Table of contents

The Askey-Wilson algebra AW(3) is shown to serve as a 'hidden' covariance algebra for quantum algebra SL_q(2). The generators of AW(3) are chosen to be linear combinations of SL_q(2) generators with operator-valued coefficients.

The generalized form of the Kac formula for Verma modules associated with linear brackets of hydrodynamics type is proposed. Second cohomology groups of the generalized Virasoro algebras are calculated. Connection of the central extensions with the problem of quantization of hydrodynamics brackets is demonstrated.

The SOq(4) quantum algebra is used for the description of a q-analogue of the hydrogen atom. The energy spectrum and degeneracy of a q-analogue of the hydrogen atom is obtain with q being real or a phase.

The general Miura transformation (t,x,u(t,x)) to (s,y,v(s,y)): v=a(t,x,u,. . ., delta ^ru/ delta x^r), y=b(t,x,u,. . ., delta ^ru/ delta x^r), s=c(t,x,u,. . ., delta ^ru/ delta x^r) is considered which connects two evolution equations u_t=f(t,x,u,. . ., delta ⁿu/ delta xⁿ) and v_s=g(t,x,u,. . ., delta ^mu/ delta x^m). The conditions c=c(t) and m=n are proven to be necessary. It is shown that every Miura transformation, admitted by a constant separant equation u_t=f, consists of the following three transformations: (i) (t,x,u) to (t,x,w), where w=a(t,x,u,. . .,u_{x. . .x}); (ii) (t,x,w) to (t,y,v), where y=x and v=w, or y=w and v=w_x, or y=w_x and v=w_xx; (iii) a transformation of time s=c(t) and a contact transformation of (y,v). As an example, the Korteweg-de Vries equation is transformed to three new nonlinear equations, of which two have neither nontrivial algebra of generalized symmetries nor infinite set of conserved densities.

The inverse spectral method for an N*N spectral problem is studied via the delta -problem for a one-dimensional complex space. The complex mKdV equations are explicitly solved as an example.

Following Avron (1982) the authors consider the Stark effect for Bloch electrons in the case of a finite number of gaps. They prove that the ladders of resonances are given by the Wannier decoupled-band approximation and the perturbation theory. The Fermi golden rule yields the width behaviour of Buslaev and Dmitirieva (1990).

The first observations of noise-induced enhancements and phase shifts of a weak periodic signal-characteristics signatures of stochastic resonance (SR)-are reported for a monostable system. The results are shown to be in good agreement with a theoretical description based on linear-response theory and the fluctuation dissipation theorem. It is argued that SR is a general phenomenon that can in principle occur for any underdamped nonlinear oscillator.

The author demonstrates show two tilings with face-centred icosahedral symmetry can be derived from one another by using only local information. The tiling under consideration are the zonohedral tiling proposed by Socolar and Steinhardt, which is closely related to the original three-dimensional rhombohedral tiling, and the tetrahedral tiling of Danzer. Both tilings have matching rules, and his proof is based on this property.

The authors study the critical behaviour of the SU(N) generalization of the one-dimensional Hubbard model with arbitrary degeneracy N. Using the integrability of this model by Bethe ansatz they are able to compute the spectrum of the low-lying excitations in a large but finite box for arbitrary values of the electron density and of the Coulomb interaction. This information is used to determine the asymptotic behaviour of correlation functions at zero temperature in the presence of external fields lifting the degeneracy. The critical exponents depend on the system parameters through an N*N dressed charge matrix implying the relevance of the interaction of charge- and spin-density waves.

A microcanonical Monte Carlo algorithm has been developed to calculate the density of states of a two-dimensional Blume-Capel model in zero field. The full density of states is calculated for a 16*16 lattice as a function of the two spin summations which appear in the Hamiltonian. This permits the partition function and related thermodynamic functions to be evaluated for any temperature and any single site anisotropy from a single data set. Results are presented for the pseudo-critical transition temperature and the maximum in the specific heat as a function of the single site anisotropy parameter. The technique will allow the region near the tricritical point to be explored in detail and will permit the future determination of the zeros of the partition function. It should be noted that the microcanonical sampling method of Lee (1990) was found to be unusably slow for the S=1 problem in regions were the density of states had a significant slope and the microcanonical sampling described in this work is a significant improvement on his sampling method.

