Table of contents

We investigate the dynamical system derived from the SU(2) Yang-Mills-Higgs theory with sphaleron solution for the case of spatial homogeneity. Numerical studies of the Poincare surface of section and the Lyapunov characteristic exponents show that this system exhibits an order-to-chaos transition similar to one found in the SU(2) magnetic monopole solution, which strongly suggests that such a transition is characteristic of the non-Abelian field theories with non-trivially topological solitons.

The anomalous exponent, eta _p, for the decay of the reunion probability of p vicious walkers, each of length N, in d (=2- epsilon ) dimensions, is shown to come from the multiplicative renormalization constant of a p directed polymer partition function. Using renormalization group (RG) we evaluate eta _p to O( epsilon ²). The survival probability exponent is eta _p/2. For p=2, our RG is exact and eta _p stops at O( epsilon ). For d=2, the log corrections are also determined. The number of walkers that are sure to reunite is 2 and has no epsilon expansion.

We investigate invasion percolation fingers in a quenched medium in which the randomness has a gradient following a power law both in the direction of the flow and perpendicular to it. The first gradient corresponds to a pressure gradient, the second one to the density of microcracks that arises in a self-organized way around a large crack. We give an argument for the value of the fractal dimension as found in previous simulations. We calculate the roughness exponent of the fingers and find a value consistent with 1/2 as predicted by the argument.

We consider the dynamics of a simple one-dimensional model and we discuss the phenomenon of ageing (i.e. the strong dependence of the dynamical correlation functions over the waiting time). Our model is the so-called random random walk, the toy model of a directed polymer evolving in a random medium.

A continuum model of a polymer with non-zero bending energy, fluctuating without overhangs in the half plane, is considered. The exact partition function is obtained from the Marshall-Watson solution of the Klein-Kramers equation for Brownian motion in the half space. The partition function contains information on probabilities associated with the integral of a Brownian curve and reproduces Sinai's t^-5/4 result for the asymptotic first passage time density. The t^-5/2 dependence of a different passage probability implies a first-order polymer adsorption transition for short-range pinning potentials.

We have studied the phase separation of a system with a conserved order parameter in the presence of a flat surface via Monte Carlo simulations. For various quench parameters, we observed the domain growth in a surface layer. The kinetics of this ordering was studied, both in the presence and absence of bulk phase separation. In the absence of bulk phase separation, the surface domains were found to follow model A growth kinetics. In the presence of bulk phase separation, a crossover in the surface kinetics between model A and model B dynamics was observed.

A hierarchy of typical integrable, coupled Burgers systems is proposed by introducing a special spectral problem involving q dependent variables u₀, u₁, ..., u_q-1. These systems possess a hereditary structure and may reduce to a hierarchy of Burgers equations under the reduction u_i=0, 0<or=i<or=q-2. It is shown that their flows and Lax operators commute mutually but each system is not Hamiltonian.

An integral transformation generalizing the Bargmann transform is established among two sets of wavefunctions associated with the configuration and Bargmann-Fock-Segal-like representations of a relativistic harmonic oscillator. The time-independent configuration-space wavefunctions are also studied and the lack of unitarity occurring when factorizing the time dependence of the wavefunction is solved by modifying the scalar product.

Within the framework of the J-matrix method of scattering, a given multichannel potential is modelled by its restriction on a subspace spanned by a certain set of L² functions. The scattering S-matrix is found exactly for this model potential at the Harris eigenvalues which result from the diagonalization of the scattering Hamiltonian in the same subspace. These values are then analytically continued in the complex energy plane. The poles of the S-matrix are then identified with the complex resonance energies. The associated widths for each resonance are then extracted from the residues of the S-matrix elements at the designated energy.

Nambu mechanics generalizes discrete classical Hamiltonian dynamics. Using the Euler equations for a rotating rigid body, the connection between this representation and noncanonical Hamiltonian mechanics is shown. Nambu mechanics is extended to incompressible ideal hydrodynamical fields using energy and helicity in 3D (enstrophy in 2D). The Hamiltonian and the Casimir invariants of the non-canonical Hamiltonian theory determine the dynamics in a non-singular trilinear bracket.

