Table of contents

We show by numerical integration that the discretized version of simple deterministic partial differential equations with singular terms proposed by Zhang (1992) exhibit rich spatio-temporal behaviour representing a mixture of stochastic and deterministic regimes. Varying the relative weight B of the singular term we have been able to detect transitions in the global behaviour of the solutions by determining their total width w(t). In particular, we have found intermittent solutions as well as a power-law dependence of W=w(t to infinity ) on B.

A geometric generalization of force-free fields is proposed. Properties of the minimum-energy configuration are deduced and its connection with minimal surfaces is shown for the simplest case.

We consider a diatomic molecule with fixed nuclei of charges Z₁ and Z₂, and N electrons. We find upper bounds on the kinetic energy of the molecule and the energy of the electronic contribution as a function of the nuclear separation R.

We study the kinetic equation which describes the evolution of cluster-size distribution of the aggregation-fragmentation problem, the so-called generalized Smoluchowski equation for several dynamics of cluster growth. By using the method of the generating function we find analytical solutions for some cases. Besides, numerical solutions are found for the time evolution of clusters where the aggregation and fragmentation kernels do not allow a complete analytical solution.

We study the spectrum of the fractional Brownian motion of Riemann-Liouville type using two approaches, namely the double frequency spectral density and the Wigner-Ville spectrum. The bifrequency representation gives a complex-valued function which contains two distinctive terms. These terms can be identified as the diagonal and off-diagonal distribution of the spectrum in a frequency-frequency plane. The physical interpretation of these two terms is briefly discussed. A calculation of Wigner-Ville spectrum gives an alternative way of representing the spectrum of this nonstationary process in the time-frequency plane. Asymptotic approximation of the Wigner-Ville spectrum is obtained. We show that the large-time average spectrum of Riemann-Liouville fractional Brownian motion exhibits a power law in the high-frequency range.

Giving importance to the active participation of concentration fluctuations in the dynamics, a phenomenological model is proposed to describe the experimentally observed suppression of phase separation when a turbulent binary fluid mixture is quenched below its consolute point T_c. It is assumed that an inverse cascade of concentration fluctuations is acted upon by the turbulence-generated viscous shear together with a loss due to the presence of Alfven-like waves, which cuts off the inverse cascade at some scale. As a consequence, The apparent depression of critical temperature follows the power law Delta T_c approximately R^lambda with lambda =1.44 and 2.4 in two extreme theoretical limits (R=Reynolds number). This result is comparable to the experimental values of 1.4 and 2.1 in two different measurements.

We consider the propagation of a shock wave (SW) in the damped Toda lattice. The SW is a moving boundary between two semi-infinite lattice domains with different densities. A steadily moving SW may exist if the damping in the lattice is represented by an 'inner' friction, which is a discrete analogue of the second viscosity in hydrodynamics. The problem can be considered analytically in the continuum approximation, and the analysis produces an explicit relation between the SW's velocity and the densities of the two phases. Numerical simulations of the lattice equations of motion demonstrate that a stable SW establishes if the initial velocity is directed towards the less dense phase; in the opposite case, the wave gradually spreads out. The numerically found equilibrium velocity of the SW turns out to be in very good agreement with the analytical formula even in a strongly discrete case. If the initial velocity is essentially different from the one determined by the densities (but has the correct sign), the velocity does not alter significantly, but instead the SW adjusts itself to the given velocity by sending another SW in the opposite direction.

We study the static as well as the glassy or dynamical transition in the mean-field p-state Potts glass. By numerical solution of the saddle-point equations we investigate the static and dynamical transitions for all values of p in the non-perturbative regime p>4. The static and dynamical Edwards-Anderson parameter increase with p logarithmically. This makes the glassy transition temperature lie very close to the static one. We compare the main predictions of the theory with numerical simulations.

A polymer folding model on the square lattice is constructed with attractive contact interactions of strength 1/c², 0<c<1. The corresponding model on a dynamical random lattice, with freely fluctuating coordination number at each vertex, is formulated as a random two-matrix model and an expression for the partition function of a length-L chain is derived. Numerical estimates and analytical evaluation for L to infinity show a third-order collapse transition at c= square root 2-1. Geometrical critical exponents are computed in each phase and interpreted. The Knizhnik-Polyakov-Zamolodchikov two-dimensional quantum gravity scaling relations are used to predict the corresponding behaviour on the regular lattice, which lies in a different universality class from the percolation Theta -point of Duplantier and Saleur (1987).

