Table of contents

We investigate the probability for the largest segment with total displacement Q in an N-step random walk to have length L. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large-N limit. In particular, the size of the longest loop has a distribution with square-root singularity at l identical to L/N=1, an essential singularity at l=0, and a discontinuous derivative at l=1/2.

We consider a general class of boundary terms of the open XYZ spin-1/2 chain compatible with integrability. We have obtained the general elliptic solution of a K-matrix obeying the boundary Yang-Baxter equation using the R-matrix of the eight-vertex model and derived the associated integrable spin-chain Hamiltonian.

We present the hypothesis that at the theta -temperature of a single interacting polymer in two dimensions, a particular combination of universal distance amplitude ratios is given exactly by a simple formula in terms of critical indices only. This generalizes a similar claim for noninteracting self-avoiding walks, which was derived from conformal invariance considerations. We support our hypothesis with a series analysis of interacting self-avoiding walks on the square lattice, high-precision simulations of interacting self-avoiding walks on the Manhattan lattice at the exact theta -point and similar simulations of interacting trails on the square lattice.

We give an absolute stability criterion for additive neural networks with discrete time dynamics, i.e. we show that there exists a value for the gain parameter of the sigmoidal transfer function below which the system admits only one fixed point, attracting all trajectories. As an example, we compute this value in the case of random synaptic weights and a fully connected net, in the thermodynamic limit.

We derive an envelope solitary-wave solution for the following generalized nonlinear Schrodinger equation U_t+ia₂U_xx+a₃U_xxx=ib mod U mod ²U+b₁( mod U mod ²U)_x by gauge transformation.

Exact analytical expressions for second-order correlation functions of both statistically homogeneous and heterogeneous models for random surface fractals are presented. The treatment is based on random germ-grain models and set-theoretical operations. The homogeneous two-phase models are variable with respect to volume fraction (0 to 1), fractal dimension of the internal surface (2 to 3) and the geometry of the convex grains. For heterogeneous models. The characteristics of two-phase microstructures superimposed onto the homogeneous fractal substructure can also be varied in a wide range.

We develop a discrete time version of the so-called Brussels formalism in nonequilibrium statistical mechanics for continuous endomorphisms of a Banach space. We show that, if the evolution operator U and the projector P are such that PU is a compact operator and the spectral radius of (I-P)U(I-P) is strictly less than the spectral radius of U, then the formalism holds and the evolution operator is quasicompact.

We study random copolymers consisting of two kinds of monomers, with attraction between similar kinds. The mean-field analysis of this system indicates a continuous phase transition into a phase with periodic microdomain structure. It is shown that the inverse of the renormalized propagator has a minimum at non-zero wavenumbers. Consequently, there is an anomalously large contribution of fluctuations that make the disordered phase locally stable at every finite temperature. However, below a certain temperature, the ordered phase is shown to be locally stable and a weak first-order transition is possible, similar to the weak crystallization theory developed by Brazovskii.

We present a new series-expansion method for the analysis of the covering fraction Theta in a random sequential adsorption process with nearest-neighbour exclusion on a lattice. It is the first method to yield sequences of increasing lower bounds and decreasing upper bounds on Theta . It is illustrated by numerical calculations for the hexagonal lattice.

The process of phase-induced self-diffusion has been investigated for the three-dimensional Ammann-Kramer-Penrose tiling. This happens along the lines of Kalugin and Katz (1993) for the octagonal planar tiling. It is found that for any permutation within two subsets of the ten vertices in a triacontahedral cage there is a corresponding loop in phase space. There are other loops by which vertices are exchanged between interlocked triacontahedral cages. Along chains of such triacontahedral cages, percolative diffusion is possible. This self-diffusion, however, occurs along two separated sublattices. Thus it is a two-component diffusion.

Burrows and Sulston (1991) have introduced conditional block entropies H_n from information theory in order to give a quantitative measure of disorder for sequences and, if possible, a characterization of quasicrystalline sequences. We give here some properties concerning these block entropies and give an explicit formula for the sequences (H_n)_{n in N} corresponding to the Thue-Morse sequence, the Rudin-Shapiro sequence and the paperfolding sequence. We deduce from these computations that this measure of disorder does not enable us to distinguish between deterministic sequences even if they have different spectral properties.

We have extended the high-temperature specific heat series of the three-dimensional spin- 1/2 Ising model to O( upsilon ²⁶). Analysis of the new series gives alpha =0.101+or-0.004.

