Table of contents

A method for analysis of a three-dimensional radiowave propagation problem in a thin spherical inhomogeneous waveguide is presented. This method has a wider range of applications than the technique of telegraphist's equations. The solution of the three-dimensional boundary value problem is approximated by means of the solution of a two-dimensional Schrodinger equation with a periodic potential.

The q-deformed fermionic oscillator is identified with the usual fermionic oscillator by virtue of a simple transformation. So the realization of quantum algebra SU_q(2) can also be constructed from the usual fermionic oscillator.

General forms of 4*4 solutions of the Yang-Baxter equation, without the additivity of spectral parameters, are obtained. By making a parameterization of the colours in terms of the spectral parameters the authors shed new light on the coloured solution of the Yang-Baxter equation.

The authors investigate the energy-level statistics in dependence on the boson number and the underlying classical motion for a system of collective states of zero angular momentum in gamma -soft nuclei described in the framework of the O(6) dynamical symmetry of the interacting boson model. This presents a relatively complex test case for relations between classical regular/chaos, dynamical symmetry and energy-level statistics. The classical limit is integrable due to an additional constant of motion P_y, but the corresponding quantal energy-level statistics in the range of realistic parametrization is close to Gaussian orthogonal ensemble statistics. However, the boson number N_B, which plays the role of Planck constant in semiclassical approximation, appears as a control parameter of quantum chaos, and the energy level statistics asymptotically approaches Poisson statistics with increasing boson number, i.e., with decreasing volume of the unit cell of quantum space.

A class of infinite-dimensional representations of the quantum universal enveloping algebra U_q(sl(2)) is considered. Equivalence and reducibility conditions are examined.

The authors study the asymptotic kinetics of the random sequential adsorption (RSA) of a mixture of particles with a continuous distribution of sizes. Their results provide further support for the idea that the power law exponent for the approach to the jamming limit is simply related to the number of degrees of freedom of the adsorbing species. They also predict the behaviour of the mean radial distribution function in the asymptotic regime.

An exact diagonalization of the Hamiltonian for the one-dimensional quantum model consisting of an arbitrary number of isotropic XY chains connected by many-spin interactions is carried out with the help of the generalized nested Bethe ansatz. An exact solution for the ground-state energy of the considered model is obtained.

The authors present a closed-form expression for the probability distribution of the height fluctuations in the Zhang model of anomalous surface roughening. The result-which includes both the steady state behavior and the time evolution to the steady state-is based on an analogy between the (d=1+1)-dimensional 'surface' problem and a d=1 Levy flight. In the limit case of conventional ballistic deposition they obtain a Gaussian distribution for the height fluctuations. Their results are corroborated by detailed numerical simulations.

The authors study the collapse of self-attracting self-avoiding walks on a Manhattan lattice, by means of phenomenological renormalization group techniques. Their results support a recent prediction of nonuniversality for the entropic exponent gamma . The surface exponents are in the same universality class as the theta transition on undirected lattices, thus differing from those of the theta ' point. They use a two-parameter renormalization group to obtain estimates of the crossover exponent.

There is known to be a close relation between the Kolmogorov-Sinai entropy (sum of the positive Lyapunov exponents) of an ergodic dynamical system and the algorithmic complexity of encoding trajectories of the system with respect to some partition. The authors explicitly gives an encoding which demonstrates this relation for the square Sinai billiard. The encoding depends on the fact that the collision criterion for the billiard is an example of rational approximants. The method may be used to achieve very fast simulation times for the system.

The authors study the local electronic properties of a family of generalized Fibonacci lattices associated with the sequences which are given by the inflation rule (A,B) to (ABⁿ,A), where n is an arbitrary positive integer greater than one. A unified real-space renormalization-group approach is presented to calculate the local Green function and the local density of states at any given site.

The authors study the relationship of two 'q-deformed' spin-1 chains-both of them are solvable models-with a generalized supersymmetric t-J fermion model in one dimension. One of the spin-1 chains is an anisotropic VBS model for which they calculate ground state and ground-state properties. The other spin-1 chain corresponds to the Zamolodchikov-Fateev model which is solvable by Bethe ansatz and is equivalent to a certain t-J model. The two spin-1 models intersect for a certain value of the 'deformation' parameter q in a second-order phase transition.

Verma representation theory for classical Lie algebra is extended to study the representation of quantum universal enveloping algebra (quantum algebra) for the nongeneric case that q is a root of unity. On certain subspaces and quotient spaces of the Verma space, finite- and infinite-dimensional irreducible or indecomposable representations of sl_q(3)=U_q(sl(3)) are obtained in explicit matrix forms.

