Table of contents

The authors consider the O(3) gauge transformation for three-state vertex models on lattices of coordination number three. Using an explicit mapping between O(3) and SL(2), they establish that there exist exactly six polynomials of the vertex weights, which are fundamentally invariant under the O(3) transformation. Explicit expressions of these fundamental invariants are obtained in the case of symmetric vertex weights.

The author found that certain solutions of some nonlinear partial differential equations can be obtained easily by an iteration method. As examples, the Korteweg-de Vries equation (also the 'modified' version), the 1D and 2D Burgers equation and the Kadomtsev-Petviashvili equation have been studied. For the Liouville equation the author found the general solution.

Using the method developed by Nucci (1989), the author has obtained the pseudopotentials, Lax pairs, Backlund transformations and the singularity manifold equations of some variable coefficient nonlinear equations. These equations are generalized Korteweg-de Vries, generalized modified Korteweg-de Vries and generalized Boussinesq equations.

New Fujimoto-Watanabe non-constant separant evolution equations admitting generalized symmetries are shown to be connected with the Korteweg-de Vries, Sawada-Kotera, Kaup and linear equations via chains of differential substitutions. A conjecture is proposed which explains why the Ibragimov transformation to constant separant equations is a universal link in those chains.

By making use of the system of coordinates in which the separation of the variables in the Hamilton-Jacobi equation takes place, the authors find a bi-Hamiltonian structure of a two degrees of freedom Hamiltonian system corresponding to the integrable Henon-Heiles Hamiltonian H=1/2(p_q²+p₂²+Aq₁²+Bq₂²)-q₁²q₂-2 q₂³.

A new approach for finding analytical solutions of the two-dimensional sine-Gordon equation is presented. The essence of this approach is the established relation between the solutions of the one-dimensional wave equation having the form of running waves and solutions of the two-dimensional sine-Gordon equation.

The simplest parasupersymmetric model found in one-dimensional quantum mechanics is considered. It contains one bosonic and one parafermionic degree of freedom. By successive restriction of the superpotentials the author shows how the algebra of symmetry is simplified. Some discussion of the general parasupersymmetry algebras is presented.

A general method is developed for constructing representations of the Temperley-Lieb algebra, which in turn yield solutions of the Yang-Baxter equation. Several classes of multiparameter dependent R-matrices are obtained using this method, and the connection of the method with quantum groups is discussed.

By establishing a new boson realization of quantum universal enveloping algebra sl_q(2) and its representations in the non-generic case that q is a root of unity, the authors systematically construct non-generic R-matrices of sl_q(2) through the universal R-matrix. These new R-matrices are not covered by the standard R-matrices constructed in terms of quantum groups and the non-standard ones obtained by using the extended Kauffman's diagram technique.

The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.

An exactly solvable deterministic model on a hierarchical lattice is presented for the left-sided multifractality of the growth probability distribution in a generalized diffusion-limited aggregation ( eta model). The exactly renormalizable growth probability is given by a multiplicative process. It predicts the tip behaviour (the maximum growth probability is a power law) and the fjord behaviour (the minimum growth probability has a logarithmic singularity). An analytical form of a 'left-sided' generalized dimension D(q) is shown for any eta .

The probability of a planar Brownian closed curve enclosing a fixed area is rederived using a simple method of functional integration. The mean square distance between two points of the ring is calculated. When the constraint on the enclosed area increases, one passes from a diffusion regime of the Brownian ring to a normal regime, where the curve approaches a circle.

The storage properties of an optimal neural network with correlated patterns is studied allowing for a finite fraction of errors determined by the Gardner-Derrida cost function. A discontinuity in the probability distribution of the local stabilities is seen as a drastic decrease in the minimal fraction of errors. There is also an enlargement of the domain in which the replica symmetric solution is stable, allowing for higher storage capacities.

Simultaneous grading of Lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all Lie algebras admitting the chosen grading. The theory is completely different from the traditional ways of contracting representations. The graded contractions fall naturally into two classes: discrete and continuous ones related respectively to 2-cocycles and coboundaries of the grading group.

Radial integrals consisting of products of Whittaker functions of the first and second kind arise frequently in expressions for various scattering processes. In particular, the Dirac-Coulomb radial integrals for intermediate energy DWBA lepton pair production and bremsstrahlung consist of products of Whittaker functions that result in slowly converging series expressions. An approximate partial differential matrix equation in the lepton energy is obtained that can be used to numerically propagate Dirac-Coulomb radial integrals along the lepton and photon energy spectrum, resulting in substantial time savings on the computer. Radial integrals for DWBA electron pair production from 10 MeV photons are calculated as an example of the use of the approximate equation that yield three significant digit accuracy.

The statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the Hubbard-Stratonovich transformation. When used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick. The model turns out to bear a close resemblance to the Anderson model on the Bethe lattice: it possesses a delocalization transition that occurs with an increase in the 'mean connectivity' parameter. In the delocalized phase the level-level correlation function proves to have a universal (Dyson) form with the full density of states replaced by the contribution from the infinite cluster.

