Table of contents

The Lagrangian star is a germ of the system of Lagrangian submanifolds in the symplectic manifold . We investigate the symplectic group action on Lagrangian stars and construct the basic invariants of such action. The Kashiwara signature for 3-Lagrangian linear stars is generalized to the nonlinear case and the generalized contact classes for Lagrangian stars are constructed. Finally, we obtain the generic classification of simple normal forms of reduced Lagrangian stars with respect to a hypersurface.

We investigate the AC conductivity of binary random impedance networks, with emphasis on its dependence on the ratio , with and being the complex conductances of both phases, occurring with respective probabilities p and 1-p. We propose an algorithm to determine the rational h-dependence of the conductance of a finite network, in terms of its poles and of the associated residues. The poles, which lie on the negative real h-axis, are called resonances, since they show up as narrow resonances in the AC conductance of the RL-C model of a metal-dielectric composite with a high quality factor Q. This approach is an extension of a previous work devoted to the dielectric resonances of isolated finite clusters. A numerical implementation of the algorithm, on the example of the square lattice, allows a detailed investigation of the resonant dielectric response of the binary model, including the p-dependence of the density of resonances and the associated spectral function, the Lifshitz behaviour of these quantities near the endpoints of the spectrum of resonances, the distribution of spacings between neighbouring resonances, and the Q-dependence of the fraction of visible resonances in the RL-C model. The distribution of the local electric fields at resonance is found to be multifractal. This result is put into perspective with the giant surface-enhanced Raman scattering observed, for example, in semicontinuous metal films.

The fluctuation theorem of Gallavotti and Cohen holds for finite systems undergoing Langevin dynamics. In such a context all non-trivial ergodic theory issues are bypassed, and the theorem takes a particularly simple form. As a particular case, we obtain a nonlinear fluctuation-dissipation theorem valid for equilibrium systems perturbed by arbitrarily strong fields.

The ground states of the Bernasconi model are binary sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with one-valued off-peak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical theory of cyclic difference sets, we specify all values of N for which perfect sequences exist and how to construct them. For other values of N, we investigate almost perfect sequences, i.e. sequences with two-valued off-peak correlations of minimum amount. Numerical and analytical results support the conjecture that almost perfect sequences exist for all values of N, but that they are not always ground states. We present a construction for low-energy configurations that works if N is the product of two odd primes.

We study numerically the region above the critical temperature of the four-dimensional random field Ising model. Using a cluster dynamic we measure the connected and disconnected magnetic susceptibility and the connected and disconnected overlap susceptibility. We use a bimodal distribution of the field with for all temperatures and a lattice size L = 16. Through a least-square fit we determine the critical temperature at which the two susceptibilities diverge. We also determine the critical exponents and . We find that the magnetic susceptibility and the overlap susceptibility diverge at two different temperatures. This is coherent with the existence of a glassy phase above . Accordingly with other simulations we find . In this case we have a scaling theory with two independent critical exponents.

A general theoretical treatment is developed for the solution of the time-dependent coagulation equation (with constant coagulation kernel) in the presence of a source term possessing arbitrary time dependence. It is shown how the relevant nonlinear first-order differential equation can be transformed into a linear second-order equation, which can then be used to obtain the general solution of the problem together with information about its asymptotic long-term behaviour. The technique is applied to a periodic source term where it is found that the long-term behaviour of the solution exhibits the same periodicity as the source. Detailed results are derived for particular source terms.

We discuss a self-avoiding walk model of the adsorption of a copolymer at a plane surface, where the copolymer contains two types of monomers. We concentrate on the case where one type of monomer (A) has a short-range interaction with the surface and the other (B) has no interaction with the surface. We show that the sequence distribution of the comonomers affects the location of the adsorption transition and that for some sequence distributions the adsorption transition occurs at the same place as for the homopolymer poly-A. Moreover, in some circumstances the crossover exponent is identical to that of poly-A.

An approach to the problem of Taylor dispersion in the Taylor vortex is given where use of the slaving principle of Haken is made. This new approach is also used in the study of the axial dispersion of particles with inertia in the Taylor vortex. An homogenization approach to the derived dispersion equation is given, making use of the multiple scales perturbation theory.

