Table of contents

The low-lying excitations of the asymmetric XXZ spin chain are derived explicitly in the antiferromagnetic regime through the Bethe ansatz. It is found that a massless and conformal invariant phase with central charge c = 1 is separated from a massive phase by a line on which the low-lying excitations surprisingly scale with the lattice length as . The mass gap vanishes with an exponent as one approaches the massless phase. The connection with the asymmetric six-vertex model and some physical consequences are discussed.

It is shown that a self-fractional Fourier function for an angle where N and M are indivisible integers (N < M), is also a self-fractional Fourier function for angles (j = 1, 2, ...). This allows us to define a self-fractional Fourier function of order M. An eigenvalue of a fractional Fourier operator for angle is equal to , where L is an integer.

A multilinear M-dimensional generalization of Lax pairs is introduced and its explicit form is given for the recently discovered class of time-harmonic, integrable, hypersurface motions in .

The nearest-neighbour XY spin glass on a hypercubic lattice in four dimensions is studied by Monte Carlo simulations. A finite-size scaling analysis of the data leads to a finite-temperature spin glass transition at . The critical exponents are estimated to be and . The results imply that the lower critical dimensionality for the XY spin glass is less than four.

The usual fixed-size box-counting algorithms are inefficient for computing generalized fractal dimensions in the range of q < 0. In this letter we describe a new numerical algorithm specifically devised to estimate generalized dimensions for large negative q, providing evidence of its better performance. We compute the complete spectrum of the Hénon attractor, and interpret our results in terms of a `phase transition' between different multiplicative laws.

All coboundary Lie bialgebras and their corresponding Poisson - Lie structures are constructed for the oscillator algebra generated by . Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal R-matrices.

The convenience of a new thermodynamic frame for the description of anomalous diffusion is explored. Our research, which makes use of a recent new definition for entropy arising from multifractal analysis, shows that both dynamical and thermodynamical effects may contribute to non-classical diffusion.

We study the critical properties of the weakly disordered two-dimensional Ising and Baxter models in terms of the renormalization group (RG) theory generalized to take into account replica symmetry breaking (RSB) effects. Recently it has been shown that the traditional replica-symmetric RG flows in the dimension are unstable with respect to the RSB potentials and a new spin-glass type critical phenomena has been discovered (Dotsenko Vik S, Harris B, Sherrington D and Stinchbombe R 1995 J. Phys. A: Math. Gen. 28 3093; Dotsenko Vik S and Feldman D E 1995 J. Phys. A: Math. Gen. 28 5183). In contrast, here it is demonstrated that in the considered two-dimensional systems the renormalization-group flows are stable with respect to the RSB modes. It is shown that the solution of the renormalization group equations with arbitrary starting RSB coupling matrix exhibits asymptotic approach to the traditional replica-symmetric ones. Thus, to leading order the non-perturbative RSB degrees of freedom do not affect the critical phenomena in the two-dimensional weakly disordered Ising and Baxter models studied earlier.

We have carried out numerical simulations of the three-dimensional Ising spin glass model with first neighbour Gaussian couplings using three replicas for each sample of couplings. We have paid special attention to the measure of two types of Binder cumulant that can be constructed from the three possible overlaps between the replicas. We obtain new information about the possible phase transition and perform an initial analysis of the ultrametricity issue.

We regard the bond directed percolation on a square lattice as a discrete-time Markov process of a one-dimensional interacting particle system. The coefficients in series expansion of the probability of having m particles at time n - 1 are studied. We derive the difference equations for the first and the second series of coefficients and prove that these coefficients are expressed using the ballot numbers, whose special cases are known as the Catalan numbers. As a corollary of our results, we prove a part of the conjecture by Baxter and Guttmann that the correction terms are expressed as rational functions of the Catalan numbers. We also give approximations for the percolation probability using the present formula.

