Table of contents

A continuum model of a fluctuating semiflexible polymer chain confined along an axis is considered. In the regime of strong confinement, configurations with overhangs are negligible, and the partition function is determined by a partial differential equation. An extremum principle for eigensolutions is formulated. The exact solution of the differential equation for a polymer in a harmonic potential is given. A lower bound for the confinement free energy of a polymer in a tube is obtained.

A recently proved theorem is used to derive a conserved quantity associated with a velocity-dependent symmetry for Lagrangian systems. In addition, a generalization of the theorem is given.

A massive dynamic field theory is employed to calculate the critical exponent z of the three-dimensional dilute Ising model to two-loop order without expansion in square root 4-d (with result z approximately=2.191). Applying the same method to the pure Ising model at three-loop order we find z=2.022 for d=3 and z=2.124 for d=2.

The first-passage time density, psi (r, t) (defined as the probability density for the time spent by a random walker to travel (for the first time) the distance r that separates the starting site from its nearest neighbours), and the survival probability S(r, t) (i.e. the probability that a random walker who starts at a site has not been absorbed by traps located on its nearest neighbours at distance r in the time interval (0, t)), were calculated for the class of deterministic fractals in which sites are isolated from the rest of the lattice by their nearest neighbours. The large xi identical to r/( square root (2Dt)¹dw/) asymptotic expressions for these quantities are psi (r, t) approximately=A xi ^nu 2+dw/ exp(-C xi ^nu ) and h(r, t)=1-S(r, t) approximately=(A/C)(d_w-1) xi ^{- nu} 2/exp(-C xi ^nu ) with v=d_w/(d_w-1), A and C being characteristic constants for each fractal. The asymptotic expression for S(r, t) is used to justify that, for this class of deterministic fractals, the propagator or Green function is given asymptotically by P(r, t)~t^-ds/2 xi ^alpha exp(-C xi ^nu ) for large xi , with alpha = nu /2-d_f. This value of alpha differs from others proposed recently.

The 2D off-critical q-state Potts model with boundaries was studied as a factorizable relativistic scattering theory. The scattering S-matrices for particles reflecting off the boundaries were obtained for the cases of `fixed` and `free` boundary conditions. In the Ising limit, the computed results agreed with recent work.

We generalize the microcanonical algorithm developed by Creutz et al. (1986), and make a detailed comparison with the exact solution in the case of a two-dimensional Ising model at finite volume. We present a new numerical method to compute the temperature in the microcanonical ensemble. This allows us to define a `thermalization` criterion to estimate the point where the differences between canonical and microcanonical results are the smallest. This criterion is shown to work well in the case of the two-dimensional Ising system.

We study by Monte Carlo simulations the critical behaviour and the cross-over scaling of the true self-avoiding walks (TSAW) on fractal lattices above the upper marginal dimension. We estimate the Flory exponent nu which characterizes the RMS end-to-end distances of SAW on a percolation cluster at percolation thresholds both in three and four dimensions and on a DLA cluster in three dimensions. Results were in good agreement with the predictions of the known Flory formulae. We also discuss the fractal-to-Euclidean and the RW-to-TSAW cross-over scaling. Monte Carlo data appear to collapse in the scaling regions for both cases; however, we found that the scaling function for the latter is different from that on the regular lattices.

The creation and annihilation of traffic jams are studied by a computer simulation. The one-dimensional (1D) fully-asymmetric exclusion model with open boundaries for parallel update is extended to take into account stochastic transition of particles (cars) where a particle moves ahead with transition probability p_t if the forward nearest neighbour is not occupied. Near p_t=1, the system is derived asymptotically into a steady state exhibiting a self-organized criticality. In the self-organized critical state, a traffic jam (start-stop wave) and an empty wave are created at the same time when a car stops temporarily. The traffic jam disappears by colliding with the empty wave. The coalescence process between traffic jams and empty waves is described by the ballistic annihilation process with pair creation. The resulting problem near p_t=1 is consistent with the ballistic process in the context of 1D crystal growth studied by Krug and Spohn (1988). The typical lifetime <m> of start-stop waves scales as <m> approximately= Delta p_t^{-0.54+or-0.04} where Delta p_t=1-p_t. It is shown that the cumulative distribution N_m( Delta p_t) of lifetimes satisfies the scaling form N_m( Delta p_t) approximately= Delta p_t^1.1f(m Delta p_t^0.54). Also, the typical interval <s> between consecutive traffic jams scales as <s> approximately= Delta p_t^-0.5+or-0.04. The cumulative interval distribution N_s( Delta p_t) of traffic jams satisfies the scaling form N_s( Delta p_t) approximately= Delta p_t^0.50g(s Delta p_t^0.50). For p_t<1, no scaling holds.

The Hamiltonian of an Ising quantum chain with a square-root-increasing transverse magnetic field is exactly diagonalized and its spectrum determined by the zeros of Charlier polynomials. We compute also the magnetization at zero temperature as a function of the position in closed form.

