Table of contents

The connection between the Ocneanu trace on H_n(q) and Schur functions leads to a simple method for calculating the irreducible characters of the Hecke algebras H_n(q). The characters appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions.

The authors construct the regular representation quantum Lie algebra of SL_q(2) and its quotient representations and then analyse the reducible structures of these representations when q^p=1 for an integer p. They show how the indecomposable representations of SL_q(2) are obtained by a purely algebraic method.

The method of construction of boson realizations of semisimple Lie algebras formulated in a previous paper is applied to the case of the quantum group U_q(sl(2)). The realizations are expressed in terms of the usual Weyl algebra.

The authors present the results of an analysis of the squeezing of components of the (conventional) electromagnetic field in quantum group analogues of the Heisenberg-Weyl (HW) coherent state and SU(1,1) squeezed state. They find that squeezing occurs for all finite-q values not equal to unity in the HW q-coherent state, in contrast to the usual case; and also in the SU_q(1,1) case, although here less than in the usual (q=1) SU(1,1) squeezed state.

A method is shown to convert a partial summation into a residue problem using z-transforms. This method is particularly useful for evaluating partial sums whose infinite sums diverge.

There is a close relationship between the theory of coupled differential equations and supersymmetric quantum mechanics. The authors set up the bridge connecting these fields with two theorems concerning both the coupling and noncoupling cases.

The analytic structure of the solution of the driven pendulum is investigated through Painleve analysis in the complex time plane. The existence is pointed out of a two-armed infinite sheeted Riemann structure of the singularities after an exponential transformation.

The authors develop a variational approach for studying interfaces and other manifolds in a disordered quenched medium. The method may be applied to problems which range from directed polymers to the interface of an Ising model in a random magnetic field. They find that replica symmetry is spontaneously broken and the results of the Flory approximation are recovered in a simple way. Corrections to this approximation may be computed in a systematic way.

A new class of coupled soliton equations in 2+1 dimensions is proposed. They describe the interactions of a wavepacket of short waves with a single long wave on the xy plane. The N-soliton solutions of the equations are obtained explicitly. The proposed system of equations is also shown to include, as special cases, various known physical equations, indicating that the system is relevant to describe physical systems and may have wide applications in hydrodynamics and plasma physics, etc.

It is shown that the soliton-gas phenomenology breaks down for the classical statistical mechanics of the sine-Gordon model. As a main cause of the breakdown, it is pointed out that the solitons and breathers in the gas phenomenology do not precisely represent the total degrees of freedom of the system.

The authors use a simple generating function to calculate exactly the entropy of random quantum states for finite-dimensional Hilbert spaces over real, complex and quaternionic scalars. This allows them to extend their previous formula for the quantum correlation information of a state determination apparatus to include real and quaternionic von Neumann analysers.

The authors investigate the conductivity properties of a randomly diluted medium where the fraction of present bonds varies along the mean voltage drop, and reaches the percolation threshold. They obtain the scaling of the conductivity as a function of the system size for any concentration profile. They show that a transient scaling regime also appears for small system sizes and a rapidly varying concentration of bonds close to the threshold. Finally the distribution of local currents is also investigated.

The critical behaviour of fully directed self-avoiding walks (DSAW) on Sierpinski carpets is studied by numerical simulation. The authors propose a fractal cell generation method to form an infinite fractal lattice with the structure of a Sierpinski carpet. The obtained critical exponent ν_|| is independent of the fractal dimension of a Sierpinski carpet d_f but ν and ν_⊥ are dependent on d_f. The results indicate that DSAW on different Sierpinski carpets belong to different universality classes.

For neural networks with J couplings the perceptron problem for random unbiased patterns is considered. An algorithm that uses concepts of the continuous perceptron problem as well as ideas of biological optimization is proposed and investigated. The distribution of local stabilities and the critical storage capacity alpha _c are determined. While for N less than 50 the value of alpha _c is approximately 0.83, the storage capacity goes down to alpha _c=0.7 for N=255.

Grey-toned patterns are pictures composed of pixels of several shades of grey. The ability of neural networks using multistate neurons to store such patterns is systematically investigated. If conventional generalizations of Hopfield networks using analogue or soft neurons are considered, it is impossible to stabilize these grey tones. Nevertheless it is shown that it can be done with networks that use neurons which have only a discrete set of possible activities. This is demonstrated for the pseudo-inverse rule for the synaptic couplings, where only the stability of the patterns shrinks with increasing number Q of grey tones one wants to store. If the patterns are uncorrelated one can use the Hebb rule and in this case the mean field theory is presented. Applying this rule the storage capacity decreases as Q^-2 with the number of grey tones.

