Table of contents

The authors study how the three-dimensional 'air' or Pythagorean distance r(Q, Q') between two points Q and Q' on a non-equilibrium crumpled fractal surface (CS), with the topology of the plane, transforms in the internal or geodesic distance x(Q, Q')-with probability P(x, r)-after the unfolding of the CS on a plane. The probability distribution P(x, r) governing this process is examined for the first time. Among other results they find that (1) the width of P(x, r) 'diverges' for r near the ensemble average radius R of the CS and (2) (x) approximately r¹³.

A new method of calculating the growth probabilities of the DLA clusters based on the Spitzer theorem is presented. It allows very accurate determination of the probabilities of hitting the random walkers by the perimeter of the cluster, even deep in the 'fjords' where the probabilities are small. The evidence for the first-order phase transition in the plots of f(q) are found. The large fluctuations of the minimal growth probabilities between different clusters are also discussed.

The analysis of an exponentially autocatalyzed reaction-diffusion system near the Hopf bifurcation point has been carried out using a reductive perturbation approach to obtain a description in terms of the Ginzburg-Landau equation. The conditions for the occurrence of instability, in the presence and absence of diffusion, leading to Hopf bifurcation are also derived. The nature of the governing equations leads to multi-valued instability conditions and eventually results in more than one region in parameter space where instability of uniform oscillations due to diffusion is possible.

The authors propose a numerical method for extracting the spectrum of decay rates of time correlations in chaotic dynamical systems. The sum of the first p decay rates is related to the asymptotic behaviour of the time correlation of suitable exterior forms of order p. The method is applied to maps of the interval and to the Henon map.

The authors discuss the dynamics of an observable field related to a locally conserved field. They show that for a model of phase ordering dynamics and for a variation of model C critical dynamics, that the two-time momentum space correlation function decays asymptotically as a stretched exponential. They show that under some conditions the asymptotic behaviour is experimentally observable and argue that the non-exponential behaviour is not a feature specific to these models but should rather apply for any observable coupled to a locally conserved field.

The authors present the average conformational properties of a polymer chain interacting with a surface and free to move in a space of size L perpendicular to the surface. The properties are found by solving the problem by means of integral equations.

The authors consider a symmetric version of the Derrida-Gardner-Zippelius model (DGZ). It is shown that in the limit of extreme dilution this modification of the DGZ model can be solved exactly. This means that for the evolution of the main overlap they obtain analytic expressions which (in contrast to the DGZ model) constitute a chain of coupled equations.

The authors propose a new method of series analysis in which the available series coefficients are fitted to an algebraic equation. Such a proposal restricts solutions to have algebraic singularities with rational critical exponents. The advantage of the method over the method of differential approximants is that in those cases where the two methods are exact, the algebraic approximant method usually (though not invariably) requires far fewer series coefficients for its exact specification, and hence discovery. The method is therefore potentially valuable in the study of two-dimensional critical systems in particular.

The author provides an extension of the equivariant bifurcation lemma to the case of nonlinear Lie point symmetries. In this extension, the role of the fixed subspaces under the symmetry subgroups is played by some well specified flow-invariant manifolds. Some typical examples are also considered.

For Pt.I, see ibid., vol.22, p.4271 (1989). The author constructs the dynamical symmetries of U(16), the algebra related to the quantization of classical reflection asymmetric shapes constructed from lambda ^P=0⁺, 1^-, 2⁺, 3^- multipoles. Generators, Casimir operators and their expectation values and branching rules are detailed for all dynamical symmetry limits, focusing primarily on those relevant to octupole deformations and vibrations. The resulting operators allow the construction of dynamical symmetry Hamiltonians.

The authors determine reductions of irreducible representations of E₆, E₇ and E₈ according to the chain of subgroups E_kA₁T_k, k=6, 7, 8, for all A₁ subgroups of E_k. Here A₁ is either SU(2) or O(3) and T₆, T₇, T₈ are respectively the tetrahedral, octahedral and icosahedral groups. The result is the list of coinciding branching rules for E_kT_k proceeding via different subgroups A₁.

