Table of contents

We prove the following general result for Laguerre polynomials.

For all

 

provided that .

The Macdonald operators associated with the classical root systems are constructed based on the infinite-dimensional representation for solutions of the Yang - Baxter equation and the reflection equation.

A recently proposed renormalization group approach to dimensional crossover in quasi-one-dimensional quantum antiferromagnets is improved and then shown to give identical results, in some cases, to those obtained earlier.

We analyse and simulate a two-dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in a two-dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particle in a rectangular box of size and elongated shape we observe that the stationary distribution of particles corresponds to the mth Laplacian eigenfunction. For smaller elongations a > b we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the mth eigenvalue of the Laplacian with rectangular boundaries.

We discuss a new mechanism leading to a matrix product form for the stationary state of one-dimensional stochastic models. The corresponding algebra is quadratic and involves four different matrices. For the example of a coagulation - decoagulation model explicit four-dimensional representations are given and exact expressions for various physical quantities are recovered. We also find the general structure of n-point correlation functions at the phase transition.

We investigate a crossover behaviour of the conservative and non-conservative growing models in which the surface relaxation is restricted by the given value of threshold, H. The time dependence of the mean step height, which scales as , is found to be responsible for the observed crossover behaviour of the surface width. A modified scaling ansatz is also proposed to describe the crossover behaviour in the growing models of this kind.

This paper documents a study of novel fracture-toughness properties of brittle heterogeneous materials. Through simulations of the rupture process based on a lattice discretization of the material, the spatial variation of dissipated energy due to fracture is evaluated. Under certain conditions, its distribution is characterized by a multifractal spectrum . Importantly, depends not only on the initial heterogeneity present in the material but also on the nature of the externally applied load. This provides a renewed load-path-dependent definition of fracture toughness material properties. It avoids the difficulties associated with `traditional' continuum/fracture mechanics definitions where the macroscopic fracture mode must be known a priori.

The randomness of disc packings, generated by random sequential adsorption (RSA), random packing under gravity (RPG) and Mason packing (MP) which gives a packing density close to that of the RSA packing, has been analysed, based on the Delaunay tessellation, and is evaluated at two levels, i.e. the randomness at individual subunit level which relates to the construction of a triangle from a given edge length distribution and the randomness at network level which relates to the connection between triangles from a given triangle frequency distribution. The Delaunay tessellation itself is also analysed and its almost perfect randomness at the two levels is demonstrated, which verifies the proposed approach and provides a random reference system for the present analysis. It is found that (i) the construction of a triangle subunit is not random for the RSA, MP and RPG packings, with the degree of randomness decreasing from the RSA to MP and then to RPG packing; (ii) the connection of triangular subunits in the network is almost perfectly random for the RSA packing, acceptable for the MP packing and not good for the RPG packing. Packing method is an important factor governing the randomness of disc packings.

The explicit direct and inverse orthogonalization coefficients (OCs) are derived for the projected basis states of the two parametric covariant irreducible representations of (including Elliott's n = 3 case) and (Draayer), with the alternative choices of the Gram - Schmidt processes, began, respectively, from the lowest (Vergados) or highest absolute value of the intrinsic parameter K. The different direct and inverse OCs (the latter equivalent to the definite boundary isofactors, or the boundary resubducing coefficients) are found to be connected by the definite analytical continuation - contraction procedures with mutually related OCs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of . The orthogonalization coefficients considered are presented as definite algebraic-polynomial structures under the square root, with the linear factors and numerator - denominator polynomials in terms of the partition-dependent functions of Biedenharn and Louck, .

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first-order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).

We show that R-matrices of all simple quantum groups have properties which permit one to present quantum group twists as transitions to other coordinate frames on quantum spaces. This implies physical equivalence of field theories invariant with respect to q-groups (considered as q-deformed spacetime groups of transformations) connected with each other by twists. Taking into account this freedom, we study quantum spaces of the special type, with commuting coordinates but with q-deformed differential calculus, and construct invariant multidimensional Jackson derivatives. We consider a particle and field theory on a two-dimensional q-space of this kind and come to the conclusion that only one (timelike) coordinate is proved to be discretized.

We consider quantized derivation representations of Hopf algebras in some associative algebras . The algebra of quantized differential operators of bialgebra representations, which are special representations by derivations, are constructed. For the quantum enveloping algebras of Lie algebras associated with the root systems , , and we define a deformation of the exterior algebra of forms, and by applying the above results it is shown that the quantized adjoint representation of induces a bialgebra representation of in . The resulting algebra of differential operators is a deformation of the standard Koszul complex of Lie algebras. It admits an exterior derivative which is, in particular, a -module morphism. Hence cohomology groups of relative to some -module can be constructed.

The symmetry of the polynomial solutions of the Calogero - Sutherland - Moser model which corresponds to the -deformed symmetric group or the general linear group GL(N) is treated using the relationship between and the unitary group SU(N). A -deformed relation - SU(N) is studied up to N = 3.

