Table of contents

We doubt the relevance of soliton theory to the modelling of tsunamis, and present a case in support of an alternative view. Although the shallow-water equations do provide, we believe, an appropriate basis for this phenomenon, an asymptotic analysis of the solution for realistic variable depths, and for suitable background flows, is essential for a complete understanding of this phenomenon. In particular we explain how a number of tsunami waves can arrive at a shoreline.

We introduce the concept of cloning for classes of observables and classify cloning machines for qubit systems according to the number of parameters needed to describe the class under investigation. A no-cloning theorem for observables is derived and the connections between cloning of observables and joint measurements of noncommuting observables are elucidated. Relationships with cloning of states and non-demolition measurements are also analysed.

We present a dequantization procedure based on a variational approach whereby quantum fluctuations latent in the quantum momentum are suppressed. This is done by adding generic local deformations to the quantum momentum operator which give rise to a deformed kinetic term quantifying the amount of 'fuzziness' caused by such fluctuations. Considered as a functional of such deformations, the deformed kinetic term is shown to possess a unique minimum which is seen to be the classical kinetic energy. Furthermore, we show that extremization of the associated deformed action functional introduces an essential nonlinearity to the resulting field equations which are seen to be the classical Hamilton–Jacobi and continuity equations. Thus, a variational procedure determines the particular deformation that has the effect of suppressing the quantum fluctuations, resulting in dequantization of the system.

Using high precision Monte Carlo simulations and a mean-field theory, we explore coarsening phenomena in a simple driven diffusive system. The model is reminiscent of vehicular traffic on a two-lane ring road. At sufficiently high density, the system develops jams (clusters) which coarsen with time. A key parameter is the passing probability, γ. For small values of γ, the growing clusters display dynamic scaling with a growth exponent of 2/3. For larger values of γ, the growth exponent must be adjusted suggesting the ordered (jammed) state is not a genuine phase but rather a finite-size effect.

We consider the quantum Calogero model, which describes N non-distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two-body interactions proportional to the inverse square of the inter-particle distance. We elaborate a novel solution algorithm which allows us to obtain fully explicit formulae for its eigenfunctions, arbitrary coupling parameter and particle number. We also show that our method applies, with minor changes, to all Calogero models associated with classical root systems.

Given a polygonal closed curve on a lattice or space group, we describe a method for computing the writhe of the curve as the average of weighted projected writhing numbers of the polygon in a few directions. These directions are determined by the lattice geometry, the weights are determined by areas of regions on the unit 2-sphere, and the regions are formed by the tangent indicatrix to the polygonal curve. We give a new formula for the writhe of polygons on the face centred cubic lattice and prove that the writhe of polygons on the body centred cubic lattice, the hexagonal simple lattice, and the diamond space group is always a rational number, and discuss applications to ring polymers.

The flow in and around a fracture modelled as a two-dimensional permeable lens immersed in an infinite porous medium of different permeability is analytically solved by means of conformal mapping and Fourier transform. When the lens is more permeable than the surrounding medium, singularities occur at angular points for flow parallel to the lens, while velocities vanish at these points for flow perpendicular to the lens. In the opposite case, when the lens is less permeable than the surrounding medium, singularities are exchanged and flows parallel and perpendicular to the lens are regular and singular, respectively. Predictions are successfully compared with data obtained by a numerical code.

A simplified version of a time-dependent annular billiard is studied. The dynamics is described using nonlinear maps and we consider two different configurations for the billiard, namely (i) concentric and (ii) eccentric cases. For the concentric case and for a null angular momentum, we confirm that the results for the Fermi–Ulam model are recovered and the particle does not experience the phenomenon of Fermi acceleration. However, on the eccentric case the particle demonstrates unlimited energy gain and Fermi acceleration is therefore observed.

During an adiabatic pumping cycle a conventional two-barrier quantum device takes an electron from the left lead and ejects it to the right lead. Hence, the pumped charge per cycle is naively expected to be Q ⩽ e. This zero-order adiabatic point of view is in fact misleading. For a closed device we can get Q > e and even Q ≫ e. In this paper, a detailed analysis of the quantum pump operation is presented. Using the Kubo formula for the geometric conductance, and applying the Dirac chains picture, we derive practical estimates for Q.

In this paper we consider a square lattice with three types of interactions—nonlinear pair interaction between nearest neighbours, harmonic diagonal interaction and three-particle interaction. The continuum approximation combined with the reductive perturbation method allows us to derive the Kadomtsev–Petviashvili (KP) equation with the lump-soliton solution. Alternatively, the soliton solution is obtained by the pseudo-spectral method. Special attention is paid to determine the range of parameters where the obtained solution is stable. The existence and stability of soliton solutions is checked and verified in molecular dynamics simulations.

