Table of contents

We prove continuity of quantum conditional information S(ρ¹²|ρ²) with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party. This can, e.g., be used to prove the continuity of squashed entanglement.

I show that an experimental technique used in nuclear physics may be successfully applied to quantum teleportation (QT) of spin states of massive matter. A new non-local physical effect, the 'quantum-teleportation effect', is discovered for the nuclear polarization measurement. Enhancement of the neutron polarization is expected in the proposed experiment for QT that discriminates only one of the Bell states.

The key steps of intracellular virion reproduction include viral genome replication, mRNA synthesis and degradation, protein synthesis and degradation, capsid assembly and virion release from a cell. Our analysis, incorporating these steps (with no deterioration of the cell machinery), indicates that asymptotically depending on the values of the model parameters the viral kinetics either reach a steady state or are out of control due to an exponential growth of the virion population. In the latter case, the cell is expected to rapidly die or the virion growth should be limited by other steps.

Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T → 0), the size of the system going to infinity (N → ∞) while N[1 − tanh(J/k_BT)] is kept finite (J being the nearest neighbour coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions.

We present a multigroup model of the Boltzmann equations governing the transient transport regime in polar semiconductors. Efforts have been made to give an accurate description of the coupled hot-electron hot-phonon system, which allows us to study the modifications of the carrier and longitudinal optical phonon distribution functions in comparison to the usual equilibrium phonon calculations. Computations are performed for InP, taking into account all the relevant scattering mechanisms. We investigate the response of the coupled electron–phonon system to a step-like high dc electric field pulse. Moreover, we discuss the relation between our model and a matrix method.

We investigate the effects of aperiodic interactions on the critical behaviour of an interacting two-polymer model on hierarchical lattices (equivalent to the Migadal–Kadanoff approximation for the model on Bravais lattices), via renormalization-group and transfer-matrix calculations. The exact renormalization-group recursion relations always present a symmetric fixed point, associated with the critical behaviour of the underlying uniform model. If the aperiodic interactions, defined by substitution rules, lead to relevant geometric fluctuations, this fixed point becomes fully unstable, giving rise to novel attractors of different nature. We present an explicit example in which this new attractor is a two-cycle attractor, with critical indices different from the uniform model. In the case of the four-letter Rudin–Shapiro substitution rule, we find a surprising closed curve whose points are attractors of period two, associated with a marginal operator. Nevertheless, a scaling analysis indicates that this attractor may lead to a new critical universality class. In order to provide an independent confirmation of the scaling results, we turn to a direct thermodynamic calculation of the specific-heat exponent. The thermodynamic free energy is obtained from a transfer-matrix formalism, which had been previously introduced for spin systems, and is now extended to the two-polymer model with aperiodic interactions.

We consider a two-level system, , coupled to a general n level system, , via a random matrix. We derive an integral representation for the mean reduced density matrix ρ(t) of in the limit n → ∞, and we identify a model of which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for ρ(∞). We also consider an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of ρ(t) on an appropriate time scale.

We consider several different directed walk models of a homopolymer adsorbing at a surface when the polymer is subject to an elongational force which hinders the adsorption. We use combinatorial methods for analyzing how the critical temperature for adsorption depends on the magnitude of the applied force and show that the crossover exponent ϕ changes when a force is applied. We discuss the characteristics of the model needed to obtain a re-entrant phase diagram.

Interaction of a domain wall with boundaries of a system is studied for a class of stochastic driven particle models. Reflection maps are introduced for the description of this process. We show that, generically, a domain wall reflects infinitely many times from the boundaries before a stationary state can be reached. This is in evident contrast with one-species models where the stationary density is attained after just one reflection.

We propose a new exactly solvable model of strongly correlated electrons. The model is based on a d–p model of the CuO₂ plane with infinitely large repulsive interactions on Cu-sites, and it contains additional correlated-hopping, pair-hopping and charge–charge interactions of electrons. For even numbers of electrons less than or equal to 2/3-filling, we construct the exact ground states of the model, all of which have the same energy and each of which is the unique ground state for a fixed electron number. It is shown that these ground states are the resonating-valence-bond states which are also regarded as condensed states in which all electrons are in a single two-electron state. We also show that the ground states exhibit off-diagonal long-range order.

