Table of contents

We consider a special family of bc-systems of higher rank and discuss some properties of its associated anomaly.

We study a class of non-integrable systems, linear chains with homogeneous attractive potentials and periodic boundary conditions, which are not perturbations of the harmonic chain. In particular, we deal with the system H₄ with a purely quartic potential, which may be shown to be stochastic without any transition. For this model we prove the following pseudo-harmonic properties: (1) the existence of a spectrum of frequencies which are proportional to the harmonic ones, according to a well defined law; (2) the separability on average of the Hamiltonian function among normal modes with these frequencies. Moreover, as far as stochasticity and pseudo-harmonicity are concerned, H₄ is the limit of the Fermi-Past-Ulam (FPU) chain when the energy density tends to infinity. In this frame, the same results as previously obtained for the FPU chain at high energy density are proven to be independent of the presence of the harmonic potential, and to hold at arbitrarily high energies. As a byproduct, we have a stochasticity indicator based on correlations which proves to be very efficient and reliable.

Correlated Gaussian random fields provide good models for a variety of physical phenomena including metallic and ocean surfaces, vegetation, geomorphology and turbulence. Their properties have been studied extensively yet some aspects of their behaviour are still not fully understood. In particular, the statistics of the number of samples within a correlated Gaussian random field which exceeds some threshold (i.e. the support of the random field) have not previously been analysed in depth. This threshold exceedence problem is discussed herein with closed-form expressions being presented for the variance of the support of the random field. Both one-dimensional (time series) and two-dimensional (image) cases are discussed and the results are extended to a correlated gamma-distributed random field which is appropriate for many non-Gaussian applications. The results are validated by comparison with numerical simulations and an assessment against approximate methods is made which reveals the superiority of the expressions presented. Potential applications of the theory are discussed with particular reference to the radar target detection problem and the modelling of breaking waves.

We revisit the work of Dhar and Majumdar (1999 Phys. Rev. E 59 6413) on the limiting distribution of the temporal mean M_t = t^-1∫₀^tdu sign y_u, for a Gaussian Markovian process y_t depending on a parameter α, which can be interpreted as Brownian motion in the time scale t' = t^2α. This quantity, the mean `magnetization', is simply related to the occupation time of the process, that is the length of time spent on one side of the origin up to time t. Using the fact that the intervals between sign changes of the process form a renewal process on the time scale t', we determine recursively the moments of the mean magnetization. We also find an integral equation for the distribution of M_t. This allows a local analysis of this distribution in the persistence region (M_t→±1), as well as its asymptotic analysis in the regime where α is large. Finally, we put the results thus found in perspective with those obtained by Dhar and Majumdar by another method, based on a formalism due to Kac.

A generalized maximum-entropy-based approach to noisy inverse problems such as the Abel problem, tomography or deconvolution is presented. In this generalized method, instead of employing a regularization parameter, each unknown parameter is redefined as a proper probability distribution within a certain pre-specified support. Then, the joint entropies of both, the noise and signal probabilities, are maximized subject to the observed data. After developing the method, information measures, basic statistics and the covariance structure are developed as well. This method is contrasted with other approaches and includes the classical maximum-entropy formulation as a special case. The method is then applied to the tomographic reconstruction of the soft x-ray emissivity of the hot fusion plasma.

The analogue of the classical Onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite-dimensional Hilbert space. The dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of Gorini-Kossakowski-Sudarshan type. The basic Spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics. To this end, an appropriate Lie series is constructed from a particular symmetrization procedure applied to the ordinary Campbell-Hausdorff expansion. In this way, only Hermitian contributions by higher-order commutators are generated, which allow an identification with so-called generalized Onsager coefficients. The explicit derivations concentrate on second-, third- and fourth-order coefficients, whereas complete detailed expressions are worked out for second and third order. In a suitable coherence-vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the Lie algebra of SU(N). As an illustration, an application to generalized Bloch equations for two-level systems is studied, where the Onsager-like expansion can be compared with exact results for entropy production. We find that convergence is good even for rather large deviations between initial and final state if the calculation includes second- and third-order coefficients only. The formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes.