We analyse for a class of fragmentation models how the shattering transition is related to the behaviour of the size distribution function, and discuss existence criteria for its moments. For models with homogeneous fragmentation kernels exact expressions are derived for all moments in the shattering and non-shattering regimes.

Several attempts have been made recently to use wavelets transforms for extracting histograms of scaling exponents from experimental turbulence data. Similar techniques are here applied to a Gaussian signal having a Kolmogorov ⁵/₃ energy spectrum. This is an instance of the class of fractional Brownian motions, having scaling exponent ¹/₃ almost everywhere. For the Gaussian signal, a spurious non-trivial histogram is obtained by applying a Mexican hat wavelet transform analysis. On the other hand, we show that the use of complex wavelets and, even more so, the application of an optimal wavelet transform method strongly reduce the spurious fluctuations observed in the processing of the Gaussian signal.

Dynamical systems in SL(2, R) or SL(2, C) naturally appear in the transfer matrix method for quasiperiodic chains characterized by arbitrary irrational numbers. We show new subdynamical systems and invariants that are related to full diagonal and off-diagonal components of the transfer matrices; they are analogous to formulae of Chebyshev polynomials of the first and second kinds. Applying them to an electronic problem on the Fibonacci chain, we obtain sets of self-similar polynomials, quasiperiodic extension of the Chebyshev polynomials of the first and second kinds with self-similar properties. Two scaling factors of the self-similarities coincide with ones obtained by the perturbative decimation renormalization group method.

We study the percolation transition in a long-range correlated system: a self-affine surface. For all relevant physical cases (i.e. positive roughness exponents), it is found that the onset of percolation is governed by the largest wavelength of the height distribution, and thus self-averaging breaks down. Self-averaging is recovered for negative roughness exponents (i.e. power-law decay of the height pair correlation function) and, in this case, the critical exponents that characterize the transition are explicitly dependent on the roughness exponent above a threshold value. Below this threshold, the spatial correlations are no longer relevant. The problem is analytically investigated for a hierarchical network and by means of numerical simulations in two dimensions. Finally, we discuss the application of those properties to mercury porosimetry in cracks.

Dimer-dimer catalytic surface reaction of the type AB+C₂ to 1/2 A₂+C₂B is studied by Monte Carlo simulation both on square and hexagonal lattices. Various models are proposed and studied. For the case of a square lattice the three models in which we (i) ignore both the diffusion and desorption of various reactants (M1 model), (ii) consider diffusion of C species only (M2 model), and (iii) consider the diffusion of C atoms as well as their recombination and desorption (M3 Model) all give a final poisoned state for all feed concentrations. For model M1 there is a continuous crossover from one poisoned state to another, while for M2 and M3 an irreversible phase transition separates one poisoned state from the other. The diffusion of A atoms (M4 model) is found to be very crucial to the evolution of the system towards a final steady reactive state. The slightest movement of the A atoms releases the trapped vacancies and other reactants which are available for further reaction. An irreversible phase transition now separates a poisoned state from a steady reactive state. For the hexagonal lattice the M3 model already leads the system towards a steady reactive state. The role of increasing the number of nearest neighbouring sites is seen to be similar to that of the diffusion of A atoms.

We study the equilibrium properties of the real weights linear perceptron within the replica formalism framework, focusing on the effects of the normalization of the weights on the learning and generalization capabilities of the network. We also investigate the effects of static noise corrupting the training data and of dynamical noise acting on the weights during the training stage.

The problem of generalization by single-layer perceptrons is studied in the case of time-dependent rules. Lower bounds for the generalization errors within the 'single presentation of examples' case are obtained for randomly drifting rules. These bounds suggest a learning algorithm which uses knowledge of the error itself. Since this error is not readily available it has to be estimated through a mechanism of self-evaluation. The capacity of incorporating recency information into the error estimate is highly desirable. The mechanism proposed has the advantage, beyond good performance, of being self-adaptive, in the sense that it adjusts to changes in the unknown drift rate of the rule. The performance of the rule is also studied for sudden changes in an attempt to mimic the so-called Wisconsin test.