Using the Baker - Akhiezer function technique we construct a separation of variables for the classical trigonometric three-particle Ruijsenaars model (a relativistic generalization of the Calogero - Moser - Sutherland model). In the quantum case, an integral operator M is constructed from the Askey - Wilson contour integral. The operator M transforms the eigenfunctions of the commuting Hamiltonians (the Macdonald polynomials for the root sytem ) into the factorized form where S(y) is a Laurent polynomial of one variable expressed in terms of the basic hypergeometric series. The inversion of M produces a new integral representation for the Macdonald polynomials. We also present some results and conjectures for the general n-particle case.

We study the propagation of an electromagnetic wave with a high intensity in a ferromagnetic medium, in a way analogous to the mathematical theory of nonlinear geometric optics. We find that the evolution of the modulation of a quasi-monochromatic plane wave is described by a nonlinear transport equation. In the simplest case, we retrieve a result described by other models: the main nonlinear effect is a phase modulation proportional to both the time and the square of the amplitude of the wave.

But the most interesting feature is that the rapidly oscillating wave generates slowly varying waves that belong to soliton propagation modes, and that the latter react on the former by a phase factor. Two regimes occur, depending whether the slowly varying waves travel slower or faster than the modulation of the rapidly oscillating wave. An analogue to the boom of a supersonic airplane can thus be observed in the phase modulation of the incident wave.

The solutions of the Dirac equation with minimal and non-minimal coupling terms are investigated by transforming the relativistic equation into a Schrödinger-like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given. It is pointed out that techniques used in the analysis of quasi-exactly solvable potentials of non-relativistic quantum mechanics can be applied to relativistic problems as well.

By performing unitary transformations, the exact solution of the time-dependent singular oscillator is obtained. The invariant operator and the auxiliary equation are rigorously established. The non-adiabatic Berry's phase is calculated.

One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrödinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from the locality of the potential in the wave equation and represent the effect of causality in the time dependence of the scattering process.

A supersymmetric extension of the classical dispersive water wave equation is proposed. The system is shown to be tri-Hamiltonian. In particular, the second Hamiltonian structure is analogous to the N = 2 superconformal algebra. A gauge map to the even-parity super KP system is exhibited.

Gauss decomposition of the Lie algebra has been used to derive the Miura map for discrete completely integrable systems, which exactly reproduces the results obtained in a recent paper by Yang and Schmid. We have dealt with the cases of discrete mKdV, discrete sine - Gordon, discrete KdV and the Volterra system. Our approach has the advantage of yielding the Lax pair for the modified equation. It can also lead to a second stage of modified equation if the same procedure is repeated twice.

We integrate sub-families of a family of nonlinear Schrödinger equations proposed by Doebner and Goldin in the (1 + 1)-dimensional case which have exceptional Lie symmetries. Since the method of integration involves non-local transformations of dependent and independent variables, general solutions obtained include implicitly determined functions. By properly specifying one of the arbitrary functions contained in these solutions, we obtain broad classes of explicit square integrable solutions. The physical significance and some analytical properties of the solutions obtained are briefly discussed.

Lie bialgebra structures on e(2) are classified. For two Lie bialgebra structures which are not coboundaries (i.e. those which are not determined by a classical r-matrix) we solve the cocycle condition, find the Lie - Poisson brackets and obtain quantum group relations. There is one to one correspondence between Lie bialgebra structures on e(2) and possible quantum deformations of U(e(2)) and E(2).

The integrable dispersive long wave equations, especially the higher dimensional ones, are of current interest in both physics and mathematics. Obtained in this paper, via a symbolic-computation-based method, are new families of exact solutions to the 2+1-dimensional integrable dispersive long wave equations. Sample solutions from those families are presented. Solitary waves are merely a special case in one family.

A square-root anharmonic oscillator potential is analysed as a model which simulates a quasi-relativistic squeezing of the harmonic oscillator spectrum. Several eligible (namely, perturbative, variational, Hill-determinant and Riccati - Padé) methods of construction of its bound states are compared.

In this paper the following references should be:

[7] Gerry C C and Silverman S J 1983 Phys. Lett. 95A 481; 1984 Phys. Rev. A 29 1574 Gerry C C and Laub J 1985 Phys. Rev. A 32 709, 3376 Gerry C C and Togeas J B 1984 Phys. Lett. 106A 215 Gerry C C 1986 Phys. Lett. 115A 9 Mathys P and De Meyer H 1988 Phys. Rev. A 36 168 Bargmann V 1947 Ann. Math. 48 568 Gerry C C and Laub J 1984 Phys. Rev. A 30 1229

In this paper reference [7] should be:

[7] Barut A O 1971 Dynamical Groups and Generalised Symmetries in Quantum Theory (Christchurch: University of Canterbury Press) Bednar M 1973 Ann. Phys. 75 305 Mathys P and De Meyer H 1988 Phys. Rev. A 36 168 and references therein Gerry C C and Laub J 1984 Phys. Rev. A 30 1229; 1985 Phys. Rev. A 32 709, 3376

In this paper the following were incorrectly printed.

The last line of equation (32) should have been:

The first line of equation (33) should have been: