of the optimal storage capacity of neural networks is extended to study finite-temperature effects. The typical volume of the space of interactions is calculated for strongly diluted networks as a function of the storage ratio alpha , temperature T and the tolerance parameter m, from which the optimal storage capacity alpha _c is obtained as a function of T and m. At zero temperature it is found that alpha _c=2 regardless of m while alpha _c in general increases with the tolerance at finite temperatures. The authors show how the best performance for given alpha and T is obtained, which reveals a first-order transition from high-quality performance to a low-quality one at low temperatures. An approximate criterion for recalling, which is valid near m=1, is also discussed.

The entire set of binary vectors to be stored using a single-layer perceptron can be divided into two groups, one for which the output neuron state consistently equals one of the input neuron states and a second for which the output neuron state consistently negates the same input neuron. The capacity of the single-layer perceptron depends on the ratio between these two groups. This dependence is examined via statistical mechanical methods, producing the probability of obtaining a linearly separable solution for a random selection of input-output relations, for a given value of the above ratio. This probability is extremely useful for designing recurrent neural network training algorithms. These algorithms make use of the obtained results to select the most probable internal representations to be realized in such nets. Moreover, the distribution of the linearly separable binary functions enables one to obtain a good estimate for the total number of linearly separable binary functions for a certain number of input neurons, a task considered as a hard computational problem. Additional incentives for carrying out the calculation are understanding the capacity of simple nets for certain types of input-output correlations and laying the foundations for analysing some constructive training algorithms such as the tiling and upstart algorithms. All results show consistency with existing theoretical results.

The authors present several results on the finite-size corrections in the SK model within the spin-glass phase slightly below the transition temperature and along the de Almeida-Thouless (AT) line. In the first case, they study the finite-size corrections for the distribution of overlaps P(q) and for ultrametricity. Along the AT line, finite-size corrections to the free and internal energy are also obtained. Numerical results are in good agreement with their predictions.

The authors find a sequence consisting of time-dependent evolution vector fields whose time-independent part corresponds to the master symmetries of the Toda equations. Each master symmetry decomposes as a sum consisting of a group symmetry and a Hamiltonian vector field. Taking Lie derivatives in the direction of these vector fields produces an infinite sequence of symmetries.

Starting from a conformal symmetry, higher-order Poisson tensors, deformation relations and master symmetries for the Toda lattice are obtained. A hierarchy of time-dependent symmetries is also constructed. Using reduction, deformation relations previously known to hold up to a certain equivalence relation are shown to be exact.

For any polynomial perturbation of the generalized harmonic oscillator the Berry phase is explicitly computed, in all orders of perturbation theory, as a polynomial in h(cross) whose term of order zero is the Hannay angle. This construction is a direct application of the technique of quantum normal forms.

An explicit link between finite-gap solutions of the stationary axisymmetric Einstein equation found by Korotkin and Matveev (1990) is obtained.

An alternative to the Bogomolny separation for the three-dimensional Heisenberg model is studied. An example of an alternative approach to one- and two-dimensional spin field models is given. Energetic stability of ferromagnet configurations with a non-trivial Hopf index within an isotropic Heisenberg model is considered.

The authors present a detailed analysis of the nearest-neighbour spacing distribution (NNSD) for random matrices on which the non-zero elements are confined to a band around the diagonal. The vanishing of the remaining elements induces non-analytic (logarithmic) behaviour for the NNSD at very small spacings. A conjective for the precise behaviour of these logarithmic terms is presented and supported by extensive numerical computations.

Monte Carlo simulations and finite-size scaling were used to study the one-dimensional BEG (Blume-Emergy-Griffiths) model under the influence of two competing dynamics: single spin flip (Glauber dynamics at finite temperature T) and random exchange to two spins (Kawasaki dynamics at infinite temperature). The phase diagram shows first- and second-order phase transitions. The critical exponents are obtained for different values of the rates and the range of the exchanges. The authors find beta =0.325+or-0.020, gamma =0.996+or-0.004 and nu =1.71+or-0.01 for long-range exchanges and equal intrinsic rates. However, for different rates and the nearest-neighbour exchanges they obtain beta =0.07+or-0.03, gamma =1.60+or-0.06 and nu =1.53+or-0.08, thus exhibiting non-universal behaviour.

The authors find the exact localized solutions of a class of nonlinear Dirac equations perturbed by a point interaction potential, that is, any sharply peaked potential approaching the delta -function limit. A detailed analysis of the existence conditions of these localized solutions is carried out. They show that their results agree with non-relativistic predictions when both the self-coupling and the point interaction potential are weak. Lower bounds to the size of the localized solutions are presented.

In the framework of the reduced Maxwell-Bloch system the authors investigate in local form the interaction of optical soliton with an arbitrary background field composed of causal and spontaneous parts. A non-trivial phase shift of the soliton is also found.

A general method of evaluation of traces of arbitrary powers of Hermitian operators containing one- and two-body interactions and defined in a finite-dimensional model space is presented. The model space is taken as the antisymmetric and spin-adapted part of an N-fold Cartesian product of a one-electron space. The method is based on the symmetric group approach to the theory of many-electron systems.

For original paper see Bao-Heng Zhao, ibid., vol.25, p.L617 (1992). It is argued that a recent letter by Bao-Heng Zhao (1992) on the one-dimensional Schrodinger equation with a delta '-interaction is seriously flawed and hence arrives at wrong conclusions.

For original paper see Coveney et. al. ibid., vol.25, p.4947 (1992). The previous authors state, in the introductory section of their paper that they are looking only at a 'small part' of the Brussels formalism. The title of their paper does not reflect this restriction and refers to the general validity of the Brussels formalism. The criticisms are focused on a few papers published about twenty years ago. However, even restricted to these papers, their claims do not apply. The present authors point out two major misrepresentations.

For original paper see Coveney et. al., ibid., vol.25, p.4947 (1992). George and Prigogine (ibid., vol.26, p.3905, 1993) interpret the authors previous paper as destructive criticism of the Brussels formalism. However they argue that their aim was to discover conditions under which this method might be supported by rigorous analysis and that their results give a set of sufficient conditions, admittedly quite restrictive, for the validity of a part of the formalism. The Brussels school has never promulgated any agreed choice for the function space in which the formalism's operators are to act. In their work they chose a Hilbert space. This structure sufficed for the theorems they could prove, which concern only the P subspace. They agree that additional structure, such as that of a rigged Hilbert space, may be necessary to give effect to the Brussels formalism outside the P subspace, but since this additional structure would not have affected their results they do not see that its omission was a misrepresentation. Further comments are made.