In this paper we apply a new simulation technique proposed in Wang and Landau (WL) (2001 Phys. Rev. Lett.86 2050) to sampling of three-dimensional lattice and continuous models of polymer chains. Distributions obtained by homogeneous (unconditional) random walks are compared with results of entropic sampling (ES) within the WL algorithm. While homogeneous sampling gives reliable results typically in the range of 4–5 orders of magnitude, the WL entropic sampling yields them in the range of 20–30 orders and even larger with comparable computer effort. A combination of homogeneous and WL sampling provides reliable data for events with probabilities down to 10⁻³⁵.

For the lattice model we consider both the athermal case (self-avoiding walks, SAWs) and the thermal case when an energy is attributed to each contact between nonbonded monomers in a self-avoiding walk. For short chains the simulation results are checked by comparison with the exact data. In WL calculations for chain lengths up to N = 300 scaling relations for SAWs are well reproduced. In the thermal case distribution over the number of contacts is obtained in the N-range up to N = 100 and the canonical averages—internal energy, heat capacity, excess canonical entropy, mean square end-to-end distance—are calculated as a result in a wide temperature range.

The continuous model is studied in the athermal case. By sorting conformations of a continuous phantom freely joined N-bonded chain with a unit bond length over a stochastic variable, the minimum distance between nonbonded beads, we determine the probability distribution for the N-bonded chain with hard sphere monomer units over its diameter a in the complete diameter range, 0 ⩽ a ⩽ 2, within a single ES run. This distribution provides us with excess specific entropy for a set of diameters a in this range. Calculations were made for chain lengths up to N = 100 and results were extrapolated to N → ∞ for a in the range 0 ⩽ a ⩽ 1.25.

The dynamic properties of the cubic nonlinear Schrödinger equation are investigated numerically using the symplectic scheme (Euler centred scheme). We discuss the dynamic behaviour of the cubic nonlinear Schrödinger equation with varying nonlinear parameter. The results show that the system exhibits regular recurrence for weakly nonlinearity. We also illustrate that the system will exhibit varying dynamic behaviour with increasing nonlinear parameter, i.e. the system will show the homoclinic orbit (HMO) crossing, quasi-recurrence, pseudorecurrence, irregular motion or stochastic motion for a strongly nonlinear constants.

We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two dual sets of generalized reflection equations with associated consistent fusion procedures. Products of their respective solutions yield commuting quantum traces.

Padé approximants are used to find approximate vortex solutions of any winding number in the context of Gross–Pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials. Rational function and generalized rational function approximations of axisymmetric solitary waves of the Gross–Pitaevskii equation are obtained in two and three dimensions. These approximations are used to establish a new mechanism of vortex nucleation as a result of solitary wave interactions.

An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focused on Laplace pairing, Sweedler cohomology for 1- and 2-cochains and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi-exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorizable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda–Calogero system, with a corresponding rational potential, is also discussed.

The twisted XXZ chain alias the six-vertex model is investigated at roots of unity. It is shown that when the twist parameter is chosen to depend on the total spin an infinite-dimensional non-Abelian symmetry algebra can be explicitly constructed in all spin sectors. This symmetry algebra can be identified with the upper or lower Borel subalgebra of the sl₂ loop algebra. The proof uses only the intertwining property of the six-vertex monodromy matrix and the familiar relations of the six-vertex Yang–Baxter algebra.

Factorized dynamics in soliton cellular automata with quantum group symmetry is identified with a motion of particles and anti-particles exhibiting pair creation and annihilation. An embedding scheme is presented showing that the D⁽¹⁾_n-automaton contains, as certain subsectors, the box–ball systems and all the other automata associated with the crystal bases of non-exceptional affine Lie algebras. The results extend the earlier ones to higher representations by a certain reduction and to a wider class of boundary conditions.

