Table of contents

A model of a self-attracting walk is proposed. Mean square displacement of a particle xi grows with time t as a power law xi ² approximately t^{2 nu} with the exponent nu <1/2. It is shown that nu =1/(2D-D_b) where D is the fractal dimension of the cluster consisting of the visited sites and D_b is the fractal dimension of its boundary. A compact cluster with fractal boundaries was obtained by a computer simulation for d=2. The derived value of nu is in a good agreement with that obtained by computer simulation.

We present a general self-consistent procedure using lattice Green's functions to find the stress redistribution due to multiple ruptures in a discrete elastic medium. We use this method to study a dipole version of the quasi-static crack-propagation model for earthquakes proposed by Chen et al. (1991) Unlike the earlier treatment we do not assume that ruptures occur independently. We obtain the stresses self-consistently and this changes the scaling behaviours of the frequency-size (seismic moment, energy) distribution. We make a careful comparison of our results with recent analysis of seismological data.

The fermionic model is considered equivalent in the critical region to the two-dimensional Ising model in homogeneous magnetic field. The ground state of the fermionic Hamiltonian is obtained from the variational principle of the quantum mechanics, where probe states are taken in the BCS-like form. In this approximation the explicit representation of the scaling function is derived which is characterized by the correct values of critical indexes and the susceptibility critical amplitudes ratio C₊/C_-=12 pi .

Basic representations of A_2l⁽²⁾ and D_l+1⁽²⁾ are studied. The weight vectors are represented in terms of Schur's Q-functions. The method to get the polynomial solutions to the reduced BKP hierarchies is shown to be equivalent to a certain rule in the Maya game.

Following an idea of Bogoyavlensky (1988) it is shown that a nonlinear supersymmetric integrable system can be constructed by using the automorphisms of supersymmetric Lie algebra.

An electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional equations by means of Sommerfeld integrals. Anisotropic impedance boundary conditions are satisfied on the wedge's faces. This system of functional equations is solved by regular perturbation method. It is shown that for weak anisotropy the solution is presented by converging series which are Neumann recurrent series for linear equations with contracting operators. In a general case, the problem is reduced to linear equations with compact operators. The wave field asymptotic is computed for the region outside the vicinity of the edge of the wedge.

The critical exponent, nu , of the correlated (or coherent) length of a turbulent cluster has been related to the minimal fractal dimension, D_min, of multifractal isotropic turbulence. The relationship has turned out to be nu D_min=3/2. For the homogeneous case, D_min=3 and hence, nu =1/2 (the so-called mean-field approach of the theory of critical phenomena). For D_min approximately=2.36 (the well known turbulent value), nu approximately=0.63. This result allows one to classify this case as the critical phenomenon of the so-called 'thermal' class of universality. Transition to the 'percolation' class of universality ( nu approximately=0.9) is determined by the boundary conditions.

The paper deals with random spiral motion described in polar coordinates: the trajectory of a particle is described by the distance r from a given point as a function of the angle theta relative to a given direction. A stochastic description of the spiral shapes generated by the random variations of r and theta is suggested, based on the following assumptions: (a) the ray r is made up of additive contributions corresponding to certain jump processes; (b) the angle phi , between two jumps is a random variable selected from a given probability law g( phi ) d phi , with finite or infinite moments; (c) the contributions rho ₁, rho ₂, ... of the different jumps to the ray r are independent random variables selected from a probability law p( rho ) d rho with finite moments. An expression for the generating functional of a random spiral r( theta ) is given in the form of an infinite series which can be used to evaluate the stochastic properties of the ray r. The asymptotic properties of the random spiral shapes depend on the function g( phi ) d phi : if all moments of the angle between two jumps exist and are finite, the average shape is a linear Archimedean spiral (r( theta )) approximately theta as theta to infinity , and the dispersion of the ray increases linearly with the angle ( Delta r²( theta )) approximately theta as theta to infinity . If g( phi ) has a long tail g( phi ) approximately phi ^-(1+H) as phi to infinity with 1>H>0, the average shape is a nonlinear (fractal) Archimedean spiral (r( theta )) approximately theta ^H as phi to infinity and the fluctuations of the ray have an intermittent behaviour ( Delta r²( theta )) approximately theta ²H as phi to infinity . A complete analysis is possible in a Markovian-like case for which the angle between two jumps is exponentially distributed. In this case a closed expression tor the generating functional is available and all cumulants of the ray can be computed exactly: the mth cumulant which expresses the correlations among the rays at different angles theta ₁, ..., theta _m is proportional to the minimum angle min ( theta ₁, ..., theta _m) and the intermittent behaviour is missing. An alternative stochastic description is suggested based on the assumption that the number of jumps in a given angle interval is distributed according to an inhomogeneous Poisson law. This model is also analytically tractable; it is less restrictive in the sense that the average shape corresponds to a broader class of spirals including the linear and the Fractal Archimedean and the logarithmic ones; however, it cannot be used to describe the intermittent behaviour.