The method devised by Cerkaski (1987) has been used for the group Sp(2n) to find some 6-j symbols for which a single multiplicity index is required. This approach possesses some advantages over that based on Racah's techniques for SO(3): the formulae for the 6-j symbols are comparatively simple, and they automatically exhibit the symmetries of Jucys (1973). Elementary means are used to show that the eigenvalues of Casimir's operator C₂ for Sp(2n) are identical to those of C₂ for O(2d) if the authors take d=-n and replace the irreps of Sp(2n) by their transposed counterparts for O(2d), where d is integral or half-integral. This enables the formulae for Sp(2n) to be used to generate expressions for some 6-j symbols for the orthogonal groups O(2d) and SO(2d). Illustrations are provided by SO(2l+1) and O(4).

Suppose A and B are symmetric 3*3 matrices. Rotate one of them by sigma in SO(3) and add; the eigenvalues of the resulting matrix A+B^sigma vary over a set determined by the eigenvalues of A and B. In the case where one of A and B has a repeated eigenvalue the authors find the image of SO(3) Haar measure d sigma on this set, which describes the coupling of different rigid rotors.

The problem of the stability of the numerical solution of the one-dimensional Schrodinger equation with symmetrical potential V(x) is considered. This problem is illustrated by the example y"+(E-x²)y=0 with y(0)=1 and y(x) to 0 when x to infinity , for which the conventional shooting method using the Numerov integrator fails for E=-1, to find y(x) beyond x=5. It is shown that these values are reached by using a different procedure for shooting with the same Numerov integrator. This procedure starts the integration at any 'large' value L of x, and steps backward towards x=0. The method is applied to another value of E for which an exact value of y(x) is available. This tests shows that the accuracy of the computed values of y(x) is independent of the choice of L. Thus the method does not improve the eigenvalue computation, but it allows the determination of the solution y(x) for large x when such values are needed.

The Mello-Pereyra-Kumar theory is extended to arbitrary spin. Classification yields three universality classes. The average over disorder is related to geometrical concepts. The solution of the resulting differential equation is discussed, localization lengths and universal conductance fluctuations are computed.

n marbles with zero velocity are pushed away by a wall moving at constant velocity. The collisions are treated using an inelasticity coefficient eta and the equations of motion are numerically integrated until no more marbles collide with the wall. The principal investigated quantity is the relative kinetic energy E_k of the bounced marbles. A model of independent collision waves is presented which predicts with a good precision the maximum of E_k and the value at which the marbles stick to the wall. It is found that, for large n, the only important parameter is gamma =(1- eta )n. Equivalent results are found when a marble is thrown against the column.

Electromagnetic fields of the toroidal solenoid with a time-dependent current are studied. Their properties and the possible practical applications of the results obtained are discussed.

Through the use of coherent states describing the action-angle variables the authors establish a strict parallelism between Berry's phase and Hannay's angle. In particular they define a geometrical transport for classical tori which leads to Hannay's angle. As in the case of Berry's phase, the transport can be associated with the minimization of a distance.

The authors introduce a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time-dependent Hermitian operator. Rigorous upper bounds are derived for two different 'time-ordered' approximation schemes and for errors resulting from approximating a time-ordered exponential by an ordinary exponential operator. The properties and the usefulness of the product formula algorithms are examined by applying them to the problem of Zener tunnelling. The most efficient algorithm is employed to solve the time-dependent Schrodinger equation for a wavepacket incident on a time-modulated rectangular barrier.

The authors derive, by a biorthonormal state approach, the analogy of Berry's phase factor for open, non-conservative systems, for both adiabatic and non-adiabatic evolution. In the latter case, a (non-unitary) evolution operator method is exploited. An application is given to the optical supermode propagation in the free-electron laser.

The exact propagator of the author's dynamic system is presented and confirmed by expanding it in terms of the energy eigenfunctions and eigenvalues, which agree with those obtained from the corresponding Schrodinger equation. For the case of a rational wedge the propagator can be expressed as a sum over 'classical paths', but with the modified Van-Vleck formula. The author also evaluates the density matrix and the partition functions.

A deformation approach resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation. This completes the programme of Robnik (1984) to develop a purely algebraic method for the quantization of non-integrable Hamiltonian systems about a point of stable equilibrium.

The dynamics of an envelope soliton under the action of δ-kicks is investigated in the framework of a nonlinear Schrodinger equation. It is shown that soliton dynamics may be described by a 'standard map' for a sufficiently large number of δ-kicks. The estimation of the soliton lifetime and radiation level is derived by means of the inverse scattering transform.

