Table of contents

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Geometric Numerical Integration of Differential Equations'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with symplectic or multisymplectic integration; volume-preserving integration; symmetry-preserving integration; integrators that preserve first integrals, Lyapunov functions, or dissipation; exponential integrators; integrators for highly oscillatory systems; Lie-group integrators, etc. Papers on geometric integration of both ODEs and PDEs will be considered, as well as application to molecular-scale integration, celestial mechanics, particle accelerators, fluid flows, population models, epidemiological models and/or any other areas of science. We believe that this issue is timely, and hope that it will stimulate further development of this new and exciting field.

The Editorial Board has invited G R W Quispel and R I McLachlan to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are the following:

• The subject of the paper should relate to geometric numerical integration in the sense described above.

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

• Papers should be original; reviews of a work published elsewhere will not be accepted.

The guidelines for the preparation of contributions are as follows:

• The DEADLINE for submission of contributions is 1 September 2005. This deadline will allow the special issue to appear in late 2005 or early 2006.

• There is a strict page limit of 16 printed pages (approximately 9600 words) per contribution. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length. Further advice on publishing your work in Journal of Physics A: Mathematical and General may 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 e-mail to jphysa@iop.org, quoting `JPhysA Special Issue—Geometric Integration'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the web site 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 the electronic code on floppy disk if available and quoting `JPhysA Special Issue—Geometric Integration'.

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

This 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 study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrödinger equation with the time-dependent potential given by the Weierstrass elliptic -function where the modular parameter τ plays the role of (imaginary) time. In the scaling limit, the equation transforms into a 'non-stationary Mathieu equation' for the vacuum eigenvalues of the Q-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painlevé III equation.

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.

The problem of self-tuning a system to the Hopf bifurcation in the presence of noise and periodic external forcing is discussed. We find that the response of the system has a non-monotonic dependence on the noise strength, and displays an amplified response which is more pronounced for weaker signals. The observed effect is to be distinguished from stochastic resonance. For the feedback we have studied, the unforced self-tuned Hopf oscillator in the presence of fluctuations exhibits sharp peaks in its spectrum. The implications of our general results are briefly discussed in the context of sound detection by the inner ear.

Maxwell's multipoles are a natural geometric characterization of real functions on the sphere (with fixed ℓ). The correlations between multipoles for Gaussian random functions are calculated by mapping the spherical functions to random polynomials. In the limit of high ℓ, the 2-point function tends to a form previously derived by Hannay in the analogous problem for the Majorana sphere. The application to the cosmic microwave background (CMB) is discussed.

In this paper we have studied the upper bound of the time derivative of information entropy for non-Markovian and thermodynamically closed system(s) using reduced model theory (RMT). The upper bound is calculated on the basis of the Fokker–Planck equation and the Schwartz inequality principle. Our calculation shows that the upper bound exhibits extremal nature in the variation of system parameters such as noise correlation time, dissipation strength. The present calculation also considers how the upper bound does change if we increase the number of auxiliary variables involved in the RMT.

The separability theory of Hamiltonian systems on Riemannian manifolds is reviewed and developed. Particular attention is paid to the systems generated by the so-called special conformal Killing tensors, i.e. Benenti systems. Then, infinitely many new classes of separable systems are constructed by appropriate deformations of Benenti class systems.

The differentiation formula

is derived, where K_n−1/2(z) is a modified spherical Bessel function and a is an arbitrary constant.

In this paper, we generalize Cartan's work on Riemannian locally and globally symmetric spaces to locally and globally symmetric Berwald spaces. We prove that a Berwald space is locally symmetric if and only if the flag curvature is invariant under parallel displacements and a locally symmetric Berwald space is locally isometric to a globally symmetric Berwald space.

A derivation of the basis of states for the superconformal minimal models is presented. It relies on a general hypothesis concerning the role of the null field of dimension 2κ − 1/2. The basis is expressed solely in terms of G_r modes and it takes the form of simple exclusion conditions (being thus a quasi-particle-type basis). Its elements are in correspondence with (2κ − 1)-restricted jagged partitions. The generating functions of the latter provide novel fermionic forms for the characters of the irreducible representations in both Ramond and Neveu–Schwarz sectors.