Several recent works have shown that the one-dimensional fully asymmetric exclusion model, which describes a system of particles hopping in a preferred direction with hard core interactions, can be solved exactly in the case of open boundaries. Here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices. With this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules. They obtain several explicit forms for these non-commuting matrices which are, in the general case, infinite-dimensional. Their approach allows exact expressions to be derived for the current and density profiles. Finally they discuss briefly two possible generalizations of their results: the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles.

The authors describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3^N/4 times a polynomial in N, and so the approach is greatly superior to direct counting techniques. They have enumerated walks of up to 39 steps. As a consequence, they are able to accurately estimate the critical point, critical exponent, and critical amplitude.

The authors describe a new algebraic technique, utilizing transfer matrices, for enumerating self-avoiding trails on the square lattice. They have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding walk universality class. Assuming that trails behave like A lambda ⁿn¹¹32/, they find lambda =2.72062+or-0.000006 and A=1.272+or-0.002.

Linear diffusion in a system of noninteracting fermion oscillators is constructed using the methods of statistical dynamics. The temperature distribution is shown to obey the heat equation C_v(T) delta T/ delta t= lambda div (C_v(T) grad T)/2 where C_v= delta (E)/ delta T is the heat capacity/molecule. An example shows that the system violates the 'principle of minimal entropy production' at a stationary state. The model confirms the similar conclusion drawn by M.J. Klein (1956).

The phase diagram of two different nonequilibrium three-state systems is here studied by means of MFRG and computer simulations; critical exponents are obtained by a finite-size scaling analysis of the MC data. A symmetry argument used by Grinstein et al. to predict the critical behaviour of two-state nonequilibrium systems (with up-down symmetry) is shown to apply to these three-state systems (with symmetry of interchange between two of those states).

The ordering dynamics of a system with a non-conserved order parameter is considered following a quench into the ordered phase from high temperature. Newman and Bray (1990) have set up an expansion in powers of 1/n for the O(n) model to obtain correlation functions and the two-time exponent lambda , but their calculation contains a simplifying assumption which is incorrect. Tight upper and lower bounds for lambda are obtained as a function of the space dimension d. These bounds exclude the result of Newman and Bray, although the dependence of lambda on d is qualitatively very similar. Comparison with simulations shows that the first-order 1/n calculation does not agree with numerical results as well as previously thought.

Generalized deformations of the fermionic algebra are studied. The polynomial representations of these algebras are constructed. All the deformation schemes can be realized by the same polynomial basis (using the Bargmann representation), thus proving that all deformed fermionic algebras are isomorphically equivalent to the non-deformed fermionic algebra.

The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.

The logic of constructing and interrelating classical realizations of the dual space of SO(3) is exhibited. Analogies with different methods of quantization of gauge theories are pointed out. The analysis is applied to SU(3) to obtain existing realizations and to construct new ones. The Gelfand-Zetlyn basis for the irreducible representations of SU(3) is explicitly realized using polynomials in four variables and positive or negative integral powers of a fifth variable. Another realization uses a spinor of SO(6)*SO(3,1). These are the analogues of Schwinger-Bargmann construction for SU(2).

Multimatrix models for which the index set has a group structure and the interaction obeys a 'zero curvature' condition can be deformed related to central extensions of this group. The deformed multimatrix models lead to statistical systems defined on random graphs with a topological action. It is shown, how these topological theories on graphs can be used to weight graphs according to topological conditions.

Explicit results for particular coupling coefficients for the most degenerate representations of SO(n) are given. The isoscalar factors which allow a recursive calculation of arbitrary coupling coefficients from those of SO(3) are derived. 6j- and 9j-symbols for most degenerate representations are also briefly discussed.

