Table of contents

A class of games with two players, who base their actions on the results of the previous round, is considered. These games are generalizations of the Iterated Prisoner's Dilemma. The best counterstrategy for player 1, given a strategy for player 2, is found by treating these games as Markov processes. Several transitions between such best strategies are found and shown to be akin to first-order phase transitions. The roles of a number of special strategies are elucidated. It is shown that there is a strategy for player 2 that never loses, even if this player does not know what kind of game is being played.

The axial next-nearest-neighbour Ising (ANNNI) model is studied for thin films of up to L = 10 layers, with a distinct phase diagram for each film thickness. The systematics of the ordered phases, as obtained from mean-field theory, Monte Carlo simulations and low-temperature expansions, is discussed. Results are compared to those for the ANNNI model in the limit L⟶∞.

We consider a general N-degrees-of-freedom nonlinear system which is chaotic and dissipative, and show that the nature of chaotic diffusion is reflected in the correlation of fluctuation of the linear stability matrix for the equation of motion of the dynamical system whose phase space variables behave as stochastic variables in the chaotic regime. Based on a Fokker-Planck description of the system in the associated tangent space and an information entropy balance equation, a relationship between chaotic diffusion and the thermodynamically inspired quantities such as entropy production and entropy flux is established. The theoretical propositions have been verified by numerical experiments.

We calculate numerically the exact relaxation spectrum of the totally asymmetric simple exclusion process (TASEP) with open boundary conditions on lattices up to 16 sites. In the low- and high-density phases and along the nonequilibrium first-order phase transition between these phases, but sufficiently far away from the second-order phase transition into the maximal-current phase, the low-lying spectrum (corresponding to the longest relaxation times) agrees well with the spectrum of a biased random walker confined to a finite lattice of the same size. The hopping rates of this random walk are given by the hopping rates of a shock (a domain wall separating stationary low- and high-density regions), which are calculated in the framework of a recently developed non-equilibrium version of Zel'dovich's theory of the kinetics of first-order transitions. We conclude that the description of the domain wall motion in the TASEP in terms of this theory of boundary-induced phase transitions is meaningful for very small systems of the order of ten lattice sites.

The classical two-dimensional anisotropic triangular nearest-neighbour Ising (ATNNI) model is studied by the density matrix renormalization group (DMRG) technique when periodic boundary conditions are imposed. Applying the finite-size scaling to the DMRG results, a commensurate-disordered (CD) phase transition line as well as temperature and magnetic critical exponents are calculated. We conclude that the CD phase transition in the ATNNI model belongs to the same universality class as the ordered-disordered phase transition of the Ising model.

The Ising system with a small fraction of random long-range interactions is the simplest example of small-world phenomena in physics. Considering the latter both in an annealed and in a quenched state we conclude that: (a) the existence of random long-range interactions leads to a phase transition in the one-dimensional case and (b) there is a minimal average number p of these interactions per site (p<1 in the annealed state, and p≃1 in the quenched state) needed for the appearance of the phase transition. Note that the average number of these bonds, pN/2, is much smaller than the total number of bonds, N²/2.

The spatial distribution of persistent spins at zero temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ(t)~t^Z, is identified such that for length scales r<<ξ(t) the persistent spins form a fractal with dimension d_f; for length scales r>>ξ(t) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z(2-d_f) with θ = 0.209±0.002 of Jain (Jain S 1999 Phys. Rev. E 59 R2493), Z = 1/2 and d_f~1.58.

Information dynamics is discussed from the point of view of microscopic thermal flow. In the fields of learning, association, storage and so on, the concepts of neural networks (NNWs) have been widely used. Their neurodynamics have been investigated as a stochastic process of an infinite neuron system, using the replica method. This approach includes unsettled points; regimes where replica symmetry (RS) solutions and replica symmetry breaking (RSB) solutions are valid, low-dimensional behaviour regimes, and so forth.

First, to make them clear, using supersymmetry (SUSY) fields the dynamics of NNWs are investigated for a family of NNWs interacting among m neurons. The dynamics of the system are supposed to be specified according to the Langevin dynamics with Gaussian white noise (i.e. a random influence from the surroundings) under an environmental parameter β (such as the inverse temperature). The results obtained without ambiguity are as follows: the RS solutions are valid in the regime where our solutions satisfy the fluctuation-dissipation theorem (FDT), while the RSB solutions appear in the SUSY-breaking regime. As a function of the environmental parameter, the system displays transitions from usual (ergodic) phases to phases with broken ergodicity.

Secondly, the information dynamics of NNW is derived as the microscopic thermal flow.

The algebraic-geometric approach is extended to study evolution equations associated with the energy-dependent Schrödinger operators having potentials with poles in the spectral parameter, in connection with Hamiltonian flows on nonlinear subvarieties of Jacobi varieties. The general approach is demonstrated by using new parametrizations for constructing quasi-periodic solutions of the shallow-water and Dym-type equations in terms of theta-functions. A qualitative description of real-valued solutions is provided.

We show that SU(n) Bethe ansatz equations with arbitrary `twist' parameters are hidden inside certain nth-order ordinary differential equations, and discuss various consequences of this fact.

