Table of contents

New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.

We determine numerically the relevant spectrum of scaling indices for the fractal diagram of the standard map. Infinite partitions, related by the Gauss transformation and an appropriate measure had to be used in order to obtain convergent results. This choice of the measure and the partitions is motivated by the method of modular smoothing.

The density of complex eigenvalues of random asymmetric matrices is found in the large-N limit. The matrices are of the form where A is a matrix of independent, identically distributed random variables with zero mean and variance . The limiting density is bounded. The area of the support of cannot be less than . In the case of commuting with its conjugate, is expressed in terms of the eigenvalue distribution of the non-perturbed part .

An alternative description of the Lie superalgebra osp(2n + 1/2m) in terms of generators and relations is given. The generators, called Green generators, are the root vectors of osp(2n + 1/2m), corresponding to the orthogonal roots.

We study the out-of-equilibrium dynamics of several models exhibiting ageing. We attempt to identify various types of ageing systems using a phase space point of view. We introduce a trial classification, based on the overlap between two replicas of a system, which evolve together until a certain waiting time, and are then totally decoupled. In this way we investigate two types of systems, domain growth problems and spin glasses, and we show that they behave differently.

Classical diffusion in a random medium involves an exponential functional of Brownian motion. This functional also appears in the study of Brownian diffusion on a Riemann surface of constant negative curvature. We analyse in detail this relationship and study various distributions using stochastic calculus and functional integration.

We present a model of spheres moving in a high-dimensional compact space. We relate it to a mixed matrix model with a O(N) invariant model plus a P(N) invariant perturbation. We then study the low pressure regime by performing a diagrammatic expansion of this matrix model. Finally, we show the results from numerical simulations and we present evidence for a glassy regime at high pressures.

The spin-1 XY chain in a transverse field is studied using finite-size scaling. The ground-state phase diagram displays a paramagnetic, an ordered ferromagnetic and an ordered oscillatory phase. The paramagnetic - ferromagnetic transition line belongs to the universality class of the 2D Ising model. Along this line, universality is confirmed for the finite-size scaling functions of several correlation lengths and for the conformal operator content.

We investigate the problem of selecting an informative subsample out of a neural network's training data. Using the replica method of statistical mechanics, we calculate the performance of a heuristic selection algorithm for a linear neural network which avoids overfitting.

The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe - Peierls approximations, using the replica method.

We investigate the properties of the one-step replica-symmetry-breaking (1RSB) solution for a perceptron learning from examples with weight mismatch where the entropy zero line crosses the Almeida - Thouless (AT) line of the RS solution. For a small number of examples we find the optimal 1RSB solution which has the maximum free energy, non-negative entropy and satisfies the stability condition, the AT criterion for the 1RSB solution. The transition from RS to 1RSB is continuous or discontinuous depending on whether the RS AT line is above or below the zero entropy line. However, for a relatively large number of examples, the 1RSB solution which maximizes the free energy becomes unstable, and should be replaced by higher-step RSB solutions. We also obtain the AT line for the 1RSB solution.

The thermodynamic behaviour of a magnetic system coupled to external fields is studied in a situation in which the paramagnetic and diamagnetic terms are treated on an equal footing. It is shown by explicit calculation that the linear response functions satisfy certain inequalities and in particular that the paramagnetic and diamagnetic response functions satisfy separate inequalities. The magnetic moment is found to be gauge invariant only if both the paramagnetic and diamagnetic terms are included.

The relaxational dynamics of a (1 + 1)-dimensional directed polymer in random potential is studied by Monte Carlo simulations. A series of temperature quench experiments is performed changing waiting times . A clear crossover from quasi-equilibrium behaviour to off-equilibrium behaviour appears in the dynamical overlap function whose scaling properties are very similar to those found in the three-dimensional spin-glass model. In the part, the fluctuation dissipation theorem of the first kind which relates the response function to the tilt field with the conjugate correlation function, is found violated. These ageing effects are brought about by the very slow growth of the quasi-equilibrium domain driven by successive loop excitations of various sizes, which form complex network structures.

The dual q-Hahn polynomials in the non-uniform lattice are obtained. The main data for these polynomials are calculated (the square of the norm of the coefficients of the three-term recurrence relation, etc), as well as the lattice representation as a q-hypergeometric series. The connection with the Clebsch - Gordan coefficients of the quantum algebras and is also given.