The authors develop and analyse certain character reductions for the infinite-dimensional unitary discrete series representations of the non-compact symplectic group Sp(2n,R). The group reductions considered are Sp(2k,R) contains/implies Sp(2,R)*O(k) and the more general Sp(2nk,R) contains/implies Sp(2n,R)*O(k). They use Schur function techniques to derive succinct formulae involving certain infinite series of Schur functions. The results are relevant to the study of many-body systems with interactions of bilinear form and to the description of various quantum phenomena including collective behaviour.

The author introduces the piecewise linear functions sin_Kx and cos_Kx representing K straight line segments with vertices on sinx and cosx respectively. The author Fourier expands these functions, discusses their properties, and derives a number of identities which follow from the expansion of the functions themselves and their integrals or derivatives. The motivation to study sin_Kx and cos_Kx comes from the computer simulations of the Rayleigh-Taylor instability in which the eigenmodes sinx and cosx are represented by sin_Kx and cos_Kx, K+1 being the number of nodes used in the simulations. The author finds that the harmonics generated by a finite K representation occur only at multiples of K plus or minus one.

The quantum Minkowski spacetime has real structure and this seems to be contradictive to the differential calculus in it. Dual differentiations are introduced to solve this problem. This duality can be extended to differential calculus in any C*-algebra.

The usual construction of coherent states allows a wider interpretation in which the number of distinguishing state labels is no longer minimal; the label measure determining the required resolution of unity is then no longer unique and may even be concentrated on manifolds with positive co-dimension. Paying particular attention to the residual restrictions on the measure, the authors choose to capitalize on this inherent freedom and in formally distinct ways, systematically construct suitable sets of extended coherent states which, in a minimal sense, are characterized by auxiliary labels. Interestingly, they find these states lead to path integral constructions containing auxiliary (essentially unconstrained) path-space variables. The impact of both standard and extended coherent state formulations on the content of classical theories is briefly examined, the latter showing the existence of new, and generally constrained, classical variables. Some implications for the handling of constrained classical systems are given, with a complete analysis awaiting further study.

The authors derive the complete spectrum of the Landau-Lifshitz equation using the Hirota method. Subsequently, they perform a phase shift analysis of spin-wave-kink and spin-wave-breather collisions, respectively. Finally, they use this result in order to derive the spin wave density of states in the presence of an arbitrary number of solitons.

Using a time-dependent scattering theory for light the authors obtain expressions for the time that light spends in a finite dielectric medium surrounded by vacuum, the so-called dwell time. Given an incoming wavepacket, they focus upon the light scattered in an arbitrary direction. In view of the similarity between Maxwell's equations and the Schrodinger equation, some useful results are derived for the case of Schrodinger potential scattering as well. They show that the dwell time of a Schrodinger particle is a derivative of the phase shifts with respect to the potential, rather than to energy as is the case for the phase-delay time. They indicate the relation to absorption arguments. Because the potential for classical waves is energy dependent, derivatives of the phase shift with respect to potential enter into the phase-delay time for classical waves.

Scattering of Gaussian wavepackets by impenetrable solenoids is studied using both analytical and numerical methods. A formula for the asymptotic (t to infinity ) angular distribution of the scattered packet is derived and used to discuss the physical meaning of cross sections, the optical theorem, and the classical limit. Angular distributions obtained from this formula are found to be in good agreement with angular distributions calculated from numerical solutions of the Schrodinger equation.

Space and time cavities with internally trapped electromagnetic fields are considered as wave-particle models of extended time-like and space-like particles. It is shown that in an N-dimensional (N=1,2,3) space-cavity photons undergo a conversion into a system composed of a bradyon at rest and N infinite-speed transcendent tachyons moving circularly about the bradyonic constituent. The time-like Klein-Gordon equation and its space-like counterpart for inner and outer fields associated with trapped radiation have been derived.