Table of contents

We offer a clear numerical demonstration of the Berry-Robnik level spacing distribution in a dynamical regime in which regular and irregular regions coexist in classical phase space. In order to achieve this we had to go very deeply into the semiclassical limit, which, so far, has only been possible for an abstract dynamical system, namely the quantized standard map on a torus. We have performed an extensive numerical analysis of the quasi-energy spectra and of the eigenstates in the two-dimensional phase-space representation. We have confirmed the validity of Percival's conjecture(1973) that the eigenstates can be clearly classified either as regular or irregular where the small set of mixed-type states vanishes extremely slowly as we approach the far semiclassical limit. It has been verified that the assumptions (statistical independence) implicit in the Berry-Robnik theory are indeed satisfied giving rise to the observed excellent agreement between theory and experiment. The same high quality agreement is also observed in our comparison of the semiclassical theoretical (Seligman and Verbaarschot, 1985) and numerical delta statistics.

The equivalence between the Poisson representation and the Fock space formalism, both used to derive coarse-grained equations from the master equation for non-equilibrium systems, is explicitly demonstrated.

The fraction of samples spanning a lattice at its percolation threshold is found via simulations of bond and site-bond percolation to have a universal value of about 0.42 in three dimensions. The ratio of shift (of the finite to infinite size threshold) to width of the threshold distribution is also compatible with universality. Lattices with up to 1001³ sites and/or very good statistics were needed to obtain clear results in these Monte Carlo tests; preliminary studies with smaller lattices were inconclusive.

The one-dimensional contact process is analysed by a cluster approximation. In this approach, the hierarchy of rate equations for the densities of finite length empty intervals are truncated under the assumption that adjacent intervals are not correlated. This assumption yields a first-order phase transition from an active state to the adsorbing state. Despite the apparent failure of this approximation in describing the critical behaviour, our approach provides an accurate description of the steady-state properties for a significant range of desorption rates. Moreover, the resulting critical exponents are closer to the simulation values in comparison with site mean-field theory.

We present results from a quantum and semiclassical theoretical study of the rho _xy and rho _xx resistivities of a high mobility 2D electron gas in the presence of a dilute random distribution of tubes with magnetic flux Phi and radius R, for arbitrary values of k_fR and F=e Phi /h. We report on novel Aharonov-Bohm-type oscillations in rho _xy and rho _xx, related to degenerate quantum flux tube resonances, that satisfy the selection rule (k_fR)²=4F(n+1/2), with n an integer. We discuss possible experimental conditions where these oscillations may be observed.

Two methods are described for obtaining exact travelling monotone shock wave solutions to the two-dimensional Korteweg-de Vries-Burgers equation. They are compared with two methods that have been reported recently by other authors.

We perform the singularity analysis of four new generalized bilinear equations by Hu (1993), of which two possess two-soliton solutions and the other two possess even three-soliton solutions. All four equations fail to pass the Painleve test for integrability, exhibiting non-dominant movable logarithmic singularities at highest resonances of generic branches.

A multiplicative Poisson bracket on GL(N) can be restricted onto SL(N) if the determinant det is central on GL(N). To decide whether the det is central, a simple criterion is derived in the r-matrix language with the help of a general formula for Poisson brackets between determinants of arbitrary minors.

Generalized symmetries are sought for certain coupled nonlinear Schrodinger equations. It is shown that all Lie-Backlund symmetries but one are equivalent to Lie point symmetries.

We present a much improved calculation of the dynamic critical exponents of diffusion d_w, d_s, and the combination d_f/d_w for the DLA (diffusion-limited aggregate) on square and simple cubic lattices using two independent approaches, one based on the exact enumeration for the displacement and velocity autocorrelation and the other on the eigenspectrum of random walks on the DLA. The two methods give consistent results, which, however, clearly rule out the scaling relation d_s=2d_f/d_w first proposed by Alexander and Orbach (1982), similarly to the case of the Eden tree reported previously.

