Table of contents

We have tested the generality of the stiffness instability mechanism recently proposed by Fisher and Jin (1993) for the critical wetting phase transition in three-dimensional systems with short-ranged forces. We extend the analysis of Fisher and Jin to a class of Aukrust-Hauge type models and find that the stiffness instability is specific to the case where the unrenormalized transition is precisely second order (i.e where the mean-field specific heat critical exponent is precisely zero). A linear functional renormalization-group analysis of this Aukrust-Hauge class of models yields exactly the same dramatic fluctuation-induced non-universal critical exponents that have previously been predicted. We conclude by considering possible implications of this result on future Ising model simulations and on the basis of this propose a numerical test for the validity of existing renormalization-group analyses of continuum effective interfacial Hamiltonians.

We consider the Glauber dynamics of the q-state Potts model in one dimension at zero temperature. Starting with a random initial configuration, we measure the density r_t of spins which have never dipped from the beginning of the simulation until time t. We find that for large t, the density r_t has a power-law decay (r_t approximately t^{- theta} ) where the exponent theta varies with q. Our simulations lead to theta approximately=0.37 for q=2, theta approximately=0.53 for q=3 and theta to 1 as q to infinity .

A composite quantum model on a lattice which describes the system of q-bosons interacting with U_q(su(2)) spin impurities is introduced and solved exactly under periodic boundary conditions. In one limit the model is shown to become a new exactly soluble quantum system on a lattice which can be interpreted as a q-deformed version of the quantum Dicke model. In the limit of infinitesimal spacing the model is further reduced to a multilevel version of the previously introduced continuum-limit Dicke model. For spin 1/2 the previous results for this particular case are confirmed.

A new universal R-matrix for the quantum Heisenberg algebra H(1)q is obtained by imposing the analyticity in the deformation parameter. Despite the non-quasitriangularity of this Hopf algebra the quantum group induced from it coincides with the quasitriangular deformation already known.

(p,q)-integration is defined for the (p,q)-oscillator. This is used to prove a completeness relation for the coherent states of the (p,q)-oscillator. The (p,q)-analogue of the Bargmann-Fock representation is also discussed.

A kinetic version of random Apollonian packing model is introduced. In this model, droplets nucleate spontaneously, grow at a uniform rate and stop growing upon collisions. The fractal dimension, D_f of the pore space is found to be equal to D_f=d(1-exp(2-(2^d+2-2)/(d+2)), a result which is confirmed exactly in 1D and numerically in 2D.

Finding the optimal separation of two clusters of normalized vectors corresponds to training thresholds and weights in a neural network of maximum stability. In order to achieve this, two local iterative algorithms are presented which treat threshold and weights all in one, avoiding the need to calculate any intermediate 'test' quantities. Convergence is proved, and the separation/stability obtained is shown to match theoretical predictions and to be superior to existing algorithms.

A mode in a confined planar region can contain evanescent waves in its plane-wave superposition. It would seem impossible to construct such a mode by continuation of an external scattering superposition. Which must contain only real plane waves. However, evanescent waves can be expressed as the singular limit of an angular superposition of real plane waves. This is surprising because. In the direction perpendicular to that in which it decays. An evanescent wave oscillates faster than the free-space wavenumber; thus the singular superpositions lie in the class of 'superoscillatory' functions, which vary faster than any of their Fourier components. The superposition is the limit of an exact (i.e. non-paraxial and non-singular) Gaussian beam on its evanescent side slopes. Far from the axis, the beam possesses, on each side, a line of phase singularities (nodal points) which organize the global energy current. The three-dimensional generalization provides an explicit elementary construction of a superoscillatory function.

We review the problem of fractional statistics as it applies to two current areas of interest in condensed-matter physics: the fractional quantum Hall effect (FQHE), and high-temperature superconductors (HTSC). In the case of the former, we emphasize Haldane's recent definition (1991) of a fractional exclusion principle, and show a relation between this idea and the standard definition of fractional statistics in terms of a complex exchange phase. We show that a fractional exclusion principle is both appropriate and useful for the quasiparticles in the FQHE. In the case of the HTSC (where Haldane's novel definition has not been pursued), we review the experimental status of the 'anyon superconductivity' model for the HTSC. Here we find much less support for the hypothesis that the excitations are anyons. We also argue that the past neglect of Haldane's fractional exclusion principle makes tile resulting theory inconsistent.

