Table of contents

The continuum model describing growth with conservation of particle number requires an infinite number of nonlinear terms for a one-dimensional substrate. By considering an N-vector generalization of this growth model, we show that the dynamic critical exponent z = 4 in the limit. This spherical limit provides the only universal exponents for one-dimensional growth models with conserved particle number but non-conserved noise.

The Lax representation in terms of matrices is constructed for separable multiply-periodic systems splitting on two tori. Hyperelliptic Kleinian functions and their reduction to elliptic functions are used.

A stochastic process generating biologically-relevant RNA folding pathways induces a time-dependent probability measure in conformation space. This measure determines the information content which is used here as a marker for the steps that make folding a robust process. For real RNA molecules, the amount of information generated is shown to reach its absolute maximum within the experimentally-relevant timescale, well below the thermodynamic limit of ergodic time.

We study learning from stochastic examples by perceptrons with Ising weights in the framework of statistical mechanics. Under the one-step replica symmetry breaking ansatz, the behaviour of learning curves are classified according to some local property of the rules by which examples are drawn. The conditions for the existence of perfect learning together with other behaviours of the learning curves are given precisely. The results agree with those obtained by Seung (1995) using a refined annealled approximation.

We investigate the possibility of simulating exchange effects in a two-dimensional fermion liquid within a model of distinguishable particles. For this purpose, we intend to reproduce both bulk and microscopic properties of the system of N indistinguishable particles, as predicted by the Landau theory of Fermi liquids adapted to describe a single atomic monolayer, by means of a Hartree calculation of the ground state, the single particle spectrum and the effective interaction of a system of N distinguishable particles interacting via a coordinate-dependent plus momentum-dependent interaction. It is shown that for given original two-body interactions of the smoothed square or Lennard-Jones shape, it is possible to construct an image system possessing the same microscopic properties as the original quantum liquid within a substantial range of single particle momenta.

Using computer enumerations and the algebraic approximant method of series analysis, several new exact results have been found for moments of width, perimeter and loops for directed animals on the square and triangular lattices. A proof using q-series is given for exact solutions for widths on both the square and triangular lattices.

We have studied the phase transition behaviour of the random field Ising model in the presence of a transverse (or tunnelling) field. The mean field phase diagram has been studied in detail, and in particular the nature of the transition induced by the tunnelling (transverse) field at zero temperature. A modified hyperscaling relation for the zero-temperature transition has been derived using the Suzuki - Trotter formalism and a modified `Harris criterion'. A mapping of the model to a randomly diluted antiferromagnetic Ising model in uniform longitudinal and transverse field is also given.

We study a simple nonlinear mapping with a strange nonchaotic attractor characterized by a singular continuous power spectrum. We show that the symbolic dynamics is exactly described by a language generated from a suitable inflation rule. We derive renormalization transformations for both the power spectrum and the autocorrelation function, thus obtaining a quantitative description of the scaling properties. The multifractal nature of the spectrum is also discussed.

Employing techniques recently developed in the context of the Fokker - Planck approach to electron transport in disordered systems, we calculate the conductance length correlation function for quasi one-dimensional 1D wires. Our result is valid for arbitrary lengths L and . In the metallic limit the correlation function is given by a squared Lorentzian. In the localized regime it decays exponentially in both L and . The correlation length is proportional to L in the metallic regime and saturates at a value approximately given by the localization length as .

We solve the O(n,1) nonlinear vector model on the Bethe lattice and show that it exhibits a transition from ordered to disordered state for . If the replica limit is taken carefully, the model is shown to reduce to the corresponding supersymmetric model. The latter was introduced by Zirnbauer as a toy model for the Anderson localization transition. We argue thus that the non-compact replica models describe correctly the Anderson transition features. This should be contrasted to their failure in the case of the level correlation problem.

The survival probability, , of a diffusing prey (`lamb') in the proximity of N diffusing predators (a `pride of lions') in one dimension is investigated. When the lions are all to one side of the lamb, the survival probability decays as a non-universal power law, , with the decay exponent proportional to . The crossover behaviour as a function of the relative diffusivities of the lions and the lamb is also discussed. When , the lamb survival probability exhibits a log-normal decay, .