The statistical-mechanical properties of particles forming dimers are compared with the results obtained on a two-dimensional lattice through a random sequential addition (RSA) experiment. The relation between the most likely fraction of dimers filling the lattice at equilibrium in the disordered phase and the corresponding non-equilibrium jamming limit attained through RSA is explored. Some implications relevant for the physics of the liquid state are also discussed.

The mathematical properties of the generating function S(x) for the number s_n of directed column-convex lattice animals on a square lattice with a given directed-site perimeter n are investigated. In particular, it is shown that S(x) can be expressed exactly in terms of algebraic hypergeometric functions. A detailed investigation of the asymptotic behaviour of s_n as n to infinity is carried out by applying the Darboux method to the hypergeometric formula for S(x). It is also demonstrated that s_n satisfies a four-term recurrence relation. Finally, it is noted that the techniques used to analyse the lattice animal generating function S(x) can be applied to any other generating function which satisfies a cubic algebraic equation.

The infinite-range four-state clock spin glass, in the presence of an arbitrary magnetic field, is investigated by the replica method. The appropriate replica-symmetry breaking scheme is discussed for this model. It is shown that two distinct Almeida-Thouless lines are possible, depending on the direction of the applied field. The full phase diagram of the model, showing its instability regions, is presented and discussed.

We investigate two randomly frustrated 1D or quasi-1D Ising systems with diluted disorder, namely the ferromagnetic chain in a random field and the ladder spin glass. We present an analytical study of the susceptibilities (linear chi , nonlinear chi ₃, and higher orders), which characterize the response to a uniform external field at finite temperature. Both models admit a continuum description in the scaling regime of low temperature and low impurity concentration, where the susceptibilities obey power laws, similarly to usual critical phenomena, albeit unlike the mean-field theory of spin glasses. We obtain explicit expressions for the scaling functions of chi and chi ₃, and an estimate for the essential Lee-Yang singularity.

We consider the Hopfield model of neural networks in which the patterns, as well as the spins, are dynamical variables. The characteristic time scales of the dynamics of the spins and the patterns are assumed to be widely separated such that the spins completely equilibrate at the time scale at which the elementary changes in the patterns take place. We study the situation in which each type of variable thermalizes at different temperatures, respectively, T and T'. In this case, such a system is described in terms of the traditional replica formalism in which the number of replicas n=T/T' is still the finite parameter. The complete phase diagram of the model in the space of the parameters T, alpha and n is obtained. If the parameter n is negative, the model is argued to present some similarities with the unlearning training algorithm. In this case a substantial increase in size of the retrieval phase in the plane (T, alpha ) is found.

The full replica-symmetry-breaking (RSB) solution of the Hopfield model at zero temperature (T=0) is investigated. By using the RSB scheme by de Dominicis, Gabay and Orland, a free-energy functional in the so-called Sompolinsky gauge and variational equations are formulated. The resulting equations are conveniently defined for numerical analysis since a singularity at T=0 is formally avoided. Elaborate numerical analyses provide a corrected storage capacity, order parameter functions and the frozen field distribution both in the spin-glass phase and the ferromagnetic retrieval phase.

We study on-line learning with a momentum term for nonlinear learning rules. Through introduction of auxiliary variables, we show that the learning process can be described by a Markov process. For small learning parameters eta and momentum parameters alpha close to 1, such that gamma = eta /(1- alpha )² is finite, the time-scales for the evolution of the weights and the auxiliary variables are the same. In this case Van Kampen's expansion can be applied in a straightforward manner. We obtain evolution equations for the average network state and the fluctuations around this average. These evolution equations depend (after rescaling a of the time and fluctuations) only on gamma : all combinations ( eta , alpha ) with the same value of gamma give rise to similar behaviour. The case with alpha constant and eta small requires a completely different analysis. There are two different time-scales: a fast time-scale on which the auxiliary variables equilibrate and a slow time-scale for the change of the weights. By projection on the space of slow variables the fast variables can be eliminated. We find that, for small learning parameters eta and finite momentum parameters alpha , learning with momentum is equivalent to learning without a momentum term with a rescaled learning parameter eta = eta /(1- alpha ). Simulations with the nonlinear Oja learning rule confirm the theoretical results.