It was shown by Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263) that N-parameter reductions of the Benney equations correspond to particular N-parameter families of conformal maps. In recent papers (Baldwin and Gibbons 2003 J. Phys. A: Math. Gen.36 8393–417, Baldwin and Gibbons 2004 J. Phys. A: Math. Gen.37 5341–54), the present authors have constructed examples of such reductions where the mappings take the upper half p-plane to a polygonal slit domain in the λ-plane. In those cases, the mapping function was expressed in terms of the derivatives of Kleinian σ functions of hyperelliptic curves, restricted to the one-dimensional stratum Θ₁ of the Θ-divisor. This was done using an extension of the method given in Enolskii et al (2003 J. Nonlinear Sci.13 157) extended to a genus 3 curve (Enolski V Z and Gibbons J Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, in preparation). Here, we use similar ideas, but now applied to a trigonal curve of genus 4. Fundamental to this approach is a family of differential relations which σ satisfies on the divisor. Again, it is shown that the mapping function is expressible in terms of quotients of derivatives of σ on the divisor Θ₁. One significant by-product is an expansion of the leading terms of the Taylor series of σ for the given family of (3, 5) curves; to the best of the authors' knowledge, this is new.

Using three different approaches, we analyse the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach comprises the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis and the third method is more numerical, using integer arithmetics. These three methods have their own domain of application, but they give corroborating results, and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions.

The solution of several instances of the Schrödinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math.27 585–600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Grünbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not.10 461–84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.

We present a new ultradiscretization approach which can be applied to discrete systems, the solutions of which are not positive definite. This was made possible, thanks to an ansatz involving the hyperbolic-sine function. We apply this new procedure to simple mappings. For the linear and homographic mappings, we obtain ultradiscrete forms and explicitly construct their solutions. Two discrete Painlevé II equations are also analysed and ultradiscretized. We show how to construct the ultradiscrete analogues of their rational and Airy-type solutions.

We consider a perturbed integrable system with one frequency, and the approximate dynamics for the actions given by averaging over the angle. A classical qualitative result states that, for a perturbation of order ε, the error of this approximation is O(ε) on a time scale O(1/ε), for ε → 0. We replace this with a fully quantitative estimate; in certain cases, our approach also gives a reliable error estimate on time scales larger than 1/ε. A number of examples are presented; in many cases, our estimator practically coincides with the envelope of the rapidly oscillating distance between the actions of the perturbed and of the averaged systems. Fairly good results are also obtained in some 'resonant' cases, where the angular frequency is small along the trajectory of the system. Even though our estimates are proved theoretically, their computation in specific applications typically requires the numerical solution of a system of differential equations. However, the time scale for this system is smaller by a factor ε than the time scale for the perturbed system. For this reason, computation of our estimator is faster than the direct numerical solution of the perturbed system; the estimator is also rapidly found in the cases when the time scale makes impossible (within reasonable CPU times) or unreliable the direct solution of the perturbed system.

By using Lie's invariance infinitesimal criterion, we obtain the continuous equivalence transformations of a class of nonlinear Schrödinger equations with variable coefficients. We construct the differential invariants of order 1 starting from a special equivalence subalgebra . We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schrödinger equations which can be mapped, by means of an equivalence transformation of , to the well-known cubic Schrödinger equation. We also provide the explicit form of the transformation.

We recall the importance of recognizing the different mathematical nature of various concepts relating to -symmetric quantum theories. After clarifying the relation between supersymmetry and pseudo-supersymmetry, we prove generically that nonlinear pseudo-supersymmetry, recently proposed by Sinha and Roy, is just a special case of -fold supersymmetry. In particular, we show that all the models constructed by these authors have type A two-fold supersymmetry. Furthermore, we prove that an arbitrary one-body quantum Hamiltonian which admits two (local) solutions in closed form belongs to type A two-fold supersymmetry, irrespective of whether or not it is Hermitian, -symmetric, pseudo-Hermitian and so on.

Correlation functions C(t) ∼ ⟨ϕ(t)ϕ(0)⟩ in ohmically damped systems such as coupled harmonic oscillators or optical resonators can be expressed as a single sum over modes j (which are not power-orthogonal), with each term multiplied by the Petermann factor (PF) C_j, leading to 'excess noise' when |C_j| > 1. It is shown that |C_j| > 1 is common rather than exceptional, that |C_j| can be large even for weak damping, and that the PF appears in other processes as well: for example, a time-independent perturbation ∼ leads to a frequency shift ∼C_j. The coalescence of J (>1) eigenvectors gives rise to a critical point, which exhibits 'giant excess noise' (C_j → ∞). At critical points, the divergent parts of J contributions to C(t) cancel, while time-independent perturbations lead to non-analytic shifts ∼^1/J.