The evolution of macroscopic order parameters in separable, stochastic neural networks, which becomes deterministic in the thermodynamic limit, can be completely described by the inverse 'temperature' beta , the embedding matrix A, and the set of initial conditions (m₀). Using mainly the techniques of bifurcation theory we present evidence that the qualitative behaviour is governed by one relevant eigenvalue of the matrix A. We show that a variety of bifurcation phenomena can occur as beta is varied, including quasi-periodic solutions displaying mode locking in the discrete time case. We illustrate our results with numerical simulations of, for reasons of computational intensity, the discrete time case only.

We investigate, through statistical mechanics techniques, the problem of finding N-tuples J=(J_i,J₂,...,J_N) with a fixed fraction k of non-vanishing entries that solve a set of P= alpha N linear equations. In particular, we determine the region in the plane ( alpha ,k) where these solutions exist with a probability of 1 in the limit of large N. We also obtain at insight into the microscopic structure of these solutions by calculating the distribution of the probability of an arbitrary non-vanishing entry J_i. Moreover we evaluate analytically the performances of several easy-to-implement heuristic algorithms and compare them with the optimal solution, a task that is suited very well to the statistical mechanics approach.

We consider a two-dimensional cellular automaton whose rule consists of two subrules. The first, applied synchronously, is a local rule inspired from the 'game of life', with a larger neighbourhood. The second, applied sequentially, describes the motion of a fraction m of individuals. Such rules appear to be useful for modelling complex systems in ecology, such as natural populations of animals, in which the motion of the individuals is believed to play an important role. If the motion is long-range, the density of individuals exhibits a sequence of period-doubling bifurcations and behaves chaotically when m is large enough. If the motion is short-range (i.e. restricted to first neighbours), patterns become inhomogeneous. Spatial correlations decay with a finite correlation length xi of the order of square root m. We observe the formation of domains, of mean width xi , with a chaotic behaviour of the local density of individuals, but the collective behaviour is stationary (the global density tends to a fixed value when the lattice size is much larger than xi * xi ).

This article contains the theoretical study of a weak, slowly oscillating electric field across a wedge of non-chiral smectic C liquid crystal. Smectic C continuum theory is used to derive the governing equations. By use of the method due to Melnikov (1963) these equations are then examined for the possibility of exhibiting chaotic solutions. The results are discussed in relation to possible experimental observations and material constants.

We study the energy level statistics of the generic Hamiltonian systems in the transition region between integrability and chaos and present the theoretical and numerical evidence that in the ultimate (far) semiclassical limit the Berry-Robnik (1984) approach is the asymptotically exact theory. However, before reaching that limit, one observes phenomenologically a quasi-universal behaviour characterized by the fractional power-law level repulsion and globally quite well described by the Brody (or Izrailev) distribution. We offer theoretical arguments explaining this extremely slow transition and demonstrate it numerically in improved statistics of the Robnik billiard and in the standard (Chirikov) map on a torus.

One-dimensional profiles f(x) can be characterized by a Minkowski-Bouligand dimension D and by a scale-dependent generalized roughness W(f, epsilon ). This roughness can be defined as the dispersion around a chosen fit to f(x) in an epsilon -scale. It is shown that D=lim_{epsilon to 0}(2-In W(f, epsilon )/In epsilon ) holds for profiles nowhere differentiable. This establishes a close connection between the roughness and the fractal dimension and proves that D=2-H for self-affine profiles (H is the roughness or Hurst exponent). Two numerical algorithms based on the roughness, one around the local average (f(x))_epsilon (usual roughness) and the other around the local RMS straight line (a generalized roughness), are discussed. The estimates of D for standard self-affine profiles are reliable and robust, especially for the last method.

Models or theories that are stable, in the sense that they do not change in a qualitative manner under a small change of parameters, have a higher probability of having a wider range of validity. This also seems to be true for the fundamental theories of nature. Using the deformation theory of algebras, we review the stabilizing deformations leading from non-relativistic to relativistic and from classical to quantum mechanics. Unlike previous treatments, both deformations are carried out on a finite-dimensional algebra setting. One then finds that the resulting relativistic quantum algebra is itself unstable and admits a two-parameter stabilizing deformation. Taking into account reasonable physical constraints to identify the deformed variables, a new algebra is then proposed as the stable algebra of relativistic quantum mechanics in tangent space. This is isomorphic to the algebra of the pseudo-Euclidian group in five dimensions.

A q-analogue of the root system is constructed for this algebra which is similar to the root system of classical Lie algebras. It is then used to construct, in detail, a class of representations of the algebra of functions on the quantum group GL_q(n), and a q-boson realization of the generators of GL_q(n) is given. I also construct infinite-dimensional Hilbert-space representations of this algebra. The main result of this paper is stated in proposition 8.

