Table of contents

In two-mode Fock space a new completeness relation for the q-analogue of the single-complex-variable-state is proved by virtue of some q-deformed combinatorial identities, which are also derived here for the first time.

The authors introduce a multiparametric generalization of the non-standard R matrix if sl(n). Applying the method of Faddeev-Reshetikhin-Takhtajan (1990) to the R matrix, they construct the quantum group associated to it. The new feature of this quantum group is that upon a certain choice of the parameters, there can be nilpotent elements in it which is a sign of some hidden superstructure.

The authors give an explicit expression of the recurrency formulae of canonical realization for quantum enveloping algebras U_q(sl(n+1, C)). In these formulae the generators of the algebra U_q(sl(n+1, C)) are expressed by means of n-canonical q-boson pairs one auxiliary representation of the algebra U_q(gl(n, C)).

General methods for separation of coupled differential equations are applied in coupled-channel scattering theory with appropriate asymptotic conditions. It is shown that the problems with thresholds cannot be separated. Finally it is pointed out that separation of the coupled-channel s-wave problem without thresholds can be realized by the use of generalized Darboux transformations.

The generating function for the product of the associated Laguerre and Hermite polynomials is formulated and used to determine the coordinate-space wave function of the displaced number states of the harmonic oscillator for the general case of a complex displacement parameter. The benefits, in this context, of an extended form of the associated Laguerre polynomials are advertised.

The authors consider the following nonlinear coupled partial differential equations which describe a biochemical model system: delta x/ delta t=x²y-Bx+K delta ²x/ delta r²; delta y/ delta t=-x²y+Bx+K delta ²y/ delta r² where B is a parameter and K is the diffusion constant for the concentrations x and y. Seeking a wave solution, the authors obtain a nonlinear ordinary differential equation in one dynamical variable only. This ODE is analysed for the Painleve property and then they are able to find a one-parameter family of solutions which are of biological interest.

The arguments in favour of the possibility of the parity-invariant description of the interaction between electric and magnetic charges are re-examined. It is shown that P-invariance is possible if one generalizes the framework of the standard Abelian gauge coupling to a U(1)(X)U(1) coupling.

Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of the usual quantum mechanical phase space. The case of a slightly non-commutative space is considered in detail. A free particle in such a space is found to behave as if it is a charge particle moving in a self-generated magnetic field.

It is shown that the quantization of two Hamiltonians generating (with respect to two different symplectic structures) the same Hamiltonian vector field, may have completely different spectra.

The authors have investigated the current distribution in the two-component deterministic percolation model in which the ratio h of poor to good conductance is regarded as a small parameter. It is found that the minimum current I_min(h) scales anomalously with h: I_min(h)/I_min(1)=exp(-constant(ln h)²)H(hL^phi ), where L is the size of the network, H a function describing the crossover from fractal to homogeneous behaviours and phi the crossover exponent. The exponential prefactor is quite similar to the behaviour of left-sided multifractality in diffusion limited aggregations. It is found that phi coincides with the crossover exponent for all multifractal moments of the current distribution.

A method is presented which yields the exact solution of the generating function for the first-passage probability and related observables, for a one-dimensional random walk with random hopping rates on each site. A geometrical and statistical-mechanical interpretation of the problem is suggested and is used for obtaining the asymptotic properties of the occupation and first-passage probabilities. The class of values with the property of self-averaging is found.

The authors determine the diffusion constant K for unbiased, discrete-time random walks on infinitely long planar strips of finite width with regularly spaced, stepped edges that model, albeit in a simple and regular manner, some of the features of the corrugated surfaces of diffusion channels in porous media. They derive a formal expression for K and use it to compute K exactly for a range of values of the strip width and the step width, the latter serving as a measure of the roughness of the boundaries. They also consider the role of specific boundary conditions at the edges by determining K for both myopic and blind random walkers in each case. The results shed some light on the effects of spatially inhomogeneous transverse boundaries on the coefficient of diffusion in the longitudinal direction.

A two-dimensional q-state Potts model with vacancies and four-spin interactions is studied. The parameter space of the model contains a critical and a tricritical manifold. Moreover for 0<or=q<or=9/4 a multicritical point is found which is the locus where the tricritical transition changes from first- to second-order. At this multicritical point, which can be located exactly, the model is solvable. The authors compute the value of the central charge and a number of critical exponents.

The authors explore the self-dual plane of a two-dimensional Potts model with vacancies and four-spin interactions, using finite-size scaling of transfer-matrix results. Since the transfer matrix construction is based on the random-cluster representation of the Potts model, its application is not restricted to an integer number q of Potts states. For q<or=4 the self-dual plane contains a critical region belonging to the ordinary Potts universality class, and a region of first-order transition points. The two regions are separated by a line of which a range consists of Potts tricritical points, while another range is of a first-order nature. The point separating these two ranges is, for 0(q<or=9/4, identified with the multicritical point of which an exact solution has recently been given by Nienhuis et al. (1979).