For pt. I see ibid., vol.23, p.4185 (1990). The SU_q(2) algebra is realized by means of both the Poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q-deformed oscillators of two different types. The structures of the Hopf algebra and the quantum Yang-Baxter equation are also discussed on a quantum level. A set of j-representations of the quantum algebra SU_q(2) is constructed based on the 'type-II' q-oscillators. Multi-deformations of the oscillators of the two types and multi-deformed algebras expressed in Poisson brackets as well as in Lie brackets are proposed.

Vector coherent states are defined for the positive discrete series irreducible representations of the non-compact orthosymplectic superalgebras osp(P/2N,R), where P=2M or 2M+1. An orthonormal Bargmann-Berezin basis, symmetry-adapted to osp(P/2N,R)so(P)(+)sp(2N,R)so(P)(+)u(N), is constructed and used to develop the K-matrix theory for osp(P/2N,R). A general method is provided for determining the conditions of existence of star representations (and of grade star representations in the osp(2/2N,R) case), and the branching rule for their decomposition into a direct sum of so(P)(+)sp(2N,R) irreducible representations. As a by-product, it also enables the matrix elements of the odd generators between basis states of lowest-weight so(P)(+)u(N) irreducible representations to be calculated in a straightforward way.

For pt.I see ibid. vol.23, p.5383 (1990). The techniques, presented in the first paper of the present series, for determining the conditions for the existence of star or grade star positive discrete series irreducible representations of osp(P/2N,R) (P=2M or 2M+1), and the branching rule for their decomposition into a direct sum of so(P)(+)sp(2N,R) irreducible representations, as well as for constructing explicit matrix realizations, are illustrated with a few selected examples. The latter include the most general irreps of osp(1/2N,2), osp(2/2,R), osp(3/2,R), osp(4/2,R), osp(2/4,R), and the most degenerate irreps of osp(2/2N,R). In addition, all the information necessary for dealing with other cases amenable to a full analytic treatment is provided.

Generalized Backlund transformations are applied to derive links between a large number of different types of nonlinear diffusion equations, including many which are of physical significance. Some new exactly linearizable forms are determined.

A Painleve analysis of the two-dimensional Burgers equation is carried out and used to obtain a restricted Backlund transformation that maps a subclass of the solutions of the 1+2D Burgers equation onto a linear heat-like equation. Alternatively, the Backlund transformation can be expressed as a map onto the derivative of the one-dimensional Burgers equation in appropriate dependent and independent variables. The singularity analysis also yields a further class of solutions obtained by solving a Schwarzian differential equation.

The one-dimensional single-particle Schrodinger eigenvalue equation is represented by a generalized matrix eigenvalue equation of the form A Psi =Ea Psi where A is a symmetric matrix, a is a symmetric positive-definite matrix and Psi is a column vector. The local error in making the representation is proportional to delta ⁴ where delta is the step size. Several methods of solution which make use of the sparsity of the matrices are investigated.

If the state space of a quantum system is finite dimensional it may be considered as the carrier space of an irreducible unitary representation of some compact Lie group G. In this case the set of operators may be put in one-to-one correspondence with a class of functions defined on the group parameters (Q representation). For these functions a binary relation is derived that corresponds to the product of two operators. The general formalism is applied to G=SU(2) (spin coherent states) and this result is used to derive a product formula for the non-compact Heisenberg-Weyl group (coherent oscillator states.).

Both trigonometric and rational solutions to the spectrum-dependent Yang-Baxter equation are presented for the minimal representations of the quantum exceptional E₆ and E₇ universal enveloping algebras.

Shows that (non-adiabatic) Berry phases in time-independent systems arise for fundamentally different reasons than those for time-dependent systems. For time-independent Hamiltonians the Berry phases are seen to only depend on the chosen cyclic initial state, the Hamiltonian merely providing the appropriate period. The author also discusses two time-dependent examples where the time dependence arises from different sources, namely a two-level atom in an intense laser and a harmonic oscillator with a periodic forcing term.

It is shown that a stochastic process described by a complex Langevin equation leads to real-time quantum mechanics. The author also derives a relation between a transition probability in the theory and a transition probability amplitude in real-time quantum mechanics in the framework of path-integral formulations. Finally, taking a harmonic oscillator case as an example, the Fokker-Planck equation is solved exactly, and a non-dissipation property of the stochastic process is pointed out.

The presence of an external electromagnetic wave leads to an important change on the Coulomb interaction between two quantum charged particles and gives rise to long-range radiative forces. These effects result in the modification of Rutherford scattering.

The regularized non-Abelian chiral Jacobian factor in the path-integral formulation of fermions interacting with background vector and axial vector fields is investigated at finite temperature in arbitrary even dimensions. After separating out the terms with normal parity, it is shown that the self-consistent non-Abelian chiral Jacobian is the same at T=0 and T not=0.

