Table of contents

We discuss the phenomenon of pre-acceleration in the light of a method of successive approximations used to construct the physical order reduction of a large class of singular equations. A simple but illustrative physical example is analysed to gain more insight into the convergence properties of the method.

We discuss classical Fourier - Gauss transforms of a three-parameter family of the continuous Al-Salam - Chihara polynomials . It is shown that they are related to both the continuous big q-Hermite and the q-Hermite polynomials.

The effect of dispersive transport in the single-species reaction-diffusion models of coagulation and annihilation is considered. This transport is modelled through a fractal-time random walk, in which the stepping-times of the walker (a typical particle) follows a renewal process characterized by a pausing time distribution proportional to a stable law at long times, , with (the fractal dimension of the time). This leads to a sublinear mean squared displacement for the particles: . The decay of the concentration of particles, A(t), is obtained for all space dimensions d, and for the whole course of the reactions. The obtained results are exact for short and long times, with the long time asymptotics for d = 1, for d = 2 and for . The effect of highly non-homogeneous space distributions of particles is also considered. It is found that a fractal segregation of dimension (with ) in the initial distribution of particles in the space leads to for d = 1, for d = 2 and for , and for . This shows a subordination phenomenon in the combination of space- and time-fractal distributions.

The propagation of the first displacement maximum of a semi-infinite wavetrain in a two-dimensional random-fibre network is analysed. Model calculations and numerical simulations are used for demonstrating that two qualitatively different wavefront velocities appear in the network. A transient wave, which travels fast and whose amplitude decreases exponentially, dominates the short-time behaviour when the bending stiffness of the fibres is small and the driving frequency is high. This mode can be described by a one-dimensional model. The transient-wave mode propagates even if the bending stiffness of the fibres vanishes, in which case the normal sound velocity is zero. The usual, and slower, effective medium mode always dominates at late times. It also dominates at short times if the driving frequency is low and/or the bending stiffness of the fibres is relatively high.

We study analytic and numerical solutions for a model for the dynamics of cluster growth with fragmentation. The model is restricted to a process involving a monomer - cluster reaction and the cluster of N particles cannot adsorb particles, it can only split up, that is, the cluster is unstable, and for mathematical convenience we propose that it is split up mainly into monomers. In this model both coagulation and fragmentation processes scale with cluster size. Both sourceless and with-source evolutions are discussed. Our analysis shows that in the with-source case the final evolution for the concentrations is of the form for and linear on t for . By comparison, when there is no restriction to the maximum size for polymers and no dissociation the evolution goes asymptotically as .

Applying the Coulomb fluid approach to the Hermitian random matrix ensembles, universal derivatives of the free energy for a system of N logarithmically repelling classical particles under the influence of an external confining potential are derived. It is shown that the elements of the Jacobi matrix associated with the three-term recurrence relation for a system of orthogonal polynomials can be expressed in terms of these derivatives and therefore give an interpretation of the recurrence coefficients as thermodynamic susceptibilities. This provides an algorithm for the computation of the asymptotic recurrence coefficients for a given weight function.

We also show that a pair of quasilinear partial differential equations, obtained in the continuum limit of the Toda lattice, can be integrated exactly in terms of certain auxilliary functions related to the initial data, and in our formulation in terms of integrals of the logarithm of the weight function. To demonstrate this procedure we give some examples where the initial data increases along the half line.

Combining identities of the theory of orthogonal polynomials and certain Coulomb fluid relations, a second-order ordinary differential equation (with coefficients determined by the Coulomb fluid density) satisfied by the polynomials is derived. We use this to prove some conjectures put forward in previous papers. We show that, if the confining potential is convex, then near the edges of the spectrum of the Jabcobi matrix, orthogonal polynomials of large degree is uniformly asymptotic to Airy function.

We extend the standard calculation of the number of metastable states (fixed points) so as to take into account cyclic attractors as well. We calculate analytically the average number of cycles in the Little model as a function of the fraction of flipping spins (1-q)/2. We find that the cycles of length two are by far the most common type of attractors. In particular, for small values of the storage parameter, the inverse cycles (q=-1) are the dominant attractors.

We investigate a non-equilibrium reaction - diffusion model and equivalent ferromagnetic spin- spin chain with alternating coupling constant. The exact energy spectrum and the n-point hole correlations are considered with the help of the Jordan - Wigner fermionization and the interparticle distribution function method. Although the Hamiltonian has no explicit translational symmetry, the translational invariance is recovered after a long time due to the diffusion. We see the scaling relations for the concentration and the two-point function in finite-size analysis.

We found that two thin hard needles can collide hundreds of times without any additional forces. The number of collisions is sensitive to the initial conditions, especially the rotational phase of each needle. Long continuation of the chattering collision takes place at a limited region in the parameter space. The scattering angle was almost constant for the collision number in a frequent chattering collision, while the change in the collision number in a short chattering collision brings about a sizeable perturbation on the scattering process.