We take as a starting point the ground-state electron density in two-electron model atoms in which Coulomb confinement in the He atom is first replaced by harmonic restoring forces. Switching off electron–electron interactions, one readily constructs a third-order differential equation for the ground-state electron density, as in the recent work of March and Ludeña (2004 Phys. Lett. A 330 16). We then switch on two different model interactions, first in the so-called Hookean atom going back to Kestner and Sinanoglu (1962 Phys. Rev.128 2687), in which e²/r₁₂ is retained as in He, and secondly in the Moshinsky (1968 Am. J. Phys.36 52) atom in which Kr²₁₂/2 is switched on. Some analyticity properties of the low-order linear homogeneous differential equations which result are next studied. He-like atomic ions are then treated in the limit of large atomic number Z. In this latter case, one identifies both the electron–nuclear cusp, or equivalently Kato's theorem, and the corresponding electron–electron cusp in the ground-state spatial wavefunction Ψ(r₁, r₂). A final comment concerns quantum information and entanglement in relation to the recent work of Amovilli and March (2004 Phys. Rev. A 69 054302).

We study separable (i.e., classically correlated) states for composite systems of spinless fermions that are distinguishable. For a proper formulation of entanglement formation for such systems, the state decompositions for mixed states should respect the univalence superselection rule. Fermion hopping always induces non-separability, while states with bosonic hopping correlation may or may not be separable. Under the Jordan–Klein–Wigner transformation from a given bipartite fermion system into a tensor product one, any separable state for the former is also separable for the latter. There are, however, U(1)-gauge invariant states that are non-separable for the former but separable for the latter.

In the context of Feynman's derivation of electrodynamics, we show that noncommutativity allows particle dynamics other than the standard formalism of electrodynamics.

Consider a surface enclosing a fixed volume, described by a free energy depending only on the local geometry; for example, the Canham–Helfrich energy quadratic in the mean curvature describes a fluid membrane. The stress at any point on the surface is determined completely by geometry. In equilibrium, its divergence is proportional to the Laplace pressure, normal to the surface, maintaining the constraint on the volume. It is shown that this source itself can be expressed as the divergence of a position-dependent surface stress. As a consequence, the equilibrium can be described in terms of a conserved effective surface stress. Various non-trivial geometrical consequences of this identification are explored. In a cylindrical geometry, the cross-section can be viewed as a closed planar Euler elastic curve. With respect to an appropriate centre the effective stress itself vanishes; this provides a remarkably simple relationship between the curvature and the position along the loop. In two or higher dimensions, it is shown that the only geometry consistent with the vanishing of the effective stress is spherical. It is argued that the appropriate generalization of the loop result will involve null stresses.

In many mathematical and physical contexts, spinors are treated as Grassmann odd valued fields. We show that it is possible to extend the classification of reality conditions on such spinors by a new type of Majorana condition. In order to define this graded Majorana condition we make use of pseudo-conjugation, a rather unfamiliar extension of complex conjugation to supernumbers. Like the symplectic Majorana condition, the graded Majorana condition may be imposed, for example, in spacetimes in which the standard Majorana condition is inconsistent. However, in contrast to the symplectic condition, which requires duplicating the number of spinor fields, the graded condition can be imposed on a single Dirac spinor. We illustrate how graded Majorana spinors can be applied to supersymmetry by constructing a globally supersymmetric field theory in three-dimensional Euclidean space, an example of a spacetime where standard Majorana spinors do not exist.

The problem of the self-interaction of a quasi-rigid classical particle with an arbitrary spherically symmetric charge distribution is completely solved up to the first order in the acceleration. No ad hoc assumptions are made. The relativistic equations of conservation of energy and momentum in a continuous medium are used. The electromagnetic fields are calculated in the reference frame of instantaneous rest using the Coulomb gauge; in this way the troublesome power expansion is avoided. Most of the puzzles that this problem has aroused are due to the inertia of the negative pressure that equilibrates the electrostatic repulsion inside the particle. The effective mass of this pressure is −U_e/(3c²), where U_e is the electrostatic energy. When the pressure mass is taken into account the dressed mass m turns out to be the bare mass plus the electrostatic mass m = m₀ + U_e/c². It is shown that a proper mechanical behaviour requires that m₀ > U_e/3c². This condition poses a lower bound on the radius that a particle of a given bare mass and charge may have. The violation of this condition is the reason why the Lorentz–Abraham–Dirac formula for the radiation reaction of a point charge predicts unphysical motions that run away or violate causality. Provided the mass condition is met the solutions of the exact equation of motion never run away and conform to causality and conservation of energy and momentum. When the radius is much smaller than the wavelength of the radiated fields, but the mass condition is still met, the exact expression reduces to the formula that Rohrlich (2002 Phys. Lett. A 303 307) has advocated for the radiation reaction of a quasi-point charge.