Non-generic contributions to the quantal level density from parallel segments in billiards are investigated. These contributions are due to the existence of marginally stable families of periodic orbits, which are structurally unstable, in the sense that small perturbations, such as a slight tilt of one of the segments, destroy them completely. We investigate the effects of such perturbations on the corresponding quantum spectra, and demonstrate them for the stadium billiard.

We consider the resummation of the semiclassical Selberg zeta function for quantized maps on compact phase space, specifically the quantized Baker's map. In particular, we demonstrate that the semiclassical periodic-orbit expansion leads to an effectively finite polynomial for the spectrum of the quantum map. However, the coefficients are not self-inverse (as required by unitarity). An extension of the semiclassical approximation by including corrections due to the discontinuities of the classical map improves the situation. The improved polynomial is closer to being self-inverse but the eigenphases remain complex. Slightly better results are obtained if the functional equation is imposed on the semiclassical expansion.

General Yang-Baxterization of reflection equations for R with two distinct eigenvalues is presented. This procedure is used to construct reflection K-matrices of reflection equations for the A_n-1 vertex models. (Diagonal) K-matrices by de Vege and Gonzalez-Ruiz(1993) for the A_n-1 vertex model are reproduced. Furthermore, some new off-diagonal K-matrices for the A_n-1 vertex model are obtained.

Previous work in the neutron-proton quasi-spin formalism to express shell-model Hamiltonian matrix elements in terms of a set of reduced matrix elements is extended to several interacting shells.

Irreducible unitary representations of the group CM(3), the 'three-dimensional collective motion group', which is the semidirect product of a six-dimensional Abelian group T₆ and SL(3,R), are constructed. A countable basis is identified in the carrier space of each representation. On each SL(3,R) orbit, elements of the Lie algebra cm(3) are represented as differential operators. The relationship of the Bohr model and the CM(3) model is discussed.

A regular way to define an additive coproduct (or coaddition) on q-deformed differential complexes is proposed for quantum groups and quantum spaces related to the Hecke type R-matrices. Several examples of braided coadditive differential bialgebras (Hopf algebras) are presented.

We derive the icosahedrally-symmetric octahedral tiling of Danzer(1991), denoted by T^(D), locally from the tiling T^(2F) of Kramer et al(1989), obtained by icosahedral projection from the root lattice D₆. Moreover, we determine all windows such that the tiling T^(D) can be obtained by projection from the 6D root lattice D₆. We reconstruct all vertex configurations of the tiling T^(D) using the tools of the projection method.

We study the second central extension of the one spatial dimensional Galilei group G. We find that the kinematic quantity 'jerk' is connected with the physics of one of the co-adjoint orbits of the first central extension of G, as much as acceleration is associated with the physics of one of the co-adjoint orbits of G.

This paper shows that a Lie algebroid structure on a smooth vector bundle A to ( pi over)Q gives rise to a Lie algebroid structure on the bundle TA to (T pi over)TQ, called the tangent Lie algebroid. The analysis uses global arguments. A Lie algebroid A is equivalent to a certain Poisson structure on A*, and the tangent bundle of any Poisson manifold has a tangent Poisson structure. The tangent Poisson structure on TA* is then dualized to produce the tangent Lie algebroid structure on TA. Local calculations are used, and formulae for local brackets are given.

We introduce a new construction for bilinear invariant forms on Lie algebras based on the method of graded contractions. The general method is described and the Z₂, Z₃ and Z₂(X)Z₂ contractions are found. The results can be applied to all Lie algebras and superalgebras (finite or infinite dimensional) which admit the chosen gradings. We consider some examples: contractions of the killing form, toroidal contractions of sl(3,C) and we briefly discuss the limit to new Wess-Zumino-Witten actions.