The method and details of the calculation of the three-loop contribution to the anomalous dimension of the diffusion coefficient of the model of a random walk in a potential random field with long-range correlations are presented. Contrary to earlier conjectures, this contribution does not vanish identically. A new method of calculation of multi-loop Feynman graphs with complicated numerator structure is suggested. It leads to simpler integrals in a space of higher dimensionality, which are computed using the recursion relations of the uniqueness method.

The replica method has previously been used to calculate the semicircular averaged eigenvalue spectrum of the Gaussian orthogonal ensemble of real symmetric N*N random matrices in the limit where N to infinity . The authors develop a perturbative scheme which, within this same replica framework, is used to calculate the corrections within this semicircular band of eigenvalues to order 1/N and 1/N². A new and straightforward self-consistency argument is presented and used to derive the shape of the averaged eigenvalue spectrum when N is large but finite and the scaling behaviour of this averaged eigenvalue spectrum near the band edges is demonstrated in a straightforward fashion. Some comments are made on the relation of the results to those of field theoretical calculations in zero dimensions.

The authors consider self-avoiding walks on the square lattice which are confined to lie in or on the boundary of a square with vertices at (0, 0), (0, L), (L, 0) and (L, L). They ask for the number of such walks which begin at the origin and end at the vertex (L, L), especially in the large L limit. Similarly they ask for the mean number of steps in such walks as a function of L. At fixed L the authors also associate a fugacity with the number of steps of the walk and ask how the system behaves as a function of this fugacity. They provide some rigorous results, in particular proving that there is a phase transition at some particular value of the fugacity, and supplement these with the analysis of series data for the problem.

The authors investigate the trapping of particles on regularly multifurcating Cayley trees by using random walk methods. In order to find a good approximation for the survival probability of the walker, they use a cumulant expansion, and obtain an exact expression for the variance of the range of nearest-neighbour random walks; the expressions are corroborated to high accuracy by simulation calculations. The method may turn out to be useful for more complex problems, such as walks on ultrametric spaces.

The authors study a generalization of the solid-on-solid (SOS) model recently proposed for the roughening transition of reconstructing and non-reconstructing FCC (110) solid surfaces. The generalized model is expressed in terms of Ising variables representing nearest-neighbour atomic column height differences. The model is solved exactly for the order-disorder transition, obtained in the limit where all possible Ising configurations are allowed. In the opposite limit, where the local height conservation rule is imposed to recover the BCSOS (or six-vertex) symmetry, finite-size transfer-matrix calculations yield a phase diagram where the reconstruction transition (corresponding to the order-disorder transition in the other limit) is smeared out, and a roughening transition occurs at a higher temperature. The phase boundaries obtained in the two limits (Ising and BCSOS) are compared. The model Hamiltonian contains a single parameter ( lambda ), which connects in a natural way these limiting cases ( lambda =0 and respectively). The authors study by finite-size calculations a simplified version of this Hamiltonian, named the lambda -model, which just interpolates between the simple Ising model and Rys's F-model. In particular, they analyse the behaviour of the correlation length and the heat capacity peak for the whole range of values of lambda , and the step energy for lambda large enough.

The magnetization per spin for a class of models with the Husimi-Temperley type interaction satisfies the Burgers equation with a diffusion coefficient 1/2N, where N is the number of particles. The role of time is played by the dimensionless interaction parameter, and the role of spatial coordinate by the dimensionless field variable. Both thermodynamic scaling and finite-size scaling for the magnetization near the critical point are derived from a family of self-similar solutions of the corresponding Burgers equation. The models are specified by the initial conditions which are chosen to correspond in the high-temperature region to a vanishing interaction constant and in the low-temperature region to an infinitely large interaction constant.

The authors calculate a Landau mean-field phase diagram for the wetting transition on a cylindrical substrate as a function of the distance from coexistence. Although both the bulk field and the curvature cause the wetting layer to remain finite, they affect the wetting phase diagram in different ways.

For bilinear equations of the form P(D)f.f=0 the author finds all possibilities for rewriting g²P(D)f.f-f²(D)g.g=0 in the form Q(D)f.g=0. This is the first step in finding a Backlund transformation.

Recently Loewe and Sanhueza (ibid., vol.23, p.553, (1990)) examined supercritical effects caused by a δ potential acting on a Dirac particle in one dimension and also by a δ-shell potential in three dimensions. Based on the observation that supercritical effects are absent for the δ potential, they suggested that the effects depend on the spatial extension of the potential. The authors point out that the effects are absent for a class of non-local separable potentials in one dimension. The ranges of the potentials can be chosen arbitrarily; the δ potential is a special case.