Table of contents

We consider a one-parameter family of third-order nonlinear recurrence relations. Each member of this family satisfies the singularity confinement test, has a conserved quantity, and moreover has the Laurent property: all of the iterates are Laurent polynomials in the initial data. However, we show that these recurrences are not Diophantine integrable according to the definition proposed by Halburd (2005 J. Phys. A: Math. Gen.38 L1). Explicit bounds on the asymptotic growth of the heights of iterates are obtained for a special choice of initial data. As a by-product of our analysis, infinitely many solutions are found for a certain family of Diophantine equations, studied by Mordell, that includes Markoff's equation.

We employ symbolic calculation to perform a systematic study of the accuracy of split-step Fourier transform methods for the time-dependent Gross–Pitaevskii equation (GPE). Provided the most recent approximation for the wavefunction is always used in the nonlinear atom–atom interaction energy, every split-step algorithm we have tried has the same-order time stepping error for the nonlinear GPE and for the linear Schrödinger equation.

The QCD partition function in the external stationary gluomagnetic field is computed in the third order in external field invariants in arbitrary dimension and arbitrary covariant gauge. The contributions proportional to third order invariants in gluon field strength are shown to be dependent on the covariant quantum gauge fixing parameter α.

We introduce a new kind of continuous-variable-type entangled pure states, called single-mode excited entangled coherent states (SMEECSs), which are obtained through actions of a creation operator of a single-mode optical field on the ECSs. We study the mathematical properties and entanglement characteristics of the SMEECSs, and investigate the influence of photon excitations on quantum entanglement. It is shown that the SMEECSs form a type of cyclic representation of the Heisenberg–Weyl algebra. It is found that the photon excitations seriously affect the entanglement character of the SMEECSs. We also show how such states can be produced by using cavity QED and quantum measurements.

In a recent paper (Bogoyavlenskiy V A 2002 J. Phys. A: Math. Gen.35 2533), an algorithm aiming to generate isotropic clusters of the on-lattice diffusion-limited aggregation (DLA) model was proposed. The procedure consists of aggregation probabilities proportional to the squared number of occupied sites (k²). In the present work, we analysed this algorithm using the noise reduced version of the DLA model and large-scale simulations. In the noiseless limit, instead of isotropic patterns, a 45° (30°) rotation in the anisotropy directions of the clusters grown on square (triangular) lattices was observed. A generalized algorithm, in which the aggregation probability is proportional to k^ν, was proposed. The exponent ν has a nonuniversal critical value ν_c, for which the patterns generated in the noiseless limit exhibit the original (axial) anisotropy for ν < ν_c and the rotated one (diagonal) for ν > ν_c. The values ν_c = 1.395 ± 0.005 and ν_c = 0.82 ± 0.01 were found for square and triangular lattices, respectively. Moreover, large-scale simulations show that there is a nontrivial relation between the noise reduction and anisotropy direction. The case ν = 2 (Bogoyavlenskiy's rule) is an example where the patterns exhibit the axial anisotropy for small and the diagonal one for large noise reduction.

We study the out-of-equilibrium dynamics of the spherical ferromagnet after a quench to its critical temperature. We calculate correlation and response functions for spin observables which probe length scales much larger than the lattice spacing but smaller than the system size, and find that the asymptotic fluctuation–dissipation ratio (FDR) X^∞ is the same as for local observables. This is consistent with our earlier results for the Ising model in dimension d = 1 and d = 2. We also check that bond observables, both local and long range, give the same asymptotic FDR. In the second part of the paper the analysis is extended to global observables, which probe correlations among all N spins. Here, non-Gaussian fluctuations arising from the spherical constraint need to be accounted for, and we develop a systematic expansion in to do this. Applying this to the global bond observable, i.e. the energy, we find that non-Gaussian corrections change its FDR to a nontrivial value which we calculate exactly for all dimensions d > 2. Finally, we consider quenches from magnetized initial states. Here, even the FDR for the global spin observable, i.e. the magnetization, is nontrivial. It differs from the one for unmagnetized states even in d > 4, signalling the appearance of a distinct dynamical universality class of magnetized critical coarsening. For lower d, the FDR is irrational even to first order in 4 − d and d − 2, the latter in contrast to recent results for the transverse FDR in the n-vector model.

The Raise and Peel model is a recently proposed one-dimensional statistical model describing a fluctuating interface. The evolution of the model follows from the competition between adsorption and desorption processes. The model is non-local due to the possible occurrence of avalanches. At a special ratio of the adsorption–desorption rates, the model is integrable and many rigorous results are known. Off the critical point, the phase diagram and scaling properties are not known. In this paper, we search for indications of phase transition studying the gap in the spectrum of the non-Hermitian generator of the stochastic interface evolution. We present results for the gap obtained from exact diagonalization and from Monte Carlo estimates derived from temporal correlations of various observables.

