Table of contents

Dynamical zeta functions provide a powerful method to analyse low-dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand, even simple one-dimensional maps can show an intricate structure of the grammar rules that may lead to a non-smooth dependence of global observables on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one-dimensional map of the real line. Using the Baladi–Ruelle generalization of the Milnor–Thurnston kneading determinant, we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero.

We study a credit-risk model which captures effects of economic interactions on a firm's default probability. Economic interactions are represented as a functionally defined graph, and the existence of both cooperative and competitive business relations is taken into account. We provide an analytic solution of the model in a limit where the number of business relations of each company is large, but the overall fraction of the economy with which a given company interacts may be small. While the effects of economic interactions are relatively weak in typical (most probable) scenarios, they are pronounced in situations of economic stress, and thus lead to a substantial fattening of the tails of loss distributions in large loan portfolios. This manifests itself in a pronounced enhancement of the value at risk computed for interacting economies in comparison with their non-interacting counterparts.

We analyse the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite-size, mean-field limit. The mean-field equations for particle densities are written in terms of Ricatti equations with the steady-state current J as a parameter. These equations are solved both analytically and numerically. Upon imposing the boundary conditions set by the injection and extraction rates, the currents J are found self-consistently. We find a number of cases where analytic solutions can be found exactly or approximated. Results for J from asymptotic analyses for slowly varying hopping rates agree extremely well with those from extensive Monte Carlo simulations, suggesting that mean-field currents asymptotically approach the exact currents in the hydrodynamic limit, as the hopping rates vary slowly over the lattice. If the forward hopping rate is greater than or less than the backward hopping rate throughout the entire chain, the three standard steady-state phases are preserved. Our analysis reveals the sensitivity of the current to the relative phase between the forward and backward hopping rate functions.

We find a representation of the row-to-row transfer matrix of the Baxter–Bazhanov–Stroganov τ₂-model for N = 2 in terms of an integral over two commuting sets of Grassmann variables. Using this representation, we explicitly calculate transfer matrix eigenvectors and normalize them. It is also shown how form factors of the model can be expressed in terms of determinants and inverses of certain Toeplitz matrices.

We study the quantum analogue of stirring of water inside a cup using a spoon. This can be regarded as a prototype example for quantum pumping in closed devices. The current in the device is induced by translating a scatterer. Its calculation is done using the Kubo formula approach. The transported charge is expressed as a line integral that encircles chains of Dirac monopoles. For simple systems, the results turn out to be counter-intuitive, e.g. as we move a small scatterer 'forward' the current is induced 'backwards'. One should realize that the route towards quantum-classical correspondence has to do with 'quantum chaos' considerations, and hence assumes greater complexity of the device. We also point out the relation to the familiar S-matrix formalism which is used to analyse quantum pumping in open geometries.

By solving the Euler–Lagrange equations, we determine the elastic curves in the two-dimensional sphere which are circular at rest. We characterize the family of closed critical curves by a rational condition and the existence of closed elastic curves for the different possible values of their curvature at rest is proved. In the final part of the paper, we utilize a numerical approach to get a better understanding of the space of closed critical curves. In this manner we analyse their shape, uniqueness and minimizing properties.

Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie algebras . This procedure is extended to contractions of isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Lie algebras , providing an additional analytical method to obtain their invariants. Further, matrix formulae for the invariants of other inhomogeneous Lie algebras are presented.

Integral symmetry of the confluent Heun equation reduces the central two-point connection problem for the reduced confluent Heun equation to the asymptotical problem. Some heuristic formulae for the transition matrix are obtained and compared with results of the computational calculations.

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particularly important function, related to the τ-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for a convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized.

We construct a boundary Lagrangian for the supersymmetric sine-Gordon model which preserves (B-type) supersymmetry and integrability to all orders in the bulk coupling constant g. The supersymmetry constraint is expressed in terms of matrix factorizations.

Ray theory is used to solve a sequence of elastic wave scattering problems, either within an isolated solid or else within two elastic solids in contact along a planar interface, with time-harmonic forcing of a Gaussian type. By appealing to the ideas behind Gaussian beam summation, we then integrate the resulting Gaussian beam solutions to produce explicit, closed-form expressions for the cylindrical fields that radiate into the elastic media as a result of Lamb-type localized boundary forcing. This approach furnishes something usually impossible from ray analyses of diffraction problems—namely the associated far-field diffraction coefficients or 'directivities'—and does so without the need to resort to delicate steepest descents and branch cut manipulations of cumbersome and complex-valued integral transform solutions. Reference is also made to applications to some problems involving non-localized interfacial forcing.

