Table of contents

We investigate the critical behaviour of the fully packed O(n) loop model on the square lattice in which each vertex is visited once by a loop. A transfer-matrix analysis shows that this model can be interpreted as a superposition of a low-temperature O(n) model and an solid-on-solid (SOS) model, as for the fully packed model on the honeycomb lattice. However, not all of the critical exponents are the same for both lattices. In contrast, the fully packed model on the triangular lattice appears to behave as a pure low-temperature O(n) model.

A technique for solving the Lippmann - Schwinger equation in momentum space based on Chebyshev polynomials is proposed. It is found that exact results are obtained for standard low-energy nuclear separable potentials with an extremely low number of meshpoints. The technique naturally leads to a class of analytically solvable separable potentials. The application of the Chebyshev technique to two local standard N - N potentials, Yukawa and Reid, is discussed.

We suggest a new method, named the knitting ansatz, to generate solutions to the Yang - Baxter equation with lower-dimensional representations of the braid group. To support our ansatz, we work out an example of a new -matrix constructed according to this idea, with two braid group representations of familiar 6-vertex type with different q-parameters.

We consider self-avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary. We numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions. The phase diagram is elucidated with the aid of some rigorous arguments.

The dynamics of on-line learning in neural networks with continuous units is dominated by plateaux in the time dependence of the generalization error. Using tools from statistical mechanics, we show for a soft committee machine the existence of several fixed points of the dynamics of learning that give rise to complicated behaviour, such as cascade-like runs through different plateaux with a decreasing value of the corresponding generalization error. We find learning-rate-dependent phenomena, such as splitting and disappearing of fixed points of the equations of motion. The dependence of plateau lengths on the initial conditions is described analytically and simulations confirm the results.

Electrical and optical properties of binary inhomogeneous media are currently modelled by a random network of metallic bonds (conductance , concentration p) and dielectric bonds (conductance , concentration 1-p). The macroscopic conductivity of this model is analytic in the complex plane of the dimensionless ratio of the conductances of both phases, cut along the negative real axis. This cut originates in the accumulation of the resonances of clusters with any size and shape. We demonstrate that the dielectric response of an isolated cluster, or a finite set of clusters, is characterized by a finite spectrum of resonances, occurring at well defined negative real values of h, and we define the cross section which gives a measure of the strength of each resonance. These resonances show up as narrow peaks with Lorentzian line shapes, e.g. in the weak-dissipation regime of the RL - C model. The resonance frequencies and the corresponding cross sections only depend on the underlying lattice, on the geometry of the clusters, and on their relative positions. Our approach allows an exact determination of these characteristics. It is applied to several examples of clusters drawn on the square lattice. Scaling laws are derived analytically, and checked numerically, for the resonance spectra of linear clusters, of lattice animals, and of several examples of self-similar fractals.

We consider the surface critical behaviour of diagonally layered Ising models on the square lattice where the inter-layer couplings follow some aperiodic sequence. The surface magnetization is analytically evaluated from a simple formula derived by the diagonal transfer matrix method, while the surface spin - spin correlations are obtained numerically by a recursion method, based on the star - triangle transformation. The surface critical behaviour of different aperiodic Ising models is found in accordance with the corresponding relevance - irrelevance criterion. For marginal sequences the critical exponents are continuously varying with the strength of aperiodicity and generally the systems follow anisotropic scaling at the critical point.

We investigate the retrieval phase diagrams of an asynchronous fully connected attractor network with non-monotonic transfer function by means of a mean-field approximation. We find for the noiseless zero-temperature case that this non-monotonic Hopfield network can store more patterns than a network with monotonic transfer function investigated by Amit et al. Properties of retrieval phase diagrams of non-monotonic networks agree with the results obtained by Nishimori and Opris who treated synchronous networks. We also investigate the optimal storage capacity of the non-monotonic Hopfield model with state-dependent synaptic couplings introduced by Zertuche et al. We show that the non-monotonic Hopfield model with state-dependent synapses stores more patterns than the conventional Hopfield model. Our formulation can be easily extended to a general transfer function.