In recent works, we have proposed a stochastic cellular automaton model of traffic flow connecting two exactly solvable stochastic processes, i.e., the asymmetric simple exclusion process and the zero range process, with an additional parameter. It is also regarded as an extended version of the optimal velocity model, and moreover it shows particularly notable properties. In this paper, we report that when taking optimal velocity function to be a step function, all of the flux-density graph (i.e. the fundamental diagram) can be estimated. We first find that the fundamental diagram consists of two line segments resembling an inversed-λ form, and next identify their end-points from a microscopic behaviour of vehicles. It is notable that by using a microscopic parameter which indicates a driver's sensitivity to the traffic situation, we give an explicit formula for the critical point at which a traffic jam phase arises. We also compare these analytical results with those of the optimal velocity model, and point out the crucial differences between them.

We study some of the basic properties of a generalized Cauchy process indexed by two parameters. The application of the Lamperti transformation to the generalized Cauchy process leads to a self-similar process which preserves the long-range dependence. The asymptotic properties of spectral density of the process are derived. Possible application of this process to model relaxation phenomena is considered.

A classical 2D clock model is known to have a critical phase with Berezinskii–Kosterlitz–Thouless (BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed a method called 'level spectroscopy', which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D six-clock model which can be mapped to the quantum self-dual 1D six-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view.

We examine the dynamics of a network of genes focusing on a periodic chain of genes, of arbitrary length. We show that within a given class of sigmoïds representing the equilibrium probability of the binding of the RNA polymerase to the core promoter, the system possesses a single stable fixed point. By slightly modifying the sigmoïd, introducing 'stiffer' forms, we show that it is possible to find network configurations exhibiting bistable behaviour. Our results do not depend crucially on the length of the chain considered: calculations with finite chains lead to similar results. However, a realistic study of regulatory genetic networks would require the consideration of more complex topologies and interactions.

We use Lie point symmetries of the (2+1)-dimensional cubic Schrödinger equation to obtain new analytic solutions in a systematic manner. We present an analysis of the reduced ODEs, and in particular show that although the original equation is not integrable they typically can belong to the class of Painlevé-type equations.

This paper considers four-dimensional manifolds upon which there is a Lorentz metric h and a symmetric connection Γ which are originally assumed unrelated. It then derives sufficient conditions on h and Γ (expressed through the curvature tensor of Γ) for Γ to be the Levi-Civita connection of some (local) Lorentz metric g and calculates the relationship between g and h. Some examples are provided which help to assess the strength of the sufficient conditions derived.

Apparently new summations in terms of well-known special functions are deduced for hypergeometric-type series containing a digamma function as a factor. As a by-product of this investigation new reduction formulae for the Kampé de Fériet function F^0:2;1_2:1;0[x, x] are obtained.

This paper provides explicit techniques to compute the exponentials of a variety of anti-Hermitian matrices in dimension 4. Many of these formulae can be written down directly from the entries of the matrix. Whenever any spectral calculations are required, these can be done in closed form. In many instances only 2 × 2 spectral calculations are required. These formulae cover a wide variety of applications. Conditions on the matrix which render it to admit one of three minimal polynomials are also given. Matrices with these minimal polynomials admit simple and tractable representations for their exponentials. One of these is the Euler–Rodrigues formula. The key technique is the relation between real 4 × 4 matrices and the quaternions.

The method of q-oscillator lattices, proposed recently in Bazhanov and Sergeev 2005 (Preprint hep-th/0509181), provides the tool for a construction of various integrable models of quantum mechanics in (2 + 1)-dimensional spacetime. In contrast to any one-dimensional quantum chain, its two-dimensional generalizations—quantum lattices—admit different geometrical structures. In this paper, we consider the q-oscillator model on a special lattice. The model may be interpreted as a two-dimensional lattice Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe ansatz: the p-particles' wavefunction is the sum of plane waves. Consistency conditions is the set of 2p equations for p one-particle wave vectors. These 'Bethe ansatz' equations are the main result of this paper.

We propose an analytic formula for the non-local Fisher information functional, or electronic kinetic correlation term, appearing in the expression of the kinetic density functional. Such an explicit formula is constructed on the basis of well-founded physical arguments and a rigorous mathematical prescription.

We investigate local distinguishability of quantum states by use of convex analysis of joint numerical range of operators on a Hilbert space. We show that any two orthogonal pure states are distinguishable by local operations and classical communications, even for infinite-dimensional systems. An estimate of the local discrimination probability is also given for some families of more than two pure states.