We study generalization in a large fully connected committee machine with continuous weights trained on patterns with outputs generated by a teacher of the same structure but corrupted by noise. The corruption is due to additive Gaussian noise applied in the input layer or the hidden layer of the teacher. Contrary to related cases, in the presence of input noise the generalization error epsilon _g is not minimized by the teacher`s weights. For small values of the load parameter alpha the student is in a permutation-symmetric phase. As alpha increases three additional phases emerge. The large- alpha theory of the stable phase is similar to the tree committee machine. In particular, at zero temperature in the presence of noise epsilon <sub>g</sub> does not approach its minimal value epsilon <sub>min</sub> and the student`s weights do not converge to those of the teacher. For a positive temperature epsilon _g- epsilon _min decays as a power of alpha , the exponent being the same as in the corresponding case of the tree. However, for all values of alpha an at least metastable phase exists which is permutation symmetric with respect to the teacher.

Recent studies of optimization in neural networks trained with noisy data have shown that replica-symmetric solutions are unstable in the low-noise region of parameter space. We calculate the 1-step replica-symmetry broken solution in this region, which joins the replica-symmetric solution continuously at the de Almeida-Thouless line. These solutions yield satisfactory agreement with simulations for the aligning field distribution, better than those given by the replica-symmetric ansatz.

The Hopf algebra dual form for the non-standard uniparametric deformation of the (1+1) Poincare algebra iso(1,1) is deduced. In this framework, the quantum coordinates that generate Fun_w(ISO(1,1)) define an infinite dimensional Lie algebra. The T-matrix formalism is used to derive a universal R-matrix for both U_wiso(1,1) and Fun_w(ISO(1,1)). It is also shown how these results can be generalized for the triangular deformations of (1+1) Poincare and Galilei algebras that include a spacetime dilation generator.

We show that the distribution of the percolation threshold in a large finite system does not converge to a Gaussian when the size of the system goes to infinity, provided that the two widely accepted definitions of correlation length are equivalent. The shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length. The result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v. All results are rigorously proven in the 2D case. Generalizations for three dimensions are also discussed.

We give a systematic discussion of the relation between subsingular vectors of Verma modules over semisimple Lie algebras G and differential equations which are conditionally G-invariant. This is extended to the Drinfeld-Jimbo q-deformation U_q(G) of G. We treat in detail the conformal algebra su(2,2), its complexification sl(4) and their q-deformations. The conditionally invariant equations are the d`Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel`fand-Gel`fand (1971). We also give the q-difference analogues of these equations.

Trying to extend a local definition of a surface of a section, and the corresponding Poincare map to a global one, one can encounter severe difficulties. We show that global transverse sections often do not exist for Hamiltonian systems with two degrees of freedom. As a consequence we present a method to generate the so-called W-section, which by construction will be intersected by (almost) all orbits. Depending on the type of tangent set in the surface of the section, we distinguish five types of W-sections. The method is illustrated by a number of examples, most notably the quartic potential and the double pendulum. W-sections can also be applied to higher dimensional Hamiltonian systems and to dissipative systems.

Noether`s symmetry transformations for higher-order Lagrangians are studied. A characterization of these transformations is presented, which is useful for finding gauge transformations for higher-order singular Lagrangians. The case of second-order Lagrangians is studied in detail. Some examples that illustrate our results are given; in particular, for the Lagrangian of a relativistic particle with curvature, Lagrangian gauge transformations are obtained, though there are not Hamiltonian gauge generators for them.

We present a Baxterization of a two-colour generalization of the Birman-Wenzl-Murakami (BWM) algebra. Appropriately combining two RSOS-type representations of the ordinary BWM algebra, we construct representations of the two-colour algebra. Using the Baxterization, this provides new RSOS-type solutions to the Yang-Baxter equation.

Ordering properties of boson operators have been very extensively studied, and q-analogues of many of the relevant techniques have been derived. These relations have far reaching physical applications and, at the same time, provide a rich and interesting source of combinatorial identities and of their g-analogues. An interesting exception involves the transformation from symmetric to normal ordering, which, for conventional boson operators, can most simply be effected using a special case of the Campbell-Baker-Hausdorff (CBH) formula. To circumvent the lack of a suitable q-analogue of the CBH formula, two alternative procedures are proposed, based on a recurrence relation and on a double continued fraction, respectively. These procedures enrich the repertoire of techniques available in this field. For conventional bosons they result in an expression that coincides with that derived using the CBH formula.

The rich dromion structures for a (2+1)-dimensional KdV equation are revealed. The dromions in a high dimensional integrable model may have a free shape in one or more directions. Multi-dromion solutions can be driven by perpendicular line, non-perpendicular line and curved line ghost solitons.

It is shown that for a (D+1)-dimensional scalar field system with an arbitrary potential whose Fourier representation exists in the sense of tempered distributions, the effective potential and multi-particle-state energies by the Bogoliubov transformation technique are identical to those calculated by the Gaussian wavefunctional approach. However, the Bogoliubov transformation technique differs from the latter as it can be applied to quantizing a static soliton without difficulty.