The equipartition-of-energy principle has been tested for a simple model system of weightless granular media comprising inelastic spheres being shaken at a steady state in a uniaxially vibrating box. Granular dynamics simulations are reported and compared with a theoretical model based on the axioms of classical kinetic theory. For inelastic particles, both frictionless and with rotational degrees of freedom, the shaker reaches a steady state. Both the theoretical model and the simulation results show that equipartition always prevails at sufficiently small vibration amplitudes, irrespective of the frequency, over the whole density range. The independence of the reduced energy on frequency is an exact scaling result. When the shaker amplitude increases to the same order as a density-dependent characteristic path length, there are significant deviations from equipartition behaviour. In all the simulations, a Maxwell - Boltzmann distribution of velocities closely prevails within each degree of freedom, even when equipartition is not obtained between the longitudinal and transverse directions. The steady-state energetics of the saw-tooth shaker are evaluated analytically by means of an energy balance using exact kinetic theory and the known hard-sphere fluid collision frequency via the equation of state. This gives rise to scaling laws which enable granular-`thermodynamic' and transport properties of the fluidized granular material to be determined from the corresponding-state thermodynamic and transport properties of the classical hard-sphere fluid in thermal equilibrium.

In searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards (with Dirichlet boundary conditions) with respect to the boundary data (the normal derivative), we have performed instead various numerical tests for the Robnik billiard (a quadratic conformal map of the unit disk) for 600 shape parameter values, where we have looked at the sensitivity of the energy levels with respect to the shape parameter. We show the energy level flow diagrams for three stretches of fifty consecutive (odd) eigenstates, each with index 1000 to 2000. In particular, we have calculated the (unfolded and normalized) level curvature distribution and found that it continuously changes from a delta distribution for the integrable case (circle) to a broad distribution in the classically ergodic regime. For some shape parameters the agreement with the GOE von Oppen formula is very good, whereas we have also cases where the deviation from GOE is significant and of physical origin. In the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the Berry - Robnik (1984) surmise. Here the agreement with theory is not good, partially due to the localization phenomena which are expected to disappear in the semiclassical limit. We stress that even for classically ergodic systems there is no global universality for the curvature distribution, not even in the semiclassical limit.

Using the Kubo - Martin - Schwinger (KMS) condition and exchange algebras we discuss how to compute correlation functions for the conformal exchange field theories. In particular, we derive the Verlinde algebra from modular invariance of the conformal field theories. We also show how the Verlinde matrices can be related to the KMS matrices.

k-point correlations of complex zeros for Gaussian ensembles of random polynomials of order N with real coefficients (GRPRC) are calculated exactly, following an approach of Hannay for the case of Gaussian random polynomials with complex coefficients (GRPCC). It is shown that in the thermodynamic limit of Gaussian random holomorphic functions all the statistics converge to their GRPCC counterparts as one moves off the real axis, while close to the real axis the two cases are essentially different. Special emphasis is given to one- and two-point correlation functions in various regimes.

We discuss two questions related to the finite-time behaviour of a sequence that generates a self-similar system. The first one concerns the way to approach the self-similarity exponent associated with the system, from the analysis of the sequences observed during a finite time. After this approximation procedure we established a criterion to decide whether the analysed sequence can be considered to be a finite-time subsequence of a generating sequence for a self-similar system. In the second part of this work, assuming the asymptotic self-similarity for the system generated by a sequence, we study the conditions ensuring the appearance of anomalous scaling on the structure functions, due to finite-time effects. We use this method to show that, in the case of a real turbulent sequence, anomalous scaling is not incompatible with asymptotic self-similarity.

We study the predictability of a theoretical model for earthquakes, using a pattern recognition algorithm similar to the CN and M8 algorithms known in seismology. The model, which is a stochastic spring-block model with both global correlation and local interaction, becomes more predictable as the strength of the global correlation or the local interaction is increased.

The two-dimensional surface of a sphere can be parametrized by coordinates representing two charged pions acting as Goldstone bosons of a broken symmetry. We construct in full concrete detail, and in a general class of coordinate systems, all the relevant structure forming a framework for this low-energy effective theory.

We introduce new classes of Schrödinger equations with time-dependent potentials which are transformable to the free particle equation through non-local transformations. These non-local transformations arise when considering the potential systems of the Schrödinger equation. Explicit formulae are given for the potentials and the corresponding solutions related to the solutions of the free particle equation.

The vacuum expectation value of the energy-momentum tensor of a real scalar field in the presence of an arbitrary scalar background field is considered. The problem of renormalization is treated in detail. In the special case of a background field depending on one coordinate only, we give an explicit integral representation for the renormalized vacuum energy. Three explicit examples illustrate the use of this representation as well as some properties of the vacuum energy density. We find that a twice continuously differentiable background potential leads to a continuous energy density.