The relationship between standard fractional Brownian motion (FBM) and FBM based on the Riemann-Liouville fractional integral (or RL-FBM) is clarified. The absence of stationary property in the increment process of RL-FBM is compensated by a weaker property of local stationarity, and the stationary property for the increments of the large-time asymptotic RL-FBM. Generalization of RL-FBM to the RL-multifractional Brownian motion (RL-MBM) can be carried out by replacing the constant Hölder exponent by a time-dependent function. RL-MBM is shown to satisfy a weaker scaling property known as the local asymptotic self-similarity. This local scaling property can be translated into the small-scale behaviour of the associated scalogram by using the wavelet transform.

We analyse the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-½ Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L^-3 to the energy and the corrections of order L^-2 and L^-3 to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L^-1 occur, unmodified by logarithms (except of course for the leading log L term in the specific heat); and the energy expansion contains only odd powers of L^-1. In the specific-heat expansion any power of L^-1 can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.

Using the representation of E_q(2) on the non-commutative space zz*-qz*z = σ; q<1, σ>0 summation formulae for the product of two, three and four q-Kummer functions are derived.

Eigenfunctions of the whispering-gallery type in elliptic cavities are considered. Asymptotic expansions for resonances are derived from the uniform asymptotic expansions of Mathieu functions and modified Mathieu functions constructed by applying the Langer-Olver method. These asymptotic expansions are improved by including exponentially small terms which lie beyond all orders of the perturbative series and can be captured by carefully taking into account Stokes's phenomenon. A classification of resonances along the four irreducible representations of _2v (the symmetry group of the elliptic cavity) is provided, and the splitting up of resonances is then understood in connection with the breaking of O(2)-symmetry (invariance under any rotation).

This paper employs higher-order winding numbers to generate Hamiltonian motion of particles in two dimensions. The ordinary winding number counts how many times two particles rotate about each other. Higher-order winding numbers measure braiding motions of three or more particles. These winding numbers relate to various invariants known in topology and knot theory, for example Massey and Milnor numbers, and can be derived from Vassiliev-Kontsevich integrals. The invariants can be regarded as complex-valued functions of the paths of the particles. The real part gives the winding number, whereas the imaginary part seems uninteresting.

In this paper, we set the imaginary part to be a Hamiltonian for particle motions. For just two particles, this gives the familiar motion of two point vortices. However, for three or more particles, the Hamiltonian generates more complicated intertwining patterns. We examine the dynamics for the case of three particles, and show that the motion is completely integrable. The intertwining patterns correspond to periodic braids; closure of these braids gives links such as the Borromean rings. The Hamiltonian provides an elegant method for generating simple geometrical examples of complicated braids and links.

We present a method for evaluating plethysms of Schur functions that is conceptually simpler than existing methods. Moreover the algorithm can be easily implemented with an algebraic computer language. Plethysms of sums, differences and products of S-functions are dealt with in exactly the same manner as plethysms of simple S-functions. Sums and differences of S-functions are of importance for the description of multi-shell configurations in the shell model. The number of variables in which the S-functions are expressed can be specified in advance, significantly simplifying the calculations in typical applications to many-body problems. The method relies on an algorithm that we have developed for the product of monomial symmetric functions. We present a new way of calculating the Kostka numbers (using Gel'fand patterns) and give, as well, a new formula for the Littlewood-Richardson coefficients.

We study the dynamics of a nematic liquid crystal in a shear flow by employing the gradient of the Landau-de Gennes free-energy function on second-rank tensors, modified by constant and rotational terms. We predict configurations of equilibria and periodic solutions found in numerical simulations and explain certain anomalous nongeneric continua of equilibria. The existence of these continua shows that the model is structurally unstable.

The problem of the propagation of surface waves over deep water is considered. We present a rigorous approach towards the only known (non-trivial) explicit solution to the governing equations for water waves - Gerstner's wave. Some properties of this solution, and how these relate to some basic conclusions about water waves that may be observed experimentally, are discussed.

A generalization of the results of Rasetti and Zanardi concerning avoiding errors in quantum computers by using states preserved by evolution is presented. The concept of dynamical symmetry is generalized from the level of classical Lie algebras and groups, to the level of a dynamical symmetry based on quantum Lie algebras and quantum groups (in the sense of Woronowicz). An intrinsic dependence of the concept of dynamical symmetry on the differential calculus (which holds also in the classical case) is stressed. A natural connection between quantum states invariant under a quantum group action, and quantum states preserved by the dynamical evolution, is discussed.

We study measure perturbations of the Laplacian in L²(²) supported by an infinite curve Γ in the plane which is asymptotically straight in a suitable sense. We show that if Γ is not a straight line, such a `leaky quantum wire' has at least one bound state below the threshold of the essential spectrum.