Wavefunctions of one and two-dimensional quantum systems can be parametrized by a finite number of zeros lying in phase space. We study correlations of these zeros for fully chaotic systems in terms of a statistical model based on random polynomials. Excellent agreement is found for the two-point correlation function and nearest-neighbour spacing distribution of this model and the results obtained for wavefunctions of dynamical systems. We conjecture that these correlation functions are valid for any chaotic system after rescaling the phase-space distances (unfolding). Some consequences for the distribution of zeros due to time-reversal symmetry are also discussed.

Thermodynamics of the quantum Toda lattice is studied based on Gutzwiller's quantization condition and Yang - Yang's thermodynamic formulation. It is shown that Gutzwiller's quantization condition becomes identical to the Bethe ansatz equation in the thermodynamic limit. We have calculated the thermodynamic averages of the energy, higher-order conserved quantities and the specific heat. The average energy and the specific heat become close to those of the classical Toda lattice in the high temperature limit, while they are similar to those of a harmonic chain in the low temperature limit.

We obtain the asymptotic behaviour of the nearest-neighbour level spacing distribution for a q-deformed unitary random matrix ensemble.

The optimal algorithm for on-line learning in the tree K-parity machine is studied. We introduce a set of recursion relations for the relevant probability distributions, which permit study of the general K case. The generalization error curve is determined and shown to decay to zero for large as even in the presence of noise. There is no critical noise level. The dynamics of on-line learning is studied analytically near the origin. In the absence of previous knowledge, the learning dynamics has a fixed point at . Previous knowledge is needed in at least K - 1 branches for the learning to take place.

Some of the ground states of Baxter's IRF model in the square lattice are constructed. It is shown that the model has an infinite series of ground states which correspond to monodisperse close-packed triangles of fixed orientation and varying size.

We show that the problem of a random walk with a boundary attractive potential may be mapped onto the free massive bosonic quantum field theory with a defect line. This mapping allows us to recover the statistical properties of random walks by using boundary S-matrix and form factor techniques.

A mean-field approach for epidemic processes with high migration is suggested by analogy with non-equilibrium statistical mechanics. For large systems a limit of the thermodynamic type is introduced for which both the total size of the system and the total number of individuals tend to infinity but the population density remains constant. In the thermodynamic limit the infection rate is proportional to the product of the proportion of individuals susceptible to infection and the average probability of infection. The limit form of the average probability of infection is insensitive to the detailed behaviour of the fluctuations of the number of infectious individuals and may belong to two universality classes: (1) if the fluctuation of the number of infectives is non-intermittent it increases with the increase of the partial density of infectives and approaches exponentially the asymptotic value one for large densities; (2) for intermittent fluctuations obeying a power-law scaling the average probability of infection also displays a saturation effect for large densities of infectives but the asymptotic value one is approached according to a power law rather than exponentially. For low densities of infectives both expressions for the average probability of infection are linear functions of the proportion of infectives and the infection rate is given by the mass-action law.

The multiple scattering of scalar waves in diffusive media is investigated by means of the radiative transfer equation. This approach, which does not rely on the diffusion approximation, becomes asymptotically exact in the regime of most interest, where the scattering mean free path is much larger than the wavelength . Quantitative predictions are derived in that regime, concerning various observables pertaining to optically thick slabs, such as the mean angle-resolved reflected and transmitted intensities, and the width of the enhanced backscattering cone. Special emphasis is put on the dependence of these quantities on the anisotropy of the cross section of the individual scatterers, and on the internal reflections due to the optical index mismatch at the boundaries of the sample. The large index mismatch regime is studied analytically, for arbitrary anisotropic scattering. The regime of very anisotropic scattering, where the transport mean free path is much larger than the scattering mean free path , is then investigated in detail. The relevant Schwarzschild - Milne equation is solved exactly in the absence of internal reflections.

We construct quark mixing matrices within a group theoretic framework which is easily applicable to any number of generations. Familiar cases are retrieved and related, and it is hoped that our viewpoint may have advantages both phenomenologically and for constructing underlying mass matrix schemes.