The thermodynamic limit for the square-triangle random-tiling model is considered. The analytic solution of the Bethe-ansatz equations found recently by Widom (1993) is obtained. The analytic expression for entropy density as a function of the fraction of the plane occupied by triangles alpha _t is derived for the case alpha _t>or= 1/2 , i.e. for the phason gradients having 6-fold symmetry, including 12-fold symmetric phase as the limiting case. The exact values for phason elastic constants are also found.

We determine the asymptotic level spacing distribution for the Laguerre ensemble in a single-scaled interval, (0,s), containing no levels, E_beta (0,s), via Dyson's Coulomb-fluid approach. For the alpha =0 unitary Laguerre ensemble, we recover the exact spacing distribution found by both Edelman (1988) and Forrester (1993), while for alpha not=0, the leading terms of E₂(0,s), found by Tracy and Widom (1994), are reproduced without the use of the Bessel kernel and the associated Painleve transcendent. In the same approximation, the next leading term, due to a 'finite-temperature' perturbation ( beta not=2), is found.

A method is developed to treat the constraints in the spin wave theory and a new Hermitian Bose transformation decorated by ail constraints is found which is different from those of Holstein-Primakoff (1940) or the Dyson-Maleev (1957). The transformed Hamiltonian includes both dynamic and kinematic interaction and it has been confined in physical proper space automatically. A model Hamiltonian which has completely same eigenvalues is presented and a scheme of approximation is described.

In this article, based on a recent theorem by Lieb et al. (1968), we shall prove two theorems on the momentum distribution functions of the half-filled Hubbard model on a d-dimensional simple cubic lattice in a mathematically rigorous way. More precisely, we shall first show that the half-filled positive-U and negative-U Hubbard models have the same momentum distribution functions n up arrow (q) and n down arrow (q). Then, we will show that n_sigma (q) are symmetric functions about the value n= 1/2 . Finally, we shall briefly discuss some possible applications of these theorems to the further numerical investigations on the ground state of the Hubbard model at half-filling.

Boundary height distributions of the two-dimensional Abelian sandpile model are studied in the self-organized critical state. All height probabilities are calculated explicitly both at open and closed boundaries. The leading asymptotic term of the corresponding correlation functions is observed to behave as r^-4 when r to infinity . On the basis of conformal field theory predictions the bulk height correlators are shown to have the same critical exponents as boundary ones. All heights seem to be identified with appropriate counterparts of the local energy operator in the zero-component Potts model.

We investigate the diffusion motion of a Brownian particle which is acted upon by both a friction force with memory effect and a noise with long-range correlation effects. The noise is expressed as f(X,t) approximately X^{- sigma} F(t), sigma >0, where X and t are the displacement and time, respectively, and F(t) has the long-time correlation effect <F(0)F(t)> approximately t^{- beta} ,0< beta <1, beta =1,1< beta <2. The generalized Langevin equation, corresponding Fokker-Planck equation and its solution at large time are established. A variety of anomalous diffusion patterns are derived. Due to the long-range correlation effects, the effective diffusion coefficient is dependent on both the displacement and time, and the probability density For finding the Brownian particle at displacement X and time t is non-Gaussian distribution. When this model is applied to diffusion on fractals, O'Shaughnessy and Procaccia's results (1985) can be naturally derived.

In the framework of a generalized statistical mechanics introduced recently by Tsallis (1993), we derive a generalized form of the fluctuation-dissipation theorem, which expresses a relation between extended susceptibilities and equilibrium fluctuations. To achieve this, we consistently propose a generalized functional form For the instantaneous distribution function. The present theorem recovers as particular cases the corresponding generalized relations already obtained for the specific heat in terms of the generalized energy fluctuations and for the susceptibility of a magnetic system under the action of a uniform magnetic field.

An analytical expression of the pair distribution function is derived for the car-parking problem and for the random sequential adsorption of K-mers onto a one-dimensional lattice. Both on a lattice and in the continuum limit, super-exponential decay of the distribution function is observed. A comparison of the spatial correlations with those at equilibrium demonstrates the influence of the irreversibility of the RSA process.

The dynamics of a random sequential adsorption process on a quasi-one-dimensional lattice with three rows is solved exactly. The long-time behaviour of the coverage density is rho (t)=¹/₃-2 exp(-(²/₃+t))/9. It is shown that the number of connected lattice animals increases like n!²3/, indicating that most two-dimensional lattice animals are non-compact.

In this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typical properties with the quasiperiodic Fibonacci chain. For a subclass we explicitly construct the atomic surface. Moreover, scaling properties of the energy spectrum are discussed in relation to the dynamics of trace maps.

