Table of contents

The q-deformation of symmetric functions is introduced leading to q-analogues of many well known relationships in the theory of symmetric functions. q-deformed scalar products are developed and used to define q-dependent symmetric functions. The symmetric functions commonly associated with the names Hall-Littlewood, Schur and Jack are all special cases of the q-deformation of Macdonald's (1988) new symmetric functions P_lambda (s,t). A q-analogue of the spin and ordinary characters of S_n is given and illustrated by the explicit calculation of examples of q-deformed characters. The methods used are closely parallel to those of quantum groups.

The problem of reflection and refraction of electromagnetic waves on the interface of dielectric media is dealt with according to the generalized variational principle and some properties of the electromagnetic waves near the interface are deduced from the transformation properties of the constrained system under the transformation of coordinates. These lead to the equation of motion of the centre of energy which shows that the transverse shift is in existence.

The Schrodinger equation with interaction -Ze²/(r+ beta ) is shown, for general values of the parameters, to be reducible to the confluent Heun equation.

A discretized form of the one-dimensional Dirac equation is solved to study relativistic band structures. Properties of periodic continued fractions are used to find the dispersion relation inside allowed bands. The Mathieu potential is considered and it is found that relativistic effects cause a reduction of the width of allowed bands and gaps.

Shows that the Darboux-Bargmann method can be used to derive Backlund transformations and construct explicit soliton solutions for certain inhomogeneous nonlinear Schrodinger-type equations.

Considers numerical simulations of the Langevin equations recently proposed by Abraham et al. (1990) to model the spreading of a fluid on a substrate. A dependence upon the temperature is found in both the dynamical and the static properties. The contact angle, as a function of the parameters, is calculated in the partial wetting case. A precursor film with constant speed is observed in the complete wetting phase. The fluid above the precursor film is found to behave in time as t^alpha , with alpha function of the height of the fluid. At low temperature, alpha is smaller than 1/2 and larger at high temperature. The profile of the fluid is found to be close to a parabola, at low temperature.

Two models of the wetting dynamics of a non-volatile liquid are introduced. Both models exhibit a transition between partial and complete wetting regimes. The first model, which can also be viewed as an inhomogeneous surface growth model, is studied in a two-dimensional capillary geometry. For complete wetting the profile of the fluid surface, apart from a precursor film, then assumes a stationary shape, which scales with the system size. A precursor film forms even at the transition, which one can locate analytically, although here it moves only as t^chi , where chi approximately=1/2. This first model permits only fluctuations parallel to the wall over which the fluid spreads. Monte Carlo simulations of the second model show that the inclusion of fluctuations perpendicular to the wall can lead to qualitatively new phenomena.

The authors study the distribution of heights in the self-organized critical state of the Abelian sandpile model on a d-dimensional hypercubic lattice. They calculate analytically the concentration of sites having minimum allowed value in the critical state. They also calculate, in the critical state, the probability that the heights, at two sites separated by a distance r, would both have minimum values and show that the lowest-order r-dependent term in it varies as r^-2d for large r.

Studies a two-state version of the sandpile model of self-organized criticality. Instead of a critical height of stability as in the sandpile model, the author introduces a hard core repulsion among different particles at the same position. In the case of a collision particles hop randomly to the nearest neighbours. Critical exponents obtained by numerical simulation show strong disagreement with the values theoretically predicted for the sandpile model but they are close to the numerical values of other models.

The authors have shown that the renormalization group fixed point for mutually avoiding directed random walks in (d+1) dimensions is exactly 2 pi in where in =2-d, with no higher order terms in in . Since the whole calculation can be carried out to all orders exactly, the model is also significant from a pedagogical point of view.

The lacunarity is computed for randomly diluted two-dimensional lattices in terms of the chemical distance. The crossover behaviour and the question of universality are discussed.

Presents simulations supporting the conjecture that properly defined exterior perimeters of 2D critical percolation clusters have the same fractal dimension, D=4/3, as self-avoiding random walks. In addition, the authors discuss possible generalizations.

A class of recently introduced irreversible multilayer adsorption models without screening is analysed. The basic kinetic process of these models leads to power law behaviour for the decay of the jamming coverage as a function of height. The authors find the exact value for the power law exponent. An approximate analytical treatment of these models and previous Monte Carlo simulations are found to be in good agreement.

