Table of contents

We consider mean first passage time (MFPT) of random walks from one end to the other of a segment of a Sinai lattice, with a reflecting left boundary and an absorbing right boundary. Random fields are located at each site and can accept values 1/2 +or- epsilon with equal probability. We investigate the nature of the distribution of MFPT over Sinai fields, employing multifractal formalisms. We calculate the fractal dimension D(0)) and find it varies nearly linearly with epsilon , the strength of Sinai disorder. We then study the scaling behaviour of the partition function. To this end we make a scaling ansatz and fit it to our exact results on finite lattices, which yields the scaling exponents tau (q). We report results on the scaling exponents for various values of the disorder parameter.

Eigenvalues of the commuting family of transfer matrices are expected to obey the T-system, a set of functional relations, proposed recently. Here we obtain the solution to the T-system for C₂⁽¹⁾ vertex models. They are compatible with the analytic Bethe ansatz and Yang-Baxterize the classical characters.

A new kind of coherent states (CSs) for Lie superalgebra osp(l/2,R) are introduced. Some properties of these states are discussed. It is shown that they can encompass the Glauber CSs, su(1,1) CSs and squeezed states within a common formalism. The D-algebra differential operator realization of the osp(1/2,R) generators on the CSs and their diagonal projectors are constructed.

Applying the first Noether theorem to the Liouville equation u_xy=exp u, we find all (namely, a continuum of) non-trivial conservation laws of this equation. Then we find five new zero-curvature representations of the Liouville equation (by 2*2 traceless matrices) which contain, respectively, 1, 1, 2, 2 and 3 essential parameters. Finally, we show that all known zero-curvature representations of the Liouville equation are equivalent (in a definite sense) to matrices of conservation laws.

Bosonic Fock representations of the Virasoro algebra and the affine Lie algebras attached to classical Lie algebras with non-zero centres are explicitly presented. Based on these, Fock representations of the affine-Virasoro algebra are constructed. An automorphism of the affine-Virasoro algebra is also given.

The Burgers model for flux infiltration into a semi-infinite Soil is solved for generic space-dependent initial moisture and time dependent supply rate. General formulae in terms of quadratures are obtained for the moisture content and for the time to ponding of the soil.

A classical canonical transformation can convert the Poincare group generators from the usual 'instant form' into expressions associated with a light-cone. However, there is no consistent Hamiltonian dynamics for quantum wave functions on a light-cone, because the radial momentum operator is not self-adjoint.

The probability density of a non-relativistic particle is written as the sum of two non-negative density functions. The Schrodinger equation is equivalent to continuity equations (with source terms) for these functions. The equations are interpreted as describing two modes of the particle, which continually transform into each other.

The method of minimization of dimension for systems of first-order partial differential equations (PDEs) is extended analogously to systems of functional differential equations (FDEs). This method is applied to an exact first-order system of FDEs for the grand partition function of a one-dimensional classical fluid giving an alternative derivation of the pair potentials found by Baxter, for which exact thermodynamics can be obtained. These potentials satisfy a constant-coefficient ordinary differential equation (ODE). The method also gives the eigenvalue problem for the thermodynamics in these cases which is illustrated by deriving it explicitly for the simplest case, which is the exponential potential. The connection is derived between the finite system with external field to which the method applies and the infinite system without external field. This clarifies some points in Baxter's work and sheds some light on possible extensions of it.

The q-deformed boson gas at finite temperature is investigated for the quantum deformation parameter q to be complex (s=ln q=a+ib). It is shown that the real part of the energy density distribution corresponds to the Planck energy density distribution for a suitable range of the parameter b, which restricts the maximum number of quantum states in the model of the system. The spectra of q-oscillator are different for different (a, b)values. The classical dimensionless constant, characterizing the Wien shift law, is presented as a function of (a,b) and, in the spirit of the theory of q-deformed boson gas, it is possible to interpret it as the radiation of a non-ideal black-body or a real body. The role of the imaginary part of oscillator energy is also studied.

