Table of contents

The O(n) loop model on the honeycomb lattice with mixed ordinary and special boundary conditions is solved exactly by means of the Bethe ansatz. The calculation of the dominant finite-size corrections to the eigenspectrum yields the mixed boundary scaling index and the geometric scaling dimensions describing the universal surface critical behaviour. Exact results follow in the limit n=0 for the polymer adsorption transition with a mixed adsorbing and free boundary. These include the new configurational exponent gamma _l=85/64.

We investigate the noisy, discrete, single lane highway traffic model proposed by Nagel and Schreckenberg (1992) and demonstrate by computer simulations that, as a function of the car density, there is a dynamical transition from laminar flow to jammed traffic in the system related to the divergence of the relaxation time of average car velocity. The critical density is close to the position of the maximum in the fundamental diagram. We give estimates of the critical exponents related to this jamming transition. The critical density of the geometrical ((1+1)-dimensional percolation) transition of jammed regions can be rather far from the jamming point depending on the strength of the noise which indicates that cooperativity leading to the jamming transition can be long range in character.

A direct and exact method for calculating the density of states for systems with localized potentials is presented. The method is based on explicit inversion of the operator E-H. The operator is written in the discrete variable representation of the Hamiltonian, and the Toeplitz property of the asymptotic part of the infinite thus obtained matrix is used. Thus, the problem is reduced to the inversion of a finite matrix.

It is found that the Hamiltonian of the S= 1/2 isotropic Heisenberg chain with N sites and elliptic non-nearest-neighbour exchange is diagonalized in each sector of the Hilbert space with magnetization N/2-M, 1<M<or=(N/2), by means of double quasiperiodic meromorphic solutions to the M-particle quantum Calogero-Moser problem on a line. The spectrum and highest-weight states are determined by the solutions of the systems of transcendental Bethe-ansatz-type equations which arise as restrictions on particle pseudomomenta.

We calculate the storage capacity of a perceptron for correlated Gaussian patterns. We find that the storage capacity alpha _c can be less than 2 if similar patterns are mapped onto different outputs and vice versa. As long as the patterns are in a general position we obtain, in contrast to previous works, that alpha _c>or=1 in agreement with Cover's theorem. Numerical simulations confirm the results.

A new supersymmetric model for electrons with generalized hopping terms and Hubbard interaction on a one-dimensional lattice is solved by means of the Bethe ansatz. We investigate the phase diagram of this model by studying the ground state and excitations of the model as a function of the interaction parameter, electronic density and magnetization. Using arguments from conformal field theory we can study the critical exponents describing the asymptotic behaviour of correlation functions at long distances.

The ground-state energies of the antiferromagnetic XXZ model at a given spin are determined on chains with an odd number N of sites. Analytical and numerical solutions of the Bethe ansatz equations are compared for the N-even and N-odd case. The scaling properties of the ground-state energies enable the determination of the zero-temperature susceptibility. For the isotropic case, we analyse the logarithmic terms in the low-field limit.

We study the statistical mechanics of a D-dimensional array of Josephson junctions in the presence of a magnetic field on a lattice of side 2. In the high-temperature region the thermodynamical properties can be computed in the limit D to infinity . A conjectural form of the thermodynamic properties in the low-temperature phase is obtained by assuming that they are the same as an appropriate spin-glass system, based on quenched disordered couplings. Numerical simulations show that this conjecture is very accurate in one regime of the magnetic field, while it is probably slightly inaccurate in a second regime.

A family of nonequilibrium kinetic Ising models, introduced earlier, evolving under the competing effect of spin flips at zero temperature and nearest-neighbour random spin exchanges, is further investigated. By increasing the range of spin exchanges and/or their strength the nature of the phase transition 'Ising-to-active' becomes of (dynamic) mean-field type and a first-order tricritical point is located at the Glauber ( delta =0) limit. Corrections to the mean-field theory are evaluated up to sixth order in a cluster approximation and found to give good results concerning the phase boundary and the critical exponent beta of the order parameter which is obtained as beta approximately=1.0.

We study the generalization ability of multilayer neural networks with tree architecture for the case of random training sets. The first layer consists of K spherical perceptrons with binary output. A Boolean function B computes the final output from the K values produced by the first layer. We first calculate the learning behaviour of Gibbs learning in the case of learnable rules, where teacher and student have the same architecture and the Boolean function B is permutation symmetric with respect to the hidden units. In the asymptotic case of high loading alpha to infinity (as usual alpha is the loading parameter) we find that the generalization error vanishes in the same way for all B; the reason being is that there are two effects cancelling each other. In the opposite limit for small alpha we find qualitatively different behaviour, i.e. some networks undergo a phase transition. We show how these differences in the behaviour depend on certain characteristics of the Boolean function B. We then study the Bayes algorithm, which generalizes according to the majority decision of a certain ensemble of machines, and find that the parity function plays a special role: if the teacher is a parity-machine there always exists a single student parity-machine that generalizes as well as the Bayes algorithm.