Non-local symmetries of the equations u₀=f(u)u₁₁₁, w₀=g(w₁₁)w₁₁₁ are investigated. Equations which admit non-local linearization are described, and formulas for generating solutions derived. Non-Lie ansatz u=h(x) phi ( omega )+f(x) phi ( omega )+g(x) is used for reduction of some nonlinear equations.

A non-standard way of representing an evolution equation in the form of a system is proposed. This representation allows us to investigate all the different classes of third-order integrable evolution equations simultaneously. Using this approach, a preliminary classification of these equations is made.

A unified framework for defining Lie and covariant derivatives of spinor fields is presented, which is applicable without restriction on the spacetime connection. The results obtained previously by other authors are analysed and compared with the outcomes of this new formalism.

In this paper we study a class of (1+1)-dimensional integrable systems called the k-constrained KP hierarchy. We first show by using the Wronskian technique that the first flow of the 2-constrained KP hierarchy possesses solutions in generalized double Wronskian form; then we give a conjecture on the Wronskian structure for solutions of the general k-constrained KP hierarchy.

Necessary conditions for the existence of compressive solitary-wave solutions of a partial differential equation derived by Scott and Stevenson (1986) which describes the two-phase fluid flow in a medium compacting under gravity are derived. It is shown that for compressive solitary-wave solutions to exist which satisfy certain boundary conditions it is necessary that n=m>1 where n and m are the exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix, respectively, to the voidage. The effect of the value of the exponents n and m on the shape of the solitary wave is investigated by using existing analytical solutions and new numerical solutions.

A three-dimensional wave equation which describes large deformations of a cord (thin rope) is obtained. It is shown that the wave equation has an N-solitary wave solution propagating with the same velocity and maintaining their initial shapes. In addition, the N-solitary wave solution of a factorized form is also derived through an application of one-spiral solitary wave solution, which is a time-dependent extension of the one-spiral curve of an elastic wire discussed by H Tsuru J. Phys. Soc. Japan 56 (1987) 2309.

A new class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation is investigated. The obtained waves require two spatial dimensions for their propagation, i.e. they generalize solutions of the (2+0)-dimensional sine-Gordon equation. The parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency. The obtained solutions describe a new class of Josephson waves whose velocity is smaller than the Swihart velocity. If omega =0 the running waves are reduced to the self-consistent phase, current and magnetic field distributions in a large two-dimensional Josephson junction. The self-restriction coefficient for the Josephson current corresponding to one of the structures is calculated.

Exact implicit solutions with functional parameters of the (2+1)-dimensional Harry Dym (HD) equation are constructed by the delta -dressing method. The interrelation between the HD and modified Kadomtsev-Petviashvili (mKP) equations is established within the framework of the delta -dressing method. Knowledge of the mKP eigenfunctions allows the solutions of the (2+1)-dimensional HD equation to be represented in a simple parametric form.

The non-isospectral symmetries of a general class of integrable hierarchies are found by generalizing the Galilean and scaling symmetries of the Korteweg-de Vries equation and its hierarchy. The symmetries arise in a very natural way from the semi-direct product structure of the Virasoro algebra and the affine Kac-Moody algebra underlying the construction of the hierarchy. In particular, the generators of the symmetries are shown to satisfy a subalgebra of the Virasoro algebra. When a tau-function formalism is available, the infinitesimal symmetries act directly on the tau-functions as moments of Virasoro currents. Some comments are made regarding the role of the non-isospectral symmetries and the form of the string equations in matrix-model formulations of quantum gravity in two dimensions and related systems.

Within the framework of Feynman path integration, expectation values of quantum mechanical operators may be exactly obtained for a class of time-dependent problems. Attention is focused on the two-dimensional motion of a charged particle in a perpendicular magnetic field with a time-dependent driving force. A harmonic oscillator potential is included to ensure that the corresponding density matrix is properly defined although some expectation values are defined without it. This potential is at least a mathematical convenience. Some discussion concerning the conditions under which steady states may be attained is also included.

