Table of contents

The partition function of the Baxter - Wu model is exactly related to the generating function of a site-colouring problem on a hexagonal lattice. We extend the original Bethe ansatz solution of these models in order to obtain the eigenspectra of their transfer matrices in finite geometries and general toroidal boundary conditions. The operator content of these models is studied by solving numerically the Bethe-ansatz equations and by exploring conformal invariance. Since the eigenspectra are calculated for large lattices, the corrections to finite-size scaling are also calculated.

It is shown that any function from can be represented as a linear combination of M self-fractional Fourier functions of order M, which are orthogonal to each other. Each of them contains a selection of Hermite - Gauss modes of the generator function. The physical meaning of the synthesis of self-fractional Fourier functions is discussed.

For a system of two spin- particles and for given four observables, two for each and non-commuting, there exists a unique state which admits Hardy's non-locality. Hence, no mixture state admits Hardy's non-locality.

We extend the Nagel - Schreckenberg stochastic cellular automata model for single-lane vehicular traffic to incorporate quenched random deceleration probabilities. We show, by computer simulations, that at low densities this model displays queueing of cars with a power-law probability distribution of gaps between the cars while at high densities the behaviour of the model is similar to the jammed phase of the standard Nagel - Schreckenberg model. The approach to the steady state is characterized by the same critical exponents as for the coarsening process in the simple exclusion processes with random rates, recently investigated independently by Krug and Ferrari, and Evans. The numerical values of the exponents for gap distributions are in agreement with the analytical conjecture of Krug and Ferrari, which implies that the models belong to the same universality class.

All polynomial solutions of the WDVV equations for the case n = 4 are determined. We find all five solutions predicted by Dubrovin, namely those corresponding to Frobenius structures on orbit spaces of finite Coxeter groups. Moreover we find two additional series of polynomial solutions of which one series is of semi-simple type (massive). This result supports Dubrovin's conjecture if modified appropriately.

Phase diagrams as a function of anisotropy D and magnetic field H are obtained for discommensurations and surface states for a model antiferromagnet, equivalent to a mean-field approximation, in which H is parallel to the easy axis. The surface spin-flop phase exists for all D. We show that there is a region where the penetration length of the surface spin-flop phase diverges. Introducing a discommensuration of even length then becomes preferable to reconstructing the surface. The results are used to clarify and correct previous studies in which discommensurations have been confused with genuine surface spin-flop states.

We report the occurrence of critical probabilities associated with the maximum of diversity and the maximum number of fragments (clusters) on a two-dimensional square lattice. Some scaling relations of these two variables are observed in accordance with work on fragmentation processes.

Using the method introduced by Grisaru et al, boundary S-matrices for the physical excitations of the open Hubbard chain with boundary fields are studied. In contrast to the open supersymmetric t-J model, the boundary S-matrix for the charge excitations depend on the boundary fields though the boundary fields do not break the spin-SU(2) symmetry.

The effects of error propagation in the reproduction of diploid organisms are studied within the population genetics framework of the quasispecies model. The dependence of the error threshold on the dominance parameter is fully investigated. In particular, it is shown that dominance can protect the wild-type alleles from the error catastrophe. The analysis is restricted to a diploid analogue of the single-peaked fitness landscape.

The dynamics of the avalanche width in the recently proposed evolution model is described using a random walk picture. In this approach the critical exponents for avalanche distribution and avalanche average time are found to be the same as in the previous mean-field approximation whereas the critical value of the fitness is in perfect agreement to previous numerical estimates. A continuous time random walk picture is studied as a possible way to improve the mean-field treatment.

We begin by revisiting the so-called Caldirola - Kanai Hamiltonian and discuss its inherent ambiguity: does it represent a dissipative harmonic oscillator (HO) subject to a friction force, or does it describe an HO with a time-dependent mass (TDM)? Although classically both descriptions do coexist, in the quantum domain the solution of the Schrödinger equation (or Heisenberg equations of motion) with a TDM does not present inconsistencies, however, the dissipative Hamiltonian shows violation of the Heisenberg uncertainty principle. This violation is avoided by introducing a stochastic force in the equations of motion, which will take care of the fluctuations due to the environment. Once the distinction between the dissipative and amplifying Hamiltonian is made clear, we consider the problem of the quantum TDM HO subject to dissipation, showing that both phenomena may be merged and described by a single Hamiltonian, the amplifying - dissipative Hamiltonian. We obtain the solutions of the Heisenberg equations of motion for the canonical momentum and position; next, we specialize on the weak damping limit and analyse the effects of the amplifying - dissipative process on the mean values of the physical variables.

We consider the behaviour of multi-state neural networks averaged over an extended monitoring period of their dynamics. Pattern reconstruction by clipping the activities is proposed, leading to an improvement in retrieval precision.

We investigate the XY model with ferromagnetic nearest and antiferromagnetic next-to-nearest neighbours couplings (ANNNXY model). We formulate the study of both phases of the model in terms of different Coulomb gas models. The non-frustrated ferromagnetic phase is thus transformed into a usual Coulomb gas with one species, whereas the frustrated phase is translated into a Coulomb gas with three species interacting with each other in an anisotropic lattice. We then generalize the Kosterlitz - Thouless renormalization group equations for both phases by treating the Ising and Kosterlitz - Thouless order parameters in an independent way. This enables us to discuss the nature of the transition (Ising and (or) Kosterlitz - Thouless) in the frustrated phase.