It has been shown that the topological characterization of an equivariant system should preferably be achieved by working in a fundamental domain generated by the symmetry properties appearing in the phase space. In this paper, we discuss the case when the equivariance of the studied system is taken into account to study the evolution of the population of periodic orbits when a control parameter is varied. The Burke - Shaw system is considered here as an example. It is shown that the equivariance of this system may be used to reduce the multimodal first-return map in a Poincaré section to a unimodal map. A relationship between four-symbol sequences and two-symbol sequences is given. The non-trivial evolution of the orbit spectrum of a multimodal map is then predicted from the much simpler unimodal map to which the multimodal map reduces.

We consider the one-dimensional asymmetric exclusion process with an impurity. This model describes particles hopping in one direction with stochastic dynamics and a hard core exclusion condition. The impurity hops with a rate different from that of the normal particles and can be overtaken by these particles. We solve this model exactly and give its phase diagram. In one of the phases the system presents a shock, i.e. a sharp discontinuity between a region of high density of particles and a region of low density. Density profiles and relevant exponents are explicitly calculated. These exact results for systems of finite size are consistent with anomalous diffusion laws observed in infinite systems.

We show that in supervised learning from a supplied data set Bayesian model selection, based on the evidence, does not optimize generalization performance even for a learnable linear problem. This is demonstrated by examining the finite size effects in hyperparameter assignment from the evidence procedure and the resultant generalization performance. Our approach demonstrates the weakness of average case and asymptotic analyses. Using simulations we corroborate our analytic results and examine an alternative model selection criterion, namely cross-validation. This numerical study shows that the cross-validation hyperparameter estimates correlate more strongly than those of the evidence with optimal performance. However, we show that for a sufficiently large input dimension the evidence procedure could provide a reliable alternative to the more computationally expensive cross-validation.

The problem of directed polymers in a random medium is generalized to include a killing probability. The model is then studied using results of the theory of extreme statistics. These reveal that, in a region of parameters, the thermodynamics of polymers with killing in finite dimensions is similar to that of the mean field solution of directed polymers on a Cayley tree. The implication of these results for the behaviour of directed polymers in high dimensions is also discussed.

Various q-state Potts models on Bethe lattices are investigated when q is non-integer. Models with 0 < q < 1 have been shown to be related to spin glasses as well as polymer models. We find that such systems exhibit period doubling cascades, chaos, etc. Even in the case of a one-dimensional system (a Bethe lattice with branching ratio equal to one) the system has interesting behaviour when 0 < q < 1 as shown by Glumac and Uzelac through looking at Lee - Yang zeros. We study these systems from a completely different approach to that of dynamical systems.

Local parametric statistics of zeros of Husimi representations of quantum eigenstates are introduced. It is conjectured that for a classically fully chaotic system one should use the model of parametric statistics of complex roots of Gaussian random polynomials which is exactly solvable as demonstrated below. For example, the velocities (derivatives of zeros of Husimi function with respect to an external parameter) are predicted to obey a universal (non-Maxwellian) distribution

where is the mean square velocity. The conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom. Dynamical formulation of the `zero - flow' in terms of an integrable many-body dynamical system is given as well.

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wavelengths smaller than the order of the disk radius we recover the usual semiclassical approximation; however, for wavelengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.

An exhaustive analysis of the shape properties of discrete self-avoiding random walks is presented. The dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self-avoiding walks are studied. By means of high-precision Monte Carlo simulations it is shown that the characteristics of the shape of self-avoiding random walks, when regarded as a function of the walk length, present the expected asymptotic behaviour. The differences with conventional random walks depend upon the observable considered: the probability distributions of the principal inertia eigenvalues of self-avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks, while in the cases of the asphericity or ratios of inertia eigenvalues, the distributions for self-avoiding walks are somewhat more peaked than their counterparts for conventional random walks. Analytical expressions for the probability distributions of inertia moment ratios and the two-dimensional asphericity are given. For common random walks there is an excellent agreement between these analytical distributions and the Monte Carlo data. In this work it is established that this concordance is maintained if the same analytical distributions are applied to the self-avoiding case. This means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self-avoiding walks.

