Table of contents

The antipodal correlations between opposing sectors of large critical percolation clusters are negative, as expected, but vanish in the infinite size limit approximated by an inner square much smaller than the whole lattice but much larger than the nearest-neighbour distance. Antipodal correlations are a new numerical measure of fractal lacunarity, that is, of texture: zero correlation expressed that the lacunarity is neutral in percolation clusters.

Several works have recently reported 1/f noise in the number of particles in some physical systems where white noise is imposed on the boundary. We show that these results should not hold in general because they were not derived with a correct definition of the power spectrum.

We combine random walks, growth and decay, and convection, in a Monte Carlo simulation to model 1D interface dynamics with fluctuations. The continuum limit corresponds to the deterministic Fisher equation with convection. We find qualitatively the same type of asymmetry, as well as velocity difference, for interface profiles moving in opposite directions. However, a transition apparent in the mean-field (continuum) limit is not found in the Monte Carlo simulation.

Using molecular dynamics (MD) simulations, we find the formation of a heap in a system of granular particles contained in a box with oscillating bottom and sidewalls. The simulation includes the effect of static friction, which is found to be crucial in maintaining a stable heap. We also find another mechanism for heap formation in systems under constant vertical shear. In both systems, heaps are formed due to a net downward shear by the sidewalls. We discuss the origin of net downward shear for the vibration-induced heap.

Collective diffusion of particles is studied in a three-dimensional lattice-gas model with site-energy disorder. The distribution of the site energies is exponential, and the diffusion coefficient of independent particles may vanish. It is demonstrated by numerical simulations that a coefficient of collective diffusion exists. An estimate of the diffusion coefficient for smaller particle concentrations suggests a power-law dependence on concentration, consistent with the simulations. An effective-medium calculation of the coefficient of collective diffusion gives good agreement with the data at smaller particle concentrations and rough agreement at larger concentrations.

A method to construct various s= 1/2 isotropic spin models with exact dimer ground states is proposed. Constructed models are usually of much simpler forms than those produced by the method using Lowdin's projection operators. They include a series of one-dimensional Heisenberg models which consist of linearly decayed exchange interactions and are natural extensions of the Majumdar-Ghosh model. A two-dimensional spin model with four spin interactions is also constructed to have the same spatial symmetry as a simple square lattice.

Two-parameter (p,q) deformation of a parabosonic algebra underlying the two-particle Calogero model is considered. The Fock-space representation, Bargmann-Fock representation, coherent states, spectrum-generating algebra and a hidden supersymmetry are discussed. The paraboson algebra induces a centreless Virasoro-type algebra with a doubling of the space of generators. A two-parameter deformation of this Virasoro-type algebra is studied.

The known periodic self-Fourier-Fresnel function is comb(x,y). We show that the 2D periodic functions assembled by the Dirac comb with cell and aperture functions are the approximate self-Fourier-Fresnel functions if the cell function is compact and the cell and aperture functions are even and mutually Fourier transformable.

We develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations. Using it we obtain a number of new two-dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions.

Hawking's zeta-function regularization procedure (1977) is shown to be rigorously and uniquely defined, thus putting an end to the spreading lore about different difficulties associated with it. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method-and other analytical procedures-are clarified and corrected.

We show by means of a low-temperature series expansion that a large, possibly infinite number of interface layering transitions are observed for the Abraham model in dimensions d>or=3.

A hierarchical cluster-cluster aggregation computer model is introduced which allows one to build random fractal aggregates on a d-dimensional lattice with a fractal dimension fixed a priori. The algorithm works iteratively by sticking aggregates of the same number of particles at the correct centre-to-centre distance in order to recover the desired scaling. With the more efficient versions of the model, any fractal dimension ranging from 1 up to a d-dependent upper limit D_M(d) can be obtained. One estimates D_M(2) approximately=1.80+or-0.03 and D_M(3) approximately=2.55+or-0.04. Calculations up to d=8 show that the ratio D_M(d)/d decreases as d increases.