We analyse the finite-size scaling of two-dimensional U(1) lattice gauge theories with a theta -term interaction. The finite-size corrections to the specific heat, Binder-Landau and U₄ cumulants agree with the expected asymptotic behaviour for first-order phase transitions. However, we find that the leading correction to the position of the extremal points of these quantities is not universal. On the other hand, the finite-size corrections to the mass gap behave as for second-order phase transitions. In particular, the curves corresponding to different-size approximations do not cross in the vicinity of the transition points. The feature is associated to the existence of a divergent correlation length and holds for a wider class of models.

By parametrizing the t-j model we present a new electron correlation model with one free parameter. In one dimension, this model is of SU_q(1 mod 2) symmetry. The energy spectra are shown to be modulated by the free parameter in the model. The solution and symmetric structures of the Hilbert space, as well as the Bethe ansatz approach, are discussed for special cases.

We consider a one-dimensional Kronig-Penney model with randomly placed dimer impurities. We show that this model has infinitely many resonances (zeroes of the reflection coefficient) giving rise to extended states, instead of the one allowed resonance arising in random tight-binding models with paired correlated on-site energies. We present exact transfer-matrix numerical calculations supporting, both realizationwise and on average, the conclusion that the model has a very large number of extended states, which can be relevant in several physical contexts.

In this paper we investigate the transport properties for rigid spherically symmetric macromolecules, having a segment density distribution falling off as r^{- lambda} . We calculate the rotational and translational diffusion coefficient for a spherically symmetric polymer and the shear viscosity for a dilute suspension of these molecules, starting from a continuum description based on the Debye-Brinkman equation. Instead of numerical methods for solving equations we use perturbative methods, especially methods from boundary-layer analysis. The calculations provide simple analytical formulae for the shear viscosity eta , and the translational and rotational diffusion coefficients D_T and D_R. The results can also be applied to suspensions of other porous objects, such as aggregates of colloidal particles in which D=3- lambda is called the fractal dimension of the aggregate.

The directed compact percolation cluster model of Domany and Kinzel (1984) is considered in the presence of a wall which is parallel to the growth direction and hence restricts the lateral growth of the cluster in one direction. The critical exponents are found to depend on whether the wall is wet or dry. In the former case the model is solved exactly for all the standard percolation functions and the critical behaviour is found to be the same as that for cluster growth with no wall present. With this boundary condition the cluster is completely attached to the wall and the model may also be viewed as one of symmetric compact cluster growth. In the case of a dry wall the cluster may repeatedly leave and return to the wail as it grows and in this case the percolation probability has been derived exactly by Lin (1992) and found to have a critical exponent different from that of the bulk. Lin's result is rederived and an exact formula for the percolation probability is found for a more general model in which the cluster growth is biased either towards or away from the wall. It is found that the unbiased case is special in that any bias away from the wall recovers the bulk critical exponent and a bias towards the wall produces a problem in the same class as the wet-wall model.

Discontinuous irreversible phase transitions (IPTs) from active states to absorbing (poisoned) states in a trimolecular irreversible reaction model, which involves one monomer and two different dimers, are studied by means of Monte Carlo simulations. Evaluated dynamic critical exponents strongly suggest that each first-order IPT has its own universality class.

A linearly separable Boolean function is teamed by a diluted perceptron with optimal stability. A different level of dilution is allowed for teacher and student perceptron. The learning algorithms used were the optimal annealed dilution and Hebbian dilution. The generalization ability, i.e. the probability to recognize a pattern which has not been learned before, is calculated in replica symmetry.

The two-dimensional inhomogeneous zeta-function series (with homogeneous part of the most general Epstein type) Sigma _{m,n in Z}(am²+bmn+cn²+q) ^-s is analytically continued in the variable s using zeta-function techniques. A simple formula is obtained which extends the Chowla-Selberg formula to inhomogeneous Epstein zeta-functions. The new expression is then applied to solve the problem of computing the determinant of the basic differential operator that appears in an attempt at quantizing gravity by using the Wheeler-De Witt equation in (2+1)-dimensional spacetime with the torus topology.

The concept of the Wronskian determinant is generalized and shown to give rise to a projection operator which can be used in basis-function expansion schemes. This result is applied to relativistic single-site scattering theory and it is shown that formulae previously obtained on an ad hoc basis can be understood as special cases of a general Wronskian scattering identity. Finally, the form of the negative-energy scattering matrix is discussed.

We construct the q-deformed analogue of the completely antisymmetric tensors and the corresponding q-determinants det_q T for the quantum groups SO_q(N), Sp_q(n). The construction is based on the existence of the volume form in the algebra of exterior forms on the corresponding quantum spaces. We show that det_q T is central in Fun(SO_q(N)) (respectively Fun(Sp,(n))) is group-like under the Hopf algebra comultiplication, and that its square is 1.