Studies the four-state antiferromagnetic Potts model with next-nearest-neighbour ferromagnetic interactions on the simple cubic lattice by the Monte Carlo method. The ratio J₂/J₁ of the strength of the next-nearest-neighbour interactions to that of the nearest-neighbour interactions is to be 10. The author analyses the finite-size effects of the energy, the specific heat, the fourth-order cumulant, the magnetization, the susceptibility and the effective transition temperatures. With the help of the finite-size scaling theory the author concludes that the transition of the model is assigned to be of first order.

For the analysis of the release of fission product aerosols from reactor cores in a severe accident, it is necessary to understand the process of homogeneous nucleation of droplets from a supersaturated vapour, including the dependence on system pressure, which can vary greatly depending on the type of accident. A model of the effect of carrier gas on the nucleation process is presented, based on the changes to the energy distribution of nucleating vapour molecule clusters induced by collisions with carrier gas molecules. The rate of nucleation is altered since the cluster decay rate is strongly energy dependent. The approach is compared with previous treatments of the problem, illustrating the importance of using an equilibrium cluster energy distribution which goes beyond a Gaussian approximation, and clarifying previous confusion between cluster temperature and cluster energy in the literature. Calculations of nucleation rates are performed for n-nonane in argon and water vapour in air.

A systematic study of the Q-state Potts model of neural networks, extended to include biased patterns, is made for extensive loading alpha . Mean-field equations are written down within the replica symmetric approximation, for general Q and arbitrary temperature T. For the Q=3 model and two classes of representative bias parameters, the storage capacity and retrieval quality at zero temperature are discussed as functions of the bias, taking into account the Mattis retrieval state and the lowest symmetric states. The T- alpha diagram is obtained and the stability properties of the retrieval state are analysed at finite temperatures. A comparison is made with the biased Hopfield model.

The authors study neural network models in which the initial interaction matrix elements are drawn from an arbitrary probability distribution and in which patterns are subsequently stored by eliminating the frustrated bonds, generalizing a proposal by Kinzel (1985). They show that the optimal choice for the a priori distribution corresponds to choosing uniform ferromagnetic initial interactions. For the optimal model they study analytically the dynamical behaviour, the equilibrium properties, the sizes of the domains of attraction and a number of information-theoretical performance measures.

Small amplitude motion in the chaotic maser model with finite temperature is studied in the context of a mean field approximation. The equilibrium state is constructed for both the normal and superradiant phases. The linear response of the model is obtained in these two cases and analytical expressions for the variation of the corresponding frequencies with temperature are derived. Their behaviour both in the integrable and chaotic limit is discussed. Sum rules are shown to be fulfilled and relative percentages turn out to be temperature dependent. Finally the authors obtain the dynamical stability conditions for the chaotic maser model at finite temperature.

Coherent states for the positive mass representations of the Poincare group in the 1+1 dimensions have been obtained previously, using the fact that these representations are square integrable moduli of the subgroup of time translations. The method is extended by combining sections from the coset space into the group with homeomorphisms of the coset space (these maps are called quasi-sections). Then the generalized construction is applied to the zero mass representations of the (1+1)-dimensional Poincare group, which are square integrable moduli of subgroup of light-like translations. The resulting coherent states, indexed as before by points in phase space, yield a resolution of the identity of the Krein space of the zero mass representations (the first explicit example of such a structure), and it turns out that they coincide with the familiar wavelets based on the 'ax+b' group.

It is widely known that the Kepler problem admits SU(2,2) as a dynamical group. The author aims to show that SU(2,2) is also a dynamical group for the MIC-Kepler problem, a generalization of the Kepler problem. It is already known that the symmetry groups for the MIC-Kepler problem are SO(4), SO₀(1,3), and E(3), according to whether the energy is negative, positive, or zero. It is shown that the double cover of the respective symmetry groups, SU(2)*SU(2), SL(2,C), and SU(2)*R³, a semi-direct product, are realized as subgroups of SU(2,2). Isoenergetic orbit spaces are also studied, which are defined to be quotient manifolds of the respective energy manifolds by the respective Hamiltonian flows. Each of the isoenergetic orbit spaces is shown to be realized as a (co-)adjoint orbit of the symmetry group. In addition, use of the isoenergetic orbit spaces is discussed. In fact, a certain class of perturbed MIC-Kepler problems are shown to induce dynamical systems on the isoenergetic orbit space. If the energy is negative, the generic isoenergetic orbit space is diffeomorphic with S²*S², so that the Euler number of S²*S² provides the number of singular points for the reduced perturbed system, and in turn that of closed orbits for the perturbed MIC-Kepler problem.

The authors develop a method for calculating the Cartan forms in the parametrization of group transformations which has a simple composition law and satisfies the condition called the naturalness. The method is applied to gauge theories for finding the explicit transformation laws of gauge fields under finite local transformations of groups, and to chiral field theories for setting up chiral field Lagrangians. The explicit forms of finite transformations of gauge fields and their nonlinear realizations for gravity and supergravity are found. The expressions of the Cartan forms, the Lagrangians of the principal chiral fields and the Goldstone fields for the unitary groups U(2), SU(2), U(3), SU(3) and the coset spaces associated with these groups are derived. The Lagrangians obtained are distinguished from the previous known ones by new types of nonlinearity.