A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir elements is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the algebra and the oscillator algebra are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the (1 + 1) Poincaré algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

A method for constructing explicit exact solutions to nonlinear evolution equations is further developed. The method is based on consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high-order ordinary differential equation. The method is then applied to a free boundary problem based on the process of precipitant-assisted protein crystal growth.

We introduce an integrable inhomogeneous generalization of the Volterra lattice, allowing one to treat linear and nonlinear multiplicative noises and to discuss statistical properties of the exact solutions.

Extending the idea of Rigaut's asymptotic fractals issuing a turnover from a constant to a power-law behaviour towards smaller scales, we extend this idea to asymptotic fractals with both lower and upper turnover points, i.e. the fractal region is terminated, towards larger scales, by another constant value. This behaviour is typical for natural fractals, such biological cell boundaries.

Here, we present a new analytic function describing this bi-asymptotic behaviour. The introduced parameters can be directly interpreted. We show the advantage of this function in fitting processed images of natural and mathematical fractals, in comparison with standard procedures: the determined dimension is significantly more accurate. A program package with this new function is available.

This paper deals with the mechanics and control for multi-particle systems from a geometric point of view. The centre-of-mass system is viewed as a principal fibre bundle with structure group SO(3), the base space of which is called the internal or shape space. A natural connection and a natural Riemannian metric are both defined on the centre-of-mass system. The equations of motion for the multi-particle system are derived in the Lagrangian formalism adapted to the bundle structure, and then reduced with the conserved total angular momentum. In contrast with this, the control problem is studied with non-holonomic constraints, i.e. with the vanishing total angular momentum, and equations of motion are determined for an optimally controlled multi-particle system. The resultant equations derived in each of the mechanical and control systems are to be compared.

We introduce the extended Duffin-Kemmer-Petiau (DKP) oscillator obtained by combining two relativistic quantum oscillator models. In a study analogous to Kukulin, Loyola and Moshinsky's work on extended Dirac oscillators, we investigate whether this extended version has oscillator shells controllably independent from the spin-orbit coupling. This extended DKP oscillator is found to be exactly solvable for natural parity states. We calculate and discuss both the natural- and unnatural-parity eigenspectra of its spin-1 representation.

Euclidean systems include poly- and monochromatic wide-angle optics, acoustics, and also infinite discrete data sets. We use a recently defined Wigner operator and (quasiprobability) distribution function to set up and study the phase-space evolution of these models, subject to differential and difference equations, respectively. Infinite data sets and two-dimensional monochromatic (Helmholtz) fields are thus shown by their Wigner function on a cylinder of direction and location; the Wigner function for polychromatic wavefields has `c-number' coordinates of (two-dimensional) wavenumber and position.

The following physical situation is investigated. A dielectric cylinder with a given surface charge density is immersed in electrolite. The electrolite is treated via the two-dimensional nonlinear Poisson-Boltzmann equation. In the case of single-line charge located on the cylindrical surface, analytical expressions for the electric field as well as for the space charge distribution in the electrolite are derived. The matter investigated here is related to the problem of the structure of the electric potential emanating from DNA.

Bifurcation diagrams and plots of Lyapunov exponents in the r- plane for Duffing-type oscillators

exhibit a regular pattern of repeating self-similar `tongues' with complex internal structure. We demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a Poincaré map of the system in action-angle variables in the limit of large driving force or, equivalently, small driving frequency. This map approaches the one-dimensional form

as derived in paper I.

This second paper describes our approach to calculating the various constants and functions introduced in paper I. It gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of Duffing-type equations. Finally we show how to extend the range in the parameter space where the map is applicable.

We have studied a model of self-attracting walk proposed by Sapozhnikov using the Monte Carlo method. The mean-square displacement and the mean number of visited sites are calculated for one-, two- and three-dimensional lattices. In one dimension, the walk shows diffusive behaviour with . However, in two and three dimensions, we observed a non-universal behaviour, i.e. the exponent varies continuously with the strength of the attracting interaction.

Problems concerning the value of the exponent in the relationship describing a self-attracting walk are considered. It is suggested that further study is needed to understand how the exponent depends on the coupling energy of the walk.