We investigate a two-parameter quantum deformation of the universal enveloping orthosymplectic superalgebra U(osp(2/2)) by extending the Faddeev-Reshetikhin-Takhtajan formalism to the supersymmetric case. It is shown that U_p,q(osp(2/2)) possesses a non-commutative, non-co-commutative Hopf-algebra structure. All the results are expressed in the standard form using quantum Chevalley basis.

We consider a specific differential realization of the su(1,1) algebra and use it to explore such algebraic structures associated with shape-invariant potentials. Our approach combines elements of various methods of solving the Schrodinger equation, such as supersymmetric quantum mechanics (or the factorization method), algebraic techniques and special-function theory. In fact, it amounts to reformulating transformations mapping the Schrodinger equation into the differential equation of orthogonal polynomials in group-theoretical terms. Our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings. The procedure presented here implicitly contains a similar treatment of the compact su(2) algebra as well. Possible generalizations of this approach (involving different realizations of the su(1,1) algebra, other algebraic structures and larger classes of potentials) are also outlined.

A q-deformed version of standard quantum mechanics in the coordinate Schrodinger picture is obtained by replacing the ordinary coordinate derivative by the so-called q-discrete derivative as the representative of the momentum operator. The chosen q-discrete derivative is symmetric with respect to the exchange of q and q^-1. Under the usually adopted assumptions a q-deformed Schrodinger equation is derived for a harmonic oscillator. The complete set of eigenfunctions can be explicitly constructed as special q-functions and the corresponding energy eigenvalues are identical to those obtained by Biedenharn in his pioneering work (1989). This q-deformed oscillator exhibits a rich novel structure including dynamical symmetry and in the limit q to 1 it reveals some hitherto unknown features of the harmonic oscillator eigenfunctions.

Expressions are given for the Casimir operators of the exceptional group F₄ in a concise form similar to that used for the classical groups. The chain B₄ contained in/implied by F₄ contained in/implied by D₁₃ is used to label the generators of F₄ in terms of the adjoint and spinor representations of B₄ and to express the 26-dimensional representation of F₄ in terms of the defining representation of D₁₃. Casimir operators of any degree are obtained and it is shown that a basis consists of the operators of degree 2, 6, 8 and 12.

An asymptotic formula for the zeros, z_n, of the entire function e_q(x) for q<<1 is obtained. As q increases above the first collision point at q₁* approximately=0.14, these zeros collide in pairs and then move off into the complex z plane. They move off as (and remain) a complex conjugate pair. The zeros of the ordinary higher derivatives and of the ordinary indefinite integrals of e_q(x) vary with q in a similar manner. Properties of e_q(z) for z complex and for arbitrary q are deduced. For 0<or=q<1,e_q(z) is an entire function of order 0. By the Hadamard-Weierstrass factorization theorem, infinite product representations are obtained for e_q(z) and for the reciprocal function e_q^-1(z). If q not=1, the zeros satisfy the sum rule Sigma _n=1^infinity (1/z_n)=-1.

The dielectric behaviour of a suspension of conducting spherical particles surrounded by low conducting shells, with fixed charges on the inner side, in respect of diffusive effects has been investigated. The results describe both alpha and beta dispersion and reduce, in the corresponding limiting cases, to those previously presented by Garcia et al. (1985), Pauly and Schwan (1959) and Schwarz (1962). The alpha dispersion is shown to be strongly dependent on charge density, shell thickness and diffusion effects, as well.

We address the problem of finding, by Painleve analysis only, the Backlund transformation of partial-differential equations (PDEs) having two families of movable singularities with opposite principal parts, such as the modified Korteweg-de Vries (MKdV), sine-Gordon or nonlinear Schrodinger equations. This first paper gives an almost algorithmic method which extends the singular-manifold method of Weiss(1986) that is unable to handle these equations. First, with only one singular manifold at a time, we obtain the Darboux transformation. Second, we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective Riccati system with undetermined coefficients; the Darboux transformation then generates a very small number of determining equations, admitting a unique solution. Equivalent to the Lax pair of the Zakharov-Shabat-Ablowitz-Kaup-Newell-Segur scheme by the canonical linearization of the Riccati system. The method is here applied to the MKdV and sine-Gordon equations.

We construct a multi-component Burgers equation by means of a hereditary symmetry. This n-component system has n parameters.