The relationship between two approaches to the study of non-equilibrium field theories, namely quantum statistical mechanics and thermo field dynamics, is investigated. The formalism of superoperators acting in the Liouville space of density matrices is used to provide a detailed translation between the two approaches. It is found that thermo field dynamics is exactly equivalent to a restricted version of quantum statistical mechanics, in which the initial density matrix is constrained to be Gaussian. The dissipative perturbation theory developed within thermo field dynamics is translated into the language of statistical mechanics, and is found to be equivalent to that devised by the author using the statistical-mechanical closed-time-path method, except that the latter theory is not restricted to Gaussian initial states.

A novel correlation-function approach, making use of many-particle densities and terminating their hierarchy by employing Kirkwood's superposition principle, is applied to the bimolecular annihilation reaction A+B to O. The roles of initial concentrations, space dimension and ratio of the reactant diffusivities in the modification of the reaction rate by many-particle effects are compared with computer simulations. The many-particle effects cause the reaction A+B to 0 to exhibit spatial self-organization phenomena even in the absence of direct A-A or B-B interactions. It is argued that this result casts serious doubts on the validity of the Hanusse-Tyson-Light theorem in synergetics.

The spin probability distribution at equilibrium in a multicomponent mean-field model of a ferromagnetic slab made of N rows is considered. Everywhere, but at the critical point, the spin fluctuations converge, as the number of spins pet row tends to infinity , to Gaussians with covariance equal to the thermodynamic susceptibility matrix. At criticality, abnormal coherent fluctuations along the positive eigenvector of the interaction matrix are found, whose limit distribution is an exp(-x⁴) law.

The asphericity of a flexible monodisperse f-star polymer in d=3 is calculated in the framework of the standard two-parameter model in the asymptotic limit of infinite molecular weight. In contrast to previous analytical investigations we employ an asphericity measure which takes into account that the size and shape of the polymer coils are strongly correlated. We consider ideal non-interacting (NEV) star polymers as well as molecules with excluded-volume (EV) interactions by means of renormalized perturbation theory. We show that in both cases the mean asphericity parameter takes a universal numerical value when the molecular weight tends to infinity. The same holds for the corresponding asymptotic distribution functions, which are also proved to be universal functions. These universal properties depend only on the topology, i.e. the number of arms f. For NEV stars we obtain a decrease of the asphericity with increasing number of arms. The numerical values are in excellent agreement with results of previous computer simulations. From the extrapolation of the epsilon -expansion results for EV stars we conclude that the influence of the EV on the shape asymptotics qualitatively depends on the number of arms. For f=1 and j=2, i.e. a linear chain, we reproduce a previous finding that the EV enlarges the asphericity of the chains by about 5%. For f>3, however, we observe a decrease of the asphericity due to the EV which is intuitively appealing because of the strong steric repulsion for large values of f. These results are also in good agreement with previous computer experiments.

A new real space renormalization group transformation combining ideas from mean-field and finite-size scaling theories is presented. Application to the two-dimensional site directed percolation problem gives better values for the percolation threshold and critical exponent of longitudinal correlation length than those obtained previously with other real space renormalization group approaches. When applied to one-, two- and three-dimensional) Ising models, the results are comparable to the ones from previous mean-field-like renormalization group transformations. This method can easily be applied to other systems having second-order phase transitions.

The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the q-state Potts model to order z⁵⁶ (i.e. u²⁸), z⁴⁷, z⁴³, z³⁹, z³⁹, z³⁹, z³⁵, z³¹ and z³¹ for q=2, 3, 4, ..., 9 and 10 respectively. These series are used to test techniques designed to distinguish first-order transitions from continuous transitions. New numerical values are also obtained for the q-state Potts model with q>4.

We show that the random-field Ising model may be tuned to obtain two distinct scenarios of phase diagram topology. Explicit evidence for this is presented in a numerically exact analysis in three dimensions and on a Bethe lattice, which allows us to investigate the effects of temperature, coordination number, and asymmetry in the field distribution.

The hysteretic response of an Ising system to a periodically varying transverse (tunnelling) field has been investigated. Numerical and approximate analytical techniques have been used to solve the mean-field equation for the dynamics, in the presence of a heat bath at a finite temperature. The scaling behaviour of the hysteresis loop area and the phase diagram for the dynamic (phase) transition have been obtained.

