Table of contents

For the quantum algebra U_q(su(2)), a q-analogue of the usual spin coherent state is constructed. With help of coherent states the q-deformed Dyson and Holstein-Primakoff realizations of the quantum algebra U_q(su(2)) are given. A transformation matrix, which turns the Dyson mapping onto the Holstein-Primakoff, is presented.

A spectral parameter dependent solution of the graded Yang-Baxter equation is obtained, which in U_q(gl(m/1))(X)End(V) with V being the vector module of U_q(gl(m/1)). Invariants of this quantum supergroup are constructed using this solution and a general method developed in an earlier publication.

Light from a fast moving point object does not arrive at an observer's eye from the direction where the object currently is, but from the direction of its retarded position. If stereoscopic vision is used however, so that distance can be judged, the object does not generally appear to be at its retarded position, but at a different distance away, depending on the orientation of the 'eye pair'. These effects are analysed geometrically, invoking elementary plane projective geometry.

The author studies a tight-binding model Psi _n+1+ Psi _n-1+ lambda _nV_n Psi _n=E Psi _n, where V_n is a Fibonacci array of two potentials and lambda _n=F ^-12/n^{- alpha} . For alpha =1/2, this model is asymptotically equivalent at large n to a continuous Schrodinger equation with an electric field F and a quasiperiodic potential. He finds that for this value of alpha , almost all states are changed from critical to extended states as the field strength is increased. However, some states still remain critical and coexist with most extended states. He also discusses the result for other values of alpha .

A recent suggestion of integrability of Schamel's (1973) modified Korteweg-de Vries equation for ion acoustic waves in a two-component plasma is discussed. Various arguments, including ones based on the truncated Painleve expansion, are presented which support the existence of special structure for this equation. However, the evidence for complete integrability of Schamel's equation is not yet conclusive.

By an exact real space renormalization group method, the authors obtain the spectral dimension of the Vicsek snowflake fractal. The result seems to cast some doubt on the range of validity of the Einstein relation on the deterministic fractal.

The method of Ott, Grebogi and Yorke (1990) is extended to control transient chaos. The controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor. The time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime. The number of controlled trajectories, however, decreases the maximum perturbation according to a power law. Applicability to experimental situations and comparison with permanent chaotic cases are discussed.

The emergence of new levels of complexity that often accompanies the transition from few- to many-body systems is clearly illustrated by the progression of equilibrium states of N charges on the surface of a sphere as N increases. The characteristics of these electrostatic equilibrium states in the range 2<or=N<or=65 can be examined in detail by equilibrating 1000 randomized initial configurations for every value of N. As N increases, the equilibrium states undergo a succession of structural changes. For instance, a state with non-zero dipole moment appears at N=11, an enantiomeric or mirror image state appears at N=15, and a robust metastable state appears at N=16. For values of N exceeding 50, clusters of four or more metastable states with energies within 0.01% of each other are the dominant pattern. In analogy with some other complex systems, these energetically similar states have strikingly different asymmetric configurations.

A pore space model is presented to describe the transport of mass within pore packings of chromatographic columns. The particle pore structure is conceptually represented by a network of capillaries subjected to bond percolation disorder. The efficiency of the system is assessed in terms of a mass dispersivity factor (h) which accounts for the band broadening of the peak response. The results show a large dispersivity at a percolation fraction near the threshold which reflects the critical properties of the structure with respect to connectivity and tortuosity.

For pt III see ibid., vol. 24, p.1427 (1991). In the q-deformed oscillator system, the Hopf algebraic structures of SU_q,_{(inf) not=0}(2) and SU_q,(2) are constructed in the scheme of geometric quantization. The classical and quantum representation spaces are analyzed with respect to whether or not q is a root of unity.

The method of symmetry adaptation of wavefunctions with respect to any semisimple symmetry chain originating from a SU(l+1) algebra for completely symmetrical representations (N) of SU(l+1) is extended to the case of arbitrary representations (N₁ , N₂. . .N_l+1) of SU(l+1) with N₁>or=N₂>or=. . .>or=N_l+1 and N₁+N₂+. . .+N_l+1=N.