Given a spectrum of positive numbers ( lambda _m) from which a xi -function Z(s)= Sigma _m lambda ^-sm can be constructed, the reorganization of series of the type F(s,t) identical to _m lambda ^-s_mf( lambda _mt) into power series in t is examined in detail using the method of xi -function resummation. For summand functions f( lambda _mt) having power series expansions in lambda _mt with infinite radius of convergence, and which satisfy other conditions of a rather general nature, the author finds that F(s,t) can be reorganized to F(s,t)=_n a_nt^bn+_k c_kt^dk In t+R(s,t) where R(s,t) vanishes exponentially as t to 0. The numbers a_n, b_n, c_n, d_n can all be computed in terms of the xi -function Z(s). R(s,t) is difficult to evaluate, but important general features of this function can be determined. The power series expansion of F(s,t) can be regarded as a generalization of the heat kernel expansion (for which f( lambda _mt)=exp(- lambda _mt) and s=0) to nonzero complex variable s (which is useful) and to many other summand functions f( lambda _mt). Remarkably, the xi -function resummation method can be applied as easily to divergent series F(s,t) as it can to convergent ones. The method is therefore both a rearrangement procedure for convergent series, and a summation prescription for divergent series.

The Racah-Wigner algebra for the quantum group SU_q(2) is developed to derive explicit expressions for the q-analogues of the Van der Waerden, Racah, Wigner and Majumdar forms of the 3-j coefficient given in terms of sets of basic hypergeometric functions. Interrelationships between the members of a given set of ₃ phi ₂ are established using the reversal of series or the q to q^-1 operation. Starting with the Van der Waerden set, using three transformations of ₃ phi ₂s, 12 other sets including the Racah, Wigner and Majumdar sets, have been obtained. In the simpler case of the q-analogue of the 6-j coefficients, two sets of ₄ phi ₃s, related to each other by reversal of series are obtained.

Using a second-order formalism, a complete separation of variables in the Dirac equation for a free particle in nonstatic orthogonal curvilinear coordinates of the form t=f(u,v), x=g(u, v), y, z, is presented. It is shown that the Dirac equation is separable in eight nonequivalent coordinate systems where the Klein-Gordon equation separates. Exact solutions of the Dirac equation in the systems of coordinates obtained are presented.

A stochastic algorithm is used to simulate the immiscible displacement of a wetting fluid by a non-wetting fluid in a porous medium represented by a two-dimensional network of interconnected capillaries. As the interface advances trapping of the displaced fluid occurs thereby creating isolated islands of the displaced fluid. The number of islands of size s is found to scale approximately as s^{- alpha} , where alpha depends on the capillary number and the viscosity ratio. The effects of capillary forces on the island size distribution are also studied.

The formalism of the Wigner distribution function is reviewed. In addition to the Liouville equation, which expresses the time rate of change of this function in terms of its Moyal bracket with the Hamiltonian, and its expression as a projection operator, a third equation is proposed with the aid of an auxiliary variable s, to which a formal solution is constructed in terms of known quantum-mechanical eigenfunctions and eigenvalues. In addition, an ab initio solution to the three equations in terms of an error function is found for the free particle in one dimension. Two views are advanced: the orthodox, that this new equation is merely a consistency requirement, and the speculative, that the measurement process has something to do with the choice of s.

The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.

The authors discuss the XXZ spin chain with certain boundary terms and the representation of quantum group SU_q(2). It is shown that Bethe ansatz states are highest-weight states of SU_q(2). With a generic q they construct the irreducible representations of the quantum group. For q being a root of unity, they show that there are new Bethe ansatz states, which coincide with null states of SU_q(2). By taking certain limits the authors can derive the state mod b), which is necessary for constructing the indecomposable but reducible representations of SU_q(2) and for the completeness of the state space. In this case the Hamiltonian may not be completely diagonalized.

The logarithmic perturbation theory is modified slightly in order to deal with excited states. Instead of considering a real wavefunction describing the physical stationary state, the authors consider a complex wavefunction at the same energy, by mixing in the ghost state. For excited bound states, the former has nodes, while the latter is guaranteed not to have any nodes, and can be represented simply as exp(-G), to which the logarithmic perturbation method can be applied in a straightforward manner. The physical entities (the energy corrections) are independent of the amount of mixing of the ghost state. The connection to the Green function method is also shown. The freedom to mix in the ghost state allows the authors to justify an ad hoc approach whereby the simple version of the logarithmic perturbation theory is applied to excited bound states. The formalism is illustrated with simple examples.

Canonical transformations whose generators are linear in the basic operators P_j and arbitrary in the canonically conjugate operators Q_j are explicitly constructed. It is shown that they correspond to gauge transformations and changes of variables. Some applications are mentioned in the one-dimensional case.

With chiral theories (and gravity) in mind, the authors use block renormalization-group (RG) methods and RG equations to examine the proposal by de Alfaro, Fubini and Furlan (1984) that symmetry-breaking can be induced by the choice of conformal-invariant measures.