The author investigates the existence of an infinite period-doubling sequence in the following class of semi-symplectic maps: a two-dimensional (2D) nonlinear constant Jacobian map with an infinite period-doubling sequence, linearly and weakly coupled with a 2D linear map. The author introduces the property of semi-symplecticity as the generalization of symplecticity that incorporates uniform dissipation. The author defines Krein signatures for pairs of complex eigenvalues and show that they play the same role as in the symplectic case. The Krein signature alternates in the period-doubling sequence of a 2D constant Jacobian map, whereas the signatures of iterates of a 2D linear map form (almost always) an uncorrelated row. With these results the authors shows that any finite coupling strength destroys the infinite period-doubling sequence of the conservative maps of this class in two ways: firstly, there are 'bubbles of instability': secondly, and far more importantly, there are period-doubling bifurcations in which the newborn period-doubled orbits are unstable. So crucial parts of the period-doubling sequence are experimentally invisible. For the dissipative maps of this class the same conclusions hold, but only if the coupling strength is strong enough.

A generalization of the description of a quantum system in the time interval between two measurements is presented. A new concept of a generalized state is introduced. Generalized states yield a complete description of a quantum system when information about the system is available both from the past and from the future. The formalism of generalized states provides a natural language for describing many peculiar situations. In particular, situations in which one can ascertain the result of a measurement of any one of several non-commuting variables are analysed. 'Weak' measurements on quantum systems described by generalized states are discussed. The relation between 'weak' and 'strong' measurements is investigated.

The authors consider an electron on a triangular lattice with two magnetic fluxes. Using a C*-algebra formalism a perturbation theory around rational fluxes is developed. This semiclassical type of analysis leads to the Wilkinson-Rammal formula for the energy eigenvalues. The analytic result is verified numerically with high accuracy up to second order. Crossing of eigenvalues near band edges is shown analytically as well as numerically.

The author considers an O(n)invariant continuum model of a ferromagnet in d dimensions with quenched temperature-like disorder and determine the domain of the Griffiths phase (1969) in terms of the parameters of this model. Using the Ornstein-Zernike approximation with respect to thermal fluctuations he investigates the temperature region above the upper boundary of this domain (TT_G) for a bounded distribution of disorder, and work out the connection with the corresponding Lifshitz problem (1964). Using different assumptions for the disorder he gives an expression for the Griffiths singular part of the free energy and show that an essential singularity develops in the limit T T_G in zero magnetic field.

A percolation model, spiral percolation, in which a rotational constraint is operative is studied by the finite-size scaling method. The critical percolation probability p_c and the critical exponents v, beta , gamma , tau , sigma and also the fractal dimension D of the spanning cluster are determined. Evidence is obtained for a scaling form of the cluster distribution function.

Directed percolation in higher dimensions serves as an example of how the distribution of overhangs gives information on the critical exponents of the system. Overhangs appear as jumps in the position of the front formed when a gradient in the control parameter is imposed along one of the spatial directions. By analyzing the overhang distribution in (d+1)-dimensional directed percolation, the authors determine the critical exponents beta and v_{perpendicular to} for d2. In higher dimensions, the overhang distribution is insensitive to the critical region.

A new solvable case of O(n)-model on a hexagonal lattice is presented. It is shown that the model has a phase transition at n=2. This phase transition can be interpreted as phase transition between percolating and non-percolating phases.

The authors study the correlation between two spins diametrically opposite in a random Ising loop of 2r spins with random nearest-neighbour couplings. As a function of the inverse temperature beta , this correlation undergoes random sign changes with a density rho _r( beta ). They determine this density for an arbitrary distribution P(K) of the coupling constants for (i) r to infinity and (ii) the scaling limit beta to infinity , r to infinity at r/ beta fixed. For r to infinity , the total number of zeros on the beta -axis grows asymptotically as 1/2 1/21 zeta (3)-1/4 pi ^21/2 log r. As an application, the density of zeros is calculated for the spin-spin correlation in a double infinite Ising chain with random bonds, having, with a probability p, an infinite transverse coupling between each pair of corresponding sites. A connection is made with predictions from spin glass theory.

An explicit expression is derived for the three-variable generating function P(x, y, z)= Sigma _m>or=1, Sigma _n>or=₁ Sigma _r>or=₁ x²ⁿy^2mz^rc_n,m,r, where c_n,m,r is the number of convex polygons with horizontal width n, vertical height m and area r.

Hopfield-like neural networks (1982) with modified interactions are studied by a mean-field theory. The modification of interactions is achieved during a special thermally noised iterative procedure. The resulting couplings have an intermediate form between the Hebb-like learning rule and the pseudo inverse one. Replica-symmetric free energy of the model is obtained. Statistical properties of the model depend on three parameters: reduced number of the stored patterns alpha , reduced number of iteration steps of the modification procedure lambda and the temperature T. The phase diagram in the space of these parameters is obtained. The network can retrieve patterns at T=0 for alpha < alpha_c(lambda), where alpha_c(0) approximately= 0.14 and alpha_c(lambda to infinity) approximately= 1.07. As alpha decreases below alpha₀(lambda) the FM retrieval states become ground states of the system, where alpha₀(0) = 0.05 and alpha₀(lambda to infinity) = 2/pi.

The author presents an operator method for the calculation of partial sums and infinite series in closed form. It consists of expressing the sum as the value at origin of a generating function which is given by a differential operator acting on a simple function. When exact expressions are not available, the method is suitable for obtaining approximate results.

The behaviour of the Fokker-Planck equation in the limit of large times, which has become controversial recently, is discussed. It is shown on the basis of a theorem given by Risken (1984) that every solution of the Fokker-Planck equation must tend to the same in this limit and any information on the initial data will be lost.

The Green function for a spin-1/2 charged particle in the presence of an external plane wave electromagnetic field is calculated by algebraic techniques in terms of the free-particle Green function.