Table of contents

The spatiotemporal behaviour of the spread of influenza in France has been studied, and algebraic spatial correlations (with exponent ) spanning the whole territory have been found to be present as soon as the number of reported cases begins to increase, about 15 - 25 weeks before the peak of the epidemic. This result is surprising, as one would expect long-range correlations, if any, only in the vicinity of the maximum incidence, whereas our observations suggest that there exists an underlying non-trivial spatial structure at the very beginning of the observed epidemic. The observed long-range correlations are in fact present in the spatial distribution of the population. Correlations in the number of cases normalized by local population density are characterized by . This suggests that the spread of the epidemic is statistically uniform in space over a complex substrate that already contains the observed long-range correlations.

Various methods of constructing an orthonomal set out of a given set of linearly independent vectors are discussed. Particular attention is paid to the Gram - Schmidt and the Schweinler - Wigner orthogonalization procedures. A new orthogonalization procedure which, like the Schweinler - Wigner procedure, is democratic and is endowed with an extremal property is suggested.

We consider the scaling Lee - Yang model. It corresponds to the unique perturbation of the minimal CFT model . This model is not unitary. We are using an expression for form factors in terms of symmetric polynomials in order to obtain a closed expression for the correlation function of the trace of the energy-momentum tensor. This expression is a determinant of an integral operator. Similar determinant representations were proven to be useful not only for quantum correlation functions but also in matrix models.

In this paper we apply some theoretical predictions that arise in the mean-field framework for a large class of infinite-range models to structural glasses and we present a first comparison of these predictions with numerical results.

We find evidence for the energy density relaxation at low temperatures consisting of two steps: the convergence to some metastable state with a mean-field-like behaviour and the decay of metastable states by activated processes. We define an appropriate distance d between states and we study the corresponding equilibrium distribution that results in being highly nontrivial in the glassy phase.

The methods of fluctuation electrodynamics and the molecular cohesive force theory have been used to obtain an expression for the attractive force density between two absorbing media varying in temperature and separated by a nonabsorbing plane-parallel layer. The spectral density of the attractive force was calculated as the stress tensor projection on the exterior normal to the solid surface. The mean-square characteristics of the fluctuation field of these media were sought using the generalized Kirchhoff's law and the Green function of the related regular problem. A variety of solutions have been obtained depending on a temperature relationship of the interacting solid media. It is shown that for solids of the same temperature the obtained expression yields a formula describing the force of attraction in equilibrium.

We derive a self-duality relation for a one-dimensional model of branching and annihilating random walkers with an even number of offspring on nearest-neighbour sites. With the duality relation and by deriving further exact results in some limiting cases involving fast diffusion we obtain new information on the location and nature of the phase transition line between an active stationary state (non-zero density) and an absorbing state (extinction of all particles), thus clarifying some so far open problems. In these limits the transition is mean-field-like, but on the active side of the phase transition line the fluctuation in the number of particles deviates from its mean-field value. We also show that well within the active region of the phase diagram a finite system approaches the absorbing state very slowly on a time scale which diverges exponentially in system size. In the absence of particle diffusion the branching process (with infinite annihilation rate) is strongly non-ergodic.

We determine the joint density of secular coefficients and of the traces of powers for random unitary matrices from Dyson's circular ensembles. A byproduct are workable `unitarity conditions', necessary and sufficient for all roots of a polynomial to be unimodular and different. These findings will be useful for combinations of random-matrix theory with semiclassical approximations in calculating spectral properties of quantum systems whose classical limit is globally chaotic. In a first such application we show that reasonable quasi-energy spectra result by throwing dice à la random-matrix theory to complement restricted knowledge (rigorous or from short periodic orbits) of the secular polynomial of a unitary Floquet matrix.

We generalize quantum teleportation to, what we call, statistical teleportation utilizing previous results on distant preparation, and on the basic ingredient entities of an entangled composite-system state vector. Our main result is `the central theorem', establishing a simple necessary and sufficient condition for the crucial entity: the event that the sender of a pure quantum state has to measure in the first step of the two-step (and two-laboratory) teleportation procedure. We derive numerous consequences especially for deterministic teleportation (a special case of statistical teleportation), which is a direct generalization of the known quantum teleportation. Detailed further generalization to proper and improper mixtures is investigated. Finally, it is shown that extension to teleportation with nonlinear distant preparation is not possible unless the idea of teleportation is essentially changed.

We calculate a temperature-dependent part of the one-loop thermodynamic potential (and the free energy) for charged massive fields in a general class of irreducible rank 1 symmetric spaces. Both low- and high-temperature expansions are derived and the role of non-trivial topology influence on asymptotic properties of the potential is discussed.

The general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.

For the eigenvalues of p complex Hermitian matrices coupled in a chain, we give a method of calculating the spacing functions. This is a generalization of the one-matrix case which has been known for a long time.

The superintegrability of Calogero-Moser-Sutherland hyperbolic model and nonsuperintegrability of trigonometric one are shown.

Quantization of classical dynamical systems with a Poisson structure on homogeneous Kähler manifolds is considered. The quantization follows the method invented by Berezin and represents the unitary transition operator as a quasiclassical path integral in the coherent-state basis. In case the coherent-state manifold appears as a (degenerate) rank-one co-adjoint orbit of the symmetry group, an explicit representation of the transition amplitude in terms of classical data can be derived for large values of the highest weight, which corresponds to the quasiclassical approximation. This representation is further shown to perfectly agree, in contrast to some earlier approaches, with the known exact results and may provide non-trivial asymptotics of physical relevance.

We investigate a hydrogen-like atom (or any other system with a Coulomb potential) confined to a space which is bounded by a paraboloid. The nucleus of the atom resides at the focus of the paraboloid and we require the electronic wavefunction to vanish on the paraboloid. We derive an exact implicit analytic solution to the problem and also explicit analytic expressions for the wavefunctions and eigenenergies in the so-called strong-shift regime. We also discuss the influence of the boundary on the permanent dipole moments of the eigenstates. Finally, we investigate this system in WKB-approximation and give the Bohr-Sommerfeld quantization rule which is different from the usual rule due to the new boundary condition.

Using the transfer-matrix technique to describe transport properties in multichannel-multilayer systems, a three-term non-commutative matrix recurrence relation is deduced and solved. The matrix polynomials obtained in this way allow one to write compact expressions for the n-cell transmission amplitudes, from channel to channel i. In the one-dimensional, one-channel limit, the non-commutative polynomials reduce to the well known Chebyshev orthogonal polynomials. To illustrate the role of these polynomials in the resonant tunnelling and channel-mixing behaviour, we discuss one- and two-channel examples.

The Hirota bilinear difference equation is generalized to discrete space of arbitrary dimension. Solutions to the nonlinear difference equations can be obtained via Bäcklund transformation of the corresponding linear problems.

Coherent states are studied within the framework of complex measurable processes. It is shown that squeezed states can be accommodated within this framework and all the relevant results relating to squeezed states are derived. A generalization to complex coordinates is also provided and it is shown to lead to the Wigner distribution in a natural way.

The complete list of Poisson-Lie structures on Poincaré and Euclidean groups in three dimensions is presented. Some new solutions for inhomogenous are given.

The number of rook arrangements on the Sierpinski lattice is presented as a set of recurrence formulae. The upper and lower bounds of this number are also found.