Table of contents

We study an adsorption-desorption process of rods on a line. The desorption rate is infinitely small, so that each desorption event is instantly followed by the insertion of one or two new rods. Due to the latter possibility, the system evolves continuously to a close-packed state. The asymptotic kinetics of the densification process is analysed with the aid of gap distribution functions. If it is assumed that the system relaxes completely to equilibrium after each density increment, we show that 1- rho (t) approximately 1/ln(t), while an improved description yields 1- rho (t) approximately 1/ln(2t(ln2t)²). The range of validity of the asymptotic expressions is established by comparison with the results of a numerical simulation of the process.

We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.

Using a formal series symmetry method for the Davey-Stewartson equation (DSE), we find that there exist two sets of infinitely many formal series symmetries of the DSE. However, different from the Kadomtsev-Petviashvili equation (KPE), we failed to get infinitely many truncated symmetries. Only six truncated symmetries can be obtained simply from the formal series symmetries.

A new Sakata-Taketani equation describing vector mesons in interaction with a constant magnetic field is proposed. It leads to real energies as opposed to relativistic equations, which have already been pointed out, associated with this particular interacting context. In particular, the connection between this new proposal and a specific model coming from parasupersymmetric quantum mechanics is studied.

The existence of an electric current driven by the gradient of the magnetic field in an idealized quantum Hall effect device is predicted as a consequence of the anomalous magnetic moment of the electron (g-2 not=9) along with the exact Landau ground-state degeneracy for the two-dimensional Pauli Hamiltonian with an arbitrary magnetic field.

In the present paper an extended version of the Fulton-Gouterman transformation (FGT) is applied to an archetypical electron-phonon system which also involves electronic spin (the two-sites-two-spins-one-particle (221 model)). Two alternatives of the Fulton-Gouterman concept are given: one corresponding to the original transformation (non-exponential) and one to an exponential form. The latter appears to be interesting with regard to recent discussions of squeezing and anti-squeezing effects in superconductivity since it offers itself to a multi-particle generalization. Some remarks; on the physical background are also made.

We formulate the problem of determining the Hausdorf dimension, d_f, of the Apollonian packing of circles as an eigenvalue problem of a linear integral equation. We show that solving a finite-dimensional approximation to this infinite-order matrix equation and extrapolating the results provides a fast algorithm for obtaining high-precision numerical estimates for d_f. We find that d_f=1.305 686 729(10). This is consistent with the rigorously known bounds on d_f, and improves the precision of the existing estimate by three orders of magnitude.

The invariance of equations for self-affine surface growth to reparametrization under the Abelian group of shift transformations h(x, t) to h(x, t)+l is used to bound the form of nonlinear terms and related kinetic coefficients in relaxational surface growth equations. For conserved growth small but relevant diffusive terms second-order in the driving can always be expected. It is also shown that the asymptotic growth distributions in d>2 can be expected to be skew and are not derivable from a Hamiltonian description.

We present a new finite-size scaling method for the random walks (RW) superseding a previously widely used renormalization group approach, which is shown here to be inconsistent. The method is valid in any dimension and is based on the exact solution for the two-point correlation function and on finite-size scaling. As an example, the phase diagram is derived for the random walk in two dimensions with a surface-bulk interaction where the system has either a surface or a defect line. We also discuss an initial calculation of the corresponding phase diagram for the case of a critically diluted lattice.

We propose a simple geometrical method which enables us to link the topological properties of a random walk on the double-punctured plane and the conformal field theory characterized by the central charge c=-2 and the conformal dimension Delta =-1/8. We discuss briefly the connection between the topological invariants obtained from the conformal methods and the algebraic Alexander invariants for the simplest non-trivial braid B₃.

We show that electrons hopping over quasiperiodic tilings give rise to a modified Kondo effect, obeying a power-law behaviour in place of the standard logarithmic behaviour.

The concept of ferrimagnetism was first proposed by Neel to explain why some materials have a macroscopic magnetization but no ferromagnetic long-range order, when the temperature T is lower than a phase transition temperature T_c. In this article, based on a theorem of Lieb and Mattis, we show in a mathematically rigorous way that the global ground states of the generalized antiferromagnetic Heisenberg model on a bipartite lattice with unequal sublattice points have both ferromagnetic and antiferromagnetic long-range orders with the latter being predominant. Our rigorous results conform to Neel's theory.

A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities xi approximately exp( xi _n(K_c⁽ⁿ⁾-K)-1) are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient xi n approximately 2ln(J/2y₀)/(n+1) pi is found when the order of approximation n increases. The true critical exponent nu of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent nu ₀, nu < nu ₀=1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent eta _c, of the spin-spin correlation function at the critical point is found to be eta _c=1/(4+1/ pi ²).

Optimal separation of two clusters of normalized vectors can be performed in a neural network with adjustable threshold and weights, which is trained to maximum stability. Generalization from arbitrarily selected training clusters to a given bipartitioning of input space is studied. The network's threshold becomes a global optimization (and order) parameter, This causes the generalization ability to increase rapidly with the distance of the cluster separation plane from the origin. Separation is shown to be stochastic for small and deterministic for large training cluster sizes.