We propose a new kind of two-parameter (p, q)-deformed Heisenberg and parabose algebra, which reduces to the Heisenberg algebra for the p = 1 case and to parabose algebra for q = −1 case. Corresponding to the two-parameter deformed oscillator, we also introduce a new kind of (p, q)-deformed derivative which relates to the ordinary derivative and q-deformed derivative in an explicit manner. We study the structure of Fock-like space of the new (p, q)-deformed oscillators and derive a formal solution for the eigenvalue equation of the Hamiltonian.

The paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two physical examples illustrate effectiveness and accuracy of the presented theory.

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V = igx³. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P.

In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac oscillator. Supersymmetric quantum mechanical and shape-invariance methods are used to derive both the energy spectrum and wavefunctions in the momentum representation. As for the conventional Dirac oscillator, there are neither negative-energy states for E = −1, nor symmetry between the and cases, both features being connected with supersymmetry or, equivalently, the ω → −ω transformation. In contrast with the conventional case, however, the energy spectrum does not present any degeneracy pattern apart from that associated with the rotational symmetry. More unexpectedly, deformation leads to a difference in behaviour between the states corresponding to small, intermediate and very large j values in the sense that only for the first ones supersymmetry remains unbroken, while for the second ones no bound state exists.

First a set of coherent states à la Klauder is formally constructed for the Coulomb problem in a curved space of constant positive curvature. Then the flat-space limit is taken to reduce the set for the radial Coulomb problem to a set of hydrogen atom coherent states corresponding to both the discrete and the continuous portions of the spectrum for a fixed ℓ sector.

We study the stationary nonlinear Schrödinger equation, or Gross–Pitaevskii equation, for a single delta potential and a delta-shell potential. These model systems allow analytical solutions, and thus provide useful insight into the features of stationary bound, scattering and resonance states of the nonlinear Schrödinger equation. For the single delta potential, the influence of the potential strength and the nonlinearity is studied as well as the transition from bound to scattering states. Furthermore, the properties of resonance states in a repulsive delta-shell potential are discussed.

Using the geometric engineering method of 4D quiver gauge theories and results on the classification of Kac–Moody (KM) algebras, we show by explicit examples that there exist three sectors of infrared CFT₄s. Since the geometric engineering of these CFT₄s involves type II strings on K3 fibred CY3 singularities, we conjecture the existence of three kinds of singular complex surfaces containing, in addition to the two standard classes, a third indefinite set. To illustrate this hypothesis, we give explicit examples of K3 surfaces with H⁴₃ and E₁₀ hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine A_r and T_(p,q,r) algebras going beyond the hyperbolic subset. Such hierarchical surfaces have a remarkable signature that is manifested by the presence of poles.

The blackbody radiation problem within classical physics is reviewed. It is again suggested that conformal symmetry is the crucial unrecognized aspect, and that only scattering by classical electromagnetic systems will provide equilibrium at the Planck spectrum. It is pointed out that the several calculations of radiation scattering using nonlinear mechanical systems do not preserve the Boltzmann distribution under adiabatic change of a parameter, and this fact seems at variance with our expectations in connection with derivations of Wien's displacement theorem. By contrast, the striking properties of charged particle motion in a Coulomb potential or in a uniform magnetic field suggest the possibility that these systems will fit with classical thermal radiation. It may be possible to give a full scattering calculation in the case of cyclotron motion in order to provide the needed test of the connection between conformal symmetry and classical thermal radiation.

The conformal symmetry on the instanton moduli space is discussed using the ADHM construction, where a viewpoint of 'homogeneous coordinates' for both the spacetime and the moduli space turns out to be useful. It is shown that the conformal algebra closes only up to global gauge transformations, which generalizes the earlier discussion by Jackiw et al. An interesting five-dimensional interpretation of the SU(2) single-instanton is also mentioned.