We obtain second-order equations of degree four (six), for travelling wave solutions of the KdV (Sawada-Kotera/Kaup) equations, which reduce to first-order equations for monotone solitary waves. For the KdV equation, the singular solutions of this equation with an asymptotic value b consist of the well known solution and a new solution with a non-zero asymptotic value depending on the wave speed. We show that the well known solitary wave solutions are determined uniquely as the singular solutions with asymptotic value b = 0, which are also stationary with respect to the wave speed.

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincaré section. We show that there are topological obstacles for its existence such that only in the cases of and such a Poincaré section can exist.

We give a gauge invariant formulation of N = 2 supersymmetric abelian Toda field equations in N = 2 superspace. Superconformal invariance is studied. The conserved currents are shown to be associated with Drinfeld - Sokolov type gauges. The extension to non-abelian N = 2 Toda equations is discussed. Very similar methods are then applied to a matrix formulation in N = 2 superspace of one of the N = 2 KdV hierarchies.

We demonstrate that the left (and right) invariant Maurer - Cartan forms for any semi-simple Lie group enable solutions of the Yang - Mills equations to be constructed on the group manifold equipped with the natural Cartan - Killing metric. For the unitary unimodular groups the Yang - Mills action integral is finite for such solutions. This is explicitly exhibited for the case of SU(3).

A construction of conservation laws for chiral models (generalized -models) on a two-dimensional spacetime continuum using differential forms is extended in such a way that it also comprises corresponding discrete versions. This is achieved via a deformation of the ordinary differential calculus. In particular, the nonlinear Toda lattice results in this way from the linear (continuum) wave equation. The method is applied to several further examples. We also construct Lax pairs and Bäcklund transformations for the class of models considered in this work.

Some classical neural network systems including the Hartline - Ratliff system, the Linsker system, and the general sigmoid dynamics, are reconsidered within a more general class of dynamical systems. For synchronous dynamics the existence, uniqueness, local and global stability of stationary points is investigated. For asynchronous dynamics a convergence theorem is proved. The application of the theory of quasimonotone flows leads to some insights so far not widespread in network theory.

We consider the classical problem of linearizing a vector field X around a fixed point. We adopt a non-perturbative point of view, based on the symmetry properties of linear vector fields.

A general theory is presented of the classical and quantum mechanics of singular, non-autonomous, higher derivative systems. It is shown that adding a total derivative to a Lagrangian does not materially affect either, (a) the canonical analysis of the system, or (b) its quantum mechanics.

The multiplicity of occurrence of the adjoint representation in the decomposition of the square of any finite-dimensional irreducible representation of any compact simple Lie group is shown to be equal to the number of non-vanishing components of the Dynkin label of . The resolution of this multiplicity into contributions to the symmetric and antisymmetric squares of is discussed, with complete results being found for all of the classical and some of the exceptional simple Lie groups, and partial results culminating in conjectures for the remaining exceptional groups.

Irreducible representations of Brauer algebras are discussed in the non-standard basis. A method for evaluating subduction coefficients (SDCs), i.e. the transformation coefficients between standard and non-standard bases of Brauer algebras, is outlined. Non-trivial SDCs of for are derived. Racah coefficients of O(n) and Sp(2m) can be derived from subduction coefficients of Brauer algebras by using the Schur - Weyl duality relation between and O(n) or Sp(2m).

Racah coefficients of O(n) and Sp(2m) are derived from subduction coefficients of Brauer algebras by using the Schur - Weyl duality relation between and O(n) or Sp(2m). It is found that there are two types of Racah coefficients according to irreps of O(n) or Sp(2m) with or without trace contraction. It is proved that Racah coefficients with no trace contraction in the irreps are trivial and the same as those of unitary groups U(n), which are rank n-independent, and those with trace contraction usually are n-dependent. Racah coefficients with trace contraction for the resulting irreps with are tabulated.

The rigorous difference equations for the elements of any order of transfer matrices of coupled waves and the renormalization relations for their recursion coefficients under homogeneous and inhomogeneous rescaling are obtained. Incidentally, the generalization of Abelés theorem to arbitrary-dimensional square matrix formulae are deduced.