We study finite energy topologically stable static solutions to a global symmetry-breaking model in 3 + 1 dimensions described by an isovector scalar field. The basic features of two different types of configurations are studied, corresponding to axially symmetric multisolitons with a topological charge n and unstable soliton–antisoliton pairs with a zero topological charge.

The classical problem of thermally developing Poiseuille flow in the presence of viscous dissipation (the 'Graetz–Brinkman problem') is reviewed in this paper. Taking into account that the traditional assumptions of a uniform entrance temperature and of Poiseuille velocity profile are mutually exclusive concepts when the frictional heat generation is significant, in the present paper a non-uniform entrance temperature profile is considered. This non-uniform 'initial condition' is not prescribed, but it is deduced from the energy balance equation in a consistent way as the fully developed temperature profile of the Poiseuille flow under isothermal boundary conditions. Both for the dependence of the local Nusselt number on the Brinkman number and the developing temperature field, substantial differences have been found compared with the traditional case. The consequences of this sensitive dependence on the initial condition are discussed in detail.

An error occurred in figure 4 of this paper. For the corrected figure, please refer to the PDF.

It is difficult not to be amazed by the ability of the human brain to process, to structure and to memorize information. Even by the toughest standards the behaviour of this network of about 10¹¹ neurons qualifies as complex, and both the scientific community and the public take great interest in the growing field of neuroscience.

The scientific endeavour to learn more about the function of the brain as an information processing system is here a truly interdisciplinary one, with important contributions from biology, computer science, physics, engineering and mathematics as the authors quite rightly point out in the introduction of their book. The role of the theoretical disciplines here is to provide mathematical models of information processing systems and the tools to study them. These models and tools are at the centre of the material covered in the book by Coolen, Kühn and Sollich.

The book is divided into five parts, providing basic introductory material on neural network models as well as the details of advanced techniques to study them. A mathematical appendix complements the main text. The range of topics is extremely broad, still the presentation is concise and the book well arranged. To stress the breadth of the book let me just mention a few keywords here: the material ranges from the basics of perceptrons and recurrent network architectures to more advanced aspects such as Bayesian learning and support vector machines; Shannon's theory of information and the definition of entropy are discussed, and a chapter on Amari's information geometry is not missing either. Finally the statistical mechanics chapters cover Gardner theory and the replica analysis of the Hopfield model, not without being preceded by a brief introduction of the basic concepts of equilibrium statistical physics. The book also contains a part on effective theories of the macroscopic dynamics of neural networks. Many dynamical aspects of neural networks are usually hard to find in the existing textbook literature, so that this discussion will be very much appreciated.

The book is of an exceptionally high quality in all aspects. In my view, the style of presentation and the inclusion of recent aspects of the topic alone make the book a welcomed addition to the existing literature. It is well structured and the material covered was chosen with care. While focusing on quantitative aspects of the subject, the authors adopt a comprehensive style of presentation, being precise, but not pedantic. The student who is not familiar with the field might find the breadth of the book overwhelming at first, but will soon appreciate its pedagogical value. All mathematical derivations are performed and explained step by step for the student to follow, and they are illustrated by many concrete examples and results from computer simulations in well-presented and clear figures. If a student wants to get his hands on the mathematical tools of neural networks theory then this book is a good place to learn from. A set of instructive and valuable exercises complements each chapter (hints are given, but maybe it would have been nice to provide additional brief sample solutions in an appendix). I very much enjoyed the outlook sections at the end of each of the five parts, putting the material covered into its historical context and providing further references.

In summary, students of a quantitative discipline will find in this book a clear and self-contained introduction to the subject, lecturers might use it to design postgraduate courses, and finally it will provide a valuable reference for researchers working in the area. This book can be expected to be an asset for all types of readers, even if they already own a book on neural networks. Anyone with a serious interest in the theoretical aspects of the field would be making a mistake not to have a copy on their shelves.