We study, using dimer and Monte Carlo approaches, the critical properties and finite-size effects of the Ising model on honeycomb lattices folded on the tetrahedron. We show that the main critical exponents are not affected by the presence of conical singularities. The finite-size scaling of the position of the maxima of the specific heat does not match, however, with the scaling of the correlation length, and the thermodynamic limit is attained faster on the spherical surface than in corresponding lattices on the torus.

Domain growth in the three-dimensional random-field Ising model is investigated by Monte Carlo simulation. The time dependence of the average domain size is monitored following a quench to a low temperature. The logarithmic growth predicted by the theories of Villain (1984) and of Grinstein and Fernandez (1984) is observed. The dynamical scaling of the structure factor is also studied and found to be well satisfied. The field strength dependence of the scaling function is found to be weak.

We present a general method to calculate the connected correlation function of random Ising chains at zero temperature. This quantity is shown to relate to the survival probability of some one-dimensional, adsorbing random walker on a finite interval, the size of which is controlled by the strength of the randomness. The correlation length is exactly calculated for different random field and random bond distributions.

A model for antiferromagnetic re-entrant systems is developed. The underlying mechanism for the re-entrance is based on competing interactions between Ising and Potts variables. Renormalization-group calculations are carried out using a modified Migdal-Kadanoff approximation. The resulting global phase diagram is presented and important features are discussed. The model is then applied to liquid crystals, producing qualitative agreement with the behaviour of certain re-entrant liquid crystal systems.

At a continuous transition into a non-unique absorbing state, particle systems may exhibit non-universal critical behaviour, in apparent violation of hyperscaling. We propose a generalized scaling theory for dynamic critical behaviour at a transition to an absorbing state, which is capable of describing exponents which vary according to the initial configuration. The resulting hyperscaling relation is supported by simulations of two lattice models.

A scheme is presented which allows for the computation of the topological pressure and power spectra associated with equilibrium measures from a given time series. Its convergence properties are demonstrated on numerical data sets obtained from the Lozi and the Henon map. Numerical evidence for sharp peaks in the power spectrum of the non-hyperbolic phase is given and supported by estimates using periodic-orbit expansions.

A ring cavity system showing chaotic behaviour is constructed with three hybrid Michelson interferometers unidirectionally coupled to one another. This is for the first step in the study of spatiotemporal chaos. This system exhibits spatiotemporal chaotic behaviours of the type described by Ikeda and Otsuka. It shows local bifurcation phenomena, such as pitchfork bifurcation and saddle-node bifurcation, as well as Hopf bifurcation. Also, an isola-type solution mode was found for the system and a target-shooting method was used to make the system reach this isola mode. We have observed an interior crisis in this isola mode when there is a delay in the feedback signal to the nonlinear elements. Both the computer simulations and experimental work were performed to investigate these phenomena.

The intricate connection between the distribution of exceptional points and the spectral behaviour of Hamiltonian systems is investigated. A method to determine the distribution of exceptional points for a Hamilton operator with systematic degeneracies is described. Its implementation is demonstrated considering the hydrogen atom in a strong magnetic field. For this case, the distribution of exceptional points is a function of the scaled energy only.

The multicomponent q-coherent states associated with the quantum algebra sl_q(3) are presented. The expansions are discussed for arbitrary operators in terms of the multicomponent q-coherent states of sl_q(3). An inhomogeneous differential realization of sl_q(3) in this multicomponent q-coherent state space is obtained.

It is well known that the 9-j coefficient has 72 symmetries and that its simplest known form is the triple-sum series which exhibits none of these symmetries. A study of the expressions for the stretched 9-j coefficients (stretched in the sense that that one or more of its six triangle inequality components are stretched) via the triple-sum series shows the existence of the ₂F₁(1), ₃F₂(1) and the ₄F₃(1) summation theorems and reveals some new summation theorems as well.