It is shown that the propagator in single-file systems may quite generally be deduced from the propagator in the corresponding unrestricted case. Irrespective of its structure in the unrestricted case, the propagator of single-file systems in the long-time limit approaches a Gaussian distribution. It is found that the exponent of the time dependence of the mean-square displacement is reduced by a factor of two, compared with the unrestricted case. Examples for different models of particle propagation are considered and compared with literature data. For one model the theoretical result is compared with a molecular dynamics simulation.

Using ideas of Herbert and Till (1983), analogues of discrete energy levels of electrons in quantum optics are introduced into quantum transport theory. For simple one-dimensional devices inelastic scattering with optical phonons is modelled by a reservoir. A master equation is derived following the usual procedures in quantum optics. The hot electron kinetic equation of Herbert (1984), with corrections, is deduced as a first approximation. Our formalism is applied to a simplified tunnelling diode structure.

We study the critical properties of the weakly disordered p-component random Heisenberg ferromagnet. It is shown that if the specific-heat critical exponent of the pure system is positive, the traditional renormalization group (RG) flows at dimensions D=4- epsilon , which are usually considered as describing the disorder-induced universal critical behaviour, are unstable with respect to replica-symmetry breaking (RSB) potentials such as those found in spin glasses. It is demonstrated that the RG flows involving RSB potentials lead to fixed points which have a structure known as the one-step RSB, and there exists a whole spectrum of such fixed points. It is argued that spontaneous RSB can occur due to the interactions of the fluctuating fields with the local non-perturbative degrees of freedom coming from the multiple local minima solutions of the mean-field equations. However, it is not clear whether or not RSB occurs for infinitesimally weak disorder. Physical consequences of these conclusions are discussed.

The master equations for the random neighbour Bak-Sneppen model are solved explicitly.

The relation between discrete topological field theories on triangulations of two-dimensional manifolds and associative algebras was worked out recently. The starting point for this development was the graphical interpretation of the associativity as flip of triangles. We show that there is a more general relation between dip-moves with two n-gons and Z_n-2-graded associative algebras. A detailed examination shows that flip-invariant models on a lattice of n-gons can be constructed from Z₂- or Z₁-graded algebras, reducing in the second case to triangulations of the two-dimensional manifolds. Related problems occur naturally in three-dimensional topological lattice theories.

A universal quasi-triangular R-matrix for the non-standard quantum (1+1) Poincare algebra U_ziso(1,1) is deduced by imposing analyticity in the deformation parameter z. A family U_wg_mu of 'quantum graded contractions' of the algebra U_ziso(1,1)(+)U_-ziso(1,1) is obtained. Quantum analogues of the two-dimensional Euclidean, Poincare and Galilei algebras enlarged with dilations are contained in U_wg_mu as Hopf subalgebras with two primitive translations. Universal R-matrices for these quantum Weyl (similitude) algebras and their associated quantum groups are constructed.

Irreducible representations of Brauer algebras are constructed by using the induced representation and the linear equation method. As examples, some matrix representations of Brauer algebras D_f(n) with f<or=5 are presented.

We give representations for the Zamolodchikov-Faddeev algebra (ZFA) in the elliptic case and prove the self-consistency of this algebra in such a case. We also study the implications of our approach for those algebras which are relevant to the extension of the q-deformed affine A_n⁽¹⁾ algebra.

In the study of level spacing functions for the eigenvalues of random matrices, Mahoux and Mehta (1993) studied the functions S(t), A(t) and B(t) related to certain Fredholm determinants. These functions can be expressed in terms of the fifth (or the third) Painleve transcendents. The asymptotic behaviour for t to infinity of these functions and their derivatives with respect to a parameter were derived by them except for an unknown constant. Using Jimbo's (1982) method of monodromy preserving deformations, we connect the behaviour of the fifth Painleve transcendent at t=0 and at t= infinity , thus determining the unknown constant.