Harper's equation, a model for Bloch electrons in a magnetic field, has a band spectrum when the dimensionless magnetic field beta is a rational number p/q. This paper considers the definition of generalized Wannier functions, which can be used to represent the Bloch bands of the spectrum of the rational Harper equation by means of a von Neumann lattice. One representation of these Bloch bands can be extended to the irrational case, and taking matrix elements in this basis leads to a renormalization-group transformation acting on the Hamiltonian. The results in the present paper considerably extend a previous analysis of this renormalization-group transformation, in that the formalism is suitable for systematic calculations of the renormalized Hamiltonian, and that the transformation preserves the rotational symmetry of the Harper Hamiltonian in phase space.

In this paper we give a class of integro-differential evolution equations which generalizes the ILW_k hierarchy. We show the generalized Lax pairs, the bilinear representations and the rational and N-soliton solutions for these integro-differential equations. We also give the Hamiltonian formalism for the simplest set of equations among these integro-differential equations, which is the coupling of the ILW equation and the time-dependent Schrodinger equations.

We propose a family of symplectic maps providing finite-difference approximations for the Wojciechowski and Rosochatius systems. The Lax pair representations are found and complete integrability is proved.

The random-phase approximation is rigorously proved for a particular model of modulated diffusion consisting in a 1D integrable Hamiltonian map whose phase is modulated by an external Markov noise. Continuity of the diffusion coefficient in the limit of vanishing noise amplitude is also investigated.

Intelligent states of the SU(1,1) Lie group are investigated using the analytic representation in the unit disk of the SU(1,1) coherent-state basis. By developing this representation, we study a special class of states, which are both intelligent and coherent. These states can be created using Hamiltonians, for which SU(1,1) is the dynamical symmetry group.

N-loop solitons are constructed by means of N-cusp soliton solutions of the Harry Dym equation. The proposed approach is based on a novel link between the 'loop soliton equation' and the complex Harry Dym equation considered on a special curve.

Recent work has shown that wavelet-based numerical schemes are at least as effective and accurate as standard methods and may allow an 'easy' implementation of a spacetime adaptive grid. Up to now, wavelets which have been used for such studies are the 'classical' ones (real Daubechies' wavelets, splines, Shannon and Meyer wavelets, etc) and were applied to diffusion-type equations. The present work differs in two points. Firstly, for the first time we use a new set of complex symmetric wavelets which have been found recently. The advantage of this set is that, unlike classical wavelets, they are simultaneously orthogonal, compactly supported and symmetric. Secondly, we apply these wavelets to the physically meaningful cubic and quintic nonlinear Schrodinger equations. The most common method to simulate these models numerically is the symmetrized split-step Fourier method. For the first time, we propose and study a new way of implementing a global spacetime adaptive discretization in this numerical scheme, based on the interpolation properties of complex-symmetric scaling functions. Second, we propose a locally adaptive 'split-step wavelet' method.

An exact analogue of the method of averaging in classical mechanics is constructed for self-adjoint operators. It is shown to be completely equivalent to the usual Rayleigh Schrodinger perturbation theory but gives the sums over intermediate states in closed-form expressions. The anharmonic oscillator and the Henon-Heiles system are treated as examples to illustrate the quantum averaging method.

We prove that after the transformation (q,p) to ( lambda q, lambda p), where 0< lambda <1, every Husimi function with appropriate renormalization remains in the class of Husimi distributions. We discuss some implications of this fact on the problem of the phase distribution of a quantum state. We show that Wigner and P functions do not have this property.

Explicit path integration is carried out in a space with a ring-shaped topological defect using toroidal coordinates. The toroidal Aharonov-Bohm experiment is taken as an example.

We study the scattering problem on a graph consisting of a line with a finite-length appendix. The two parts are coupled through boundary conditions depending on three parameters; the motion on the line is free while the appendix supports a potential. The appendix bound states give rise to a ladder of resonances; we construct the resolvent and solve the corresponding pole condition for a weak coupling. In general, the condition only admits an analytic solution in particular cases. We find the pole positions numerically for a linear potential and show that the poles eventually return to the real axis when the coupling strength increases.

An extension of the classical Darboux transformations is applied to the one-dimensional Dirac equation in order to construct von Neumann-Wigner potentials allowing embedded eigenvalues. These potentials lead to a novel type of scattering problem with a trivial S-matrix composed of vanishing reflection coefficients and a trivial transmission coefficient. Related topics like the underlying symmetry of the Dirac equation and the connection with positon solutions of nonlinear evolution equations are discussed.