We consider the classical algebra from the integrable system viewpoint. The integrable evolution equations associated with the algebra are constructed and the Miura maps and consequent modifications are presented. Modifying the Miura maps, we supply a free field realization for the classical algebra. We also construct the corresponding Toda-type integrable systems.

The total delay time for a particle tunnelling through a non-rectangular potential barrier is calculated as an explicit function of the potential energy U(x), on a rigorous mathematical basis. The resulting expression is valid for sufficiently smooth potentials. It represents a physically well defined quantity that is measured in the experiment directly. In particular, an explicit expression for the tunnelling time related to the potential is obtained. It is valid if the particle's wavelength, taken far away from the barrier, is small compared with the barrier's effective length l, irrespective of the value of the barrier penetration integral. Exact sufficient conditions for the validity of Connor's parabolic connection formulae are established.

Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).

This paper analyses quantum mechanics in multiply-connected spaces. It is shown that the multiple connectedness of the configuration space of a physical system can determine the quantum nature of physical observables, such as the angular momentum. In particular, quantum mechanics in compactified Kaluza - Klein spaces is examined. These compactified spaces give rise to an additional angular momentum which can adopt half-integer values and therefore, may be identified with the intrinsic spin of a quantum particle.

In this paper we present an analytical formula for the inversion of symmetrical tridiagonal matrices. The result is of relevance to the solution of a variety of problems in mathematics and physics. As an example, the formula is used to derive an exact analytical solution for the one-dimensional discrete Poisson equation with Dirichlet boundary conditions.

We develop a renormalization group analysis for a Hamiltonian which is a periodic function of and ; this is a model for Bloch electrons in a magnetic field, and includes Harper's model as a special case. The eigenfunctions are Bloch waves when , the ratio of to the area of the unit cell, is a rational number p/q. The renormalization procedure produces an effective Hamiltonian for a subset of the spectrum, which can be expanded in . The zeroth-order term has been obtained in earlier papers; here we obtain the first-order term of this expansion in terms of the q-dimensional vectors which define the rational Bloch states (k and are Bloch wavevectors).

The effective Hamiltonian is not invariant under gauge transformations multiplying the vector . We show that the infinitesimal gauge transformation is equivalent to evolving under a gauge Hamiltonian obtained (to lowest order in ) by quantizing .

The two-parametric quantum superalgebra is consistently defined. A construction procedure for induced representations of is described and allows us to construct explicitly all (typical and non-typical) finite-dimensional representations of this quantum superalgebra. In spite of some specific features, the present approach is similar to a previously developed method which, as shown here, is applicable not only to the one-parametric quantum deformations but also to the multi-parametric ones.

In this paper, the (2 + 1)-dimensional sine - Gordon equation (2DSG) introduced by Konopelchenko and Rogers is investigated and is shown to satisfy the Painlevé property. A variable coefficient Hirota bilinear form is constructed by judiciously using the Painlevé analysis with a non-conventional choice of the vacuum solutions. First the line kinks are constructed. Then, exact localized coherent structures in the 2DSGI equation are generated by the collision of two non-parallel ghost solitons, which drive the two non-trivial boundaries present in the system. Also the reason for the difficulty in identifying localized solutions in the 2DSGII equation is indicated. We also highlight the significance of the asymptotic values of the boundaries of the system.

Due to a production error `sinh' and `cosh' were incorrectly replaced by `sin h' and `cos h', respectively, throughout the article. IOP Publishing wish to apologize to Dr Takebe for this mistake. The following equations are the correct versions.

P 6686:

p 6687:

p6689:

p6691:

p6693:

p6694:

p6695:

(see [18] for details of calculations). This means that all conserved quantities such as momentum P(x) or energy over the ground states are split into two terms:

p6696:

The main contribution comes from the momentum P of the particle as iPN, the free phase:

p6697:

p6698:

p6704:

Lemma C.2. For 0<a<b, a series

is positive for . Here the term n = 0 is understood as a/b.

Proof. Define a function f(y;x) by

f(0;x) = a/b.

p6705:

By Poisson's summation formula we have

Errors also occurred in Proposition 1.2.2. The correct version is as follows:

Proposition 1.2.2. Each component of acts on the intertwining vector as follows.

where and W is the Boltzmann weight of SOS type [5, 9]:

This proposition is proved in the same way as (3.7) of [34].