Kinetic growth walks (KGW) on the Manhattan lattice have previously been shown to be equivalent to the static problem of interacting self-avoiding walks on that lattice at the theta -temperature. Here, we illustrate how a complete set of exponents for the static problem, including the crossover exponent phi and surface exponents at the ordinary and special transition points, may be obtained from simulations of kinetic walks. In the process we find that phi m 0.430+or-0.006 which encompasses the conjectured value 3/7 approximately=0.42857 for the theta -point on an isotropic lattice. Our numerics confirm a predicted set of exponents for both the bulk and surface transitions in addition to results such as the exact internal energy at the bulk transition. Furthermore, we point out that a recently examined variant of the Nienhuis O(n) model can be mapped onto theta -point walks on the Manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures.

We measure the lacunarity of regular fractals in the limit of an infinite fractal lattice using a suitable graph counting method. The calculations are performed for a recent expression proposed for lacunarity that explores the scaling properties of the mass distribution of the fractal set on cells of radius r. We present results for various Sierpinski carpets and for the Sierpinski gasket.

We obtain the interfacial tension of the general solvable N-state chiral Potts model. It has exponent mu =(N+2)/(2N) in both the horizontal and vertical directions, in agreement with the scaling relation 2 mu =2- alpha .

In multiparameter percolation models, a directional critical exponent may be considered for each direction of approach in the parameter space to a point on the critical surface. An elementary coupling argument shows that for each point on the critical surface the directional critical exponents are equal for all directions strictly in the positive orthant. Using the substitution method, the set of directions with equal directional critical exponents can he extended beyond the orthant. The anisotropic triangular lattice bond percolation model is used to illustrate the method.

We study the dynamics of a stepped crystal surface during evaporation, using the classical model of Burton, Cabrera and Frank(1951), in which the dynamics of the surface is represented as a motion of parallel, monatomic steps. The validity of the continuum approximation treated by Frank is checked against numerical calculations and simple, qualitative arguments. The continuum approximation is found to suffer from limitations related, in particular, to the existence of angular points. These limitations are often related to the adatom detachment rate which is higher on the lower side of each step than on the upper side ('Schwoebel effect').

We have derived a closed form expression for the transverse Fourier transform G(0, 0; Q) (or surface structure factor) of the surface spin-spin correlation function near a complete wetting transition from Landau theory. Whilst G(0,0;Q) contains isolated singularities (poles) in the complex wavevector plane it does not have a simple Omstein-Zernike (OZ) form. Instead, the function exhibits two limiting OZ-like behaviours characteristic of intrinsic and coherent capillary-wave-like fluctuations depending on the value of the scaling variable Q mod H mod /sub ///^-Vco. We also discuss the decay of surface correlations in real space and identify the appropriate singular (long-ranged) contribution. In contrast to the second-moment correlation length the true correlation length at the wall diverges as mod H mod /sub ///^{-V infinity} in the limit of complete wetting.

A model of unsupervised learning is studied, where the environment provides N-dimensional input examples that are drawn from two overlapping Gaussian clouds. We consider the optimization of two different objective functions: the search for the direction of the largest variance in the data and the largest separating gap (stability) between clusters of examples respectively. By means of a statistical-mechanics analysis, we investigate how well the underlying structure is inferred from a set of examples. The performances of the learning algorithms depend crucially on the actual shape of the input distribution. A generic result is the existence of a critical number of examples needed for successful learning. The learning strategies are compared with methods different in spirit, such as the estimation of parameters in a model distribution and an information-theoretical approach.