Both trigonometric and rational solutions to the Yang-Baxter equation associated with the spinor representations of the quantum B_l universal enveloping algebras are obtained. The corresponding representations of the braid group and the link polynomials are also computed through a standard method. The quantum Clebsch-Gordan matrix, the quantum projectors and the solutions associated with the spinor representation of the quantum B₃ are presented explicitly.

A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmann's scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.

Screening and vertex operators for u(n) are given in terms of generators of the group left action on the flag manifold U(n-1)/SU(n). Their matrix elements in a basis of holomorphic sections of a vector bundle are computed.

For pt.II see ibid., vol.23, p.5371 (1990). The q-oscillator of type one and the classical and the quantum q-deformations of SU(2) algebra realized through the q-oscillators are studied in the case of q being roots of unity, i.e. q^k=1. The q-excitations are found to be non-bosonic. In particular, when k is of rank 4 and 6, the q-excitations are shown to be fermionic and parafermionic respectively.

Integrals of several spherical Bessel functions occur frequently in nuclear physics. They are difficult to evaluate using standard numerical techniques, because of their slowly decreasing oscillatory form. The authors derive an analytic expression for the infinite integral of three spherical Bessel functions. They then use this result, together with the closure relation for spherical Bessel functions, to show how in principle one can derive an analytic expression for the integral of any number of spherical Bessel functions. They demonstrate this by deriving an analytic expression for the integral of four spherical Bessel functions. As with all of these analytic formulae, the results require that all angular momenta corresponding to the spherical Bessel functions can be coupled together to give an overall scalar quantity and conserve parity. The authors discuss the numerical accuracy and stability of this procedure.

Basing on a direct method developed by Clarkson and Kruskal (1989), the Kadomtsev-Petviashvili (KP) equation has been reduced to three types of (1+1)-dimensional partial differential equations which are equivalent to the three types of the similarity reduction equations obtained by the classical Lie approach but with different independent variables. More arbitrary functions which have been missed by the classical Lie approach have been included in the solutions of the KP equation. For instance, the third type of reduction obtained by the direct method can be divided into three subcases and the third type of solution of the KP equation obtained by the classical Lie approach is only a special case of one subcase of the authors' results.

The authors consider the interaction of k-loops with an adsorption surface and prove that the polymer model of k-loops has exactly the same reduced free energy as that for self-avoiding walks interacting with a surface.

The hydrogen atom in a semi-infinite space limited by a paraboloidal boundary, with the nucleus at the focus, is studied as a model of an atom on the surface of a solid. The energy eigenvalues, hyperfine structure, electric dipole moment and pressure of the system are evaluated as the paraboloidal boundary approaches the nucleus up to the position at which the atom becomes ionized.

Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.

An overview of isomorphic correspondences between spin algebra and algebras of functions on the sphere is given. In addition, the authors show that in one case the induced product between the spin function and another function is expressible by the action of a first-order differential operator correcting the ordinary product. This deformation leads to the geometric quantization of classical spin. The classical limit is studied for a class of correspondences.

The authors analyse the nonlinear Schrodinger model (the one-dimensional Bose gas) by means of conformal field theory in conjunction with a newly developed cutoff procedure. This makes it possible to obtain exact expressions for current correlators in the model. A number of new results are presented, including the expression for the two-point current correlator on a finite-size strip and the general n-point current correlator on the plane. All these results have the full time dependence and are presented to order 1/g, where g is the strength of the repulsive delta -function interaction.

The author shows that the finite-temperature equal-time field correlators of the exactly integrable (1+1)-dimensional gas of impenetrable fermions in a magnetic field can be expressed as the first Fredholm minor of a completely integrable linear operator. This result enables the author to derive differential equations for quantum correlators and to obtain analytical formulae for long-distance asymptotics. In particular, a formula for correlation length as a function of temperature and magnetic field is presented. His analysis reveals the presence of two length scales, which separate short-distance, intermediate and extreme asymptotic domains. The importance of the cross-over region is emphasized.

In the conformal theory the magnetization operator of the critical 3-state Potts or self-dual Z₃ model is degenerate at level 6. Solving a sixth-order differential equation, the authors calculate the bulk 4-spin correlation function and the spin-spin correlation function in the half space. The spin-spin correlation function of the semi-infinite self-dual Z_N model is also obtained for arbitrary N. Both free- and fixed-spin boundary conditions are considered.