It is shown that operators, commuting with the Hamiltonian of the Hartmann potential form the quadratic Hahn algebra QH(3). The structure of this algebra and its finite-dimensional representations are described. An analysis of these representations is applied to obtain all the relevant physical results: energy spectrum, degree of degeneration and overlap functions.

The concept of an exchange algebra has recently been introduced by Rehren and Schroer (1989) in the context of two-dimensional conformal field theories to give an algebraic setting to both the dynamics and the locality requirement. Labelling the conformal families with two indices and assuming an interpolating scheme for one of the fields, it is shown that the braiding matrices for a subset of fields in Zamolodchikov's and Fateev's (1986) parafermionic theories containing all the order parameters are identical to those of the diagonal minimal models. The authors recover the full spectrum of these theories' modulo integers from the phase condition of the exchange algebra even though the subset does not include the parafermionic currents.

Dyson (1972) introduced two types of Brownian-motion ensembles of random matrices for studying approximate symmetries in complex quantum systems. The magnitude of symmetry breaking plays the role of a fictitious time t>or=0. The authors study the eigenvalue correlations in the circular-type ensembles which serve as models for the evolution operators of quantum maps with chaotic classical limits. In two cases involving time-reversal symmetry breaking they evaluate explicitly the eigenangle-density correlation functions of all orders for all t and for all values of the matrix dimensionality N. The general case is described by a hierarchic set of relations among the correlation functions. As a function of t, the transition in the correlations is found to be rapid for large N, discontinuous for N to infinity . As a function of a local parameter Lambda , which measures the mean square symmetry-admixing matrix element in units of the local average spacing, the transition is found to be smooth. The same Lambda -dependent results were found earlier for the Gaussian-type ensembles which serve as models for the Hamiltonian operators of autonomous chaotic systems.

Presents an analysis of the exact solution for the statistical mechanics of the one-dimensional ferromagnetic ANNNI chain under an external magnetic field by the transfer matrix method. Expressions in closed form are derived for the free energy, magnetic moment, disorder line, susceptibility and the temperature and field derivatives of susceptibility. The variation of the wavevector and correlation length with the external field shows interesting features.

The authors study the dimensions (mean-square radius of gyration and mean span) of self-avoiding polygons on the simple cubic lattice with fixed knot type. The approach used is a Monte Carlo algorithm which is a combination of the BFACF algorithm and the pivot algorithm, so that the polygons are studied in the grand canonical ensemble, but the autocorrelation time is not too large. They show that, although the dimensions of polygons are sensitive to knot type, the critical exponent (v) and the leading amplitude are independent of the knot type of the polygon. The knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour.

A one-dimensional model of hard rods with a long-range cosine interaction is solved exactly in some canonical ensemble cases. It exhibits a crystalline phase at low temperatures as a result of a spontaneous symmetry breaking of the order parameter which represents an effective collective mode. When the wavelength of the attractive potential is shorter than and commensurate with the rod length, the rods behave like point particles in a reduced volume; when the wavelength is comparable to the system size, the effective potential is equivalent to a quenched random field.

A Monte Carlo study is presented for the transport of particles interacting with a nearest-neighbour interaction in a two-dimensional percolating system which is connected by a source at the one end and by a sink at the opposite end. Using mobile particles as carriers, permeation of a quantity such as charge (or mass) from source to sink is studied in a density gradient. The RMS displacement of carriers shows a nondiffusive power law behaviour. The permeability coefficient for the charge transport depends non-monotonically on the carriers concentration far above the percolation threshold and becomes constant near the percolation threshold; at a constant carriers concentration, it increases continuously on increasing the site concentration.

Explicit solutions of a linear rate equation lead to an improved understanding of fragmentation with discrete and continuous mass loss. Discrete mass loss provides a general expression for the overall mass loss rate which accounts for mass loss in the shattering regime, where runaway fragmentation for small particles produces a phase of zero-mass particles. An explicit solution for the recession regime, where small particles lose large fractions of their mass to surface recession, shows that the total number of particles increases with time until the typical particles become small enough to lose all of their mass (and disappear) rather than to break. A general series solution is presented for fragmentation with continuous and discrete mass loss. A proof that continuous mass loss precludes dynamic scaling is presented.

An information theoretic measure is introduced to compare the disorder in non-periodic sequences. It is shown that the measure correctly distinguishes quasiperiodic and aperiodic sequences which have been deduced from earlier studies using diffraction patterns, although it is often necessary to use a set of measures, depending on the order of the source used. The particular sequences studied are the Thue-Morse sequence and the generalizations of the golden mean sequence commonly studied in connection with quasicrystals.

The authors examine the interaction of weakly embedded c-animals with an adsorption surface and show that such animals have the same reduced free energy as that for weakly embedded trees. By assuming the existence of theta ₀( omega ), the critical exponent for trees, they show that theta _c( omega ) exists and theta _c( omega )= theta ₀( omega )-c for all omega .