We introduce an inferential approach to unsupervised learning which allows us to define an optimal learning strategy. Applying these ideas to a simple, previously studied model, we show that it is impossible to detect structure in data until a critical number of examples have been presented-an effect which will be observed in all problems with certain underlying symmetries. Thereafter, the advantage of optimal learning over previously studied learning algorithms depends critically upon the distribution of patterns; optimal learning may be exponentially faster. Models with more subtle correlations are harder to analyse, but in a simple limit of one such problem we calculate exactly the efficacy of an algorithm similar to some used in practice, and compare it to that of the optimal prescription.

Learning in a specific type of multilayer network referred to as a K=2 parity machine is studied in the limit that both the system size N and the number of examples m become infinite while the ratio alpha =m/N remains finite. The machine consists of K=2 hidden units with non-overlapping receptive fields each of size N/2. The output is the sign of the product of the two hidden units for each input We investigate incremental learning by empirically using a least-action algorithm in the following two learning paradigms. In the first, it is assumed that each example is transmitted perfectly to a student. We show that an ability to generalize emerges as the rescaled length of the connection vector l reaches a critical value l_c. Further, we show that a student can identify the target exactly in the limit alpha to infinity , where the prediction error epsilon decreases to zero as epsilon approximately 0.441 alpha ^-13/. In the second paradigm, we examine what happens if each teacher signal is reversed to the opposite sign at a noise rate l. For small lambda , it is found that the prediction error converges to a finite value of O( square root lambda ) in O( lambda ^-32/) iterations. However, for a noise rate beyond a critical value lambda _c approximately 0.175, the student cannot acquire any generalization ability even as alpha to infinity .

It is well known that one can store up to 2N uncorrelated patterns in a simple perceptron with N input neurons. Changing the architecture of the net by adding a hidden layer may enhance the storage capacity. Learning in multilayered networks, however, is difficult and slow compared to perceptron learning. In this work a different approach is taken. A large hidden layer with N' neurons is used onto which the patterns are mapped according to a one-to-one code which is fixed beforehand. Only the connections from the hidden layer to the output unit are modified by learning. Here we show how to treat the correlations which are introduced large. Moreover, our results shed new light on the optimal capacity problem for single-layer perceptrons. We find the optimal capacity to be determined by the dimension of the space spanned by the input patterns, rather than by the size of the input layer.

We consider a simple dynamical system in three different ways, demonstrating that dynamic entropy behaviour can be radically different depending on the perspective. Namely, the Boltzmann-Gibbs entropy of the entire (invertible) system may be constant, increasing or decreasing as a function of time. However, by taking a trace of an invertible dynamical system we may either obtain a system in which the entropy is continuously decreasing or an exact (non-invertible) factor may be obtained which shows a global evolution of entropy to a unique equilibrium.

Although the double pendulum is a well known system, the literature is sparse with regard to the study of its full dynamical behaviour. As a simple conservative system that displays both ordered and chaotic motion, it poses an interesting problem for the study of classical-quantum correspondence. In this work, we present its Birkhoff-Gustavson normal form and semiclassical energies obtained from the normal form. The Pade summation technique is used to re-sum the normal-form series.

The spectral form factor K( tau ) plays a crucial role in the understanding of the statistical properties of quantal energy spectra of strongly chaotic systems in terms of periodic orbits. It allows the semiclassical computation of those statistics that are bilinear in the spectral density d(E), like the spectral rigidity Delta ₃(L) and the number variance Sigma ²(L). Since Berry's (1985, 1991) work on the semiclassical approximation of the spectral rigidity in terms of periodic orbits, it is generally assumed that the periodic-orbit expression for the spectral form factor universally obeys K( tau )=1 for tau >>1. We show that for a wide class of strongly chaotic systems, including billiards with Neumann boundary conditions and the motion on some Riemann surfaces, the asymptotic behaviour of the semiclassical spectral form factor K( tau ) depends very sensitively on the averaging employed. A Gaussian averaging is preferable from a theoretical as well as from a numerical point of view to, for example, a rectangular averaging. However, we show in this paper that the Gaussian averaging leads in some cases to an asymptotic behaviour like K( tau ) approximately e^{c tau} . Where c>0 depends only on the energy E at which the statistic is considered.

Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned superdeterminant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. Some particular cases and examples are discussed.

A new solution of the Yang-Baxter equation, that is related to the adjoint representation of the quantum enveloping algebra U_qB₂, is obtained by fusion formulae from a non-standard solution.