A 'colour' representation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra, which, in contrast to the standard case, is related to the coloured braid group representation with generic values of q is presented. Explicit realizations of L^(+or-)-matrices, occurring in this coloured variant of FRT algebra, are also obtained for the U_q(gl(2)) quantized algebra. Though these realizations are found to depend manifestly on the colour parameters, the underlying quantum group structure and associated co-product are interestingly free from such dependence. This allows us to perform the Yang-Baxterization of the coloured FRT algebra successfully, which leads to the construction of an ancestor Law operator associated with a new non-additive-type quantum R-matrix. Through different realizations of this Lax operator, a new class of quantum integrable models representing 'colour' generalizations of the well known models, such as the lattice sine-Gordon model, the Ablowitz-Ladik model lattice and the derivative nonlinear Schrodinger model etc, is generated.

An extensive computer search for non-trivial zeros of the 6-j coefficients of SU(2) has been performed to find their frequency as a function of the sum of their quantum numbers and of their polynomial degree. For the zeros of degree 2, four of the seven cases already known (taking into account Regge symmetries) of two relations between quantum numbers relevant to a Pell equation have been generalized to deal with only one relation; a new case of two relations and many new cases of three relations, all with polynomial solutions. Have been found; but all these formulae only account for one zero out of seven. For degree 3, we found four new cases of two relations and some other cases of three relations with polynomial solutions; with the already known case of a Pell equation, they account for almost half of the zeros. The fast decrease of the number of zeros with the degree and the relative smallness of their quantum number is an indication that there should be no zeros of degree larger than 9.

In the framework of the variational approach to the collective-variable formalism the radiation field generated by the multidimensional domain walls is found and the dissipation of these walls is considered. Also, the drift velocity of the domain wall in an external field is calculated.

We examine the low-energy limit of the nonlinear Dirac equation (NLDE) in 1+1 dimensions with a Lorentz scalar self-interaction. Unlike the nonlinear Schrodinger equation (NLSE), which is integrable, the NLDE is known to exhibit rich dynamics of the soliton-soliton collision when the relative speed of the solitons is small. The NLDE is intrinsically different from the NLSE even when the energy involved is small. When it is modified by adding a specific correction term, however, the NLSE well reproduces the complex features of the soliton-soliton collision described by the NLDE.

We discuss the freedom in the background field (vacuum) on top of which the solitons are built. If the Hirota bilinear form of a soliton equation is given by A(D_x)G.F=0, B(D_x)(F.F-G.G)=0, where both A and B are even polynomials in their variables, then there can be a continuum of vacua, parametrized by a vacuum angle phi . The ramifications of this freedom for the construction of one- and two-soliton solutions are discussed. We find, for example, that once the angle phi is fixed and we choose u=tan^-1 G/F as the physical quantity, then there are four different solitons (or kinks) connecting the vacuum angles +or- phi , +or- phi +or- pi /2 (where pi is the defined modulo). The most interesting result is the existence of a 'ghost' soliton; goes over to the vacuum but interacts with 'normal' solitons by giving them a finite phase shift.

We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes' two-point model (1986) (which is an essential ingredient of his reformulation of the standard model of elementary particle physics) is recovered in our approach. Reductions of the universal differential calculus to 'lower-dimensional' differential calculi are considered. The 'complete reduction' leads to a differential calculus on a periodic lattice.

A number of researches have been made for the (Euclidean) Taub-NUT metric, because the geodesic for this metric describes approximately the motion of well separated monopole-monopole interaction. From the viewpoint of dynamical systems, it is well known also that the Taub-NUT metric admits the Kepler-type symmetry, and hence provides a nontrivial generalization of the Kepler problem. More specifically speaking, because of a U(1) symmetry, the geodesic flow system as a Hamiltonian system for the Taub-NUT metric is reduced to a Hamiltonian system which admits a conserved Runge-Lenz-like vector in addition to the angular momentum vector, and thereby whose trajectories turn out to be conic sections. In particular, all the bounded trajectories of the reduced system are closed. The Taub-NUT metrics is generalized so that the reduced system may remain to have the property that all of bounded trajectories are closed. On the application of Bertrand's method to the reduced system, two types of systems are found: one is called the Kepler-type system and the other the harmonic oscillator-type system. Correspondingly, two types of metrics come out: the Kepler-type metric and the harmonic oscillator-type metric. Furthermore, the symmetry of the Kepler-type system and of the harmonic oscillator-type system are studied through forming accidental first integrals. Thus the generalization of the Taub-NUT metric accomplishes nontrivial generalizations of the Kepler problem and the harmonic oscillator.