A systematic approach is proposed to give the q-deformation of basic Lie superalgebras B(m, n), B(0, n), C(1+n) and D(m, n) in their boson-fermion oscillator representation with special attention paid to the B(m, n) series.

Using a new reduction of order technique the authors obtain four conserved quantities of the classical planar Kepler problem via the symmetries of the equations of motion and an extra vector field. By passing to a suitable quotient manifold of the evolution space they deduce that negative energy orbits are closed and periodic, without having to solve the differential equation.

The symmetries of a three-body problem, e.g. representing a model of three quarks, are determined. The relevant definitions and theorems concerning symmetries, like the symmetry condition, are listed. The software that helps solve the symmetry condition by computer is discussed.

A nonlinear superposition formula of the KdV equation with a source is proved rigorously.

A nonlinear set of two coupled reaction-diffusion equations is investigated analytically to obtain travelling wave solutions. First, the series method of Hereman et al (1990), involving powers of decaying exponentials, is used to determine the width and the velocity of those waves. Further, to get a suitable analytical expression for them, another power series is introduced for which now a tanh function acts as a new variable. As a result, recursion relations can be set up. Keeping only the lowest order terms, neglecting coefficients of higher order, the author finds solutions which correspond with earlier numerical calculations.

The authors consider various formulations of what may be called q-quantum mechanics. One involves replacing the commutator that appears in the Heisenberg equations of motion by the 'quommutator': (A,B)_q=qAB-(1/q)BA. They formally integrate the equations of motion, study some simple examples and discuss various apparent difficulties such as lack of conservation of energy, lack of unitarity and the unboundedness of the norm. They also consider another formulation in which the quantum bracket of the time derivative of an operator is used instead of just the time derivative, but the commutator is unchanged, in the Heisenberg equations of motion. This nonlinear treatment while preserving conservation of energy, has its own set of difficulties due to the nonlinearities. They also suggest that the quantum parameter may function as a kind of regulator for the time evolution of the system.

The modified version of periodic orbit theory, developed by Robbins (1989) for systems which possess discrete symmetries is used to calculate parameters of broad resonances in the case of the two-dimensional hydrogenic Stark effect at energies above the zero field ionization threshold. It is shown that the contribution of single periodic orbits of the system give results that coincide with those of the comparison equation method also presented.

The authors consider the problem of finding the quantum states that minimize the products of the 'absolute value' Nth-order fluctuation of two canonically conjugate operators. A scheme is applied to the problem, wherein the exact solution state is approximated by working in a subspace consisting of the ground state plus a finite number of higher-order states, using a variational technique. The solution is first obtained for the ground state, and this is subsequently compared to the solution assuming 2*2 and 3*3 spaces. For positive N they obtain a small, smooth decrease in the value of the fluctuation as the space increases and expect the 3*3 space results to be close to the exact Nth-order fluctuation. For negative N, however, they get an abrupt decrease in the fluctuation as the space increases.

For pt.I see ibid., vol.24, p.991 (1991). The quantized Maxwell field in a halfspace exerts stresses on the boundary plane, whose zero-point fluctuations are analyzed for a perfectly-conducting surface. The stress-correlation function on the surface is determined: its Fourier transform with respect to time and distance governs the mean-square deviation of the stress averaged over finite areas (diameters of order a) and over finite times of order T. Conditions guaranteeing physically sensible results are specified on the Fourier transforms of the averaging functions, with a systematic expansion for a/cT<<1, and the first few terms of an asymptotic approximation for a/cT>>1. The small-time behaviour of the surface-averaged correlation function appears to present a paradox which is elucidated. Finally, a very simple expression is found to leading order for the mean-square fluctuating force on perfectly conducting large bodies of arbitrary shape.

The BFACF algorithm applied to polygons involves sampling on a Markov chain whose state space is the set of all polygons. In three dimensions, for the simple cubic lattice. The authors prove that the ergodic classes of this Markov chain are the knot classes of the polygons.