The eigenvalue moment method (EMM) of Handy and Bessis (1985) is reformulated with special emphasis on the importance of positivity as a general quantization criterion; thereby making EMM theory relevant in the analysis of manifestly non-positive systems such as those corresponding to multi-dimensional excited bosonic states.

Landau levels are obtained using the techniques of phase-space quantum mechanics. Wigner functions are calculated for the eigenstates of this system.

Order and disorder operators and correlation functions are studied in the context of a fermionic quantum field theory in 1+1 dimensions.

The authors study the growth probability distribution for a single cluster growth process for percolation at the percolation threshold. They find that this distribution is indistinguishable from the chemical path distribution on the percolation cluster as studied by Havlin et al. (1985). They also study the density profile of the percolation cluster. Using the multiscaling idea of Coniglio and Zannetti (1989), they calculate the fractal dimension of the percolation clusters from the density profile and report d_f=1.895+or-0.005 for two dimensions and 2.50+or-0.01 for three dimensions. However they do not find any evidence of multifractal or multiscaling behaviour.

The shortest paths of self-avoiding walks (SAWs) with bridge lengths b=1, 2, 3 and 2 are studied by exact series expansions and by Monte Carlo simulation methods on the square and simple cubic lattices. In the series work, SAW configurations of up to 24 and 14 steps are generated on the square and simple cubic lattices respectively. Assuming the shortest path S_N between two sites, which are separated by N steps along the chain, has the scaling form S_N approximately AN+BN^{1- Delta} , it is found that Delta approximately=1 and Delta approximately=1/2 on the square and simple cubic lattices respectively, independent of the bridge length. The problem is also investigated by the Mellin-Pade approximation method using Monte Carlo data on the square lattice. The latter method gives results consistent with those of the series expansion study. The authors' results are not consistent with earlier predictions.

The authors study numerically the structure of fronts generated in a directed percolation problem in two dimensions when the probability for a bond to be present decreases linearly along one direction. They discuss several cases of relative orientation of the preferential direction with respect to the gradient. They also consider the Domany-Kinzel limit case of directed percolation, and obtain analytical results. The structure of the front is analysed using the jump size distribution introduced by Hansen et al. (1990).

The first breaking current of the randomly diluted fuse network in a hierarchical diamond lattice is studied theoretically and numerically. The authors show that the prediction proposed by Duxbury et al. (1987) for Euclidean lattices should be applicable, and that it is correct: the average current necessary to break the first bond of the lattice decreases on average as the inverse of the logarithm of the system size. Its validity is shown to rely solely on the self-averaging of the conductivity with system size. Due to the peculiar geometry of the lattice, the breaking current as a function of the lattice size exhibits a series of plateaux followed by sudden variations between them. Once the first few bonds are broken, the rest of the breaking of the network is no longer comparable to that of a Euclidean lattice: the mean breaking current for the entire lattice decreases with system size much slower than the current necessary to break the first bond.

The exact solution of the CSOS models by Pearce and Seaton is analysed. This solution appears to describe a first-order transition without droplet singularities. The authors show that the system with free boundaries has no such transition, whereas the system with fixed boundary conditions has a first-order transition at an anisotropy dependent locus, unlike the findings of Pearce and Seaton. They also explain the absence of droplet singularities. Finally they present a simple Coulomb gas calculation of some critical exponents, resulting in the same expressions as given by Pearce and Seaton (1989).

The authors study the semiclassical limit of the spin boson model, where the spin remains quantized and the field is expanded around the classical limit of a large number of photons, using a phase representation due to Bialynicki-Birula and Bialynicki-Birula (1976). They derive the semiclassical Hamiltonian and prove that in the semiclassical expansion there always arises a back reaction of the atom on the field. The nearest-neighbour spacing distribution of levels is studied in the same limit and shown to be, in general, qualitatively different from the distribution for the original model. However, the comparison between the Dyson-Mehta statistics is also studied and no qualitative difference is visible for the function Delta ₃.

The author proves by induction that, with certain fixed boundary conditions, elementary cellular automaton rule 3 has a period-two limit cycle to which all states are attracted, regardless of lattice size.