The ground-state dynamics of an entropy barrier model proposed recently for describing relaxation of glassy systems, is considered. At late stages of evolution the dynamics can be described by a simple variant of the Ehrenfest urn model. Analytical expressions for the relaxation times from arbitrary initial states to the ground state are derived. Upper and lower bounds for the relaxation times as a function of system size are obtained.

The properties of the hulls of directed percolation clusters are studied. The scaling and finite-size scaling of many quantities around the percolation threshold are derived and a novel Monte Carlo algorithm, which is more than twice as fast as the standard algorithm at , has been formulated to study these properties. Simulations have been conducted that enable an estimation of all the exponents involved. In particular, the central exponent, x, relating the average hull length of clusters to their mass, has been estimated to be 0.773(4) at the percolation threshold. This same exponent is estimated to be 0.905(5) for . Thus, this second value should hold for directed animals.

The semiclassical description of billiard spectra is extended to include the diffractive contributions from orbits which are nearly tangent to a concave part of the boundary. The leading correction for an unstable isolated orbit is of the same order as the standard Gutzwiller expression itself. The importance of the diffraction corrections is further emphasized by an estimate which shows that for any large fixed k almost all contributing periodic orbits are affected. The theory is tested numerically using the annulus and Sinai billiard. For the Sinai billiard, the investigation of the spectral density is complemented by an analysis which is based on the scattering approach to quantization. The merits of this approach as a tool to investigate refined semiclassical theories are discussed and demonstrated.

We have computed geometrical characteristics of large clusters (up to 32 768 particles) obtained by a hierarchical cluster - cluster aggregation computer model in three dimensions, the off-lattice variable-D model. Using a `box-counting' method, we have calculated the fractal dimensions of the surface and the perimeter of their two-dimensional projections as a function of their fractal dimension D. By diagonalizing the radius of gyration tensor, we have obtained numerical estimates for the intrinsic anisotropy coefficients (ratios of the eigenvalues) and we have proposed analytical expressions to describe their behaviour as a function of the fractal dimension.

A quantum-field theory defined in any space having arbitrary static background structure along some axes but translation invariance along other free directions (labelled by i) is considered. This system has a renormalized vacuum-stress tensor satisfying . This relation has nothing to do with being traceless or divergence free (neither property is assumed).

A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation. This result is applied to specific classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey and Wilson, Al Salam - Carlitz and the celebrated little and big q-Jacobi.

We find the Hopf algebra dual to the Jordanian matrix quantum group . As an algebra it depends only on the sum of the two parameters and is split into two subalgebras: (with three generators) and (with one generator). The subalgebra is a central Hopf subalgebra of . The subalgebra is not a Hopf subalgebra and its co-algebra structure depends on both parameters. We discuss also two one-parameter special cases: g = h and g=-h. The subalgebra is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of . The subalgebra is isomorphic to U(sl(2)) as an algebra but has a nontrivial co-algebra structure and again is not a Hopf subalgebra of .

We study the number of bouncing ball modes in a class of two-dimensional quantized billiards with two parallel walls. Using an adiabatic approximation we show that asymptotically for , where depends on the shape of the billiard boundary. In particular for the class of two-dimensional Sinai billiards, which are chaotic, one can get arbitrarily close (from below) to , which corresponds to the leading term in Weyl's law for the mean behaviour of the counting function of eigenstates. This result shows that one can come arbitrarily close to violating quantum ergodicity. We compare the theoretical results with the numerically determined counting function for the stadium billiard and the cosine billiard and find good agreement.

A Gelfan'd - Dyson mapping is used to generate a one-boson realization for the non-standard quantum deformation of which directly provides its infinite- and finite-dimensional irreducible representations. Tensor product decompositions are worked out for some examples. Relations between contraction methods and boson realizations are also explored in several contexts. So, a class of two-boson representations for the non-standard deformation of is introduced and contracted to the non-standard quantum (1 + 1) Poincaré representations. Likewise, a quantum extended Hopf algebra is constructed and non-standard quantum oscillator algebra representations are obtained from it by means of another contraction procedure.

The consequences of replica-symmetry breaking on the structure of ultrametric matrices appearing in the theory of disordered systems is investigated with the help of representation theory, and the results are compared with those obtained by Temesvári, De Dominicis and Kondor.

A lower bound result about the approximability of the energy function of Ising spin glasses in the quantum context is determined. The ground-state problem for the three-dimensional Ising spin glasses represented on two-level grids such that the vertical interactions are at most (where n is the number of spins set on the grid), is considered. We prove that, unless , there is a constant such that every quantum approximate polynomial-time algorithm finds a solution with absolute error greater than , with high probability, infinitely often.