The conformal mapping of the Borel plane can be utilized for the analytic continuation of the Borel transform to the entire positive real semi-axis and is thus helpful in the resummation of divergent perturbation series in quantum field theory. We observe that the convergence can be accelerated by the application of Padé approximants to the Borel transform expressed as a function of the conformal variable, i.e. by a combination of the analytic continuation via conformal mapping and a subsequent numerical approximation by rational approximants. The method is primarily useful in those cases where the leading (but not sub-leading) large-order asymptotics of the perturbative coefficients are known.

Q-ball solutions are considered within the theory of a complex scalar field with a gauged U(1) symmetry and a parabolic-type potential. In the thin-walled limit, we show explicitly that there is a maximum size for these objects because of the repulsive Coulomb force. The size of the Q ball will increase with decreasing local minimum of the potential. And when the two minima degenerate, the energy stored within the surface of the Q ball becomes significant. Furthermore, we find an analytic expression for a gauged Q ball, which is beyond the conventional thin-walled limit.

The quantum deformation of the Jordanian twist _q for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_q(B) of quantum algebras depending on parameters ξ and h. In a generic point these algebras represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian twist can be applied to the standard quantization of any Kac-Moody algebra. The corresponding classical r-matrix is a linear combination of the Drinfeld-Jimbo and the Jordanian ones. The two-parametric families of Hopf algebras obtained here are smooth and for the limit values of the parameters the standard and nonstandard quantizations are recovered. The twisting element _q also has correlated limits; in particular when q tends to unity it acquires the canonical form of the Jordanian twist. To illustrate the properties of the quantum Jordanian twist we construct the hybrid quantizations for U(sl(2)) and for the corresponding affine algebra . The universal quantum -matrix and its defining representation are presented.

It is proved that for nonlinear evolution with a singular dispersion relation, Leon's extension of the inverse spectral transform is equivalent to the -dressing problem with variable normalization, at least in a subclass of dispersion relations.

Exploiting the results of the exact solution for the ground state of the one-dimensional spinless quantum gas of fermions and impenetrable bosons with the µ/x_ij² particle-particle interaction, the Hellmann-Feynman theorem yields mutually compensating divergences of both the kinetic and the interaction energy in the limiting case µ→-¼. These divergences result from the peculiar behaviour of both the momentum distribution (for large momenta) and the pair density (for small inter-particle separation). The available analytical pair densities for µ = -¼, 0 and 2 allow one to analyse particle-number fluctuations. They are suppressed by repulsive interaction (µ>0), enhanced by attraction (µ<0), and may therefore measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behaviour of the pair density or - for fermions - from the non-idempotency of the momentum distribution and its large-momenta behaviour. They are compared with each other and with reference-free, short-range correlation-measuring ratios of the kinetic and potential energies.

Recently, an extended Wirtinger inequality proved extremely useful in studying the incipient relaxation dynamics of a nematic liquid crystal cell, in the presence of a weak anchoring potential. This inequality is proved here in detail and the specific dynamical problem to which it was first applied is also recalled.

Using a simple 2D area-preserving map we study the evolution of the deviation vector ξ of two initially nearby orbits. It is known that the deviation vector in each point is aligned along a preferable direction, which in the case of regular orbits is the direction of the tangent line to the invariant curve at that point, having also a specific value defined as the `stretching number' at this point. Before the deviation vector takes its preferred direction and value it passes through a transient period. This transient period is found to be very short in the case of chaotic orbits while it is quite long for regular orbits. The initial orientation plays a minor role on the length of the transition phase except in the case when the initial ξ is almost perpendicular to the invariant curve. In this latter case the transition phase becomes quite extended. Analytic calculations suggest that as the iteration number n increases, in the case of chaotic orbits, the deviation vector tends to its preferred value exponentially, while it evolves with an n^-1 power law in the case of regular orbits. Numerical results support the analytic predictions.

The equations governing adiabatic and isothermal quantum processes involved in an ideal two-state quantum heat engine are modified when the ideality restriction is removed. We seek and study a few situations to determine the nature and magnitude of the modifications. If one confines such systems well within the classical turning point, we show how one can profitably employ the Wilson-Sommerfeld quantization rule to estimate the leading correction terms due to non-ideality. The endeavour is likely to be important in studies on practical quantum engines.