This is undoubtedly an ambitious book. It aims to provide a wide ranging, yet self-contained and pedagogical introduction to techniques of quantum many-body theory in condensed matter physics, without losing mathematical `rigor' (which I hope means rigour), and with an eye on physical insight, motivation and application. The authors certainly bring plenty of experience to the task, the book having grown out of their graduate lectures at the Niels Bohr Institute in Copenhagen over a five year period, with the feedback and refinement this presumably brings. The book is also of course ambitious in another sense, for it competes in the tight market of general graduate/advanced undergraduate texts on many-particle physics. Prospective punters will thus want reasons to prefer it to, or at least give it space beside, well established texts in the field.

Subject-wise, the book is a good mix of the ancient and modern, the standard and less so. Obligatory chapters deal with the formal cornerstones of many-body theory, from second quantization, time-dependence in quantum mechanics and linear response theory, to Green's function and Feynman diagrams. Traditional topics are well covered, including two chapters on the electron gas, chapters on phonons and electron–phonon coupling, and a concise account of superconductivity (confined, no doubt judiciously, to the conventional BCS case). Less mandatory, albeit conceptually vital, subjects are also aired. These include a chapter on Fermi liquid theory, from both semi-classical and microscopic perspectives, and a freestanding account of one-dimensional electron gases and Luttinger liquids which, given the enormity of the topic, is about as concise as it could be without sacrificing clarity. Quite naturally, the authors' own interests also influence the choice of material covered. A persistent theme, which brings a healthy topicality to the book, is the area of transport in mesoscopic systems or nanostructures. Two chapters, some fifty pages of the book, are devoted to electron transport in mesoscopic systems; the one on interacting systems is preceded by a brief account of equation of motion techniques–a relative rarity in a general text, used here to provide background to subsequent discussion of the Coulomb blockade in quantum dots.

So does it work, and will it find a niche beside other established, wide ranging texts? On the whole I think the answer has to be yes. To begin with, the book is well organised and user-friendly, which must surely appeal to students (and their mentors). The chapters are typically bite-sized and digestible. Each is accompanied by a summary/outlook, which in doing just that attempts to place the specific topic in a wider context, together with a set of problems that illustrate, and in many cases expand substantially on, the basic subject matter. A particularly healthy feature of the book is the extent to which the authors have sought where possible to include physical and/or material applications of basic theory, thereby enlivening old material and enhancing appreciation of the new. The first chapter on the electron gas, for example, introduces the reader to a range of material examples, including 2D heterostructures, carbon nanotubes and quantum dots. A chapter on the formalism of Green's functions takes time out to explain how the single-particle spectral function can be measured by tunnelling spectroscopy, while discussion of impurity scattering and conductivity is refreshed by consideration of weak localization in bulk and mesoscopic systems, and the phenomenon of universal conductance fluctuations. And so on: in a text that could readily descend to the purely formal, the authors have clearly taken seriously the task of incorporating relevant, topical applications of the underlying theory.

In a book as wide ranging as this any reviewer is of course bound to perceive the occasional deficiency. I felt for example that some aspects of the discussion of conductance in quantum dots, notably the Coulomb blockade and the Kondo effect, were not quite up to scratch—a touch unbalanced in coverage perhaps (no serious mention of renormalization or scaling, even perturbative), with the odd conceptual error creeping in, and the Kondo effect appearing more mysterious than it really is, possibly in part because consideration of it appears before the chapter on Fermi liquid theory, in which the effect is firmly rooted.

But let me not miss the bigger picture: I don't doubt this is a pretty impressive book overall, likely to have broad appeal to budding theorists and adventurous experimentalists, either as a course textbook or—for the slave to garret or lab—as a serious source for self-study. So make a space on your bookshelves.

New branches of scientific disciplines often have a few paradigmatic models that serve as a testing ground for theories and a starting point for new inquiries. In the late 1990s, one of these models found fertile ground in the growing field of econophysics: the Minority Game (MG), a model for speculative markets that combined conceptual simplicity with interesting emergent behaviour and challenging mathematics. The two basic ingredients were the minority mechanism (a large number of players have to choose one of two alternatives in each round, and the minority wins) and limited rationality (each player has a small set of decision rules, and chooses the more successful ones). Combining these, one observes a phase transition between a crowded and an inefficient market phase, fat-tailed price distributions at the transition, and many other nontrivial effects.