Domain walls, arising from the spontaneous breaking of a discrete symmetry, can be coupled to charge carriers. In much the same way as the Witten model for a superconducting cosmic string, an investigation is made here into the case of , where a bosonic charge carrier is directly coupled to the wall-forming Higgs field. All internal quantities, such as the energy per unit surface and the surface current, are calculated numerically to provide the first complete analysis of the internal structure of a surface current-carrying domain wall.

Writing the Wigner functions of any pair of harmonic oscillator eigenstates in the Bargmann representation, a direct and detailed proof is given of their convergence (in the sense of distributions) to at the classical limit , , , m-n fixed, .

It is shown that the overdetermined system of equations obtained by truncating the Painlevé expansion for the classical Tzitzeica equation admits a solution which defines a Bäcklund transformation.

Various illumination schemes of dynamic apertures are investigated. The decay patterns of the generated ultra-wide bandwidth pulses are studied and compared to postulated diffraction lengths. It is shown that such definitions of the diffraction ranges characterize the propagation of the pulses in a broad sense. We emphasize the fact that to understand how a localized pulse decays we have to resort to the structure of its temporal and spatial spectral content. An exhaustive analysis of the depletion of the spectral components of the radiated localized pulses is presented.

Quaternion measurable processes are introduced and the Dirac equation is derived from the Langevin equation associated with a two-valued process.

For each positive integer g, we construct a one-parameter family of spectral curves for symmetric charge 2g+1 SU(2) BPS monopoles. Each spectral curve is reducible, and is the union of a line with g elliptic curves. We show that such a monopole is related to a g-gap Lamé potential. Other symmetric monopoles, related to elliptic curves, are also shown to have a similar correspondence. A suggestion is made on how this observation may be of use in the construction of new spectral curves.

Exact travelling-wave solutions of the (2 + 1)-dimensional sine - Gordon equation possessing a velocity smaller than the velocity of the linear waves in the correspondent model system are obtained. The dependence of their dispersion relations and allowed areas for the wave parameters on the wave amplitude are discussed. The obtained waves contain as particular cases static structures consisting of elementary cells with zero topological charge. The self-consistent parameters of one static structure are calculated. The obtained structures require minima spatial system sizes for their existence. As an illustration the obtained results are applied for a description of structures in spin systems with an anisotropy created by a magnetic field or by a crystal anisotropy field.

Lie - Bäcklund symmetries and conservation laws are derived for weakly nonlinear magnetohydrodynamic (MHD) equations describing the interaction of the Alfvén and magnetoacoustic modes propagating parallel to the ambient magnetic field, in the parameter regime near the triple umbilic point, where the gas sound speed matches the Alfvén speed . The dispersive form of the equations can be expressed in Hamiltonian form and admit four Lie point symmetries and conservation laws associated with space-translation invariance (momentum conservation), time translation invariance (energy conservation), rotational invariance about the magnetic field B (helicity conservation), plus a further symmetry that is associated with accelerating wave similarity solutions of the equations. The main aim of the paper is a study of the symmetries and conservation laws of the dispersionless equations. The dispersionless equations are of hydrodynamic type and have three families of characteristics analogous to the slow, intermediate and fast modes of MHD and the Riemann invariants for each of these modes are given in closed form. The dispersionless equations are shown to be semi-Hamiltonian, and to possess two infinite families of symmetries and conservation laws. The analysis emphasizes the role of the Riemann invariants of the dispersionless equations and a hodograph transformation for a restricted version of the equations.

We construct the classical Poisson structures and r-matrices for some finite-dimensional integrable Hamiltonian systems obtained by constraining the flows of soliton equations in a certain way. This approach allows us to produce new kinds of classical (dynamical) Yang - Baxter structures. To illustrate this method we present the r-matrices associated with the constrained flows of the Kaup - Newell, KdV, WKI and TG hierarchies, all generated by a two-dimensional eigenvalue problem. Some of the r-matrices thus obtained depend only on the spectral parameters, but others depend also on the dynamical variables. For consistency they have to obey a classical Yang - Baxter-type equation, possibly with dynamical extra terms.

We suggest that the recent numerical-integration perturbative approach to bound states as proposed by Skála and Cízek may be generalized via its renormalizations.