A method is described for obtaining algebraic approximations to certain integrals that appears to be independent of known methods, such as Taylor, orthonormal functions, Laplace's method, etc. It resembles the numerical technique of product integration. Often the solutions are globally valid over the full range of each parameter. In some cases a trade-off can be performed between the accuracy and the functional complexity of the solution. As examples, specific integrals for estimating extinction from aerosols are resolved. These analytical solutions are applicable to scattering from spheres, spheroids, discs and infinite cylinders.

Using Lie group methods and the Painleve test, the authors analyse nonlinear diffusion reaction equations u_t=(D(u)u)+f(u) with power law diffusion coefficients (D approximately u^v) and arbitrary nonlinear reaction terms f(u) which have a wide spectrum of applications in many areas of science. The Lie-group-based similarity method leads to a classification of the reaction terms according to its symmetry properties. With the help of the adjoint representation, the optimal system of similarity reductions is calculated. To check the integrability of the partial differential equation, the existence of generalized (Lie-Backlund) symmetries is investigated. Apart from three known cases, no further cases with third-order symmetries exist. Examining the integrability of the second-order ordinary differential equations resulting from the reductions, only a few parameter combinations can be found for which the Painleve property is given. However, they are able to construct unknown integrals of motion for a much larger range of parameter values. From the integrals, exact similarity solutions may be derived. This is demonstrated by examples corresponding to the important moving-wave reduction.

The authors determine some necessary conditions for a given partial differential equation E, written in conservative form to admit a potential symmetry (PS). A PS of E is a point symmetry of the auxiliary system S_p is obtained introducing a potential as further unknown function, then a PS leads to the construction of solutions via the classical reduction method. Given a PS, they introduce an algorithm that allows us to determine a class of E-solutions which includes the ones obtained as invariant solutions under the related point symmetry of S_p. As examples, they consider a Fokker-Planck equation, a wave equation in nonhomogeneous media and a quasilinear wave equation.

In a study of the screening properties of a charged impurity located inside and near the surface of a metal subjected to a magnetic field, there arose an interesting class of Eulerian integrals, involving the Bessel function J_v(z) or J₀(z), which were expressed in closed forms by Glasser (1984). Subsequently, Wille (1988) generalized and extended Glasser's work by giving closed-form expressions for a number of Eulerian integrals involving Meijer's G-function. Motivated by these works, the authors aim at evaluating a general class of Eulerian integrals involving Fox's H-function. Their main result is shown to provide a key formula from which one can deduce each of the aforementioned integrals and also numerous other potentially useful results not contained in the earlier works; they are led, in this way, to the corrected version of one of Wille's main integral formulae.

An extended analysis of eigenvalues and wavefunctions resulting from the Hill determinant method is carried out for the one-dimensional sextic anharmonic oscillator. It is shown that, in spite of a seemingly natural ansatz of the method, irrelevant wavefunctions may appear, leading to incorrect eigenvalues. The domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics. Effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented.

A new nonconfocal generator of involutive systems is obtained, and the relation between it and Levy equation hierarchy is discussed. Furthermore, the representation of the solution for the Levy hierarchy is given.

The authors propose a model for quantizing a nonrelativistic physical system that admits a finite number of internal degrees of freedom. They assume that the configuration space of the given system carries the structure of a smooth manifold M. The kinematics of the given system is assumed to be describable by a flow model. Under certain technical restrictions momentum and position operators are seen to act on a Hilbert space of square-integrable sections of a Hermitian vector bundle over M. They determine their necessary shape up to isometric isomorphism and distinguish two types. In particular, they show that the number of inequivalent quantizations depends on the topology of the underlying configuration space.

The author proposes the system of charge and current densities realizing static and nonstatic electric solenoids. There is a nontrivial electric vector potential outside the electrical cylindrical or toroidal solenoids. It cannot be eliminated by the gauge transformation. The nonstatic solenoids emit waves of electromagnetic potentials propagating with the velocity of light. Conditions under which they can be detected are discussed.

The quantitative theoretical analysis of Tonomura experiments testing the existence of the Aharonov-Bohm effect is given. The intensities of the electrons scattered by the toroidal solenoid are computed in the framework of the Fresnel-Kirchhoff diffraction theory, for different values of the magnetic flux and positions of the observation plane.

A new quantization scheme is proposed. The similarity of geometric and covariant quantization of a certain class of constrained dynamical systems is shown. A method to calculate anomalies is developed.

In the paper of the same title (see ibid., vol.24, p.3611, (1991)), the behaviour of the free energy was investigated for an exactly soluble condensed matter system that contained a topological soliton. When lambda , the size of the soliton in units of the coherence length, was small compared with unity, the free energy was shown to behave as a nonanalytic function of lambda . The authors show that a simple approximation to the exact free energy exists which is able to reproduce the leading small lambda results and, more generally, approximate the exact free energy, with reasonable accuracy, over a wide range of lambda . Thus at zero temperature, for example, the errors the approximate free energy makes on the value of lambda and the free energy at its minimum are 18% and 8.3.% respectively.