A system of coupled nonlinear oscillators is considered. If the size N of this oscillator network is large enough, then the model can have arbitrary prescribed attractors which can be defined by some finite-dimensional flows. One can effectively find a simple interaction between the oscillators which gives these prescribed attractors.

We study disordered systems with the replica method keeping the number of replicas finite and negative. This is shown to bias the distribution of samples towards overfrustrated ones. General results on the thermodynamics of such a system is presented. The physical situation described by this finite-n approach is one where the usually quenched variables evolve on long timescales, their evolution being driven by the quasi-equilibrium correlations of the thermalized variables. In the case of neural networks this amounts to a coupled dynamics of neurons (on fast timescales) and synapses (on longer timescales). The storage capacity of the Hopfield model is shown to be substantially increased by these coupled dynamics.

We generalize to braid statistics some of Manin's work (1989) on the quantum deformation of the general linear supergroup. The key ingredient in this construction is the introduction of a non-standard transposition map Psi ( Psi ² not=1) which is defined in terms of a generalized permutation operator P_mu (p_mu ² not=1). The dual space (enveloping algebra) with P_mu -statistics is defined. Consideration of coactions on quantum spaces clarifies the resulting structure.

We consider the twisting of the Hopf structure for the classical enveloping algebra U(g), where g is an inhomogenous rotation algebra, with explicit formulae given for the D=4 Poincare algebra (g=P₄). The comultiplications of twisted U^F(P₄) are obtained by conjugating the primitive classical coproducts by F in U(C)(X)U(C), where c denotes any Abelian subalgebra of P₄, and the universal R-matrices for U^F(P₄) are triangular. As an example we show that the quantum deformation of the Poincare algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincare algebra. The interpretation of the twisted Poincare algebra as describing relativistic symmetries with clustered two-particle states is proposed.

This paper deals with the irreducible highest-weight module L(L) of quantum group U_q(B₂) When q is a root of unity. The character of L( lambda ) has been obtained in one of the cases. As a consequence, its dimension has also been obtained. In addition, a centre element of U_q(B₂) has been found in explicit form.

We show that a quantum dimension D_q( Lambda ) for a representation rho of U_q(G), a quantized universal enveloping algebra of a compact and simple Lie group G, is computed from the algebraic equations which we found recently in studying 2+1-dimensional Chern-Simons theory. We solve the equations explicitly for the typical examples of all compact and simple Lie groups. This method can be applied to super Lie groups such as SU(m,n) and OSp(m,n).

A matrix representation of the automorphism group of pure integral octonions constituting the root system of E₇ is constructed. It is shown that it is a finite subgroup of the exceptional group of G₂ of order 12096, called the adjoint Chevalley group G₂(2). Its four maximal subgroups of orders 432, 192, 192' and 336 preserve, respectively, the octonionic root systems of E₆, SO(12), SU(2)³*SO(8) and SU(8). It is also shown explicitly that the full automorphism group of the pure octonions +or-e_i (i=1,..., 7) constituting the roots of SU(2)⁷ is a group of order 1344. Possible implications in physics are discussed.

The open spinning string is investigated in Minkowski space E_1.3. By means of a reduction to the Wess-Zumino-Witten-Novikov (WZWN) model a new Hamiltonian formalism is constructed. The Poisson bracket structure of the theory is given in terms of the algebra ((sl(2,C)(X)(t^-1, t))(+)C_z)(X)P, where P is the Poincare algebra. The covariant quantization is fulfilled in D=1+3 dimensions with help of 'bosonization' methods. The resulting theory is a combination of the theory of a free fermionic field in two dimensions and the theory of a free particle in Minkowski space. Non-triviality is conditioned by the presence of a finite number of constraints. The formula for the mass spectrum is discussed.

The singular manifold equations are used to construct nonlinear integrable partial differential equations. A number of well known integrable equations (hierarchies of Korteveg-de Vries, Caudrey-Dodd-Gibbon, Kaup-Kuperschmidt, and Harry Dym) are obtained. Some new integrable equations are presented. A possible approach to classification of integrable evolution equations is discussed.

As a first step in the description of a two-dimensional electron gas in a magnetic field, such as encountered in the fractional quantum Hall effect, we discuss a general procedure for constructing an orthonormal basis for the lowest Landau level, starting from an arbitrary orthonormal basis in L²(R). We discuss in detail two relevant examples coming from wavelet analysis, the Haar and the Littlewood-Paley bases.

We investigate the pole structure of the zeta function zeta A_nu (s)= Sigma _k lambda _k^-s built out of the eigenvalues lambda _k of a Bessel operator A_nu subject to Dirichlet boundary conditions at one end of the domain. This leads us to the study of zeta A_nu on the negative real axis, where most of the singularities occur.

