Table of contents

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/\%7Eschief/side/side.html). Participants at that meeting, as well as other researchers working in the field of difference equations and discrete systems, are invited to submit a research paper to this issue.

This meeting was the seventh of a series of biennial meetings devoted to the study of integrable difference equations and related topics. The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations, just as differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as: mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, quantum field theory, etc. It is thus crucial to develop tools to study and solve difference equations. While the theory of symmetry and integrability for differential equations is now well-established, this is not yet the case for discrete equations. The situation has undergone impressive development in recent years and has affected a broad range of fields, including the theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups, symmetries of difference equations, discrete (difference) geometry, etc. Consequently, the aim of the special issue is to benefit from the occasion offered by the SIDE VII meeting to provide a collection of papers which represent the state-of-the-art knowledge for studying integrability and symmetry properties of difference equations.

Scope of the special issue

The special issue will feature papers which deal with themes that were covered by the SIDE VII Conference. These are

•Integrability testing

•Discrete geometry and visualization

•Laurent phenomena and cluster algebras

•Ultra-discrete systems

•Random matrix theory

•Algebraic-geometric approaches to integrability

•Yang–Baxter equations

•Quantum and classical integrable systems

•Difference Galois theory

Editorial policy

•The subject of the paper should relate to the subject of the meeting. The Guest Editors will reserve the right to judge whether a contribution fits the scope of the topic of the special issue.

•Contributions will be refereed and processed according to the usual procedure of the journal.

•Conference papers may be based on already published work but should either

•contain significant additional new results and/or insights or

•give a survey of the present state of the art, a critical assessment of the present understanding of a topic, and a discussion of open problems.

•Papers submitted by non-participants should be original and contain substantial new results.

Guidelines for preparation of contributions

• The deadline for contributed papers will be 15 January 2007.

•There is a page limit of 16 printed pages (approximately 9600 words) per contribution. For submitted papers exceeding this length the Guest Editors reserve the right to request a reduction in length. Further advice on document preparation can be found at www.iop.org/Journals/jphysa

•Contributions to the special issue should if possible be submitted electronically by web upload at www.iop.org/Journals/jphysa, or by email to jphysa@iop.org, quoting 'J. Phys. A Special Issue: SIDE VII'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the website for further information on electronic submissions.

•Authors unable to submit electronically may send hard-copy contributions to: Publishing Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK, enclosing electronic code on floppy disk if available and quoting 'J. Phys. A Special Issue: SIDE VII'.

• All contributions should be accompanied by a read-me file or covering letter giving the postal and email address for correspondence. The Publishing Office should be notified of any subsequent change of address.

•The special issue will be published in the paper and online version of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue.

We present a simple method for 'reverse engineering' causal networks, based on mutual information, as a correlation measure. The goal of our method is not to recover all the causal interactions in a network but rather to recover some causal interactions with a very high confidence. For this purpose, we derive an 'exact' theoretical result for the statistical significance of mutual information. Also, we give some numerical simulation results, obtained for random Boolean networks, as an idealized model of genetic regulatory networks.

We study Schramm–Loewner evolutions (SLEs) reversibility and duality using the Virasoro structure of the space of local martingales. For both problems we formulate a setup where the questions boil down to comparing two processes at a stopping time. We state algebraic results showing that local martingales for the processes have enough in common. When one has in addition integrability, the method gives reversibility and duality for any polynomial expected value.

A second-order accurate lattice Boltzmann model is presented for non-Newtonian flow. The non-Newtonian nature of the flow is implemented using a power law model. This is used to enable the accuracy of the model to be assessed and is not a limitation of the model. The second-order accuracy is demonstrated for a range of power law model parameter values representing shear thinning and shear thickening fluids. These results are compared with those of Gabbanelli et al (2006 Phys. Rev. E 72 046312) and it is noted that a higher order of accuracy and greater computational efficiency are achieved. These results demonstrate the suitability of the LBM for shear-dependent non-Newtonian flow simulations.

We study the mass transportation on a rotary by cellular automata. Various on- and off-ramps are introduced to allow particles to move in and out of the rotary. We obtain the exact results analytically. Distinct phases of the traffic flow are classified completely. Phase diagrams in the full parameter space are derived. We show that the bulk properties and the phase transitions are totally controlled by the operation of ramps. The ramps provide a means to stabilize the density difference on the rotary and thus to support the maximum flow as a distinct phase. The efficiency of transportation can be enhanced by adding more ramps to the rotary. Adding off-ramps is more effective in a low density region, while adding on-ramps is more effective in a high density region. The transition regime between free flow and congestion can be controlled by the order of ramps. When the on-ramps and the off-ramps are located alternately to each other, the intermediate phases are significantly suppressed.

