Table of contents

A q-deformed oscillator system with quantum group SL_q(l) symmetry is constructed in terms of the q-deformed boson operators. The energy level degeneracy associated with the irreducible representation of SL_q(l) is analysed, especially for the case that q is a root of unity.

Within the framework of the R-matrix formalism of the quantum group the author deforms the algebra of fermionic oscillators and studies associated canonical transformations which mix creation and annihilation operators of the deformed oscillators.

Motion of an ideal fluid is represented as geodesics on the group of all volume-preserving diffeomorphisms. An explicit form of the geodesic equation is presented for the fluid flow on a three-torus Riemannian connection, commutator and curvature tensor are given explicitly and applied to a couple of simple flows with the Beltrami property. It is found that the curvature is non-positive for the section of two ABC flows with different values of the constants (A, B and C). The study is an extension of the Arnold's results (1989) in the two-dimensional case to three-dimensional fluid motions.

By means of the transformation connecting the sine-Gordon equation and the two-dimensional Poisson-Boltzmann equation a correspondence between the self-consistent two-dimensional Poisson-Boltzmann structures and the solutions of the sine-Gordon equation representing standing waves has been established. In this way two new solutions of the sine-Gordon equation were obtained and studied and their application for description of waves into ferromagnetics was examined.

A study of electronic scattering from a nonlinear Schrodinger equation in a one-dimensional periodic chain in the presence of an applied electric field is presented. The scattering properties are measured as a function of the strength of the nonlinear cubic term, alpha . The stability, shapes and lifetimes of the Stark ladder resonances, present in the linear system, are studied as a function of alpha . It is found that these quantities are significantly modified by the nonlinearities, much more so for alpha )0 than for alpha (0.

A classification of triangular constant solutions of the Yang-Baxter equations is performed. The formulae for the Baxterization of braid group representation are used for finding trigonometric solutions of the Yang-Baxter equations from the constant ones. Some of the obtained solutions correspond to known solutions but several new ones are obtained as well. The calculations confirm that the Baxterization formulae for more than two eigenvalues of the R-matrix are not universal.

The authors study the external surface of site-percolation clusters in three dimensions by Monte Carlo simulations on a simple cubic lattice at p=p_c=0.3117. Their results show that for all cluster sizes, s, more than 99.8% of the occupied sites are on the external surface. The number of unoccupied boundary sites t is found to obey t/s=(1-p_c)/p_c+As^{- psi} with psi =0.54=1- sigma , in agreement with the prediction of a scaling argument. The same argument also predicts psi =0.60 in two dimensions and it is confirmed. The relevance of these results to dynamical percolation growth models is discussed.

Potts clusters are connected sets of nearest neighbour sites for which the Potts variable is in the same state. At criticality there exists a fractal cluster. Using arguments from the renormalization group, conformal invariance and numerical simulations, the author determines the bulk, surface, hull and red bond dimension of these clusters as a function of the number of Potts states.

The author proposes and analyses the thermodynamic Bethe ansatz equations for the first excited state in the minimal model M₅ perturbed by the subleading thermal field. Such states and the ground states have a different singular point in their scaling free energy.

The author studies the scaling behaviour of the magnetization profile m(z) of an Ising-like magnet in a parallel-plate geometry with opposite (h₁=-h₂) surface fields. In the limit of large plate separation L (and zero bulk field) m(z) is a scaled function of z/L for temperatures T_w<or=T(T_c, where T_w is the critical wetting temperature of each semi-infinite surface. The exact form of m(z) depends on whether T=T_w, T_c)T)T_w or T=T_c. The influence of the far wall on the magnetization near one surface is long-ranged and is determined (for T(T_c) by wetting critical exponents. The author shows that the results of the capillary-wave model for a local one-point function (probability distribution, energy density) may be derived by conformally mapping the corresponding quantity defined in the semi-infinite geometry at the appropriate wetting transition. The author discusses the nature of local scale invariance for wetting transitions and speculates as to why conformal invariance has application to local one-point functions.