A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can to some extent be formulated, in very much the same way we are used to in the geometrical arena underlying classical physical theories and models. In previous work, certain differential calculi on a commutative algebra exhibited relations with lattice structures, stochastics, and parametrized quantum theories. This motivated the present systematic investigation of differential calculi on commutative and associative algebras. Various results about their structure are obtained. In particular, it is shown that there is a correspondence between first-order differential calculi on such an algebra and commutative and associative products in the space of I-forms. An example of such a product is provided by the Ito calculus of stochastic differentials. For the case where the algebra A is freely generated by 'coordinates' xⁱ, i=1, ..., n, we study calculi for which the differentials dxⁱ constitute a basis of the space of 1-forms (as a left-A-module). These may be regarded as 'deformations' of the ordinary differential calculus on Rⁿ. For n<or=3 a classification of all (orbits under the general linear group of) such calculi with 'constant structure functions' is presented. We analyse whether these calculi are reducible (i.e. a skew tensor product of lower-dimensional calculi) or whether they are the extension (as defined in this paper) of a one-dimension-lower calculus. Furthermore, generalizations to arbitrary n are obtained for all these calculi.

Firstly, some similarity reductions of the complex modified Korteweg-de Vries equation (CMKdV), which arises in the asymptotic interpretation of one-dimensional plane-wave propagation in a quadratic micropolar medium are discussed. Although it is not a soliton equation solvable by inverse scattering transformation, its similarity reductions obtained by the use of Lie group methods are of mathematical interest. Secondly, the Painleve analysis developed by Weiss et al. (1983) for nonlinear partial differential equations is applied to the CMKdV equation, and the data obtained by the truncation technique yield some analytical solutions of the ordinary modified Korteweg-de Vries equation and travelling-wave solutions of the CMKdV equation which are also solutions of the similarity reduction obtained by classical Lie group analysis.

Invariants at arbitrary and fixed energy (strongly and weakly conserved quantities) for two-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of the dynamics. Using Killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function S(z). For invariants at arbitrary energy the function S(z) is a second-degree polynomial with real second derivative. The integrability condition then reduces to Darboux's condition for quadratic invariants at arbitrary energy. The four types of classical quadratic invariants for positive-definite two-dimensional Hamiltonians are shown to correspond to certain conformal transformations. We derive the explicit relation between invariants in the physical and Jacobi time gauges. In this way knowledge about the invariant in the physical time gauge enables one to write down directly the components of the corresponding Killing tensor for the Jacobi metric. We also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection with the problem of the third integral in galactic dynamics. In our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree, respectively.

A geometrical framework is presented for the treatment of a class of dynamical systems, which are modelled by a system of second-order differential equations, coupled with first-order equations linear in the derivatives. Such problems in particular make their appearance in the study of Lagrangian systems with non-holonomic constraints. Among other things, we discuss the concepts of symmetry and adjoint symmetry for such systems and identify for that purpose an appropriate notion of 'dynamical covariant derivative' and 'Jacobi endomorphism'. The intrinsic tools which are being developed further allow a direct geometrical construction of the dynamics of non-holonomic systems. The vertically rolling disc is chosen as an illustrative example for the newly proposed formalism and results.

A method for extracting radial velocity signals produced by brown dwarfs and extrasolar planets is presented, and the implications of this analysis for the prospects of detecting extrasolar planets and brown dwarfs are discussed.

Effective formulae for the single-phase periodic solution of the AB system are obtained. The solution depends on a complete set of four complex parameters (Riemann invariants). The derivation of Whitham equations which describe a slow evolution of the modulated periodic solution is presented. The general theory is illustrated by application to the problem of soliton creation on the front of a sharp pulse.

In the context of a relativistic quantum mechanics with invariant evolution parameter, solutions for the relativistic bound-state problem have been found, which yield a spectrum for the total mass coinciding with the non-relativistic Schrodinger energy spectrum. These spectra were obtained by choosing an arbitrary spacelike unit vector n, and restricting the support of the eigenfunctions in spacetime to the subspace of the Minkowski measure space, for which (X_{perpendicular to} )²=(x-(x.n)n)²>or=0. In this paper, we examine the normal Zeeman effect (in lowest order) for these bound states, which requires n_mu to be a dynamical quantity. We recover the usual Zeeman splitting in a manifestly covariant form.

A perturbation theory is constructed within the framework of the linear form of the variable phase approach. This allows one to correctly take into account the potential tail which decreases more rapidly than the centrifugal tail. It is shown how one can use this theory for an analysis of the scattering problem in the low-energy limit and in the limit of large angular momentum.