The authors describe results of numerical simulations of growth in a restricted solid-on-solid model. They first extract the exponent beta by comparing the surface width W(t) approximately t^beta with a spatially averaged height correlation function G(t). The latter is shown to give a more accurate estimate for beta , even for relatively small systems. The exponent X is obtained from the saturation of the interface fluctuations in the late time regime. Their results lead to a conjecture of dimension dependent exponents as beta (d)=1/(d+1), X (d)=2(d+2). In addition they study time dependent crossover phenomena by relaxing the local height constraint, and using a noise reduction method in the model. They demonstrate how the effect of such crossover effects may lead to spurious values of the growth exponents. Finally, they discuss their results in relation to other discrete models and the continuum growth equations.

Eigenstates of the corner transfer matrix (CTM) for the six-vertex model are constructed and their relation to the Bethe ansatz eigenstates of the XXZ Hamiltonian is discussed. In the ferromagnetic regime ( Delta >1), eigenstates with any finite number of overturned arrows are constructed. They are related to the Bethe ansatz states by Fourier transformation over spectral (rapidity) variables. This exhibits the role of the CTM as a lattice boost operator. For the antiferromagnetic regime, the relation between CTM eigenstates and those of the XXZ Hamiltonian is complicated by the filling of the sea and associated Bethe ansatz integral equations. For this case, the authors construct CTM eigenstates by first showing that the Hamiltonian ground state is also an eigenstate of the CTM. Other CTM eigenstates are constructed from excited eigenstates of the Hamiltonian by Fourier transforming over the rapidity of each dressed excitation. They discuss the relationship between CTM eigenstates and a Heisenberg algebra of bosonic oscillators.

The authors study an extension of conventional directed percolation (DP) with two species (A and B) using mean-field and Monte Carlo methods. The densities of the two species in the steady-state exhibit phase transitions which are due both to simple percolation and to competition between the species. In 1+1 dimensions as well as a simple DP transition, there is a line of first-order transitions between pure-A and pure-B phases. The phase diagram in a 2+1 dimensions agrees qualitatively with that obtained from mean-field calculations with second-order transitions between pure and mixed phases. Preliminary studies suggest that the critical exponents are in the same universality class as (one-species) DP.

Neural networks capable of encoding sets of patterns are analysed. Solutions are found by theoretical treatment instead of by supervised learning. The behaviour of 2^R (R in N) input units is studied and its characteristic features are discussed. The accessibilities for non-spurious patterns are calculated by analytic methods. Although thermal noise may induce wrong encoding, the authors show how it can rid the output of spurious sequences. Further, they compute error bounds at finite temperature.

A neural network model with optimal connections trained with ensembles of external, discrete, noisy fields is studied. Allowing for non-zero errors in the storage, novel behaviour is observed which is reflected in the model's retrieval map. Improvement in the model's content addressability is determined by comparing the maximum storage level at which there is a near 100% basin of attraction. The cases presented have the external field applied during training, during retrieval and during both with statistically equal parameters. In all three the content addressability is improved over the zero external field network, with the equal training and retrieval fields case having the largest improvement. However, the apparent domination of the retrieval over the training field suggests this simple equality is perhaps not the optimal relationship.

The authors have derived a set of differential equations describing the spatial and temporary variation of the Yang-Mills gauge field within the O(4) group under the ansatz that the spin connection in R⁴ space is simply their gauge potential and the line element of R⁴ space is expressible in a diagonalizable form. They have also obtained four sets of analytical, special solutions to the potential and field strength components. These solutions demonstrate soliton-like properties. Though the form of the soliton is more complicated than that of the usual pulse-shaped one, these solitons are found to propagate with constant velocity c. The energy momentum tensor T₄₄ is also analysed numerically; the form of T₄₄ is found to be non-dispersive while the soliton propagates. Moreover, the total energy is found always to be positive definite.

In order to study the natural spectral line shape in the framework of quantum electrodynamics the resonance photon scattering on the relativistic few-electron atom is considered. The general case of the few overlapping levels is studied. In the resonance approximation the calculation formulae for the differential and total scattering cross-sections are obtained. The equations allowing improvement of the resonance approximation are also derived.