The automorphisms of the free group F₂ form the group Phi ₂. Three involutions are shown to generate Phi ₂. The homomorphism h₂: Phi ₂ to Gl(2, Z) induced by the Abelianization h₁:F₂ to Z² is generated by three affine reflections R³ to R³. The actions of Phi ₂ induced by the homomorphisms F₂ to SU(2), SU(1, 1), Sl(2, C) are generated by algebraic involutions and geometric reflections.

The quantum algebra U_p,q(gl(2)), with two independent deformation parameters (p, q), is studied and, in particular, its universal R-matrix is constructed using Reshetikhin's method. A contraction procedure then leads to the (p, q)-deformed Heisenberg algebra U_{p, q}(h(1)) and its universal R-matrix. Using a Sugawara construction employing an infinite number of copies of these Heisenberg modes, a (p, q)-deformed Virasoro algebra is obtained. The closure property of the (p, q)-Virasoro algebra necessitates two parameters ( alpha , beta ) for the generators (L_m( alpha , beta )). While the parameter alpha may be taken as an integer, the parameter beta is continuous on a complex path and imparts an integral equation structure to the (p, q)-deformed Virasoro algebra.

Using the concept of a non-standard Hilbert space over the quantum complex plane recently introduced by Kowalski et. al. (1993), we construct a unitary q-analogue of the Weyl-displacement operator. We investigate the q-displaced vacuum states and show that they exhibit properties analogous to coherent states in the undeformed theory. They are, however, distinct from any q-coherent states previously found in the literature.

We classify the unitary irreducible representations of the q-oscillator algebra bb⁺-q²b⁺b=1 by the sign of the definite operator K=bb⁺-b⁺b: K>0 corresponds to Fock representations and K<0 to non-Fock ones, K=0 being a degenerate case. We present their link to representations of extended q-oscillator algebra aa⁺-qa⁺a=q^-N obtained earlier. We give examples of all representations. Besides standard ones, we describe a new non-Fock representation of the Macfarlane type.

Transformations which leave the q-oscillator algebra invariant are considered. In general there exists no uniqueness theorem for the q-oscillator case. It is proven, however. that the analogue of von Neumann's theorem exists for a certain class of transformations. An essential part of these transformations is defined by the relation between the elements a⁺a and N involving a constant. The latter can be interpreted as a central element of the q-oscillator algebra. The explicit form of non-unitary transformations with invariance of the algebra is obtained as well.

In this paper we use a special choice of the GL_X,_qij(2N) quantum plane and its differential calculus for a q-deformed phase space to define a modified Poisson bracket and construct the contraction rule for the q-deformed Levi-Civita symbol. We find the q-deformed phase-space variable realization of the so_q(3) and q-deformed Poincare algebras.

The classical r-matrix implied by the quantum kappa -Poincare algebra of Lukierski, Nowicki and Ruegg is used to generate a Poisson structure on the Poincare group. A quantum deformation of the Poincare group (on the Hopf *-algebra level) is obtained by a trivial quantization.

We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation delta u/ delta _x0+ Delta u=auⁿ+ epsilon f(u), where a and n are arbitrary real constants, f is a smooth function, and 0< epsilon <<1.

The determination of the shape of a scatterer by non-destructive methods, such as scattering experiments, raises the question of indetermination or ambiguity problems for the scatterer's shape, depending on the given scattering data. These ambiguity problems are discussed here by means of numerical constructions of equivalent scattering problems, i.e. we determined scatterers that produced the same scattering amplitude in given conditions (fixed energies, fixed incident angles and/or directions of receivers). This construction is given in the context of a generalized scattering theory that takes into account impedance discontinuities inside the scatterer. We started with a scattering problem defined by a discontinuous curve of arbitrary shape and defined by boundary conditions. Then, a circular curve with appropriate boundary conditions was determined such that these two scattering problems yielded the same scattering amplitude within the given conditions. We also present numerical results for two particular cases of the generalized scattering theory, calculated by means of the Nystrom method. We used these results to verify that the equivalence obtained in the Born approximation holds for the exact scattering amplitudes.

In this paper a generalization of the direct method of Clarkson and Kruskal (1989) for finding similarity reductions of partial differential equations is found and discussed for the Burgers and Burgers-Huxley equations. The generalization incorporates the singular manifold method largely based upon the Painleve property. This singular manifold can be used as a reduced variable. Furthermore, a sort of inverse procedure is hereby developed through which we find the equations that yield the vector field components associated to the symmetries of the PDE. This procedure also displays the profound relationship among the symmetries and the singular manifold as a reduced variable. The symmetries found in this way are shown to be those corresponding to the so-called non-classical symmetries by Bluman and Cole (1974), and Olver and Rosenau (1986).