In this paper, an effective algorithm to generate integrable systems is given. As a result, many new integrable equations are derived in a systematic way.

We study Z₂^(X)N graded contractions of the real compact simple Lie algebra so(N+1), and we identify within them the Cayley-Klein algebras as a naturally distinguished subset.

We consider the interband light absorption coefficient (ILAC) for a d-dimensional discrete disordered system, whose Hamiltonian consist of a translation invariant part (d-dimensional discrete Laplacian) and an off-diagonal random part. Assuming that the range R of the latter is large and that its magnitude is of the order R^-d/2 we find that R= infinity limit of the ILAC. We discuss some properties of the ILAC in this limit: its boundedness, edge singularities, its singular form in the limits of vanishingly translationally invariant part or infinite random part. We also show that the latter property is the same for the system with a diagonal smoothly distributed disorder, i.e. for the discrete Schrodinger operator whose random potential has a smooth probability distribution. This should be contrasted with the integrated density of states which is always smoother than the distribution of the random potential.

A method is described for solving the close-coupling equations that arise in non-relativistic scattering theory in the asymptotic region where the scattered particle is far removed from the residual atom or ion. Typical results are presented that indicate the convergence and accuracy of the method.

We use single-cluster Monte Carlo simulations to study the role of topological defects in the three-dimensional classical Heisenberg model on simple cubic lattices of size up to 80³. By applying reweighting techniques to time series generated in the vicinity of the approximate infinite-volume transition point K_c, we obtain clear evidence that the temperature derivative of the average defect density d(n)/dT behaves qualitatively like the specific heat, i.e. both observables are finite in the infinite-volume limit. This is in contrast to results by Lau and Dasgupta(1988) who extrapolated a divergent behaviour of d(n)/dT at K_c from simulations on lattices of size up to 16³. We obtain weak evidence that d(n)/dT scales with the same critical exponent as the specific heat. As a byproduct of our simulations, we obtain a very accurate estimate for the ratio alpha / nu of the specific-heat exponent with the correlation-length exponent from a finite-size scaling analysis of the energy.

This paper describes the following results in Nambu mechanics: the definition of simple physical systems; the result of applying the Kalnay and Tascon theorem (1978) to the study of different groupings of the n phase space variables into an s-coordinate and an (n-s)-momentum; the study of the intrinsic geometry of the curve that solves the Nambu equations of motion by explicit construction of the local coordinate system; the construction of sets of Hamiltonians that generate the same set of differential equations; a way to construct canonical transformations; a study of the intrinsic geometry of a system known to have chaotic behaviour: the Lorenz model and the correspondence of an oscillating Nambu system as the classical analogue of a simple version of the Hubbard model for superconductivity.

We consider a system of N particles which are confined to the surface of a sphere and which interact via a potential that depends logarithmically on their separation. The ground-state properties of this system are investigated for N=2 to 65. Unlike the case of Coulomb interactions this system has ground-state configurations with zero dipole moment for all N. The thermal properties of a selected set of systems in the range N=2 to 500 are determined by Monte Carlo simulation. The results are compared with analytical calculations in the small- and large-N limits.

The solution of the inhomogeneous wave equation is found. The source term is a pulse which has the velocity of light. The solution is a function which describes both transient and steady-state wave processes. The latter corresponds to Brittingham's axisymmetric focus wave mode.

We calculate the ground-state degeneracy for Schrodinger-Pauli electrons on various non-compact Riemann surfaces, as a function of a parameter representing a flux. We use the results to calculate the charge transport, and the (appropriately defined) Hall conductance, for the (degenerate) lowest Landau level of Schrodinger-Pauli electrons on these surfaces. We connect the charge transport with the Atiyah-Patodi-Singer eta -invariant for (compact) manifolds with boundaries.

A wide class of Darboux transformations of a Sturm-Liouville equation providing a unified treatment of exactly solvable models in quantum scattering theory is considered systematically. A classification of Darboux transformations is given and the relations between Darboux transformations and standard inverse scattering procedures for the radial Schrodinger equation including the matrix method of Newton-Sabatier are studied. In particular, the definitions, properties and matrix generalizations of Darboux transformations associated with Marchenko-type integral equations are studied in detail.

Yan (1991 J.Phys. A: Math. Gen. 24 4731) has drawn the conclusion that, for central fields, there exists another vector constant of motion T, in addition to the angular momentum L. It is proved here that T is in fact only equal to zero 'almost everywhere'; T is not, therefore, rigorously a constant of motion.

For original paper see ibid., vol. 26, p. 4827 (1993). It is shown that a commonly used deformation of the fermionic canonical anticommutation relations is equivalent to the undeformed relations.

For original paper see ibid., vol. 26, p. 4827 (1993). We reply to the comment by Solomon and McDermott (see ibid., vol. 27, p. 2619 (1994)) that the q-deformed fermionic oscillator is equivalent to the usual fermionic quantum oscillator, noting that our many-body Hamiltonian is truly a deformed model of strong-coupling superconductivity.