Table of contents

The author constructs quantum R matrices associated with certain non-cyclic irreducible representations of U_q(sl(2,c)) at roots of unity which are parametrized by one continuous parameter. The author finds solutions for the two- and three-dimensional representations via Baxterization. The two-dimensional case corresponds to the free fermion model of Fan and Wu (1970).

Geometrical and electrical properties of crumpled wires are investigated. Power laws connecting the geometrical and electrical properties of these random systems are studied and critical exponents are calculated. In particular the authors have obtained the resistance exponent zeta which seems to be compatible with a previous conjecture.

It is shown that an analytical solution for the solitons in an anisotropic Heisenberg model does not satisfy the essential demands for being a soliton solution.

The three-dimensional XY model in the presence of two- and threefold random anisotropy has been investigated by Monte Carlo simulation and finite-size scaling using system sizes 4³, 8³, 16³, 32³. The results are consistent with the existence of a conventional transition to a spin glass phase at T_c=2.2 and with correlation exponent v=0.67. No evidence for quasiferromagnetism is found.

A model of ultradiffusion on a one-dimensional bifurcating hierarchical structure with a hierarchical set of biasing terms is introduced. By an exact renormalization decimation procedure, a new non-universal time scaling exponent for the autocorrelation function and a new crossover for the dynamical phase transition, from ordinary to anomalous diffusion, are obtained. The authors also find a new transition for the autocorrelation function from power law to exponential decay which depends on the distribution of the biasing terms. Hierarchically biased diffusion on multifurcating hierarchical structures are also discussed briefly.

An exact algorithm of the search for energy barriers against inversion of ground states is presented. For the case when each spin is allowed to flip only once, the barriers in ferromagnetic and spin glassy 2D systems are found to scale linearly with the linear size of the system.

The finite-size scaling functions of the correlation lengths in the 2D Ising model in a magnetic field are calculated numerically. The asymptotic behaviour in the limit of large scaling variables is studied and can be used to estimate the particle masses and coupling constants of the underlying effective field theory. The results are consistent with the conjectured minimal S-matrix obtained from the conformal bootstrap.

The energy of a local minimum obtained by the simulated annealing generally depends on a time tau in which a complex system has been immersed in a heat bath. How the resultant energy E( tau ) scales with a time tau is an interesting question. The diffusion process of a point in a wiggly parabola is analysed to discuss the scaling. The model is exactly solvable and the energy is found to scale as E( tau )= in +c(ln tau )^-1. This scaling is considered rather common to general complex systems. However, the limit in obtained from practical data is not necessarily the ground state energy of a system.

A model of disordered arrays of weakly coupled superconducting grains and a related random-gauge model are discussed. Evidence, from numerical studies of the zero temperature scaling behaviour of the stiffness, for a non-zero temperature spin-glass phase in the random-gauge model and in the strong magnetic field limit of the disordered arrays is presented. The universality class of a frustrated system may depend on whether the interactions have local gauge invariance.

The author has developed a method to detect attractors of any length in large neural networks with up to 1024 neurons within a reasonable period of CPU-time. In networks with symmetric couplings only stable states and, in the case of parallel dynamics, cycles of length 2 exist. The presented simulations suggest that, in sufficiently large systems, this holds also for couplings up to a distinct value of asymmetry. Beyond this value extremely long cycles are detected and the average cycle length depends exponentially on system size.

The author discusses the ground state and some elementary excitations of the B_p non-simple laced integrable models. The zeros of the associated Bethe ansatz equations are characterized by a special distribution in the complex plane, which is different from the usual string hypothesis. The introduction of this new solution makes it possible to calculate exactly various properties in the thermodynamic limit. The finite-size corrections to the eigenspectrum strongly suggest that the B_p Wess-Zumino-Witten-Novikov models are the underlying conformal field theory for these non-simple laced models.

Grading preserving contractions of Lie algebras and superalgebras of any type over the complex number field are defined and studied. Such contractions fall naturally into two classes: the Wigner-Inonu-like continuous contractions and new discrete contractions. A general method is described for any Abelian grading semigroup and any Lie algebra or superalgebra admitting such a grading. All contractions preserving z_2-, z₃-, and z₂*z₂-gradings are found. Examples of these gradings and contractions for the simple Lie algebra A₂, affine Kac-Moody algebra A⁽¹⁾ and the simple superalgebra osp(2,1) are shown.

A general algebraic formalism for studying contextual hidden variable theories is developed. The contextuality is understood as a manifestation of an inadequacy in the algebra of quantum observables for the complete description of the system. It is shown that it is always possible to 'improve' the algebra of quantum observables, explicitly paying attention to contexts, such that this improved algebra becomes a base for a hidden variable theory.

The authors construct basis states for SU(3) and for SO(5) that are polynomials in the states of the fundamental representations; they are reduced according to the finite Demazure-Tits subgroup, which acts on basis states in the same manner that the Weyl group acts on weights.

The explicit expression of a solution that relates to the seven-dimensional representation of the Lie algebra G₂ of the quantum Yang-Baxter equation (QYBE) is obtained by applying the Yang-Baxterization procedure to the braid group representation (BGR). The result is consistent with an earlier one derived by a different method.

The general Hamiltonian for the SU(2)_q-invariant arbitrary-spin Heisenberg chain is presented. Some of these interactions are shown to satisfy braid group relations and the Temperley-Lieb algebra relations. Spin 3/2 is explored in more detail and, in this case, a general solution to the braid conditions is known in the generic sense.