Using a recently established perturbative approach we analyse a single polymer chain or a few chains floating in a good solvent contained in a finite box with periodic boundary conditions. We calculate to one-loop order the partition function and the equation of state relating segment concentration to segment chemical potential , and we discuss in detail the chain length distribution for a `field theoretic' ensemble of chains characterized by fixed . Our results obey finite size scaling and cover the whole crossover from the dilute to the dense limit, where is the critical chemical potential. The different limits evolve smoothly from one another. The theoretical results for the chain length distribution are compared with Monte Carlo simulations of self-avoiding walks on a cubic lattice. We find a good agreement between our results and the simulation data.

The problem of colour symmetries of crystals and quasicrystals is investigated from its combinatorial point of view. For various lattices and modules in two and three dimensions, the number of colourings compatible with point and translation symmetry is given in terms of Dirichlet series generating functions.

We propose a classical constrained Hamiltonian theory for the spin. After the Dirac treatment we show that due to the existence of second-class constraints the Dirac brackets of the proposed theory represent the commutation relations for the spin. We show that the corresponding partition function, obtained via the Fadeev - Senjanovic procedure, coincides with that obtained using coherent states. We also evaluate this partition function for the case of a single spin in a magnetic field.

We take advantage of different generalizations of the tangent manifold to the context of graded manifolds, together with the notion of super section along a morphism of graded manifolds, to obtain intrinsic definitions of the main objects in supermechanics such as, the vertical endomorphism, the canonical and the Cartan's graded forms, the total time derivative operator and the super-Legendre transformation. In this way, we obtain a correspondence between the Lagrangian and the Hamiltonian formulations of supermechanics.

Approximate Hamiltonians for the one-dimensional (1D) Calogero and two-dimensional (2D) anyon models in a harmonic well are constructed. These Hamiltonians are exactly diagonalizable, and their spectra interpolate linearly between the Bose statistics and the Fermi statistics. In particular, in 2D, the thermodynamics is similar to that of a system obeying a generalized exclusion principle and may be viewed as a starting approximation for the thermodynamics of anyons.

Series of extended Epstein type provide examples of non-trivial zeta functions with important physical applications. The regular part of their analytic continuation is seen to be a convergent or an asymptotic series. Their singularity structure is completely determined in terms of the Wodzicki residue in its generalized form, which is proven to yield the residua of all the poles of the zeta function, and not just that of the rightmost pole (obtainable from the Dixmier trace). The calculation is a very down-to-earth application of these powerful functional analytical methods in physics.

We study a one-dimensional anisotropic exclusion model describing particles moving deterministically on a ring with a single defect, across which they move with probability 0 < q < 1. We show that the stationary state of this model can be represented as a matrix-product state.

In this paper we study left and right `regular representations' of , an infinite-dimensional analogue of the classical Lie group with entries indexed by the integers, on a space which we take to be the projective limit of Bargmann - Fock - Segal spaces. We decompose into an orthogonal direct sum where the hat indicates the closure of the algebraic direct sum, and where each is irreducible with respect to the joint left and right action, and the isotypic component of the left or right actions with double signatures of the type . For each , is the subspace generated by the joint action on a highest weight vector with highest weight . We start by defining the group and its Lie algebra , and discuss the `infinite wedge space', F, defined by Kac, and its transpose . We then define a kind of determinant function on , prove several identities and properties of these determinants, and use these to produce intertwining maps from tensor products of F and to , thereby obtaining irreducible subspaces and highest weight vectors. We further adapt these results to representations of on the inductive limit of Bargmann - Fock - Segal spaces .

A reduction procedure for Jacobi manifolds is described in the algebraic setting of Jacobi algebras. As applications, reduction by arbitrary submanifolds, distributions and the reduction of Jacobi manifolds with symmetry are discussed. This generalized reduction procedure extends the well known reduction procedures for symplectic, Poisson, contact and co-symplectic structures.

Two quasi-bi-Hamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover, the most general Pfaffian quasi-bi-Hamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

Two infinite bands of equidistant potential curves cross each other forming patterns which are periodic both in time and in energy. The dependence of the quasi-energy on the quasi-momentum (related to the translations along the energy axis) manifests zone structure analogous to that known in solid state physics. The properties of the zones are analysed.

We first generalize the Fulton - Gouterman transformation to a multi-band and exponential form. While the exponential form admits the consideration of multiparticle systems, which will be done in future papers, here we treat the multi-band form in its application to an archetypical electron - phonon Hamiltonian, which contains the Fröhlich one as a particular limiting case. Several generalizations will be outlined.

A complete analogue of Kolmogorov's perturbation algorithm in classical mechanics is presented for perturbations of self-adjoint operators. The resulting perturbation theory is different from the usual Rayleigh - Schrödinger (or Kato - Rellich) perturbation theory.

Geometric magnetism is a post-adiabatic reaction force exerted by a fast system on a slow system coupled to it. Here it is demonstrated analytically and numerically that a heavy (slow) uncharged billiard boundary, , swerves, on the average, because of geometric magnetism exerted through elastic impacts from a light (fast) charged particle moving inside in a magnetic field, for both regular and chaotic fast motions.

The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing-ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase-space dynamics near the bouncing-ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green's function. Semiclassical contributions to the trace show an -dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing-ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly -dependent. The localized bouncing-ball states found in the billiard derive from this semiclassically stable island. The bouncing-ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing-ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics. The behaviour is generically found at the border of classically stable islands in systems with a mixed phase-space structure.

We use the theory of orthogonal polynomials to write down explicit expressions for the polynomials of the first and second kind associated with a given infinite symmetric tridagonal matrix H. The Green's function is the inverse of the infinite symmetric tridiagonal matrix (H-zI). By calculating the inverse of the finite symmetric tridiagonal matrix we can find the analytical form of the inverse of the finite symmetric tridiagonal matrix, .