With a view to finding features of the weight-space of the binary perceptron that might be instructive for training binary-synapse neural networks, the maximally-stable perceptron having binary-valued weights is compared with continuous-weight perceptrons, for universal choices of stored patterns. The fraction of synaptic-weights correctly predicted by clipping the synapses of the continuous network is calculated in the thermodynamic limit and compared with simulation results for smaller systems. Numerical experiments show good agreement with theory but, in addition, indicate that those binary synapses likely to be wrongly predicted by weight-clipping are predominantly those which are weakest in the continuous-synapse perceptron. Although not rescuing training time from growing exponentially in the system size, our results suggest ways of significantly accelerating the search for successful, albeit possibly imperfect, neural networks with discrete-valued couplings.

The controversy regarding the occurrence of self-organized criticality in the cellular automaton Game of Life has not yet been resolved, mainly due to its massive computational requirements. We consider a one-dimensional version of Life which shows essentially the same local complexity as its two-dimensional counterpart but allows a more extensive computational implementation. General cluster statistics, geometrical properties and self-organized criticality are investigated. Implications concerning higher-dimensional Life are discussed.

The storage of biased patterns is examined in neural networks with sign-constrained synapses. Every neuron has outgoing synapses which are either all inhibitory or all excitatory. For random patterns stored in such networks, it is known that the presence of a discrete gauge symmetry makes the maximal storage capacity independent of the proportion of excitatory neurons to inhibitory neurons. When the stored patterns are biased, however, this discrete gauge symmetry is broken, with the result that the maximal capacity depends on the proportion of excitatory neurons to inhibitory ones. The dependence of the capacity on the fraction of excitatory neurons in the network, f, is calculated using the space of interactions approach. It is found that the storage capacity is maximal at f=0.5; this result is true regardless of the particular value of the bias in the stored patterns. The significance of this result in the neurophysiological context is discussed.

As the nonlinear parameter, k of Zaslavsky's map with twist increases, chains of periodic points are born simultaneously at the origin and move outwards. The positions of the periodic points as a function of k is investigated, and in a limited number of cases an analytic result is found. When k is very small, it is found that there is a universal relation for the radius of the periodic points as a function of the nonlinear parameter.

We consider the effect of a continuous family of neutral (bouncing ball) orbits on the energy spectrum of the quantized stadium billiard. Using a semiclassical approximation we derive analytic expressions for standard two-point spectral measures. The corrections due to the bouncing ball orbits account for some of the non-generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured. Once the bouncing ball contributions are subtracted, the spectrum is shown to be well reproduced by the semi-classical trace formula based on unstable periodic orbits. We also study special patterns in the spectrum which are due to other non-generic features such as edge effects and 'whispering gallery' trajectories.

A random walk model for the coexistence of diffusion and accelerator modes for a chaotic two-dimensional area-preserving map is constructed and solved analytically in order to explain the time behaviour of the numerically calculated diffusion coefficient for such maps.

In the spirit of establishing analogies and differences among systems with SU(1,1) and SU(2) algebras, we study the motion of an SU(1,1) kicked top in the semiclassical approximation as given by the coherent states representation. For this sake, we have proposed a Hamiltonian with the same algebraic structure as the one studied by Haake et al. (1987) for the SU(2) case, so as to investigate the modifications undergone by the phase portrait when changing a compact into a non-compact manifold. Analogously to the problem discussed by Haake et al., we obtain one involution and the associated symmetry line where fixed points lie; however, in contrast with the SU(2) case, there exists an infinite number of solutions for every set of parameters. When increasing the strength of the kick no new stationary points are born; instead, existing fixed points simply move towards the vertex of a curve and, eventually, two of them merge together and annihilate. No other type of bifurcation is detected.

We investigate the behaviour of the pairing model within the context of quantum algebras. The pairing Hamiltonian is diagonalized exactly for different values of the deformation parameter q in systems with 8 and 50 particles. The same Hamiltonian is solved with the help of the coherent state variational method. We find that the variational method gives reliable results when compared with the exact ones. We also discuss the differences between the two deformation procedures.