An exact quantization condition is given for the one-dimensional Schrodinger operator with a homogeneous anharmonic potential q^2M. It has the form of an explicit mapping from level sequences to level sequences, involving a Bohr-Sommerfeld-like quantization step, and having the exact spectrum as fixed point. Numerical tests and an approximate linear theory both suggest, at least for the few lowest M, that the mapping has a contractive region: when an initial level sequence is only asymptotically correct to lowest order, its iterates are seen to converge term by term towards the exact eigenvalues. This type of approach ought to extend to general polynomial potentials.

The applications of parabosons by Schmutz (1980) and of supersymmetry by Jarvis and Stedman (1984) in Jahn-Teller systems are compared and contrasted. Although a parasupersymmetric Jahn-Teller system has not yet been identified, the method of Schmutz is used here to show that the E* epsilon supersymmetric Jahn-Teller Hamiltonian can be written in terms of paraboson operators.

We consider the quantum Lobachevsky space L_q³, which is defined as a subalgebra of the Hopf algebra A_q(SL₂(C)). The Iwasawa decomposition of A_q(SL₂(C)) introduced by Podles and Woronowicz allows us to consider the quantum analogue of the horospheric coordinates on L_q³. The action of the Casimir element, which belongs to the dual to A_q quantum group U_q(SL₂(C)), on some subspace in L_q³ in these coordinates leads to a second order difference operator on the infinite one-dimensional lattice. In the continuous limit q to 1 it is transformed into the Schrodinger Hamiltonian, which describes zero modes into the Liouville field theory (the Liouville quantum mechanics). We calculate the spectrum (Brillouin zones) and the eigenfunctions of this operator. They are q-continuous Hermite polynomials, which are particular cases of the Macdonald or Rogers-Askey-Ismail polynomials. The scattering in this problem corresponds to the scattering of the first two-level dressed excitations in the Z_N model in the very peculiar limit when the anisotropy parameter gamma and N to infinity , or, equivalently, ( gamma ,N) to 0.

The relativistic and quantum mechanical two-body problem with interaction has been a controversial matter for many decades. We show that two very different approaches to the problem for a particular type of interaction lead to the same final equation. Furthermore, the second of them can be solved exactly in an elementary fashion, and leads to an equally spaced spectrum of a familiar type for the square of the total energy.

A continuum SU(2S)*U(1) gauge theory for a Heisenberg antiferromagnet with arbitrary spin S is obtained. We show that the level-1 SU(2) WZW model with a certain marginally irrelevant perturbation describes the low-energy phenomena of the Heisenberg model with spin-1/2. The model with spins other than 1/2 can be described in the level-2S SU(2) WZW model with a certain relevant perturbation. We give a qualitative explanation of the Haldane conjecture for Heisenberg antiferromagnets.

We derive the current algebra of supersymmetric principal chiral models with a Wess-Zumino term. At the critical point one obtains two commuting super-affine Lie algebras as expected, but, in general, there are intertwining fields connecting both right and left sectors, analogously to the bosonic case. Moreover, in the present supersymmetric extension we have a quadratic algebra, rather than an affine Lie algebra, due to the mixing between bosonic and fermionic fields; the purely fermionic sector displays an affine Lie algebra as well.

See ibid., vol.26, p.2445 (1993). The analysis of the one-dimensional Schrodinger equation with the potential - lambda /x given is found to be incorrect. Contrary to the author's claim, the origin is indeed 'impenetrable'.

For original paper see ibid., vol. 26, p. 2445 (1993). By expressing in matrix form the one-dimensional Hamiltonians where the potentials are proportional to mod x mod ^-1 and x^-1, I show that we are dealing with two very different problems. Thus, in my viewpoint, the criticism of R G Newton of the paper mentioned in the title, which is based on results relating to mod x mod ^-1 does not apply to a potential of the form x^-1.