Now, seven years after the first paper, three of the key players—Damien Challet, Matteo Marsili and Yi-Cheng Zhang—have published a monograph that summarizes the current state of the science. The book consists of two parts: a 100-page overview of the various aspects of the MG, and reprints of many essential papers.

The first chapters of Part I give a well-written description of the motivation and the history behind the MG, and then go into the phenomenology and the mathematical treatment of the model. The authors emphasize the `physics' underlying the behaviour and give coherent, intuitive explanations that are difficult to extract from the original papers. The mathematics is outlined, but calculations are not carried out in great detail (maybe they could have been included in an appendix).

Chapter 4 then discusses how and why the MG is a model for speculative markets, how it can be modified to give a closer fit to observed market statistics (in particular, reproducing the `stylized facts' of fat-tailed distributions and volatility clustering), and what conclusions one can draw from the behaviour of the MG when different kinds of agents are added. It is this chapter that really justifies the MG as a toy model, and the authors succeed in stating, but not overstating, the case for the MG.

The final chapter is devoted to extensions and alternative interpretations of the MG that take the `minority wins' mechanism as a starting point, but consider different approaches to inductive learning. Topics include evolutionary learning schemes, neural networks, and experiments with human players. The diversity of contributions demonstrates that the minority mechanism has a wider applicability and may inspire many more papers.

Part II, as mentioned, contains reprints of 27 articles on the MG and econophysics in general that are organized along the same lines as the chapters in Part I. The selection is good; the authors resisted the temptation to place too much emphasis on their own prolific output and represent a well-rounded picture of the literature.

The book thus serves several purposes, and it serves them well: it is a well-organized, concise and comprehensive introduction to the MG and the questions econophysics is concerned with, and thus of interest to researchers and graduate students who want to get involved in the field; it is a thorough summary and literature review of the MG and therefore mandatory for those who are already active on the topic; and it serves as a case study for how a toy model can be interpreted and modified to yield insight into complex phenomena, and what answers one can and cannot expect from such models. Whether the MG will serve as a foundation for econophysics in years to come (and investment firms will indeed use the MG score of applicants as a hiring criterion, as the authors jokingly speculate) or as a stepping stone to other models, only time can tell. But in the meantime, there is much to learn from it, and this book is a good place to start.

The scope of this book is to present well known simple and advanced numerical methods for solving partial differential equations (PDEs) and how to implement these methods using the programming environment of the software package Diffpack. A basic background in PDEs and numerical methods is required by the potential reader. Further, a basic knowledge of the finite element method and its implementation in one and two space dimensions is required. The authors claim that no prior knowledge of the package Diffpack is required, which is true, but the reader should be at least familiar with an object oriented programming language like C++ in order to better comprehend the programming environment of Diffpack. Certainly, a prior knowledge or usage of Diffpack would be a great advantage to the reader.

The book consists of 15 chapters, each one written by one or more authors. Each chapter is basically divided into two parts: the first part is about mathematical models described by PDEs and numerical methods to solve these models and the second part describes how to implement the numerical methods using the programming environment of Diffpack. Each chapter closes with a list of references on its subject. The first nine chapters cover well known numerical methods for solving the basic types of PDEs. Further, programming techniques on the serial as well as on the parallel implementation of numerical methods are also included in these chapters. The last five chapters are dedicated to applications, modelled by PDEs, in a variety of fields.

The first chapter is an introduction to parallel processing. It covers fundamentals of parallel processing in a simple and concrete way and no prior knowledge of the subject is required. Examples of parallel implementation of basic linear algebra operations are presented using the Message Passing Interface (MPI) programming environment. Here, some knowledge of MPI routines is required by the reader. Examples solving in parallel simple PDEs using Diffpack and MPI are also presented.

Chapter 2 presents the overlapping domain decomposition method for solving PDEs. It is well known that these methods are suitable for parallel processing. The first part of the chapter covers the mathematical formulation of the method as well as algorithmic and implementational issues. The second part presents a serial and a parallel implementational framework within the programming environment of Diffpack. The chapter closes by showing how to solve two application examples with the overlapping domain decomposition method using Diffpack.