Motivated by recent experimental results for the step sizes of dynein motor proteins, we develop a cellular automata model for intra-cellular traffic of dynein motors incorporating special features of the hindrance-dependent step size of the individual motors. We begin by investigating the properties of the aggressive driving model (ADM), a simple cellular automata-based model of vehicular traffic, a unique feature of which is that it allows a natural extension to capture the essential features of dynein motor traffic. We first calculate several collective properties of the ADM, under both periodic and open boundary conditions, analytically using two different mean-field approaches as well as by carrying out computer simulations. Then we extend the ADM by incorporating the possibilities of attachment and detachment of motors on the track which is a common feature of a large class of motor proteins that are collectively referred to as cytoskeletal motors. The interplay of the boundary and bulk dynamics of attachment and detachment of the motors to the track gives rise to a phase where high- and low-density phases separated by a stable domain wall coexist. We also compare and contrast our results with the model of Parmeggiani et al (2003 Phys. Rev. Lett.90 086601) which can be regarded as a minimal model for traffic of a closely related family of motor proteins called kinesin. Finally, we compare the transportation efficiencies of dynein and kinesin motors over a range of values of the model parameters.

A theory of additive Markov chains with a long-range memory, proposed earlier in Usatenko et al (2003 Phys. Rev. E 68 061107), is developed and used to describe statistical properties of long-range correlated systems. The convenient characteristics of such systems, memory functions and their relation to the correlation properties of the systems are examined. Various methods for finding the memory function via the correlation function are proposed. The inverse problem (calculation of the correlation function by means of the prescribed memory function) is also solved. This is demonstrated for the analytically solvable model of the system with a step-wise memory function.

In this work we describe, compile and generalize a set of tools that can be used to analyse the electronic properties (distribution of states, nature of states, etc) of one-dimensional disordered compositions of potentials. In particular, we derive an ensemble of universal functional equations which characterize the thermodynamic limit of all one-dimensional models and which only depend formally on the distributions that define the disorder. The equations are useful to obtain relevant quantities of the system such as density of states or localization length in the thermodynamic limit.

We consider a quantum graph consisting of a ring with the Rashba Hamiltonian and an arbitrary number of semi-infinite wires attached. We describe the scattering matrix for this system and investigate spin filtering for a three-terminal device.

We present and investigate a general nonlinear growth network model which incorporates accelerated growth of nodes and edges, where the growth rates of edges and nodes are all time-dependent power-law functions. The acceleration of edges determines the proportion of the internal edges to the external edges, which play a key role influencing the structure of the network. On the other hand, the effects of the acceleration of nodes on the topology of the network are discussed in the present work. This model predicts an observable two-regime scale-free degree distribution, where the scaling exponents are γ₁ < 2 and γ₂ ≈ 3, respectively. The crossover point k_cross of the degree distribution is adjusted by the growth rates of nodes and edges. The nontrivial clustering coefficient and degree assortativity coefficient are relevant to the acceleration of nodes and edges.

The phenomena of the trapped Bose–Einstein condensates related to matter waves and nonlinear atom optics can be governed by a variable-coefficient Korteweg–de Vries (vc-KdV) model with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condensate, and such a model can also be used to describe the water waves propagating in a channel with an uneven bottom and/or deformed walls. In this paper, with the help of symbolic computation, the bilinear form for the vc-KdV model is obtained and some exact solitonic solutions including the N-solitonic solution in explicit form are derived through the extended Hirota method. We also derive the auto-Bäcklund transformation, nonlinear superposition formula, Lax pairs and conservation laws of this model. Finally, the integrability of the variable-coefficient model and the characteristic of the nonlinear superposition formula are discussed.

We study the quantum mechanical magnetic two-centre problem, i.e., quantum states of an electron within the Coulomb field of two fixed nuclear centres and a homogeneous magnetic field. From the corresponding nonrelativistic Schrödinger equation various characteristic properties are derived. These include the ordering of energy levels and the monotonicity of electronic energies as a function of the nuclear separation if the internuclear axis is parallel to the direction of the B field. For such situations we also obtain lower bounds on the equilibrium separation between the nuclei and establish decay properties of bound state wavefunctions. Moreover, the molecular virial theorem is generalized to encompass the contributions from the magnetic field.

There has been much recent interest in thermal imaging as a method of non-destructive testing and for non-invasive medical imaging. The basic idea of applying heat or cold to an area and observing the resulting temperature change with an infrared camera has led to the development of rapid and relatively inexpensive inspection systems. However, the main drawback to date has been that such an approach provides mainly qualitative results. In order to advance the quantitative results that are possible via thermal imaging, there is interest in applying techniques and algorithms from conventional tomography. Many tomography algorithms are based on the Fourier diffraction theorem, which is inapplicable to thermal imaging without suitable modification to account for the attenuative nature of thermal waves. In this paper, the Fourier diffraction theorem for thermal tomography is derived and discussed. The intent is for this thermal-diffusion based Fourier diffraction theorem to form the basis of tomographic reconstruction algorithms for quantitative thermal imaging.