We consider a perturbed Hamiltonian system with n>or=3 degrees of freedom of the form H=H₀+ epsilon H₁ and show that the properties of the average value of the perturbing function H₁ along the periodic orbits of the unperturbed integrable part H₀ supply criteria for non-integrability which restrict the allowed total number of independent integrals of motion.

We consider a class of time-dependent harmonic oscillators, H(t)=p²/2mt^alpha + m omega ²t^bq²/2, whose mass and frequency vary as non-negative powers of time. Classically they describe damping oscillators slowly decaying as negative powers of time. Using the connection between classical and quantum harmonic oscillators we find analytically the Lewis-Riesenfeld invariants, obtain the exact quantum states, and compare these with the Caldirola-Kanai oscillator.

The stability of the Rayleigh-Taylor model for an electroviscoelastic Maxwell fluid are investigated. The method of multiple scales is used in order to obtain the stability conditions. A transcendental dispersion relation is obtained at zero-order. The special case, when the two fluids have the same kinematic viscosity, is considered to relax the complexity of the transcendental dispersion relation. The solvability conditions introduce a first-order differential equation. It is found that the increase in the relaxation time lambda has a destabilizing influence. Also the increase in the kinematic viscosity in the presence of the parameter lambda yields a destabilizing effect. The increase in the kinematic viscosity in the absence of elasticity (pure viscous fluids) has a stabilizing effect.

We consider the problem of the propagation of short-duration light pulses in a two-level medium in the regime of strong interaction, near the resonance, with no restriction on the strength of the coupling between field and medium. We prove that the local variations of the population difference created by the electric field actually induce a nonlinear effect by coupling the fundamental of the field Fourier component to its harmonics. As a result, the multiscale averaging limit of the Maxwell-Bloch system is a new system of coupled nonlinear Schrodinger equations for the slowly varying envelope of the electric field. This system has one integrable limit when the electric field is circularly polarized: it is the scalar nonlinear Schrodinger equation. Moreover, it possesses three constants of the motion and a Hamiltonian formulation. Then we prove that our system is nonintegrable as it does not pass the Painleve test. Finally, a stability analysis shows that it is modulationally unstable and thus it propagates steady pulses of coherent light.

We study soliton solutions of the nonlinear equation governing pulse propagation in semiconductor-doped glass (SDG) fibres which have saturation-type nonlinearity. The phase modulation of the pulse is nonlinear and no sech-type soliton exists in SDG fibres when high-order correction terms are taken into account. As compared with optical solitons in ordinary glass fibres, the property of the solitons discussed here is somewhat different.

We discuss temperature effects on anomalies. We deal with an anomaly in a spin-chain model presented by Streater(1990), where, in a paper by Gajdzinski and Streater(1991), anomaly meltdown was found. We present a slightly different treatment, that to us seems more plausible from a physical point of view, which shows anomaly persistence. We also discuss a similar problem for U(1) Schwinger terms in local current algebras.

This article adopts a Gaussian-type propagator to find the exact wavefunction of a very general time-dependent damped harmonic oscillator with an arbitrary varying mass and with a force quadratic in velocity under the action of an arbitrary time-varying driving force. The results obtained not only generalize all the known results in the literatures but can also be applied to many interesting particular cases.

We have generalized recent results on the integer quantum Hall effect, constructing explicitly a W_{1+ infinity} for the fractional quantum Hall effect such that the negative modes annihilate the Laughlin wavefunctions. This generalization has a nice interpretation in Jain's composite-fermion theory. Furthermore, for these models, we have calculated the wavefunctions of the edge excitations, viewing them as area-preserving deformations of an incompressible quantum droplet and have shown that the W_{1+ infinity} is the underlying symmetry of the edge excitations in the fractional quantum Hall effect. Finally, we have applied this method to more general wavefunctions.

In the present investigation we assess the relevance of both classically forbidden phenomena and higher-order asymptotic contributions for the semiclassical energy quantization of a particle in the anharmonic oscillator potential V(x)=x²/2+ lambda x⁴. We propose an iterative method (Iq) for obtaining higher-order semiclassical corrections, which is similar to the WKB and phase-integral methods in the lowest order, but in higher orders the approximation differs significantly. A 'primitive' Bohr-Sommerfeld energy quantization is compared with a more complete semiclassical quantization, taking into account both complex trajectory contributions and higher-order (quantal) corrections.

The finite-radius Fourier transform of the first-order vacuum-polarization correction to the Coulomb potential of a point charge, required for a Fourier-Bessel evaluation of vacuum-polarization potentials of extended charges, is calculated by an efficient analytical method.