The use of computational modelling in all areas of science and engineering has in recent years escalated to the point where it underpins much of current research. However, the distinction must be made between computer systems in which no knowledge of the underlying computer technology or computational theory is required and those areas of research where the mastery of computational techniques is of great value, almost essential, for final year undergraduates or masters students planning to pursue a career in research. Such a field of research in the latter category is continuum mechanics, and in particular non-linear material behaviour, which is the core topic of this book. The focus of the book on computational plasticity embodies techniques of relevance not only to academic researchers, but also of interest to industrialists engaged in the production of components using bulk or sheet forming processes. Of particular interest is the guidance on how to create modules for use with the commercial system Abaqus for specific types of material behaviour.

The book is in two parts, the first of which contains six chapters, starting with microplasticity, but predominantly on continuum plasticity. The first chapter on microplasticty gives a brief description of the grain structure of metals and the existence of slip systems within the grains. This provides an introduction to the concept of incompressibility during plastic deformation, the nature of plastic yield and the importance of the critically resolved shear stress on the slip planes (Schmid's law). Some knowledge of the notation commonly used to describe slip systems is assumed, which will be familiar to students of metallurgy, but anyone with a more general engineering background may need to undertake additional reading to understand the various descriptions. Any lack of knowledge in this area however, is of no disadvantage as it serves only as an introduction and the book moves on quickly to continuum plasticity. Chapter two introduces one of several yield criteria, that normally attributed to von Mises (though historians of mechanics might argue over who was first to develop the theory of yielding associated with strain energy density), and its two or three-dimensional representation as a yield surface. The expansion of the yield surface during plastic deformation, its translation due to kinematic hardening and the Bauschinger effect in reversed loading are described with a direct link to the material stress-strain curve. The assumption, that the increment of strain is normal to the yield surface, the normality principle, is introduced. Uniaxial loading of an elastic-plastic material is used as an example in which to develop expressions to describe increments in stress and strain. The full presentation of numerous expressions, tensors and matrices with a clear explanation of their development, is a recurring, and commendable, feature of the book, which provides an invaluable introduction for those new to the subject. The chapter moves on from time-independent behaviour to introduce viscoplasticity and creep. Chapter three takes the theories of deformation another stage further to consider the problems associated with large deformation in which an important concept is the separation of the phenomenon into material stretch and rotation. The latter is crucial to allow correct measures of strain and stress to be developed in which the effects of rigid body rotation do not contribute to these variables. Hence, the introduction of 'objective' measures for stress and strain. These are described with reference to deformation gradients, which are clearly explained; however, the introduction of displacement gradients passes with little comment, although velocity gradients appear later in the chapter. The interpretation of different strain measures, e.g. Green--Lagrange and Almansi, is covered briefly, followed by a description of the spin tensor and its use in developing the objective Jaumann rate of stress. It is tempting here to suggest that a more complete description should be given together with other measures of strain and stress, of which there are several, but there would be a danger of changing the book from an `introduction' to a more comprehensive text, and examples of such exist already. Chapter four begins the process of developing the plasticity theories into a form suitable for inclusion in the finite-element method. The starting point is Hamilton's principle for equilibrium of a dynamic system. A very brief introduction to the finite-element method is then given, followed by the finite-element equilibrium equations and a description of how they are incorporated into Hamilton's principle. A useful clarification is provided by comparing tensor notation and the form normally used in finite-element expressions, i.e. Voigt notation. The chapter concludes with a brief overview of implicit integration methods, i.e. tangent stiffness, initial tangent stiffness and Newton–Raphson. Chapter five deals with the more specialized topic of implicit and explicit integration of von Mises plasticity. One of the techniques described is the radial-return method which ensures that the stresses at the end of an increment of deformation always lie on the expanded yield surface. Although this method guarantees a solution it may not always be the most accurate for large deformation, this is one area where reference to alternative methods would have been a helpful addition. Chapter six continues with further detail of how the plasticity models may be incorporated into finite-element codes, with particular reference to the Abaqus package and the use of user-defined subroutines, introduced via a `UMAT' subroutine. This completes part I of the book.

Part II focuses on plasticity models, each chapter dealing with a particular process or material model. For example, chapter seven deals with superplasticity, chapter eight with porous plasticity, chapter nine with creep and chapter ten with cyclic plasticity, creep and TMF. Examples of deep drawing, forming of titanium metal-matrix composites and creep damage are provided, together with further guidelines on the use of Abaqus to model these processes.

Overall, the book is organised in a very logical and readable form. The use of simple one-dimensional examples, with full descriptions of tensors and vectors throughout the book, is particularly useful. It provides a good introduction to the topic, covering much of the theory and with applications to give a good grounding that can be taken further with more comprehensive advanced texts. An excellent starting point for anyone involved in research in computational plasticity.

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