The effect of nonlinear dynamics on quantum correlations is studied by using the spin-kicked rotor, as a model system, which consists of a two-level system and a kicked rotor. Quantum-mechanical entangled states between the two-level system and the rotor form `tunnelling' doublets when the corresponding classical dynamics of the rotor is regular. In contrast to this, chaotic dynamics of the rotor makes chaotic entangled states after the destruction of these tunnelling doublets.

A one-dimensional boson - fermion model is considered. It is shown that the model is exactly solvable and the general Bethe eigenstates are constructed. On the basis of the Bethe ansatz equations, the ground state and the thermodynamics are also given in some closed integral equations.

The graded version of quantum Yang - Baxter relation for the one-dimensional Bariev chain of fermions is formulated. From this relation the commutativity of the transfer matrices for different values of the spectral parameter follows. Our result is consistent with the applicability of the traditional Bethe ansatz method. Further, the first non-trivial conserved current next to the Hamiltonian is constructed.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

In the spirit of some earlier work on building coherent states for the Poincaré group in one space and one time dimension, we construct here analogous families of states for the full Poincaré group, for representations corresponding to mass m>0 and arbitrary integral or half-integral spin. Each family of coherent states is defined by an affine section in the group and constitutes a frame. The sections, in their turn, are determined by particular velocity vector fields, the latter always appearing in dual pairs. Geometrically, each family of coherent states is related to the choice of a Riemannian structure on the forward mass hyperboloid or, equivalently, to the choice of a certain parallel bundle in the relativistic phase space. The large variety of coherent states obtained tempts us to believe that there is rich scope here for application to spin-dependent problems in atomic and nuclear physics, as well as to image reconstruction problems, using the discretized versions of these frames.

We find the Laughlin states of the electrons on the Poincaré half-plane in different representations. In each case we show that a quantum group symmetry exists such that the Laughlin states are a representation of it. We calculate the corresponding filling factor by using the plasma analogy of the fractional quantum Hall effect.

A deformed boson mapping of the Marumori type is derived for an underlying su(2) algebra. As an example, we bosonize a pairing Hamiltonian in a two level space, for which an exact treatment is possible. Comparisons are then made between the exact result, our q-deformed boson expansion and the usual non-deformed expansion.

For each we construct a highest weight module of the Lie algebra The highest weight vectors are specific tau-functions of the Nth Gelfand - Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors.

We investigate new polynomial hierarchies of Lax pairs which contain the polynomial pairs for the system of O(3) chiral field equations and Landau - Lifshitz equation introduced recently and give an algorithmic construction of the corresponding hierarchies of soliton equations. We compare the Landau - Lifshitz equation hierarchy obtained via a polynomial bundle with the hierarchy obtained via an elliptic bundle.

This paper develops the Möbius inversion formula for the Gaussian integers and Eisenstein's integers, and gives two applications. The first application is to the two-dimensional arithmetic Fourier transform (AFT), which is suitable for parallel processing. The second application is to two-dimensional inverse lattice problems, and is illustrated with the recovery of interatomic potentials from the cohesive energy for monolayer graphite. The paper demonstrates the potential application in the physical science of integral domains other than the standard integers.

We present a first feasible scheme for detecting the phase of a single mode of radiation with ideal RMS sensitivity versus the average number of photons . It involves pairs of alternate independent homodyne measurements of two conjugated quadratures on a weakly squeezed state at the input. Nonunit quantum efficiency of photodetectors degrades phase sensitivity leading to a power law , with increasing versus .

Quantum Lie algebras are non-associative algebras which are embedded into the quantized enveloping algebras of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The quantum Lie product on is induced by the quantum adjoint action of . We construct the quantum Lie algebras associated to and . We determine the structure constants and the quantum root systems, which are now functions of the quantum parameter q. They exhibit an interesting duality symmetry under .

We consider quantum correlation functions of the antiferromagnetic spin- Heisenberg XXZ spin chain in a magnetic field. We show that for a magnetic field close to the critical field (for the critical magnetic field the ground state is ferromagnetic) certain correlation functions can be expressed in terms of the solution of the Painlevé V transcendent. This establishes a relation between solutions of Painlevé differential equations and quantum correlation functions in models of interacting fermions. Painlevé transcendents were known to describe correlation functions in models with free fermionic spectra.