Representation of a transient electromagnetic field generated by a pulsed current moving with a uniform velocity in terms of modes in the spherical coordinates is considered. Peculiarities of the spacetime structure of these modes in relation to the observation location and time as well as to the source current pulse duration, velocity of the pulse front, and the radiator's length are investigated. Possibilities of an adequate description of the fields due to the above source are discussed.

The phase space of the classical non-periodic discrete nonlinear Schrödinger equation contains both a chaotic region and a continuous family of periodic orbits. In a bid to determine the extent to which this mixed behaviour manifests itself in the transition to quantum mechanics, we study the nearest-neighbour level-spacing distribution of the corresponding quantum energy levels. This is compared with the optimal Berry - Robnik and Brody distributions, which continuously interpolate between regular and chaotic level spacing statistics.

We consider a quantum particle constrained to move within a tube of central radius d embedded in 3-space, subject to Dirichlet boundary conditions. Taking the central axis of the tube as a reference curve and setting up a locally cylindrical polar coordinate system around this curve, we derive an exact expression for the effective potential introduced by the imposition of curvature and torsion. We then employ a minimax method to ascertain a sufficient requirement for the curvature and torsion to guarantee the existence of a bound state.

The functional equation

is associated with the appearance of spatially chaotic structures in amorphous (glassy) materials. Continuous compactly supported solutions of the above equation are of special interest. We shall show that there are no such solutions for , whereas such a solution exists for almost all . The words `for almost all q' in the previous sentence cannot be omitted. There are exceptional values of q in the interval for which there are no integrable solutions. For example, , which is the reciprocal of the `golden ratio' is such an exceptional value. More generally, if is any Pisot - Vijayaraghavan number, or any Salem number, then is an exceptional value.

In this paper we construct a new type of Haar wavelets, called -wavelets of Haar, using the arithmetics of the solutions and of the algebraic equation .

In this paper we perform the semiclassical analysis of a pair of resonances in the case of a quasi-symmetrical unstable double well. We consider two kinds of asymmetric perturbations: one supported in the infinite external well, the other one of the Stark kind. We prove that the first perturbation is able to localize each state inside one of the internal wells so that we have linear Stark effect and vanishing of the splitting at the crossing point of the two resonances. This phenomenon is critical in the ratio between the internal and external barrier lengths, and the critical value of the ratio is close to two. Possible applications to the molecular structure and to the vanishing of the inversion frequency are briefly discussed.

The Novikov - Veselov (NV) equation is considered. Using Bäcklund transformations and a nonlinear superposition formula for the NV equation, some new exact solutions to this equation are found, among which are the so-called one- and two-lump solutions.

On the basis of the `RTT=TTR' formalism, we introduce the quantum double of the Yangian for with a central extension. The Gauss decomposition of the T-matrices gives us the so-called Drinfel'd generators. Using these generators, we present some examples of both finite- and infinite-dimensional representations that are quite natural deformations of their corresponding affine counterpart.

We examine the effect of damping on the nonlinear modulation of an electromagnetic plane wave in a ferrite. Depending on the value of the damping constant, the time evolution of the amplitude of the wave is either a simple exponential decay, or is described either by a nonlinear Schrödinger (NLS) equation, or by a perturbed NLS equation. We give a new exact solution to this latter equation, and a way to compute approximate solutions.

A new multi-fermion realization of is built, whose generators have been written as functions of operators of su(2) algebra. With the help of the realization, the influence of the q-deformation of su(2) algebra on the excitation spectrum and the ground state properties of the Lipkin - Meshkov - Glick model can easily be understood. It is also shown that the variation approach on the basis of the usual su(2) coherent state still works well for the q-deformed model.

This paper concerns the potential field for Yukawa-type forces in a cylindrical geometry. A set of formulae is derived for a cylindrical body in order to reduce the integration over the body from 3D (or 6D for a pair of interacting bodies) to either a 1D integration, or an analytical formula. Using these formulae we consider analytically the potential field of two Yukawa force terms, and reanalyse Long's and Spero et al's experiments for the inverse-square law test. The results show that the - graph produces similar results to the case where a single Yukawa term is considered. The theory can also be applied to other problems in short-range force experiments that involve cylindrically shaped bodies as the test masses.