The critical dynamics of the kinetic Potts model on Koch curves and regular fractals is studied by means of the exact time-dependent renormalization-group method. Different critical dynamics are found on these two families of fractals. It is shown that the value of the dynamic critical exponent z depends on both the Potts dimensionality q and the transition rates asymmetry coefficient alpha . For Koch curves the scaling law of the dynamics exponent z=D_f+f(q, alpha )/v, while for regular fractals z=D_f+2f(q, alpha )/v, where f(q, alpha ) characterizes the dependence of the dynamics exponent z on Potts dimensionality q and the transition rates asymmetry coefficient alpha , and v is the static exponent of the correlation length.

The authors examine the spacing distribution of the eigenvalues of tridiagonal real symmetric random matrices, the elements of which are distributed according to a Gaussian law. They show explicitly that for 4*4 matrices the distribution at small spacings behaves as s log² s. They surmise that for N*N matrices the behaviour is s log^N-2S and they present numerical results which support this conjecture.

The authors apply the variational lower-bound renormalization group transformation of Kadanoff (1975) to an SU (2) lattice spin model in two and three dimensions. Even in the one-hypercube framework of this renormalization group transformation the present model is characterized by having an infinite basis of fundamental operators. They investigate whether the lower-bound variational renormalization group transformation yields results stable under different truncations of this operator basis. Their results show that for this particular spin model this is not the case.

A discrete model of an interface separating two phases is considered. The interface is defined as the boundary of a self-avoiding n-omino with volume n embedded in the hypercubic lattice. The author proves that if there is no surface tension in the interface, then it has 'non-trivial' topology with probability one in the scaling limit. Here non-trivial topology means that the interface will consist of several disjoint components and will have non-zero genus. Moreover, the total area of the n-omino will be proportional to its volume.

The growth of order in vector spin systems with nonconserved order parameter ('model A') is considered following an instantaneous quench from infinite to zero temperature. The results of numerical simulations in spatial dimension d=2 and spin dimension 2n5 are presented. For n4, a scaling regime (where a characteristic length scale L(t) emerges) is entered for sufficiently long times, with L(t)t¹2/. The autocorrelation function. A(t) decays with time as A(t)t^-(d- lambda ⁾2/, and the exponent lambda (n) agrees well with the predictions of the 1/n-expansion. The cases n=2 and 3 are more complicated, due to the non-trivial role played by topological singularities, i.e. vortices (n=2) and Polyakov solitons, (n=3). For n4, universal amplitudes and scaling functions characterizing the energy relaxation and the equal-time correlation function are identified. It is argued that for d3, where an ordered phase exists at low temperature, such universal quantities characterize the entire ordered phase.

The authors investigate statistical-mechanical organization of metastable states at finite temperatures in the naive mean-field model for spin glasses by direct numerical analysis on the equations of states of the model. The number of their solutions (metastable states) is shown to agree with the replica prediction developed by Bray and Moore (1980). Furthermore, such sophisticated spin-glass properties as the universal probability law of the nonself-averaging overlap probability of metastable states, which are initially derived for the Sherrington-Kirkpatrick model by means of the replica argument, are demonstrated to be common also to the naive mean-field model.

A new model of p-branes is introduced and studied. The general solution of this model is equivalent to a particle solution of the covariant p-brane problem which the authors have obtained earlier. The constraint algebra obtained is closed in the sense that its structure functions are field independent.

The author checks by computer simulations whether the magnetization asymptotically approaches a fixed point or a limit cycle of period two. This criterion is applied to all 65536 nearest-neighbour automata.

The long-time limit of the relaxation function for system with a broad uniform but bounded continuous distribution of relaxation times is shown to be of the Yukawa form. This is in contrast to the single or multiple exponential functions often used to extract relaxation time constant tau from fitting experimental or numerical simulation data. Although such fittings will still obtain the correct tau in the long-time limit, the author's results predict a logarithmic term, which will slow the convergence.

When the kinetic energy density in _k (defined positive definite) of a system of one-dimensional, non-interacting fermions is approximated by an ordinary function of the density rho , and of the lowest n derivatives of rho , in _k>or=0 can satisfy the differential virial theorem for arbitrary density distributions only if in _k actually depends on rho and rho ' only. Thus the case n>1 is ruled out, and one is left with the result in _k= kappa rho ³+ rho '²/(8 rho ) ( kappa >or=0) already obtained in previous work.