Finite-dimensional representations of a recently proposed q-deformation, V_q, of the Virasoro algebra V₀, for q a root of unity, are constructed. The representations have a cyclic structure, and only in some special cases are they simultaneously highest-weight representations of the SU(1,1)-like subalgebra of V_q. In the 'classical' limit only those cyclic representations that are related to regular similarity transformations that become singular in the limit q to 1 reproduce highest-weight representations of the standard Virasoro algebra V₀.

Vector coherent state theory is applied to matrix representations of Osp(2/2) in the U(1/1) basis. The branching rule of Osp(2/2)(1U/1) is derived. Finite-dimensional irreducible matrix representations of Osp(2/2) in the U(1/1) basis are discussed and matrix elements of Osp(2/2) generators are obtained by using the K-matrix technique.

A q-analogue of the supersymmetric oscillator is constructed out of q-boson and q-fermion creation and annihilation operators. For a fixed n_B+n_F=2n, (where n_B and n_F are q-boson and q-fermion occupation numbers), the irreducible representations of the q-superalgebra generated by the q-oscillator are (2n+1)-dimensional. Particular cases when q is a root of +or-1 are discussed. A realization of the quantum group SU_q(2) is obtained using a pair of 1-fermion creation and annihilation operators.

Wess and Zumino (1990) succeeded in formulating a perfectly consistent differential calculus on the quantum hyperplane. The author gives the natural extension of their scheme to superspace and discusses the various consistency checks that have been performed.

The present theoretical investigation shows that the oscillation periods and the corresponding classical action integrals, associated with motion of a given energy in the local wells of a multi-well anharmonic potential, are linearly dependent. The proof makes use of the analytic behaviour of the local momentum of the particle in the complex position variable. The theorem applies to polynomial potentials of arbitrary even degree.

A variational approach to the problem of relativistic potential scattering in a laser field is developed. The field is assumed to be slowly varying compared with the collision time and to have a well defined direction of propagation, but is otherwise arbitrary. A trial function is chosen having the correct gauge-transformation property and accounting in an approximate manner for the simultaneous interaction of the projectile with the centre of force and with the laser field. The variational calculation provides a low-frequency approximation for the transition amplitude of improved accuracy compared with those obtained in previous treatments of this problem. The analysis is first given in terms of a spin-zero wave equation and then extended to allow for the scattering of a Dirac particle. A relativistic analogue of the Kroll-Watson approximation (1973) is obtained when the field is taken to be monochromatic and certain higher-order correction terms are dropped.

The authors analyse the Fisher equation via the expanded Painleve analysis approach. They obtain its singular property, auto-Backlund transformation and analytic solutions including some interesting heteroclinic and homoclinic solutions.

The authors discuss the structure of moving fronts, wave motions and self-excitation of oscillations from the point of view of the contact transport in a gas-like medium. The generalized equation of a contact transport in a system of free moving elements located far apart from each other is formulated. The parameters of contact, properties and distribution of elements are in general some functions of time. Several important solutions of these equations are obtained and analysed. Applications to other related problems such as spread of a contagious disease and percolation are briefly discussed.

An algebraic (representation-independent) analysis is presented for the Dirac oscillator in an angular momentum basis. The analysis is based on shift operators for energy and angular momentum, and it is similar to that for a non-relativistic isotropic harmonic oscillator. The shift operators generate all the eigenkets of the Dirac oscillator from a 'vacuum' ket. The shift operations yield energy eigenvalues and certain matrix elements. The relationship to the factorization method is discussed.

Within the framework of the Fock space formalism of the nonrelativistic quantum field theory the problem of hidden variables is studied. It is shown that one can construct, both for fermionic and bosonic Fock spaces, a local completely causal underlying theory which reproduces all quantum probabilities. This is done by suitable generalization of the probability concepts.

Using an 'action-angle' coherent states formalism, introduced by the authors in a preceding paper, it is shown that Berry's phase and Hannay's angle can both be derived from the same quantum unitary transformation; their relationship is easily established in this framework.

The zero-temperature helicity modulus tensor of the frustrated XY model is calculated by analysing the spectrum of the Hessian matrix for the flux-density wave state near the centre of the Brillouin zone for the lowest band. The eigenproblem for the Hessian is formulated in terms of transfer matrices. Expressions for the derivations of the lowest band with respect to the wavevector are found in terms of matrix traces which are then evaluated.

The authors present a general model for modulated lattices particularly suitable for the investigation of the moments of the density of states and establish an interpretation of the dependence on the modulation parameter which avoids the occurrence of unphysical incommensurability effects. The final result is an exact interpretation of the influence of an inevitably finite resolution on the system characteristics. They conclude by presenting some specialized results for Harper's equation (1955) in the context of this model.

The authors investigate neural networks in the range of parameters when the ground-state energy is positive; namely, when a synaptic matrix which satisfies all the desired constraints cannot be found by the learning algorithm. In particular, they calculate the typical distribution functions of local stabilities obtained for a number of algorithms in this region. These functions are used to investigate the retrieval properties as reflected by the size of the basins of attraction. This is done analytically in sparsely connected networks, and numerically in fully connected networks. The main conclusion is that the retrieval behaviour of attractor neural networks can be improved by learning above saturation.

Random walk in a percolation cluster was studied in the simultaneous application of both DC bias and AC driving fields by Monte Carlo calculations. Amplitude and phase shift of the random walk response to the external field of B₀+B₁ sin omega t were calculated as a function of B₁, omega and B₀. Some new observations are made, which cannot be expected simply from the linear superposition of the two separate results for a constant bias field (B₀) and an AC driving field (B₁ sin omega t) respectively.