Table of contents

We report here a tangible example of the fractional exclusion statistics in three spatial dimensions. We identify the chemical constraint of O - H bonds in ice, the so-called ice rule, as the origin of this novel statistics. Then the residual entropy of the crystalline ice is recognized as arising from the fractional exclusion with the statistical interaction parameter . Assuming the residual entropy of ice reflects true accessibility to a macroscopic number of states through quantum tunnelling, we propose the possibility that Berry's phase suppresses the tunnelling splitting of ground states and maintains the finite residual entropy.

We have analysed the collective behaviour of an assembly of coupled oscillators interacting through Hebbian synaptic weights designed to store information about the phase of each element. Our study focuses on the dynamic properties of the population showing the existence of a relevant time-scale, useful for the processing of information.

The Weierstrassian , and functions are generalized to . The n = 3 and n = 4 cases have already been used in instanton solutions which may be interpreted as explicit realizations of spacetime foam and the monopole condensate, respectively. The new functions satisfy higher dimensional versions of the periodicity properties and Legendre's relations obeyed by their familiar complex counterparts. For n = 4, the construction reproduces functions found earlier by Fueter using quaternionic methods. Integrating over lattice points along all directions but two, one recovers the original Weierstrassian elliptic functions.

We derive an expression for the level spacing distribution for a quantal system displaying both regular and chaotic classical motion, assuming that the level-repulsion function for the mixed system is obtained by averaging the corresponding functions for the regular and chaotic regimes with weights given by their fractional phase-space volumes. The Liouville measures of the regular region obtained from comparing this expression with the level spacing distributions of a hydrogen atom in a uniform magnetic field agree with the corresponding values obtained in the classical-mechanical analysis of this system.

The dependence of the escape time on the deviation from the critical point is investigated for the noisy on - off intermittency. The power-law scaling behaviour is destroyed by the noise in a wide region above the critical point. In the general case the width of the region decreases logarithmically with the decrease in the noise amplitude. Below the critical point the noise induced escape time is sensitive to the statistical properties of the chaotic driving signal. Universal exponential dependence holds only for the Gaussian approximation.

We investigate by extensive Monte Carlo simulations the discrete Gaussian model for a crystalline surface on a disordered substrate. The average height - height correlation scales as at all temperatures. A transition is observed in the temperature dependence of the coefficient A(T): while for it becomes almost temperature independent below . This behaviour is consistent with the predictions of a replica-symmetry broken solution of the Gaussian variational calculations. We also explore the probability distribution P(q) of a quasi-overlap q. It exhibits a transition at with its maxima shifting away from q = 0. The size-dependence of the ratio of the fourth to second moment yields a first estimate for the correlation-length exponent , and for the dynamic exponent at : .

The kinetics of the random sequential adsorption of line segments has been studied on a disordered substrate occupied with point impurities. The coverage of the surface and the jamming limits are calculated by a Monte Carlo method. The coverage has an asymptotically exponential behaviour at low concentration of the impurities. The jamming limits depend on the concentration of the impurities p. At the jamming limits decrease as p increases. At the jamming limits increase as p increases. The one-dimensional results are in good agreement with Ben-Naim and Krapivsky's analytic results. The coverage and the jamming limits on a two-dimensional disordered lattice are similar to the one-dimensional cases. The jamming limits decrease monotonically as the length of line segments increases. The minimum locations of the jamming limits for both one and two dimensions are on the same values for a given length of the k-mer.

The spin stiffness of the repulsive Hubbard model that occurs in the hydrodynamic theory of antiferromagnetic spin waves is shown to be the same as the thermodynamically defined stiffness involved in twisting the order parameter. New expressions for are derived, which facilitate the interpretation, and connections with superconducting weight and gauge invariance are discussed.

We perform the Painlevé test (i.e. the Kowalevskaya - Gambier test or the perturbative test) for the perfect fluid Bianchi type-IX relativistic cosmological model, in order to predict some probable chaotic regimes. This technique enables us to detect every single-valued particular solution associated with a given relativistic dynamical system. Several physically interesting cases are studied. These include radiation-dominated and matter-dominated universes, the case of a cosmological constant and the case of the so-called stiff-matter. In all cases but the cosmological constant case, the models studied do not pass the Painlevé test, exhibit infinitely many movable logarithms and are therefore probably chaotic. Moreover, we show that some physically interesting single-valued particular solutions present in the vacuum case cannot subsist for a perfect fluid cosmological model. In particular, in the radiation and dust cases, we show that there cannot exist any exact and closed-form axisymmetric solution.