We discuss magnetic Schrödinger operators perturbed by measures from the generalized Kato class. Using an explicit Krein-like formula for their resolvent, we prove that these operators can be approximated in the strong resolvent sense by magnetic Schrödinger operators with point potentials. Since the spectral problem of the latter operators is solvable, one in fact gets an alternative way to calculate discrete spectra; we illustrate it by numerical calculations in the case when the potential is supported by a circle.

We derive the semiclassical limit of the coherent state propagator for systems with two degrees of freedom of which one degree of freedom is canonical and the other a spin. Systems in this category include those involving spin–orbit interactions and the Jaynes–Cummings model in which a single electromagnetic mode interacts with many independent two-level atoms. We construct a path integral representation for the propagator of such systems and derive its semiclassical limit. As special cases we consider separable systems, the limit of very large spins and the case of spin-1/2.

The technique of vector differentiation is applied to the problem of the derivation of multipole expansions in four-dimensional space. Explicit expressions for the multipole expansion of the function with r = r₁ + r₂ are given in terms of tensor products of two hyperspherical harmonics depending on the unit vectors and . The multipole decomposition of the function (r₁ ⋅ r₂)ⁿ is also derived. The proposed method can be easily generalized to the case of the space with dimensionality larger than four. Several explicit expressions for the four-dimensional Clebsch–Gordan coefficients with particular values of parameters are presented in the closed form.

Algebraic mean field theory constructs a group theoretical model of quantum systems that have a weak dynamical symmetry but may break dynamical symmetry. The strong defining condition for dynamical symmetry is that states belong to one irreducible representation space. Weak dynamical symmetry demands that the densities corresponding to the states have a constant value for each Casimir. Quantum phase transitions and other complex systems exhibit weak dynamical symmetry. Furthermore mean field theory often yields analytic formulae for expectations and energy spectra that are not feasible in representation theory. This paper develops mean field theory on any coadjoint orbit of su(4) densities. The simple Lie algebra su(4) ≃ so(6) is a 15-dimensional algebra that contains the subalgebra usp(4) ≃ so(5) and the angular momentum algebra su(2). The su(4) dual space consists of density matrices which are defined by the expectations of the su(4) generators. A coadjoint orbit is a common level surface in the dual space of the three su(4) Casimirs. A Lax pair determines the dynamics of these densities on each coadjoint orbit. Analytic solutions are reported for rotating su(4) densities in equilibrium for a particular energy function.

The macroscopic equations derived from the lattice Boltzmann equation are not exactly the Navier–Stokes equations. Here the cubic deviation terms and the methods proposed to eliminate them are studied. The most popular two- and three-dimensional models (D2Q9, D3Q15, D3Q19, D3Q27) are considered in the paper. It is demonstrated that the compensation methods provide only partial elimination of the deviations for these models. It is also shown that the compensation of Qian and Zhou (1998 Europhys. Lett.42 359) using the compensation parameter K = 1 in a D2Q9 or D3Q27 model can eliminate all the cross terms perfectly, but the deviation terms ∂_xρu³_x, ∂_yρu³_y and ∂_zρu³_z still survive the compensation.

The existence domains for one-dimensional acoustic solitons and double layers in complex (dusty) plasmas with two ion temperatures are obtained, using the fluid dynamic paradigm with a general polytropic equation of state. Dust-acoustic solitons are considered in a four-component plasma of negative dust grains, cool and very hot ions, and very hot electrons. Whereas in a dust-ion-electron plasma only negative potential solitons are supported, the presence of a second ion component allows positive potential solitons to occur as well. The existence domain in parameter space is delineated, in particular, also for the reduced three-component case in which there are no free electrons, all electrons being adsorbed onto the dust grains. Next, the ion-acoustic regime is considered. Both positive and negative potential dust-ion acoustic solitons and double layers are found, and their existence conditions in the parameter space of cool ion density and Mach number derived.

An alternative method is suggested for the description of the velocity and pressure fields in an unbounded incompressible viscous fluid induced by an arbitrary number of spheres moving and rotating in it. Within the framework of this approach, we obtain the general relations for forces and torques exerted by the fluid on the spheres. The behaviour of the translational, rotational, and coupled friction and mobility tensors in various frequency domains is analysed up to the terms of the third order in the dimensionless parameter b equal to the ratio of a typical radius of a sphere to the penetration depth of transverse waves and a certain power of the dimensionless parameter σ equal to the ratio of a typical radius of a sphere to the distance between the centres of two spheres. We establish that the retardation effects can essentially affect the character of the hydrodynamic interactions between the spheres.