New integral relations between the eigenfunctions (Bessel functions and Legendre functions) of the Helmholtz equation in spherical, cylindrical and Cartesian coordinate systems are obtained. Expansion of particular solutions in one coordinate system with respect to eigenfunctions of the same equation in another coordinate system is applied. A stationary phase method is used to find the expansion coefficients.

We show that the binomial states (BS) of Stoler et al admit the ladder and displacement operator formalism. By generalizing the ladder operator formalism we propose an eigenvalue equation which possesses the number and the squeezed states as its limiting solutions. The explicit forms of the solutions, to be referred to as the generalized binomial states (GBS), are given. Corresponding to the wide range of the eigenvalue spectrum these GBS have as widely different properties. Their limits to number and squeezed states are investigated in detail. The time evolution of BS is obtained as a special case of the approach.

We study the spectral geometry of the Hopf fibration and determine the right invariant metrics on for which there exist eigenforms of the Laplacian on which pull back to eigenforms of the Laplacian on . We show that the pull-back of the volume form on can be an eigenform of the Laplacian on with non-zero eigenvalue. We show that if is a principal bundle with a bundle metric and that if , then eigenvalues cannot change. Thus eigenvalues do not change for the fibrations and if . We also study the corresponding questions in the complex category for the fibration of the Hopf manifold .

A particle moves in one spatial dimension in an attractive symmetric negative potential vf(x), and obeys non-relativistic quantum mechanics. If the system is enclosed in a finite box then, unlike the situation in , the ground-state energy is negative only when the coupling v is sufficiently large. General upper- and lower-bound formulae are derived for the critical coupling which corresponds to . Some generalizations are derived for potentials which change sign and for unsymmetric unimodal potentials.

In this paper, we give the explicit formulae of vertex operators of for level one as operators on the Fock space. Meanwhile, we point out that the free-field realization (by one fermionic field and l bosonic fields) of highest weight module with highest weight has two irreducible modules.

We analyse earlier obtained numerical results for the knot complexity and the topological persistence length (i.e. the minimal number of steps on the lattice needed for the knot to be formed) with help of well known path integrals for semiflexible chains. In addition, we provide an estimate for the number of different knots having knot diagrams with exactly n crossings. Analytical results obtained are in satisfactory agreement with the available numerical data.

The differential calculus on the quantum Heisenberg group is constructed. The duality between the quantum Heisenberg group and algebra is proved. It is shown that the calculus considered can be obtained from 3D calculus on by contraction.

The behaviour of electronic states of one-dimensional correlated disordered systems which are modelled by a tight binding Hamiltonian is studied analytically using the invariant measure method. The approach of Bovier is generalized to include the possibility of different site energies and nearest neighbour hopping integrals inside the correlated sites or the cluster. The process is further elaborated by applying to the symmetric random trimer model which contains in it many hitherto known models of this category. An alternative mathematical definition of the exceptional energy from the invariant measure density, along with physical arguments substantiating it, is presented. Furthermore, the procedure for obtaining exceptional energies is outlined and applied to the symmetric random trimer model to derive conditions for obtaining doubly degenerate exceptional energies. The Lyapunov exponent or the inverse localization length of states around the exceptional energy is found to vary as in the leading order. n denotes the degeneracy of the exceptional energy. The density of states at the exceptional energies are calculated. We further propose that one-dimensional correlated disordered systems can be mapped to a Lloyd model in which the width of the distribution of site energies is determined by the reflection coefficient of the cluster embedded in the lattice of the other constituent. The importance of our results is discussed.

Based on the concept of joint quasiprobability of the eigenstates of non-commuting components of a spin, a criterion is proposed to characterize the states of a system of N spin-s as classical or non-classical. That criterion characterizes any state of a spin- and a factorizable state of N spin-s as classical. It correctly identifies any pure entangled state of two spin-s, an entangled eigenstate of any component of exchange symmetric spin system and squeezed spin states as non-classical.

We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .