Table of contents

The generalization of the Agranovich - Toshich representation of Paulion operators in terms of bosonic ones for the case of truncated oscillators of higher ranks is derived. We use this generalization to introduce a new constraint-free bosonic description of truncated oscillator systems. The corresponding functional integral representations for thermodynamic quantities are given and the application to investigations of long-range order in the system is discussed.

An improved mean-field theory is proposed for traffic-flow models in two dimensions, both in the absence and presence of faulty traffic lights. The present theory includes the effects of blockage of cars due to cars moving in the same direction, while previous theories included only the effects of cars moving in the perpendicular direction. In the absence of faulty lights, the theory gives a much better estimate of the critical car density, above which the traffic is in a jamming phase. In the presence of faulty lights, the theory captures all the essential features in published simulation data. It gives the correct behaviour of the velocity in the dilute limit of car density. The dependence of the critical car density on the fraction of faulty traffic lights is studied.

The continuum limit of the Bethe ansatz solution for an attractive Hubbard chain is considered in which the particle number per site n is kept finite (0<n<1). It is shown that by adjusting the hopping t, the Hubbard U and the lattice constant a in a proper way the excitations connected to unbound electrons will be relativistic with a finite mass. The higher level Bethe ansatz equations for these particles are given. The spectrum and the phase shifts of these particles show close analogy to those of the massive particles of the SU(2) chiral invariant Gross - Neveu model.

The XXZ model and the Hubbard model with boundary fields are discussed. Using the exact solutions of the present models, the finite-size corrections of the ground-state energy and the low-lying excitation energies are calculated. The partition functions are also evaluated in the scaling limit. Through this calculation, the conformal weights of the primary fields in the present model are obtained.

The experimentally tested occurrence of probabilities that do not find a representation within the usual Kolmogorov probability theory is mathematically formalized through the notion of the Bell phenomenon for probability measures and for observables. Reference is made to a physically natural definition of observables that includes both the classical and the quantum usual versions. It is shown that even a classical framework can host the Bell phenomenon, provided fuzzy observables are called into play.

We consider diagram algebras, (generalized Temperley - Lieb algebras) defined for a large class of graphs G, including those of relevance for cubic lattice Potts models, and study their structure for generic Q. We find that these algebras are too large to play the precisely analogous role in three dimensions to that played by the Temperley - Lieb algebras for generic Q in the planar case. We outline measures to extract the quotient algebra that would illuminate the physics of three-dimensional Potts models.

Recently, it was shown that certain `smart' self-avoiding trails have end-to-end distances in three dimensions. Also, the corrections to scaling of and the specific heat are qualitatively very similar to those for polymers exactly at the point. The question was thus posed whether they are in the same universality class as linear polymers at the temperature. We show that this is not the case since these trails show a first-order transition, instead of the second-order transition at the usual point. We argue that this is due to the fact that the `smartness' of these trails implies that the renormalized n-body interactions vanish identically for any finite . We conjecture that the qualitative similarity with recent simulations of polymers indicates that for n-body interactions the renormalized three-body interaction is small in real polymers.

A new summability method was tested to calculate the critical exponent of the localization length for the symplectic case derived from the nonlinear -model. Although we used the same series as Hikami and others, unlike them we were able to resum the series in two dimensions (2D) and obtain the result . Values of in dimensions seem to saturate the Harris inequality up to .

A quasiperiodically forced purely quantum spin- system can exhibit rather irregular behaviour with a continuous power spectrum. We have shown analytically and numerically, that in these regimes both the topological entropy of a corresponding symbolic sequence and the maximal Lyapunov exponent vanish.

An earlier calculation of the solution of the time-independent equation for the diffusion of coagulating particles is modified to take into account the correct dependence of the Brownian diffusion coefficient on particle size. Explicit analytic results are obtained for the spatial dependence of the particle number density N and the volume fraction of particulate matter .

We study a simple model mimicking the two-dimensional growth of a solid interface B through a liquid L in presence of particles A which are pushed by the advancing front. The model considers a short-range repulsive interaction between the particles and the advancing front (the so-called Uhlmann, Chalmers and Jackson mechanism). As particles are pushed by the advancing front, this leads to the formation of aggregates which are hindrances to the growth and which can be trapped leading to the formation of internal patterns. A transition between indefinitely growing clusters and frozen ones takes place for a critical particle fraction which is larger than the critical fraction of the corresponding epidemic model with static particles. At that critical threshold , both percolating clusters and internal patterns are numerically found to be fractal with the same dimension close to the classical percolation exponent 91/48. The correlation length exponent is found to be close to the classical percolation exponent 4/3. The criticality of the internal patterns is unexpected.