Duality relations, which associate two Markov processes in different state spaces, have been useful tools in the study of the long-term behaviour of the stochastic processes of spin systems. In 1986, Gray introduced a new duality theory which is applicable to general spin systems with attractive transition rates. The theory was developed by making full use of graphical representations. In the present paper, Gray's duality is reformulated by studying the action of the formal generators on a newly chosen duality function and the dual processes defined in the state space Upsilon =(A:A is a finite subset of Y) are discussed, where Y is a collection of finite subsets of Z^d. As applications of the submodularity of the survival probability sigma (A) of the dual processes, rigorous lower bounds of the critical values are derived for the theta -contact process, the multi-particle creation model and the sexual reproduction process.

Initially, the canonical form H(J) of the Hamiltonian H(q,p)= 1/2 p²+ 1/2 omega ₀²q²+ lambda q^alpha ( omega ₀, lambda epsilon R and alpha >2), which represents a large class of anharmonic oscillators, is expressed in a closed form. This is a new result, as is the subsequent closed form of the pulsation omega (J). The connection between series reversion and inversion, as applied to generalized hypergeometric functions, is also considered and extended to apply to arbitrary series.

The paper presents a discussion of the relation between the dynamics of a mechanical system based upon a Lagrangian admitting energy conservation and the dynamics based upon its Jacobi Lagrangian, which determines the space trajectories of the system. The basic result found in the paper is the general solution of the inverse problem, i.e. how to determine the full Lagrangian, when as a starting point, an arbitrary homogeneous Lagrangian, which is used to determine the space trajectories of a system, and an arbitrarily assigned energy function which specifies the interaction of the system are given.

Similarly to the Kadomtsev-Petviashvili (KP) equation, a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2+1 space dimensions. These symmetries constitute an infinite-dimensional Lie algebra which is a generalization to the known w_infinity algebra.

We show that the modulation of an electromagnetic wave in a ferromagnet in the presence of an external magnetic field is governed by the nonlinear Schrodinger equation. We characterize the existence of an oscillatory instability of the Benjamin-Feir type for electromagnetic propagation in a saturated ferrite. Depending on the physical parameters of the system, we establish the regions in which focusing or defocusing of the initial carrier envelope occurs. We show that one of the two plane waves, circularly polarized, allowed to propagate into a ferrite, is bunching into solitons and the other is modulated by a dark soliton.

It is explained, using a coherent states approach that classical minimal coupling in the form introduced by Sternberg (1977) may be obtained as a classical limit from quantum minimal coupling. A similar procedure is used to exhibit Weinstein's symplectic structure (1978) on the space of WKB amplitudes as a classical limit.

We study the quantum Knizhnik-Zamolodchikov equation (1984) of level 0 associated with the spin 1/2 representation of U_q(SL₂). We find an integral formula for solutions in the case of an arbitrary total spin and mod q mod <1. In the formula, different solutions can be obtained by taking different integral kernels with the cycle of integration being fixed.

We address the question as to whether relativistic effects have any influence on localization phenomena in disordered systems. Starting from the Schrodinger and Dirac equations for Kronig-Penney potentials, we derive their discrete counterparts within the framework of the linear combination of atomic orbitals and tight-binding approaches. As a specific example we subsequently focus on substitutional binary alloys. We show that taking relativity into account leads to many important differences with respect to the results obtained in the non-relativistic approximation. The predicted behaviour of localization length can be very relevant in several experimental contexts.

A simple necessary and sufficient condition which tells when an A-particle fermion wavefunction can be written as a Slater determinant is proven.

A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.