Interfacial fluctuations in complete wetting films give rise to important corrections to the macroscopic Kelvin equation for the location of the shifted first-order (condensation) transition that occurs when a fluid (or an Ising magnet) is confined between two parallel adsorbing walls. The wall separation L that enters the usual Kelvin equation must be replaced by L- phi l( mu *), where l is the thickness of the wetting film at the chemical potential mu * at which capillary condensation occurs and phi is an amplitude. For pure systems with short-ranged forces thermal wandering is insufficient to renormalize the simple geometric result phi =2 for bulk dimensions d>or=3. However, in d=2, where fluctuations are much stronger, the authors predict phi =3 throughout the weak fluctuation regime of complete wetting which includes long-ranged dispersion forces as well as short-ranged forces. Their prediction is supported by an explicit analysis of a d=2 interfacial Hamiltonian. This shows that pseudo-phase coexistence, characterized by an exponentially large transverse correlation length, occurs for an exponentially narrow range of mu * determined by the corrected Kelvin equation with phi =3.

The authors study the spiralling self-avoiding lattice walk in a random environment. Upon scaling the temperature appropriately with system size, a phase transition appears. In the low temperature phase the walk segments occupy a few low-energy positions, in the high-temperature phase they are effectively free. An analogy with the random energy model is pointed out. The average size of an N-step walk is shown to be asymptotically proportional to N^1/2 log N (as was known for the homogeneous lattice), with a coefficient that increases as the temperature is lowered. The spatial distribution of the walk segments is qualitatively different above and below the critical temperature. The model also allows for a spin glass interpretation, and as such helps to clarify the connection between the concepts of frustration and the chaoticity of the pair correlation both above and below the critical point.

A new nonlocal algorithm for the simulation of trees on the lattice Z^d is proposed. The authors study the implementation and the properties of the algorithm, and show that it is decisively better than an algorithm which performs only local moves. They use the new algorithm to investigate the properties of lattice trees in two, three, four, eight and nine dimensions.

The critical properties of the perimeters (or 'hulls') of antipercolation clusters are studied in two dimensions by Monte Carlo simulations on the triangular lattice. Two different types of hulls are constructed with the help of two kinetic walk algorithms. For the standard hull, the authors very accurately determine the size distribution exponent tau '=2.14250.0003, as well as the fractal dimension d_f^S=1.7500.001. The corresponding exponents for regular percolation hulls (15/7 and 7/4 respectively) are within the error bars of their results for antipercolation. For the reduced hull, they obtain the estimate d_f^R=1.3340.004 for the fractal dimension, a result which is again close to that found for regular percolation.

The structure of the energy spectrum of the exactly soluble Maryland model is examined in detail for both incommensurate and commensurate modulations. The study is motivated by a formal similarity with Harper's equation describing crystal electrons in a uniform magnetic field. An exact expression is derived for the characteristic polynomial which determines the energy eigenvalues, allowing explicit verification of the transition from band structure, for commensurate modulations, to a point spectrum in the commensurate limit. An application is the calculation of the total width of the bands. A semiclassical argument is used to assess the relevance of the results to models of physical interest.

In a microscopic crystallographic model with incommensurate phases, the magnetoelastic DIFFOUR model, phase transitions at the paraphase boundary are connected to bifurcations of a symplectic mapping. Special attention is paid to the ferro-antiferro transition at the paraphase boundary and its implication on the incommensurate parts of the phase diagram. It is shown that the ferro-antiferro transition at the paraphase boundary is related to the crossing of a degenerate bifurcation point in the mapping. As such, the transition has been called a bifurcation transition.

The authors study the critical capacity ( alpha _c) of multilayered networks with binary couplings. They show that, for any network presenting a tree-like architecture after the first hidden layer, no fixed internal representation is required. Using Gardner's calculations, they apply statistical mechanics to the simplest network with two layers of adaptive weights. Following the same approach as for the binary perceptron they find from the zero-entropy point a critical capacity alpha _c=0.92. They discuss the validity of this result in view of exhaustive search simulations on small networks.