A strong analogy is found between the evolution of localized disturbances in extended chaotic systems and the propagation of fronts separating different phases. A condition for the evolution to be controlled by nonlinear mechanisms is derived on the basis of this relationship. An approximate expression for the nonlinear velocity is also determined by extending the concept of the Lyapunov exponent to a growth rate of finite perturbations.

The dynamical behaviour of the Ising model with heat-bath dynamics is studied. According to our investigation of the model using the damage spreading method, the dynamical phase transition temperature is estimated as 4.52+or-0.03 for the 3D Ising model. The dynamical critical exponent z has been obtained as 2.16+or-0.04 for the 2D ferromagnet and 2.09+or-0.04 for the 3D Ising model. The scaling property of magnetization is obtained.

It is shown that there exists an isomorphism between q-oscillator systems covariant under SU_q(n) and SU_q-1(n). By the isomorphism, the defining relations of a SU_q-1(n) covariant q-oscillator system are transmuted into those of SU_q(n). It is also shown that a similar isomorphism exists for the system of q-oscillators covariant under the quantum supergroup SU_q(n/m). Furthermore, the cases of q-deformed Lie (super)algebras constructed from covariant q-oscillator systems are considered. The isomorphisms between q-deformed Lie (super)algebras cannot be obtained by the direct generalization of the isomorphism for covariant q-oscillator systems.

The radiation field of toroidal-like time-dependent current configurations is investigated. The infinitesimal time-dependent configurations are found outside which the electromagnetic strengths disappear but the potentials survive. For a number of time dependences, their finite radiationless counterparts can be found. In these cases topologically non-trivial (unremovable by a gauge transformation) electromagnetic potentials exist outside sources. The well-defined rule obtained for constructing time-dependent infinitesimal sources suggests the existence of finite non-trivial radiationless sources with an arbitrary time dependence. The latter can be used to carry out time-dependent Aharonov-Bohm-like experiments and to transfer the information. Using the Neumann-Helmholtz parametrization of the current density we represent the time-dependent electromagnetic field in a form convenient for applications.

Starting from the generalization of the Itzykson-Zuber integral for U(m/n) we study the orthogonality relations for this supergroup.

The analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods. Bender-Wu branch points, at which level crossing occurs, are calculated and labelled by a process of analytic continuation. The different resonances are the values that a single analytic function takes on different sheets of a Riemann surface whose topology is described.

By shifting the parity operator in phase space, one obtains a class of operators, which we call the Wigner operator, because its expectation value is equal to the Wigner function. Calculating the commutator of Wigner operators with different arguments, we show that the corresponding observables cannot be measured simultaneously. Introducing a new parameter, we give a trace-class generalization of this operator. The displaced number states of the harmonic oscillator constitute the natural eigenstate basis of the corresponding generalized Wigner operator. Establishing the integral representation of this operator we show that its expectation value is proportional to the Q- and P- and Wigner functions at special values of the parameter. We illustrate this connection with the coherent, number and squeezed states of the harmonic oscillator.

A wave-splitting approach used elsewhere to solve a black-hole scattering problem is adapted to the formal solution of arbitrary self-adjoint wave equations in 1+1 dimensions and yields potentially useful results in this more general case. It is then shown that the self-adjoint wave equation being solved, and the non-self-adjoint linear wave equations satisfied by the one-way component waves, are naturally related to a pair of motions of the (1+1)-dimensional Toda lattice that together comprise a motion of the Kac-van Moerbeke lattice. This provides a partial explanation for the peculiar fact that every motion of the Kac-van Moerbeke lattice can be viewed as two interpolated motions of the Toda lattice.

We show that the projection operator mod pq;ye^{i phi} )(ye^{i phi} ;pq mod , where mod pq;ye^{i phi} ) is a squeezed state, obeys a partial differential equation in which the squeeze parameter y plays the role of time. It follows that related functions, such as the probability distribution functions and the Wigner function are solutions of this equation. This equation will be called a pseudodiffusion equation, because it resembles a diffusion equation in Minkowski space. We give general solutions of the pseudo-diffusion equation, first by the method of separation of variables and then by the Fourier transform method, and discuss the limitations of the latter method. The Fourier method is used to introduce squeezing into the number states, the thermal light and the Wigner function.

New boundary conditions for classical integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice, are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank-1 ansatz for the 2*2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be that of dynamical symmetry for the A₁ and BC₂ Calogero-Moser problems. Other physical realizations of our quadratic algebra are also considered.

Based on the results of singularity confinement we derive bilinear expressions for the discrete Painleve equations. In these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions. We show that the bilinear approach provides a natural framework for the derivation of Backlund and Schlesinger transforms for the discrete Painleve equations.