Using a polynomial pencil of Lax pairs we give a new bi-Hamiltonian formulation for the Landau–Lifshitz equation hierarchy and find a new recursion operator related to it.

We introduce the general definition of the unconventional geometric phase independent of the specific physical system and show how a highly squeezed operator can approximately induce the general unconventional geometric phase, which goes beyond the original unconventional one by displacement operators (Zhu and Wang 2003 Phys. Rev. Lett.91 187902). By means of the squeezed operator concerning the cavity mode state along a closed path in the phase space, we discuss specifically how to implement approximately a two-qubit geometric phase gate in a cavity QED system with two-photon interaction between the atoms and the cavity mode, assisted by a classical field. Discussions regarding the implementation time of the gating, the possible decaying sources and the experimental feasibility are given in detail.

The group of local unitary transformations acts on the space of n-qubit pure states, decomposing it into orbits. In a previous paper we proved that a product of singlet states (together with an unentangled qubit for a system with an odd number of qubits) achieves the smallest possible orbit dimension, equal to 3n/2 for n even and (3n + 1)/2 for n odd, where n is the number of qubits. In this paper we show that any state with minimum orbit dimension must be of this form, and furthermore, such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets.

A new infinite family of examples of finite non-bicolourable configurations of rays in Hilbert space is described. Such configurations appear in the analysis of quantum mechanics in terms of Bell's inequalities and the Kochen–Specker theorem and illustrate that there is no measurable space in the background of the probability model of a quantum system. The mentioned examples are naturally parametrized by a positive integer divisible by 4 and by several complex-valued parameters, whose number depends on this integer. In order to compare two configurations with the same number of rays, a notion of deformation of a configuration is introduced. The constructed examples are then interpreted as obtained by way of deformations.

We investigate the parametric fluctuations in the quantum survival probability of an open version of the δ-kicked rotor model in the deep quantum regime. Spectral arguments (Guarneri I and Terraneo M 2001 Phys. Rev. E65 015203(R)) predict the existence of parametric fractal fluctuations owing to the strong dynamical localization of the eigenstates of the kicked rotor. We discuss the possibility of observing such dynamically-induced fractality in the quantum survival probability as a function of the kicking period for the atom-optics realization of the kicked rotor. The influence of the atoms' initial momentum distribution is studied as well as the dependence of the expected fractal dimension on finite-size effects of the experiment, such as finite detection windows and short measurement times. Our results show that clear signatures of fractality could be observed in experiments with cold atoms subjected to periodically flashed optical lattices, which offer an excellent control on interaction times and the initial atomic ensemble.

We propose a classification scheme for the complete family of 1D point interactions. To do so, we first review the solutions of the wavefunctions and Green functions of the problem. Second, we derive the exact time-dependent propagators in such a way that we can write the expressions for the K's in a very compact form. As they should, such expressions do reduce to the known results in the literature according to the potential parameter values. Then, we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets. Finally, we show that the physical features associated with the scattering process can be used to establish a classification procedure. Moreover, these physical characteristics are directly related to the potential parameters leading to the many formulae for the K's. As an application, we present numerical calculations for Gaussian wave packets.

The invention of the 'dual resonance model' N-point functions B_N motivated the development of current string theory. The simplest of these models, the four-point function B₄, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulae such as the contour-integral representation obtained by Pochhammer in 1890. However, the precise features of the expected multiple-complex-variable generalizations to B_N have not been systematically studied. Here we explore the geometry underlying the dual five-point function B₅, the simplest generalization of the Euler Beta function. The original integrand defining B₅ leads to a polyhedral structure for the five-crosscap surface, embedded in , that has 12 pentagonal faces and a symmetry group of order 120 in . We find a Pochhammer-like representation for B₅ that is a contour integral along a surface of genus 5 in . The symmetric embedding of the five-crosscap surface in is doubly covered by a corresponding symmetric embedding of the surface of genus 4 in that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in . These symmetries enable the construction of elegant visualizations of these surfaces. The key idea of this paper is to realize that the compactification of the set of five-point cross-ratios forms a smooth real algebraic subvariety that is the five-crosscap surface in . It is in the complexification of this surface that we construct the contour integral representation for B₅. Our methods are generalizable in principle to higher dimensions, and therefore should be of interest for further study.