The dynamical symmetry of the non-relativistic Kepler problem has attracted much attention in the scientific literature. In the present paper, we show that the Runge - Lenz vector, which accounts for the presence of this symmetry, has its physical origin in the generator of Lorentz transformations for the relativistic two-body problem. We reach this conclusion by considering the relativistic two-body problem, for the electromagnetic as well as for the gravitational interaction, in the approximation.

Motivated by the fact that momentum observables which are modelled in a classical theory as difference quotients of functions are directly accessible to measurements, whereas differentials are mathematical idealizations, which are obtained from difference quotients via a limiting process, we propose a quantization method in which momentum operators are given in terms of q-derivatives, i.e. a particular type of difference quotient, which is particularly suitable on . The quantization scheme is obtained from Borel quantization, in which momentum operators are given as differential operators, via a q-deformation of the kinematical algebra. It will be applied to a system localized and moving on the N-point discretization of and leads to a discrete, nonlinear Schrödinger equation. In the limit , i.e. the continuous idealization, we find evolution equations which are special cases of the nonlinear Schrödinger equation derived from Borel quantization on , which is based on the undeformed kinematical algebra. It turns out that with this procedure both the real and imaginary part of the nonlinearity can be derived, which without deformation was only possible for the imaginary part. Hence, one can learn on the situation in the continuous case if it is viewed as a limit of the (q-deformed) case.

The conserved densities of hydrodynamic-type systems in Riemann invariants satisfy a system of linear second-order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalizing the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic-type systems. We discuss periodic Laplace sequences of hydrodynamic-type systems with emphasis on the simplest nontrivial case of period 2.

For systems in Riemann invariants a complete discription of closed quadruples is proposed. They turn out to be related to a special quadratic reduction of the (2 + 1)-dimensional 3-wave system which can be reduced to a triple of pairwise commuting Monge - Ampere equations.

In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associated with the (2 + 1)-dimensional n-wave system. The 2-component Laplace transformations can also be interpreted as the symmetries of the (2 + 1)-dimensional integrable equations of Davey - Stewartson type.

Laplace transformations of hydrodynamic-type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space.

In this paper, we study Lie superalgebras of matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional superalgebras whose odd subspace is non-trivial, we find those admitting a finite-dimensional invariant module of smooth vector-valued functions, and classify all the resulting finite-dimensional modules. The latter Lie superalgebras and their modules are the building blocks in the construction of quasi-exactly solvable quantum mechanical models for spin- particles in one dimension.

A generalized Laplace transform approach is developed to study the eigenvalue problem of the one-dimensional singular potential . Matching of solutions at the origin that has been a matter of much controversy is, thereby, made redundant. A discrete and non-degenerate bound-state spectrum results. Existing arguments in the literature that advocate (a) a continuous spectrum, (b) a degeneracy of energy levels as a result of a hidden O(2) symmetry, (c) an infinite negative energy state and (d) an impenetrable barrier at the origin are found to be untenable. It is argued that a judicious use of generalized functions, coupled with some classical considerations, enables the conventional method of solving the problem to recover precisely the same results which are shown to be in accord with an accurate semiclassical analysis of the problem.

Several aspects of the lattice algebra are studied. Motivated by the fact that the Lotka - Volterra model can be written in terms of a current of the lattice Virasoro algebra (the Faddeev - Takhtajan - Volkov algebra), integrable dynamical models on the lattice have been formulated as a model associated with the lattice algebra.

We study the link between symmetry reductions and constraints of the Kadomtsev - Petviashvili equations in terms of the tau function. We propose a generalization - adapted to non-zero boundary conditions - of the standard constraints, and show a particular class of solutions (solitons).

The non-stationary Schrödinger equation in a finite basis of states is considered for the Hamiltonian matrix linearly depending on time. Exact analytical solutions of asymptotic transition probabilities are obtained for a bow-tie model, in which an arbitrary number of linear time-dependent diabatic potential curves cross at one point and only a particular horizontal curve has interactions with the others. Based on the contour integral method used, some mathematical aspects such as a possible generalization of the Whittaker functions are also briefly discussed.

We study a small piece of two-dimensional Toda lattice as a complex dynamical system. In particular, it is shown analytically how the Julia set, which appears when the piece is deformed, disappears as the system approaches the integrable limit.

We propose an algorithm computing explicit generating functions of the stable multiplicities of irreducible representations of arising in the restriction from U(n), O(n) or O(n-1) to of an irreducible tensor representation of the unitary or orthogonal group; i.e. we compute the multiplicities in a way which is independent of n and m, the weight of the label of the corresponding irrep.

Generic one-loop integrals at finite temperature, appearing in integrations over non-static modes, are given a definite expression in terms of series in powers of , where m is a mass parameter and T is the temperature.

We develop the scattering theory for the Schrödinger operator with the Coulomb potential in the space of an arbitrary dimension d. In particular, we calculate the scattering matrix and show that its spectrum covers the whole unit circle. We also compute the differential cross section and show that it coincides with the classical Coulomb scattering cross section in the dimension d = 3 only.

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