The effect of background voidage on the necessary conditions for the existence of compressive solitary wave solutions in the two-phase fluid flow of a medium compacting under gravity is investigated. It is assumed that K=K₀ phi ⁿ(1- phi )^-p and xi +4/3 eta =( xi +4/3 eta )₀ phi ^-m(1- phi )q where K is the permeability of the medium, xi +4/3 eta is the effective viscosity of the solid matrix and phi is the voidage. It is shown that for compressive solitary wave solutions to exist, which satisfy certain boundary conditions, it is necessary that the background voidage phi ₀ and the exponent n lie in two regions of the ( phi ₀, n)-plane when 0<or=p<1, and that this reduces to one region when p>or=1. Necessary conditions on the exponent m are also derived. Solitary wave solutions for specific values of n, m, p, q and phi ₀ are obtained numerically and compared.

The coupled Dirac-Einstein equations for an open Robertson-Walker universe admit a discrete spectrum of non-singular recollapsing solutions with associated finite lifetimes. This spectrum can be classified by topological quantum numbers, and the lifetime is roughly proportional to these numbers. The rather complicated structure of the spectrum is due to the dynamics of the relative phase angle with respect to the positive and negative energy components of Dirac`s spinor field. The spectrum is characterized by a band structure: the phase angle remains in each allowed band for a relatively long time but then suddenly jumps to another one.

The unitarizable irreps of the deformed para-Bose superalgebra pB_q, which is isomorphic to U_q[osp(1/2)], are classified at q being root of 1. New finite-dimensional irreps of U_q[osp(1/2)] are found. Explicit expressions for the matrix elements are written down.

A new family of stationary coherent states for the two-dimensional harmonic oscillator is presented. These states are coherent in the sense that they minimize an uncertainty relation for observables related to the orientation and the eccentricity of an ellipse. The wavefunction of these states is particularly simple and well localized on the corresponding classical elliptical trajectory. As the number of quanta increases, the localization on the classical invariant structure is more pronounced. These coherent states give a useful tool to compare classical and quantum mechanics and form a convenient basis to study weak perturbations.

A fairly general form of coupled higher-order nonlinear Schrodinger (CHNLS) equations, which includes the effect of group velocity dispersion (GVD), third-order dispersion, Kerr-law nonlinearity and describing a large class of phenomena involving soliton interactions, has been investigated using Painleve (P) singularity structure analysis in order to identify the underlying integrable models. The identified integrable models agree well with those obtained from AKNS formulation. In addition, we explicitly obtain the bright and dark N-soliton solutions for the integrable model by using Hirota bilinearization derivable from the P-analysis. The form of the bright one-soliton agrees with the result derivable from the inverse scattering analysis, while that of the remaining higher-order bright solitons and dark N-solitons are reported for the first time, by including the most general linear coupling terms.

For pt.I see Seiler et al., ibid., vol.28., p.4431 (1995). We study the symplectic approach to first-order systems with constraints from the point of view of the formal theory of differential equations. We concentrate especially on systems without first-class constraints and give a geometric interpretation of an approach recently proposed by Barcelos-Neto and Wotzasek (1992). We further study the numerical properties of this approach. We also comment on some problems concerning the application to field theories.

We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e^--Xe and Cs-Cs systems.

The extended phase space of an elementary (relativistic) system is introduced in the spirit of Souriau`s (1970) definition of the `space of motions` for such a system. Our `modification` consists in taking into account not only the symmetry (Poincare) group but also its action on the (Minkowski) spacetime, i.e. the full covariant system. This yields a general procedure to construct spaces in which the equations of motion can be formulated: phase trajectories of the system are identified as characteristics on some constraint submanifold (`mass and spin shell`) in the extended phase space. Our formulation is generally applicable to any homogeneous spacetime (e.g. de Sitter) and also to Poisson actions. Calculations concerning the Minkowski case for non-zero spin particles show an intriguing alternative: we should either accept two-dimensional trajectories or (Poisson) non-commuting spacetime coordinates.

An isolated level coupled to a continuum of levels need not decay fully into them. Ordinarily one expects exponential decay, module finite-size effects, transients and long-time power-law tails. In the phenomenon we describe the amplitude for remaining in the initial state at first drops, but it then levels off and remains O(1) indefinitely. Because of the available continuum, this is different from certain quantum localization phenomena where there is an absence of on-shell levels. The origin of the effect we describe is the existence of thresholds and band edges. A condition relating the proximity to threshold with the strength of the coupling determines whether the limited decay occurs.

First, we summarize the argument against deterministic nonlinear Schrodinger equations. We recall that any such equation activates quantum nonlocality in the sense that that information could be signalled in a finite time over arbitrarily large distances. Next we introduce a deterministic nonlinear Schrodinger equation. We justify it by showing that it is closest, in a precise sense, to the master equations for mixed states used to describe the evolution of open quantum systems. We also illustrate some interesting properties of this equation. Finally, we show that this equation can avoid the signalling problem if one adds noise to it in a precise way. Cases of both discrete and continuous noise are introduced explicitly and related to the density operator evolution. The relevance for the classical limit of the obtained stochastic equations is illustrated on a classically chaotic kicked anharmonic oscillator.