Two different origins of statistical errors in multifractal analysis by the box algorithm are investigated. The authors propose a modified box algorithm reducing the statistical errors and allowing a more accurate estimation of the region where power law scaling is present.

A new class of Lie-algebraic structures is determined within examination of bosonic oscillator systems with internal SU(n) symmetries. They are generated by the SU(n) vector invariants made up of bosonic operators and act complementarily to the SU(n) group on the Fock spaces. A full spectral analysis of the Fock spaces is given with respect to both SU(n) algebras and their complementary ones. Some physical applications of the results to composite models of many-body systems are also pointed out.

For pt.I see ibid., vol.24, p.3265, (1991). Applying the method of the q-deformed boson realization for the quantum group to the case where q is a root of unity the authors establish a set of standard basis for the representation space of the quantum universal enveloping algebra U_q(C_l) associated with a typical subalgebra chain of U_q(C_l). On this basis they systematically obtain irreducible and indecomposable representations of U_q(C_l) and its subalgebras. They discuss U_q(C₂) in detail.

The path integral over the coherent states of the U(1/1) superalgebra is constructed. It is applied to the Jaynes-Cummings model whose dynamical group is U(1/1).

The authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2+1 dimensions can be constrained to integrable hierarchies in 1+1 dimensions such that submanifolds solutions of the given systems in 2+1 can be obtained by solving the resulting integrable systems in 1+1 dimensions. The constraints of the KP hierarchy to the AKNS and Burgers hierarchies respectively are shown in detail and the results of these for the modified KP and 2+1 dimensional analogue of the Caudrey-Dodd-Gibbon-Kotera-Sawata equations to several integrable systems in 1+1 are given.

A stochastic reaction-diffusion equation, which describes the time evolution of a front of concentration in a chemical system characterized by a single active component and influenced by the presence of external noise, is solved within the small noise approximation. The spatial correlations of concentration are studied. The relationship between the spectrum of the evolution operator and the correlation function is discussed and two examples (the trigger wave and the wave between a stable and an unstable state) are discussed.

Various analytic results which combine fermionic Brownian motion with stochastic integration are described, and it is shown that a wide class of stochastic differential equations in superspace have solutions. Such solutions are then used to derive a Feynman-Kac formula for a supersymmetric system in terms of the supercharge whose square is the Hamiltonian of the system. This is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable.

A Darboux transformation converting the Jost solution relating to the (n-1)-soliton solution of the KdV equation to that to the n-soliton solution is shown to be written in the form of a pole expansion and is then found explicitly for arbitrary n. Multisoliton solutions of the KdV equation are thus generated in practice by algebraic recursive procedures.

It is shown that any invertible matrix R that solves the Yang-Baxter equations generates a set of quantum hyperplanes where differential calculi can be defined. The number of such quantum hyperplanes is given by the number of different eigenvalues of the matrix R. Several examples of two-dimensional quantum hyperplanes and differential calculi are presented. The relations of quantum hyperplanes and differential calculi are covariant WRT the quantum groups defined by the matrix R. In the generic cases the exterior differential satisfies the condition d²=0.

The author obtains exact large-order perturbation corrections to the energy shifts of the LoSurdo-Stark effect in hydrogenic atoms for both non-degenerate and degenerate states. The method consists of applying perturbation theory to a recurrence relation among properly selected moments of the eigenfunction which do not explicitly appear in the calculation. The resulting recurrence relation for the perturbation corrections is suitable for computer algebra calculation of perturbation corrections to the energy. The method is most useful to treat both separable and non-separable problems in any convenient coordinate system. The author uses spherical polar coordinates and calculates the shifts of the first two hydrogenic energy levels as illustrative examples.

The energy levels of the Schrodinger equation for the potentials V(x)= Sigma _2N=2 lambda _2Nx^2N, V(x)=-Z²x²+x^2N and V(x)=x²+ lambda x^2N/(1+gx²), with 2N=4, 6, 8,. . ., 18, 20, have been calculated by using a finite difference method for various values of lambda and the quantum number n. The obtained results are compared with previously available results.