We insist on the possibility of regarding coherent states for the supersymmetric harmonic oscillator equivalently in three different ways, just as in the usual bosonic case.

It is shown that the finite dimensional irreducible representations of the quantum matrix algebra M_q,p(2) (the coordinate ring of GL_q,p(2)) exist only when both q and p are roots of unity. In this case the space of states has either the topology of a torus or a cylinder, which may be thought of as generalizations of cyclic representations.

Quantum kinematics is revisited, as a group-theoretic quantization procedure within the regular representation of non-Abelian non-compact r-dimensional Lie groups. The set of r basic quantum-kinematic invariant operators is exhibited; generalized Heisenberg commutation relations and the structure of the closed generalized Weyl-Heisenberg algebra of the quantized group are also discussed. Then it is shown how these structures yield a complete set of r 'annihilation' and 'creation' boson operators, which give rise to several intrinsic (i.e. embedded) Lie algebras, obtained in the standard way, within the quantized group model. As a miscellaneous example, these features are discussed within the quantum-kinematic theory of the Poincare group P₊^{up arrow} (1,1), and some interesting possibilities for elementary particle theory are conjectured in the light of the attained results.

We apply the inverse spectral problem method to the class of non-Abelian nonlinear lattice equations on the finite interval. The integrable discrete nonlinear Schrodinger and discrete modified Korteveg-de Vries equations are considered as examples. In the latter case the large time asymptotics for solutions are found.

We study spinor fields on the phase space of a generic Hamiltonian system. Under linearized canonical transformations these spinors transform according to the metaplectic representation of Sp(2N). We derive a path integral for their time evolution and discuss their dynamical and geometrical properties. In particular we show that they can be interpreted as semiclassical wavefunctions for the associated Hamiltonian.

The Laplace method is considered in application to one-dimensional two-state eigenvalue problems in the example of simple models, including that of coupled parabolic terms and radial oscillators with quadratic, linear and constant coupling (basic models of Renner, pseudo-Jahn-Teller effects etc.). In all cases, the solution can be presented either as a two-state analogue of the Bohr-Sommerfeld quasiclassical quantization rule, or in more general form. Compared with the semiclassical picture of inelastic collisions in the momentum-space representation, the contour of integration of 'trajectory equations' for the case of bound states is finite, and any desired degree of accuracy can easily be achieved. Because no convergence problems arise, the need to compute the equation possessing 'physical sense' (e.g. for adiabatic amplitudes) is circumvented, suggesting that the method would be applicable also to complicated potentials and to many-state problems. Numerical tests for the energy level positions and calculations of eigenfunctions are presented; practical applications and generalizations are outlined.

The new numerical approach to calculation of modified Mathieu functions is proposed. These functions play an important role in theories of electron scattering from (highly) polarizable atoms, like alkali. The algorithms we developed show very high accuracy in a wide range of energy and polarizability, which are the two principal parameters of the problem. The numerical scheme does not lose the accuracy in the so-called 'unstable' regions, where the characteristic exponent of Mathieu functions becomes complex. This stability makes possible the analytical continuation of these methods in the complex plane of parameters.

We apply the harmonic balance technique in low orders to three types of oscillator model. For three equations of the van der Pol type with rational nonlinear terms, used to model electronic oscillators, we examine the choice between: (a) rationalizing before expanding in harmonics, and (b) obtaining Fourier coefficients of nonlinear terms. Alternative (b) is shown to be more accurate, in first order, for the models of Scott-Murata and of Walker and Connelly, which apply to circuits with an inverse tangent nonlinear component. For these equations, in the case of alternative (a), we find acceptable dependence of the second-order corrections on the bifurcation parameter. When the harmonic balance method is used to set up a semi-classical quantization treatment of a nonlinear conservative oscillator, values of energies are slightly improved by the use of alternative (b). For conservative oscillators, with cubic or fifth power forces, we compare the standard method using the acceleration equation with an alternative using the energy equation. The first is shown to be more accurate in low order.