We continue our study of non-Abelian gauge theories in the framework of the Epstein-Glaser approach to renormalization theory. We consider the case when massive spin-1 bosons are present in the theory and we modify appropriately the analysis of the origin of the gauge invariance performed in a preceding paper in the case of null-mass spin-1 bosons. Then we are able to extend a result of Dütsch and Scharf concerning the uniqueness of the standard model, consistent with renormalization theory. In fact we consider the most general case, i.e. the consistent interaction of r spin-1 bosons, and we do not impose any restrictions on the gauge group and the mass spectrum of the theory. We show that, besides the natural emergence of a group structure (as in the massless case), we obtain new conditions of a group theoretical nature, namely the existence of a certain representation of the gauge group associated to the Higgs fields. Some other mass relations connecting the structure constants of the gauge group and the masses of the bosons emerge naturally. The proof is carried out using the Epstein-Glaser approach to renormalization theory.

The nonlinear evolution equations for three-wave resonances including intrapulse dispersion are solved by a special ansatz. Several types of three-wave solitary waves and kink solutions are provided explicitly.

Certain quasicrystals will be realized in cyclotomic fields, and their isomorphism structures will be given in the case of seven-fold or 30-fold symmetry.

We develop an analytical procedure to determine orbits that a harmonically driven, damped pendulum describes in the phase plane. The theory predicts the existence of more than one solution for the same system, depending on initial conditions. Also, it predicts a stable solution around the top position of the pendulum. Trajectories obtained by numerically integrating the pendulum equation in a phase-locked condition agree with our diagrams. Some periodic solutions were found that are not orbitally stable.

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations.

We describe irreducible representations of the extended Poincaré parasuperalgebra (PPSA) which includes an arbitrary number N of parasupercharges, n (n⩽{N/2}) central charges and an internal symmetry group. We also discuss wave equations which are invariant with respect to the PPSA and propose a parasupersymmetric generalization of the Wess-Zumino model for arbitrary p and N.

We find WKB expansions for the spectra of the three-dimensional isotropic harmonic oscillator and Eckart and Morse potentials. It is shown that for all these potentials the series are convergent and, when summed, yield the exact energy spectra. These results support our working hypothesis that the WKB series for the energy spectra are convergent for all one-dimensional solvable potentials and their sums yield the exact energies.

A study is provided of the transmission of a three-dimensional electromagnetic X-wave undergoing frustrated total internal reflection on the upper surface of a multi-layered structure. The stratified structure consists of successive layers alternately allowing the transmission of evanescent and free-propagation components. It is shown that the peak of an X-wave is transmitted through these successive layers at an ultra-fast speed. Under certain conditions, the total traversal time through all successive evanescent and free-propagation sections appears to be less than zero. The peak of the transmitted pulse emerges from the stack before the incident peak reaches the front surface of the stratified structure. Conditions for the materialization of this ultra-fast multiple tunnelling of pulses are pointed out and their consequences and limitations are discussed.

This paper explores the connection between wavelet methods and an efficient computational algorithm - the discrete singular convolution (DSC). Many new DSC kernels are constructed and they are identified as wavelet scaling functions. Two approaches are proposed to generate wavelets from DSC kernels. Two well known examples, the Canny filter and the Mexican hat wavelet, are found to be special cases of the present DSC kernel-generated wavelets. A family of wavelet generators proposed in this paper are found to form an infinite-dimensional Lie group which has an invariant subgroup of translation and dilation. If DSC kernels form an orthogonal system, they are found to span a wavelet subspace in a multiresolution analysis.

Nonunitary representations of a novel realization of the su(2) algebra recently introduced for the Dirac relativistic hydrogen atom are found to be actually unitary representations of a related su(1,1) algebra.

The following important corrections need to be made to this paper.

(1) Theorem 3.3 is correct as stated. Still one should make the choice of evolution

in order to obtain the proper result in theorem 4.2.

(2) Theorem 4.2 is wrong as stated and the evolution shown is not a Hamiltonian evolution. With the choice of curve evolution given above, evolution (4.3) in the paper becomes

which is Hamiltonian. In fact, it is equivalent to the Poisson structure of the so-called complexly coupled KdV system (Wang 1998), which has a compatible symplectic structure. The proof of the theorem remains identical with the corresponding natural changes.

(3) Some minor typographical errors are found in the expression of the moving frame in theorem 2.16, where a₃³ should read

and where the central terms in a₂⁴ and a₃⁴ should be divided by |ct|³. The interested reader probably guessed these inaccuracies.

I would like to thank Dr Jing Ping Wang at the Vrij Universiteit in Amsterdam for pointing out the mistake in (2), and the connection of the correct result to the complexly coupled KdV.

Reference

[1] Jing Ping Wang 1998 Symmetries and conservation laws of evolution equations PhD Thesis Vrij Universiteit, Amsterdam

The third un-numbered equation on page 7957 (i.e. `the well known Dirac equation') should read as:

This equation has already been corrected in the online edition but appears incorrectly in the print edition.