Some of the analytical properties of the four-wave interaction equations derived previously by the present authors are studied. The equations are shown to be Hamiltonian and to have, at least, four conserved densities. Solitary wave solutions of the equations are also considered by means of a modified Hirota method.

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or -type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the interaction and its symmetrized version for a particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig - Penney analogues: the lattices have bands whose widths are asymptotically bounded and do not approach zero, while the lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the coupling constant is small enough, the lattice has no gaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero.

We describe how starting from a type of nearest-neighbour localized wavefunction cluster expansion one may conveniently build up a locally correlated orthonormal `many-body' basis for a linear-chain system. The initial correlated cluster expansion approximant to the ground state plays the role of a `vacuum' while the other members of the basis are single or multiple excitations above this. The scheme is illustrated for the case of two states per site, in application to the linear-chain Heisenberg model.

q-oscillator models are considered in two and higher dimensions and their symmetries are explored. New symmetries are found for both isotropic and anisotropic cases. Applications to the spectra of triatomic molecules and superdeformed nuclei are discussed.

Coherent states for quons are constructed as eigenstates of the annihilation operator and their properties are studied. In particular, it is shown that these states form a complete set with respect to a measure.

The earlier proposed method for exact determination of the energy spectra of Hamiltonians H and the quantities , where W is a finite-dimensional perturbation, is extended to treat a still larger class of Hamiltonians. In particular, some two- and three-dimensional problems became amenable to this approach. In addition, the expressions for the density and local densities of states are deduced. As an example, an impurity problem is considered in a simple cubic lattice with short-range and in a linear chain with long-range interactions.

Quasinormal modes (QNMs) inside a leaky cavity are characterized by eigenfunctions that are purely outgoing at infinity, and by eigenvalues that are discrete and complex. Apart from a few special cases that admit analytic solutions, the determination of the complex eigenfrequency and the associated eigenfunction of QNMs is, in general, more difficult than the corresponding case of stationary states, and the various methods have their respective strengths and weaknesses. In this paper, a suitable complete basis set, respecting the outgoing wave condition, is introduced to represent the quasinormal modes in a given system, and a matrix equation is then obtained. It is shown that truncation of this matrix system leads to an efficient and accurate method for solving the eigenvalues and eigenvectors of QNMs.

An eight-component relativistic wave equation for spin- particles is derived. It is obtained from a spin- wave equation, which contains second-order derivatives in both space and time, by a procedure involving the linearization of the time derivative. It is in Hamiltonian form and is analogous to the two-component spin-0 equation, which features the use of an indefinite inner product in the description of the solution space. It is used to produce the relativistic bound-state energy eigenvalue spectrum and wavefunctions for the hydrogen atom.

The eight-component relativistic wave equation for spin- particles derived in the preceding paper is studied from a mathematical viewpoint. It is shown that it is relativistically covariant, albeit not manifestly so. It has an enlarged solution space when compared to the Dirac equation. A derivation of the equation in an arbitrary gamma matrix representation is presented. Kronecker products are used to display the analogy between the eight-component equation and the two-component spin-0 equation. The linearization procedure used to obtain the eight-component equation from the second-order spin- equation is found to have a natural mathematical validation in the study of indefinite inner product spaces. The (relativistic) quantum mechanical formalism for the eight-component equation is constructed. This features the use of an indefinite inner product and some results are presented to show how the usual quantum mechanical formalism is generalized to account for this. The reason for the decoupling of the equation into two four-component equations in the Weyl representation of the gamma matrices is given. It is shown that only one of the decoupled parts has to be solved for any problem of a single particle coupled to an external electromagnetic field. Comments are made on the solution of the hydrogen atom presented in the preceding paper.

Functions which are covariant or invariant under the transformations of a compact linear group can be expressed advantageously in terms of functions defined in the orbit space of the group, i.e. as functions of a finite set of basic invariant polynomials. The equalities and inequalities defining the orbit spaces of all finite coregular real linear groups (most of which are crystallographic groups) with at most four independent basic invariants are determined. For each group G acting in the Euclidean space , the results are obtained through the computation of a metric matrix , which is defined only in terms of the scalar products between the gradients of a set of basic polynomial invariants of G; the semi-positivity conditions are known to determine all the equalities and inequalities defining the orbit space of G as a semi-algebraic variety in the space spanned by the variables . In a recent paper, the -matrices, for , have been determined in an alternative way, as solutions of a universal differential equation; the present paper yields a partial, but significant, check on the correctness and completeness of these solutions. Our results can be easily exploited, in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid-state theory where use is made of symmetry adapted functions.