In this paper, Levinson's theorems for Schrödinger operators in with one-point interaction at 0 are derived using the concept of winding numbers. These results are based on new expressions for the associated wave operators.

In the present paper we apply geometric methods, and in particular the reduced energy–momentum (REM) method, to the analysis of stability of planar rotationally invariant relative equilibria of three-point-mass systems. We analyse two examples in detail: equilateral relative equilibria for the three-body problem, and isosceles triatomic molecules. We discuss some open problems to which the method is applicable, including roto-translational motion in the full three-body problem.

We study how well we can answer the question 'Is the given quantum state equal to a certain maximally entangled state?' using LOCC, in the context of hypothesis testing. Under several locality and invariance conditions, optimal tests will be derived for several special cases by using basic theory of group representations. Some optimal tests are realized by performing quantum teleportation and checking whether the state is teleported. We will also give a finite process for realizing some optimal tests. The performance of the tests will be numerically compared.

We investigate a fully quantum mechanical spin model for the detection of a moving particle. This model, developed in earlier work, is based on a collection of spins at fixed locations and in a metastable state, with the particle locally enhancing the coupling of the spins to an environment of bosons. The appearance of bosons from particular spins signals the presence of the particle at the spin location, and the first boson indicates its arrival. The original model used discrete boson modes. Here we treat the continuum limit, under the assumption of the Markov property, and calculate the arrival-time distribution for a particle to reach a specific region.

The notion of low-noise channels was recently proposed and analysed in detail in order to describe noise-processes driven by the environment (Hotta M, Karasawa T and Ozawa M 2005 Phys. Rev. A 72 052334). An estimation theory of low-noise parameters of channels has also been developed. In this paper, we address the low-noise parameter estimation problem for the N-body extension of the dissipative low-noise channels. We perturbatively calculate the Fisher information of the output states in order to evaluate the lower bound of the mean-square error of the parameter estimation. We show that the maximum of the Fisher information over all input states can be attained by a factorized input state in the leading order of the low-noise parameter. Thus, to achieve optimal estimation, it is not necessary for there to be entanglement of the N subsystems, as long as the true low-noise parameter is sufficiently small and the channel is properly dissipative.

We analyse the Wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non-uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.

Supersymmetric quantum mechanics may be used to construct reflectionless potentials and phase-equivalent potentials. The exactly solvable case of the λ sech² potential is used to show that for certain values of the strength λ the phase-equivalent singular potential arising from the elimination of all the bound states is identical to the original potential evaluated at a point shifted in the complex coordinate space. This equivalence has the consequence that certain general relations valid for reflectionless potentials and phase-equivalent potentials lead to hitherto unknown identities satisfied by the associated Legendre functions. This exactly solvable problem is used to demonstrate some aspects of scattering theory.

Starting from a prescribed Hamiltonian, we construct a non-Markovian evolution equation for a non-relativistic quantum system that exchanges energy with a large reservoir. In order to create sufficient mathematical freedom, the density operator is replaced by a more flexible entity that depends on two times. If these times are chosen equal, the density operator is recovered. In deriving a non-Markovian integral equation for our bitemporal operator, it is assumed that initially system and reservoir are completely uncorrelated. Furthermore, in employing Wick's theorem for factorization of reservoir correlation functions, only those Wick contractions between reservoir potentials are retained that belong to a generalized nearest-neighbour class. The latter is established by subjecting the set of plain nearest-neighbour contractions to any cyclic permutation of reservoir potentials. Through generalizing the notion of nearest-neighbour contraction, it is ensured that the trace of the density operator is conserved. By construction, our bitemporal evolution equation agrees with the Kraus map for quantum dissipation. Moreover, a sound Markovian limit exists that reproduces the complete van Hove–Davies theory. By making use of a rotating-wave approximation and Laplace transformation, the density operator of a damped N-level atom can be computed. For large times and moderate coupling to the reservoir, the atom ends up near the state of thermal equilibrium. At zero temperature, our non-Markovian integral equation gives an exact solution for the atomic density operator.

The stress contribution to the inertia of a spherically symmetric charged particle is calculated for a generic stress tensor of spherical symmetry. It is found that it is equal to the result for the isotropic pressure case, which has been previously calculated (Medina R 2006 J. Phys. A: Math. Gen.39 3801–16).

The author wishes to correct several mistakes in the bozonization formulae of the original paper. Please see PDF for details.