Critical properties of the Calogero - Sutherland model of -type (-CS model) are studied. Using the asymptotic Bethe ansatz spectrum of the -CS model, we calculate finite-size corrections in the energy spectrum. Since the -CS model does not possess translational invariance, the finite-size spectrum acquires the contributions coming from `boundaries'. We show that the low-energy critical behaviour of the model is described by c = 1 boundary conformal field theory. Thus the universality class of the model is identified as a chiral Tomonaga - Luttinger liquid.

It is shown that Ramanujan-type continuous measures for a hierarchy of biorthogonal rational functions can be systematically built from simple cases of the Rogers - Szegö (for 0 < q < 1) and Stieltjes - Wigert (for q > 1) polynomials by using the Berg - Ismail procedure of attaching generating functions to measures.

We investigate the dynamics of a particle constrained to move on a curved quantum strip waveguide embedded in three-space, subject to Dirichlet boundary conditions. By establishing a coordinate system which allows a more general formulation of the dynamics, we demonstrate consistency with previous results and provide evidence that the presence of torsion introduces a further effective potential term. This result has implications for nanostructure device engineering.

It is shown that the 1D Hamiltonian, which is a sum of operators which generate a finite nilpotent Lie algebra and depends explicitly on time existing closed form solutions of the time-dependent Schrödinger equation, cannot fulfil in general boundary and normalization conditions on a positive semi-axis. An explanation of the controversy surrounding the solutions of the quantum bouncer model, which appeared recently in the literature, is given.

New transformations of generalized Kampé de Fériet functions of several variables which occur in physical and quantum chemical contexts are given. Unlike many of the known results for these functions, the variables are not fixed.

New classes of exactly solvable potentials are discussed within the path integral formalism. They are constructed from the hypergeometric and confluent Natanzon potentials, respectively. It is found that they allow incorporation of four free parameters, which give rise to fractional power behaviour, long-range and strongly anharmonic terms. We find six different classes of such potentials.

This paper outlines a technique for determining whether or not a given first-order ordinary differential equation (ODE) is invariant under a one-parameter Lie group of conformal symmetries. The method does not require the form of the Lie group to be specified a priori. Instead, the ODE is used to determine the infinitesimal generator of the group. Once it has been ascertained that the ODE has conformal symmetries, the method immediately yields the ODE's general solution.

We construct a universal tangle invariant on a quantum algebra. We show that the invariant maps tangle to commutants of the algebra; every (1,1)-tangle is mapped to a Casimir operator of the algebra; the eigenvalue of the Casimir operator in an irreducible representation of the algebra is a link polynomial for the closure of the tangle. This result is applied to a discussion of the Alexander - Conway polynomial and quantum holonomy in Chern - Simons theory in three dimensions.

Minimum uncertainty angular momentum states for the quantum group are constructed. They involve the eigenvalues of which are q-numbers and the quantum group analogue of the Wigner d-functions for . The result is generalized for all values of and a formula for the quantum Wigner d-function is derived. The case of q = 1 is discussed and compared with the well known results for the Wigner d-functions.

Smoluchowski's coagulation equation, with reaction rate K(x,y), describing the time evolution of a size distribution c(x,t) is studied in the presence of a mass loss term m(x) = mx (m>0). For K(x,y) = 1, c(x,t) is determined explicitly for arbitrary initial distributions. If K(x,y) = xy, we determine c(x,t) explicitly for arbitrary initial distributions and describe the behaviour of c(x,t) for large x, for all times. Here, we show that a phase transition occurs in a finite time provided . An investigation into reveals that a phase transition occurs in a finite time if and only if and . An estimate of the least upper bound for is calculated, and the behaviour of c(x,t) for large x with is presented.

For each of the two simplest Hamiltonian flows from the relativistic Toda hierarchy we introduce two integrable symplectic discretizations. All four discrete-time systems are demonstrated to belong to the same hierarchy and to exemplify the general scheme for symplectic maps on groups equipped with quadratic Poisson brackets. The initial-value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flows are found for all maps.

The mixing structure of a given quantum state (statistical operator) is considered, that is, the pure states and associated weights of which it consists. Its influence on the state-dependent quantum logical implication determined by is also studied. It is found that it has no influence at all except through the null-projector. It is shown that the latter, when it determines the state-dependent implication, in a certain sense, carries with it all the Boolean mathematical structure of projectors in the given Boolean subalgebra of quantum logic , where is the `domain' of definition of the state-dependent implication.

On the penultimate line of page L428, the exponents should read:

and .