A new fermionic extension of the KdV hierarchy was recently found by Becker and Becker (1993) in relation to two-dimensional quantum supergravity, and it was shown that the hierarchy possesses a supersymmetry which has been exploited to prove the integrability of the hierarchy by Figueroa-O'Farrill and Stanciu (1993). Here, a larger group of fermionic symmetries of the new hierarchy is demonstrated and the bi-Hamiltonian structure is expressed in terms of odd Poisson brackets which are related to the antibracket of the Batalin-Vilkovisky formalism.

We apply the fusion procedure to a quantum Yang-Baxter algebra associated with time-discrete integrable systems, notably integrable quantum mappings. We present a general construction of higher-order quantum invariants for these systems. As an important class of examples, we present the Yang-Baxter structure of the Gel'fand-Dikii mapping hierarchy that we have introduced in previous papers, together with the corresponding explicit commuting family of quantum invariants.

Bluman, Reid and Kumei (1988) have introduced the potential symmetries, i.e. special nonlocal-symmetries. The concept of a non-local symmetry is aesthetically justified by the theory of coverings. The basic idea is that solutions of the covering system imply solutions of the covered system. Bluman, Kumei and Reid never considered the symmetries lost, i.e. local symmetries of the covered system that do not extend to local symmetries of the covering system. Here, examples of coverings are given that recover these lost symmetries, demonstrating the more natural setting of coverings.

The fermion interpretation of the white noise framework is considered. As an application, Ito's product formula of Applebaum, Hudson and Parthasarathy (1987) is extended to the generalized case.

The first-order quantum correction for the characterization of spontaneous radiation is calculated by means of electron quasi-classical trajectory-coherent states in an arbitrary electromagnetic field. Well known expressions for the characterization of spontaneous radiation are obtained using quasi-classical approximation. The first-order quantum correction is derived as a functional from a classical trajectory (among which is a classical spin vector). Transitions with spin flip and without spin flip are distinguished. Those elements connected with photon kick and quantum motion characteristics are selected for first-order quantum correction. It is shown that, using an ultra-relativistic approximation, the latter may be ignored, but when using a non-relativistic approximation their contributions are approximately equal. A special trajectory-coherent representation that significantly simplifies the investigation of spontaneous radiation is proposed.

We investigate how dissipation and nonlinearity affect an electromagnetic perturbation propagating into a saturated ferromagnet in the presence of an external magnetic field. We study the problem in (1+1) and (2+1) dimensions. It is found that at lowest order of the perturbation theory, the Burgers' equation in (1+1) dimensions governs such dynamics. In (2+1) dimensions we show that the phenomena obeys a nonlinear evolution equation (non-integrable) of Burgers type. We give exact solutions which describe in (1+1) dimensions the propagation of a travelling electromagnetic wave and the coalescence of N travelling fronts and in (2+1) dimensions the propagation of a nearly one-dimensional travelling front. We establish, in terms of the physical parameters of the system, whether breaking or diffusion of the initial perturbation dominates.

Vakhnenko's equation has two families of travelling wave solutions. The method of Rowlands and Infeld (1990) is used to investigate whether these solutions are stable to long wavelength perturbations of small amplitude. The method predicts stability for both families of solutions. Some comments on the validity of the method are given.

An analytical study of the influence of the long-range atomic interactions on the properties of soliton-like excitations in a one-dimensional (1D) anharmonic chain is presented. The model chosen is a nonlinear diatomic chain in which atoms are assumed to interact via a cubic and/or quartic nonlinear short-range potential and a linear long-range Kac-Baker type pair potential. In the continuum approximation, using scaling arguments, it is shown that the coupled nonlinear difference-differential equations for the motion of the two different masses can be decoupled and reduced to a generalized Boussinesq equation which admits supersonic and subsonic acoustic kink (pulse) solitons, long-wavelength acoustic oscillating solutions of breather type and optical envelope type solitons of a nonlinear Schrodinger equation. A possible alternation of envelope and dark is found that can exist not only for acoustic mode but also for optical mode.