Applying the approach of modelling physical quantities in terms of measurable maps to the phase of a light field, we distinguish a multiple-valued phase from its representation on the unit circle and the single-valued phases in the case of one random phase variable. We study two multiple-valued random phase variables, whose multiple value forms a displaced lattice, and we compare them with their representation on the torus. We relate the phase expectations and variances to single-valued phases, whereas we associate the preferred phase and the phase dispersion with the representation on the circle. We propose alternatives for the covariance of two single-valued random phase variables. Starting from the concept of a characteristic sequence, we introduce the phase characteristics of higher order. As an application to quantum optics, we reformulate the study of phase properties of the states of one- and two-mode light fields.

All possible configurations of a system of N identical particles are classified for the case when the configuration space Q of a single particle reduces to a finite set. The classification is achieved by means of a combinatorial analogue of the duality of Weyl. Two groups: the Pauli group permuting identical particles, and the symmetric group on Q, act on the Nth Cartesian power of Q. These actions mutually commute, and their orbit structure is classified by appropriate epikernels. Moreover, each stratum of each group is invariant under the dual action, which yields a coarsening of permutation representations, acting on appropriate sets of orbits. It is shown that such a coarsening recovers the geometric symmetry for the case when particles are indistinguishable.

We study the conservation laws of both the classical and the quantum mechanical continuum type systems. For the classical case, we introduce new integrals of motion along the recent ideas of Shastry and Sutherland (SS), supplementing the usual integrals of motion constructed much earlier by Moser. We show by explicit construction that one set of integrals can be related algebraically to the other. The difference of these two sets of integrals then gives rise to yet another complete set of integrals of motion. For the quantum case, we first need to resum the integrals proposed by Calogero, Marchioro and Ragnisco. We give a diagrammatic construction scheme for these new integrals, which are the quantum analogues of the classical traces. Again we show that there is a relationship between these new integrals and the quantum integrals of SS by explicit construction. Finally, we go to the asymptotic or low-density limit and derive recursion relations of the two sets of asymptotic integrals.

The semiclassical contribution of a periodic orbit to the quantum density of states diverges when the orbit bifurcates. In this case one has to apply approximations which are uniformly valid both in and a parameter which describes the distance to the bifurcation. The form of the approximation depends on the repetition number m of the orbit that bifurcates. In a two-dimensional system, the approximations are different for m = 1 up to m = 5, and for they have the same form as for m = 5. In this article, we consider the case which occurs first when an integrable system is perturbed. A uniform approximation for the contribution to the spectral density is derived, which in the limit of large reduces to a sum of semiclassical contributions of isolated periodic orbits.

Using Grassmann variables and an analogy with two-dimensional electrostatics, we obtain the average eigenvalue distribution of ensembles of asymmetrically diluted Hopfield matrices in the limit . We found that in the limit of strong dilution the distribution is uniform in a circle in the complex plane.

We consider two independent q-Gaussian random variables and and a function chosen in such a way that and have the same distribution. For we find that at least the fourth moments of and are different. We conclude that no q-deformed convolution product, parallelling the known cases q = 0 and q = 1, can exist.

By certain variations of the phase of quasiperiodic tilings, vertices can be permuted and even transported to infinity in a diffusive way. In a preceding article, we have studied such a motion for the icosahedral Ammann - Kramer - Penrose tiling. The `quantum of diffusion' occurs in a triacontahedral cage, where atoms in sets of three and seven can be interchanged arbitrarily. Here we show that these kinetic processes can be represented by the `stellated polyhedra' and , respectively.

The values for 18-mers were incorrect. Table 1 should read:

 

Table 1. Summary of enumeration data, where N is the number of sites and M is the number of walks unrelated by symmetry. CPU time given for 128 node CM-5.

The correct version of figure 1 is printed here.

 

Figure 1. Zero entropy line and the AT line of the RS and 1RSB solution for the linear transfer function.

The first paragraph of the introduction was incorrect as published. The correct version is as follows:

`The recent development of non-commutative geometry and, in particular, the theory of quantum groups, raises the question of what happens with known models of physical systems when we pass from usual configurations to non-commutative ones. For classical mechanical systems, this means that we allow the configuration space to be a Poisson manifold (positions need not commute). The phase space corresponding to a usual configuration manifold (Poisson structure equal to zero) is its cotangent bundle. For a general Poisson manifold, the role of its phase space is played by the corresponding symplectic groupoid (if such exists, it is unique - if one restricts to oneself connected and simply connected fibres).'