We have studied the failure of 'fluctuation-selection' in a ground state manifold associated with an extensive entropy. The recursive structure of the Husimi cactus permits exact calculation, and thus avoids problems associated with non-perturbative fluctuations. The structure of the barriers separating the ground states can be determined; in all the cases we have examined, we find that the system prefers to remain a spin liquid rather than to 'localize' in an ordered state that breaks the degeneracy of the manifold.

We investigate the finite-temperature Neel transition of the anisotropic large-N CP^N-1 model with one periodic spatial dimension using the effective potential method. Long-range Neel order (LRNO) and the equation of the critical line are obtained. In 1+1 dimensions, the one-loop approximation fails badly for large but finite N by the Mermin-Wagner theorem. But for infinite N, which is not realistic. it becomes exact and the appearance of LRNO is valid. The (3+1)-dimensional CP^N-1 model is investigated as a model of antiferromagnetism in high-temperature superconductors, where the anisotropic parameter represents the weak interlayer coupling. The phase structure is very different from that of the (1+1)-dimensional infinite-N CP^N-1 model.

The storage capacity of the Hopfield network is limited to p/N <or= alpha _c=0.138 (p=number of patterns, N=number of neurons), beyond which the contribution of weakly correlated patterns surpasses that of the desired pattern. This contribution can be eliminated by introducing a threshold: a pattern correlation below this threshold is simply set to zero in the synapses. We solve the mean-field equations and derive the critical value of the threshold required to stabilize p= alpha N patterns with alpha > alpha _c.

A probabilistic automata network model for the spread of an infectious disease in a population of moving individuals is studied. The local rule consists of two subrules. The first one, applied synchronously, models infection, birth and death processes. It is a probabilistic cellular automaton rule. The second, applied sequentially, describes the motion of the individuals. The model contains six parameters: the probabilities p for a susceptible to become infected by contact with an infective; the respective birth rates b_s, and b_i of the susceptibles from either a susceptible or an infective parent; the respective death rates d_s, and d_i of susceptibles and infectives; and a parameter m characterizing the motion of the individuals. The model has three fixed points. The first is trivial, it describes a stationary state with no living individuals. The second corresponds to a disease-free state with no infectives. The third and last one characterizes an endemic state with non-zero densities of susceptibles and infectives. Moreover, the model may exhibit oscillatory behaviour of the susceptible and infective densities as functions of time through a Hopf-type bifurcation. The influence of the different parameters on the stability of all these states is studied with a particular emphasis on the influence of motion which has been found to be a stabilizing factor of the cyclic behaviour.

In the periodic orbit quantization of the stadium billiard, we show that important contributions may be due to the edge orbits, i.e. to orbits bouncing between points where the curvature of the boundary is discontinuous. We explicitly show that these edge orbits are necessary to reproduce several amplitudes of the length spectrum defined by the Fourier transform of the staircase function. In this way, we explain some features overlooked in recent experiments on microwave cavities.

A generalization of the quantized enveloping algebra of B_n (U_q(B_n)) is constructed. This algebra which we call X_q(B_n), is based on a non-standard R-matrix corresponding to the series B_n, and yields U_q(B_n) in a special case.

We present a hierarchy of generalized Abelian Higgs models on R₂, descended from the hierarchy of generalized Yang-Mills (GYM) systems on R_4p where the p=1 member is the usual Abelian Higgs model. We also study the vortex number n radial self-dual solutions numerically.

We analyse the behaviour of the mixmaster universe in the context of general relativity by applying methods from the geometric theory of dynamical systems in the complex plane and in particular by using what is known as the Painleve test. We find that the singularities of the mixmaster model in the complex plane are simple poles and the resonances are -1,2. This shows that chaotic behaviour may be absent in the mixmaster universe in general relativity. This analysis reinforces some recent numerical experiments which point to the same conclusion. We also Painleve-test the Euclidean version of the Bianchi IX spacetime which includes the gravitational analogues of the Yang-Mills instanton solutions. We find that this system also has the Painleve property and this supports previous conclusions that the Euclidean Bianchi IX field equations can be solved through Jacobi elliptic functions.

We discuss the higher-dimensional solutions of the Yang-Baxter equation and reflection equation in the elliptic case. The Hamiltonian of a spin-1 quantum anisotropic chain with non-trivial boundary term is given.