Rigorous relationships among radial and logarithmic expectation values of an N-body system are obtained by means of information-theoretic methods. Especially interesting are the uncertainty expressions in terms of the product ((r^alpha )¹ alpha /(p^beta )¹ beta /, which generalize the well known relation (r²))(p²)>or=9N²/4. Additionally, an inequality involving the uncertainty in the logarithmic radius and the logarithmic momentum is shown. For illustration, the accuracy of the inequalities found is numerically analysed in a Hartree-Fock framework for atomic systems. Finally, it is pointed out how these inequalities are improved in atoms by taking into account other physical quantities or by considering some semiempirical results on atomic information entropies.

The theory of classical relativistic spinning particles with c-number internal spinor variables, modelling accurately the Dirac electron, is generalized to particles with anomalous magnetic moments. The equations of motion are derived and the problem of spin precession is discussed and compared with other theories of spin.

It is shown that the Schrodinger equation for a non-relativistic three-body system can be written in closed form by using only nine generators of three independent O(2, 1) algebras and a few physical parameters (three masses and potential characteristics). In the case of Coulomb three-body systems such a form contains only a finite number of terms and six physical parameters (three masses and three charges). For a number of Coulomb three-body systems with unit charges (i.e. X⁺X⁺Z^- and X⁺Y⁺Z^-), optimized parameters for simple approximate wavefunctions have been obtained.

The adiabatic limit of the transition probability for two-level systems driven by real symmetric time-dependent Hamiltonians is considered. When the Hamiltonian depends analytically on time, the transition probability is governed, in simple cases, by a complex eigenvalue crossing point which is, generically, a square root branching point for the eigenvalues. In such a case, the transition probability is given by the Dykhne formula, the recently discovered geometric prefactor being equal to one. In this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n/2, n>or=1, for the eigenvalues and a zero of order m>or=0 for the Hamiltonian itself. The analysis shows that the Dykhne formula must be completed by a novel prefactor which depends on both n and m. In particular, this prefactor can take the value zero, in contrast to the geometrical prefactor. We also consider the case where the transition probability is governed by N complex eigenvalue crossing points of different orders n_j and m_j, j=1, ..., N. The end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points n_j=1 and m_j=0 considered earlier.

For pt I. see ibid, vol.23, p.1555 (1990). A method to evaluate perturbations of arbitrary spectra by one of the classical ensembles was presented in a previous paper. Its application to non-trivial problems is cumbersome and I shall show that by combining the previous results with singular perturbation theory, more complicated problems can be tackled. A scaling argument is also presented to obtain a general result that goes beyond the linear repulsion regime that we discussed previously. This result is of considerable interest as it allows us to obtain a good idea of the correlation function if we, in addition, know its long-range behaviour, which can be obtained by straightforward perturbation theory.

The eigenvalue problem for the Rabi and the E(X) epsilon Jahn-Teller Hamiltonian in Bargmanns Hilbert space of analytical functions is a system of two first-order differential equations for the two-component wavefunctions, whose entire solutions (the eigenstates) are sought. We show that each eigenstate is a terminating series in the derivatives of a scalar entire function D(z), called the generalized potential, which satisfies a higher-order differential equation. The coefficients of the terminating series depend on the physical parameters and are polynomials in the independent variable z. The coefficients are identical in all eigenstates.

A new method of obtaining many-dimensional quasi-exactly-solvable models is suggested. It is based on constructing the generating function with the help of coefficients which obey a finite difference equation. The structure of this equation is selected to obtain the closed second-order differential equation for the generating function. Under some conditions this equation can be thought of as the Schrodinger equation in curved space. For the two-dimensional case the many-parametric class of solution is found explicitly. The spherically-symmetrical case is investigated in detail. It is shown that this case contains spaces of a constant Riemann curvature of both signs.

The problem of determination of the flat coordinates for a model of a topological conformal field theory corresponding to the 'twisted' version of an N=2 superconformal Landau-Ginzburg field theory with central charge c=3 is analysed. The model is characterized by a Landau-Ginzburg superpotential of the form: W= 1/4 x⁴+ 1/4 y⁴. All possible relevant and marginal perturbations with their corresponding couplings (expressed as functions of the dimensionless flat-coordinate) are added to the above superpotential to give the perturbed topological field theory model. It is seen that the couplings can be completely determined (and hence also the dependence of the perturbations on the flat coordinate) by imposing the conditions of flatness on the space of couplings of the perturbed theory.