In this paper the Galilean, scaling and translational self-similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite-dimensional Grassmannian. The string equations of the one-matrix model correspond to the Galilean self-similarity condition for this hierarchy. We describe, in terms of the initial data for the zero-curvature 1-form of the AKNS hierarchy, the moduli space of these self-similar solutions in the Sato Grassmannian. As a by-product we characterize the points in the Segal-Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura-Hirota rational solutions of the NLS equation.

Vector solutions of the Laplace equation are obtained. Their properties and possible applications are discussed. The multipole toroidal moments appear naturally in this vector basis, removing the mystery of their origin. Conditions are found for the non-radiation of charge and current densities periodically changing with time. Electromagnetic properties of the toroidal solenoid with non-zero helicity, the influence of the latter on Aharonov-Bohm scattering, and an alternative viewpoint on the toroidal solenoid with non-trivial helicity are studied.

The modifications of the electromagnetic coupling in the presence of a strong external magnetic field are further investigated. The case of non-vanishing momentum transfers acid the induced anisotropy of space are explicitly taken into account. We confirm the presence of directional terms showing an unusual linear dependence on the external field strength. These last terms overwhelm the expected logarithmic corrections and tend to reduce the effective coupling.

We consider a perfectly reflecting plane mirror moving in the vacuum of the electromagnetic field. The motional modification of the Maxwell stress tensor is computed up to first order in the displacement of the mirror. The resulting dissipative force is shown to be related to the creation of travelling-wave low-frequency photons and to obey the fluctuation-dissipation theorem.

The optical processes of a single-atom model in the phonon field have been studied. To deal with the quantum electrodynamics phenomena in condensed matter microcavities, the influence of electron-phonon interactions has to be taken into account. As a clear picture, the effect of the lattice's oscillation on the optical transition in condensed matter microcavities is reduced to a model of a single atom in a phonon field. We deduce that in the case with a phonon participating, the Bloch equation is still satisfied. In the process of spontaneous emission for an electron interacting with a phonon, we derive the Rabi oscillation for the simple atom in the phonon field.

We discuss, in the phase time approach, quantum tunnelling in the presence of dissipation for an inverted oscillator with Caldirola-Kanai damping. The exact expressions of time delay, traversal time and effective tunnelling velocity are derived. Some paradoxical aspects of tunnelling related to the particle speed in crossing the barrier-such as the Hartmann-Fletcher effect-are briefly considered.

Pure states of a free particle in non-relativistic quantum mechanics are described, in which the probability of finding the particle to have a negative x-coordinate increases over an arbitrarily long, but finite, time interval, even though the x-component of the particle's velocity is certainly positive throughout that time interval. It is shown that, for any state of this type, the greatest amount of probability which can flow back from positive to negative x-values in this counter-intuitive way, over any given time interval, is equal to the largest eigenvalue of a certain Hermitian operator, and it is estimated numerically to have a value near 0.04. This value is not only independent of the length of the time interval and the mass of the particle, but is also independent of the value of Planck's constant. It reflects the structure of Schrodinger's equation, rather than the values of the parameters appearing there. Backflow of positive probability is related to the non-positivity of Wigner's density function, and can be regarded as arising from a flow of negative probability in the same direction as the velocity. Generalizations are indicated, to the relativistic free electron, and to non-relativistic cases in which probability backflow occurs even in opposition to an arbitrarily strong constant force.

The three-body problem in one dimension with a repulsive inverse-square potential between every pair was solved by Calogero. After mapping the three-body problem to that of a particle on a plane, the known results of supersymmetric quantum mechanics are used to solve this problem, as well as a number of new ones, algebraically. This general technique is applicable when the potential is separable in the radial and angular variables on the plane, and its supersymmetric partner is shape invariant. After discussing one example in detail, an exhaustive list of such exactly solvable potentials is given.

The algebraic structure of covariant anomalies and covariant Schwinger terms in an anomalous Yang-Mills theory is investigated. A cohomological characterization is formulated and geometrically interpreted. A new method of determining covariant anomalies and covariant Schwinger terms based on the BRS-anti-BRS complex is presented.