We present an improved version of the integrability test based on application of Lie point transformations acting on the bare 'linear problem'. The test is then applied to the inhomogeneous Heisenberg ferromagnet model equation. The integrable cases selected by this test are identical with those obtained by the standard Painleve test.

The topology-preserving representation of a rectangular part of space onto a square network of formal neurons is studied using the Kohonen algorithm. Linear stability analysis shows that there is a critical ratio for the sides of the rectangle. For larger ratios the map becomes unstable. The value of the critical ratio depends on the actual shape of the adjustment function. The problems cannot be scaled away in the case of inhomogeneously sampled input space. The results of the analysis are compared with computer simulations.

Self-similarity in binary quasiperiodic sequences generated by a projection method is shown to exist when and only when it is associated with quadratic irrational numbers and the explicit self-similarity transformation for an arbitrary quadratic number is obtained. The self-similarity transformation is shown not to be reducible to the simplest form for a class of quadratic numbers.

The quasiperiodic Fibbonacci chain can be obtained by finite displacements from a periodic chain if one introduces ordered defects that are similar to discommensurations in incommensurate crystal phases. It is shown that the chain with discommensurations can also be constructed by means of a substitution rule having the Pisot property. In superspace the defect structure is a superstructure for the two-dimensional array that corresponds to the Fibonacci chain.

For chaotic systems the semiclassical approximation to the time-dependent propagator consists of a large number of terms, some of which may involve classical trajectories near focal points (caustics). Despite this the approximation has been found to remain accurate for relatively long times. In this article, that accuracy and potential reasons for its breakdown are studied. The principal tool is the Jacobi-Morse equation, the eigenvalue equation associated with the second variational derivative of the classical action. Our explanation for the accuracy lies partly in the following considerations: (1) verification of the efficacy of phase space smearing, (2) given an explicit form of the near caustic propagator, it is seen that the arguments concerning loops in phase space may be less relevant than the determination of the amount of separation in path-space, (3) and finally, the caustics in the chaotic examples studied may not be placed for maximum mischief.

The relativistic dynamics of a charged particle when it is interacting with its own electromagnetic field is analysed. The effect of the reaction force is relatively small but it is significant in the energy balance between the field and particle. It is found that a pulse in the radiation field is formed when the initial conditions simulate the creation of a particle.

The problem of the harmonic oscillator damped proportionally to the velocity is investigated in the de Broglie-Bohm hidden-variable theory. (This system is known to be classically equivalent to a particular type of undamped oscillator with variable mass.) The de Broglie-Bohm equation of motion is solved explicitly and compared with its classical equivalent. The influence of the 'quantum potential' is discussed. In contrast to some other studies of the de Broglie-Bohm theory, the results obtained here are derived without approximation.

The time-dependent integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the nonstationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are also obtained from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.

The authors recently derived a family of nonlinear Schrodinger equations on R³ from fundamental considerations of generalized symmetry: ih(cross) delta _t psi =-(h(cross)²/2m) Del ² psi +F( psi , psi ) psi +ih(cross)D( Del ² psi +( mod Del psi mod ²/ mod psi mod ²) psi ), where F is an arbitrary real functional and D a real, continuous quantum number. These equations, descriptive of a quantum mechanical current that includes a diffusive term, correspond to unitary representations of the group Diff(M) parametrized by D, where M=R³ is the physical space. In the present paper we explore the most natural ansatz for F, which is labelled by five real coefficients. We discuss the invariance properties, describe the stationary states and some non-stationary solutions, and determine the extra, dissipative terms that occur in the Ehrenfest theorem. We identify an interesting, Galilean-invariant subfamily whose properties we investigate, including the case where the dissipative terms vanish.

Operator regularization, together with the background field method, is used to calculate the one-loop renormalization constant of Chern-Simons field theory. The result shows the existence of the famous k shift, i.e. k to k+sgn(k)C_V.

In a previous letter a nilpotent BRST operator Q_q was constructed for the q-deformation of the universal enveloping algebra of sl(2). The present addendum resolves issues of non-uniqueness in the construction of the BRST operator by imposing additional structure associated with real forms, and corrects a misprint in the previous work.