Chapter 3 is a tutorial about how to incorporate the multigrid solver in Diffpack. The method is illustrated by examples such as a Poisson solver, a general elliptic problem with various types of boundary conditions and a nonlinear Poisson type problem.

In chapter 4 the mixed finite element is introduced. Technical issues concerning the practical implementation of the method are also presented. The main difficulties of the efficient implementation of the method, especially in two and three space dimensions on unstructured grids, are presented and addressed in the framework of Diffpack. The implementational process is illustrated by two examples, namely the system formulation of the Poisson problem and the Stokes problem.

Chapter 5 is closely related to chapter 4 and addresses the problem of how to solve efficiently the linear systems arising by the application of the mixed finite element method. The proposed method is block preconditioning. Efficient techniques for implementing the method within Diffpack are presented. Optimal block preconditioners are used to solve the system formulation of the Poisson problem, the Stokes problem and the bidomain model for the electrical activity in the heart.

The subject of chapter 6 is systems of PDEs. Linear and nonlinear systems are discussed. Fully implicit and operator splitting methods are presented. Special attention is paid to how existing solvers for scalar equations in Diffpack can be used to derive fully implicit solvers for systems. The proposed techniques are illustrated in terms of two applications, namely a system of PDEs modelling pipeflow and a two-phase porous media flow.

Stochastic PDEs is the topic of chapter 7. The first part of the chapter is a simple introduction to stochastic PDEs; basic analytical properties are presented for simple models like transport phenomena and viscous drag forces. The second part considers the numerical solution of stochastic PDEs. Two basic techniques are presented, namely Monte Carlo and perturbation methods. The last part explains how to implement and incorporate these solvers into Diffpack.

Chapter 8 describes how to operate Diffpack from Python scripts. The main goal here is to provide all the programming and technical details in order to glue the programming environment of Diffpack with visualization packages through Python and in general take advantage of the Python interfaces.

Chapter 9 attempts to show how to use numerical experiments to measure the performance of various PDE solvers. The authors gathered a rather impressive list, a total of 14 PDE solvers. Solvers for problems like Poisson, Navier--Stokes, elasticity, two-phase flows and methods such as finite difference, finite element, multigrid, and gradient type methods are presented. The authors provide a series of numerical results combining various solvers with various methods in order to gain insight into their computational performance and efficiency.

In Chapter 10 the authors consider a computationally challenging problem, namely the computation of the electrical activity of the human heart. After a brief introduction on the biology of the problem the authors present the mathematical models involved and a numerical method for solving them within the framework of Diffpack.

Chapter 11 and 12 are closely related; actually they could have been combined in a single chapter. Chapter 11 introduces several mathematical models used in finance, based on the Black--Scholes equation. Chapter 12 considers several numerical methods like Monte Carlo, lattice methods, finite difference and finite element methods. Implementation of these methods within Diffpack is presented in the last part of the chapter.

Chapter 13 presents how the finite element method is used for the modelling and analysis of elastic structures. The authors describe the structural elements of Diffpack which include popular elements such as beams and plates and examples are presented on how to use them to simulate elastic structures.

Chapter 14 describes an application problem, namely the extrusion of aluminum. This is a rather\endcolumn complicated process which involves non-Newtonian flow, heat transfer and elasticity. The authors describe the systems of PDEs modelling the underlying process and use a finite element method to obtain a numerical solution. The implementation of the numerical method in Diffpack is presented along with some applications.

The last chapter, chapter 15, focuses on mathematical and numerical models of systems of PDEs governing geological processes in sedimentary basins. The underlying mathematical model is solved using the finite element method within a fully implicit scheme. The authors discuss the implementational issues involved within Diffpack and they present results from several examples.

In summary, the book focuses on the computational and implementational issues involved in solving partial differential equations. The potential reader should have a basic knowledge of PDEs and the finite difference and finite element methods. The examples presented are solved within the programming framework of Diffpack and the reader should have prior experience with the particular software in order to take full advantage of the book. Overall the book is well written, the subject of each chapter is well presented and can serve as a reference for graduate students, researchers and engineers who are interested in the numerical solution of partial differential equations modelling various applications.