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlevé I equation and of the Toda lattice equation.

We have formulated a model of an equation that governs dynamics of nonlinear waves in many non-equilibrium systems. Based on this new equation, we have reviewed the well-known Lange and Newell's criterion for modulational instability of Stoke waves. Some exact solutions of this wave equation have also been found through a proper combination of the Painlevé analysis and Hirota's bilinear technique.

The sets of integrable lattice equations, which generalize the Toda lattice, are considered. The hierarchies of the first integrals and infinitesimal symmetries are found. The properties of the multi-soliton solutions are discussed.

We propose a nonlinear integral equation (NLIE) with only one unknown function, which gives the free energy of the integrable one-dimensional Heisenberg model with arbitrary spin. In deriving the NLIE, the quantum Jacobi–Trudi and Giambelli formula (Bazhanov–Reshetikhin formula), which gives the solution of the T-system, plays an important role. In addition, we also calculate the high temperature expansion of the specific heat and the magnetic susceptibility.

In this communication, the entanglement degree due to quasi-mutual entropy of a three-level atom interacting with a single cavity field is investigated. We consider the situation for which the three-level system is initially in a mixed state, whereas the field may start from either a coherent or a squeezed state. We present a derivation of the unitary evolution operator on the basis of the dressed-state formalism taking into account an arbitrary form of nonlinearity of the intensity-dependent coupling, by means of which we identify and numerically demonstrate the region of parameters where significantly large entanglement can be obtained. Most interestingly, it is shown that features of the degree of entanglement are influenced significantly by different forms of the nonlinearity. The atom and radiation subsystems exhibit alternating sets of collapses and revivals due to the initially mixed states of the atom and radiation employed here.

We computationally investigate the complete polytope of Bell inequalities for two particles with small numbers of possible measurements and outcomes. Our approach is limited by Pitowsky's connection of this problem to the computationally hard NP problem. Despite this, we find that there are very few relevant inequivalent inequalities for small numbers. For example, in the case with three possible 2-outcome measurements on each particle, there is just one new inequality. We describe mixed 2-qubit states which violate this inequality but not the CHSH. The new inequality also illustrates a sharing of bi-partite non-locality between three qubits: something not seen using the CHSH inequality. It also inspires us to discover a class of Bell inequalities with m possible n-outcome measurements on each particle.

We investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1, 2, ..., of deformations exists for each family of shape-invariant potentials. We prove that the mth deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules , where is a codimension m subspace of ⟨1, z, ..., zⁿ⟩. In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules . By construction, these algebraically deformed Hamiltonians do not have an hidden symmetry algebra structure.

An su(1, 1) dynamical algebra to describe both the discrete and the continuum parts of the spectrum for the Morse potential is proposed. The space associated with this algebra is given in terms of a family of orthonormal functions {Φ^σ_n} characterized by the parameter σ. This set is constructed from polynomials which are orthogonal with respect to a weighting function related to a Morse ground state. An analysis of the associated algebra is investigated in detail. The functions are identified with Morse-like functions associated with different potential depths. We prove that for a particular choice of σ the discrete and the continuum parts of the spectrum decouple. The connection of this treatment with the supersymmetric quantum mechanics approach is established. A closed expression for the Mecke dipole moment function is obtained.

Using the natural connection equivalent to the SU(2) Yang–Mills instanton on the quaternionic Hopf fibration of S⁷over the quaternionic projective space HP¹ ≃ S⁴ with an SU(2) ≃ S³ fibre, the geometry of entanglement for two qubits is investigated. The relationship between base and fibre i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury–Fubini–Study metric on HP¹ between an arbitrary entangled state, and the separable state nearest to it. Using this result, an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anholonomy of the connection and entanglement via the geometric phase are shown. Connections with important notions such as the Bures metric and Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.

Bi-partite entanglement in multi-qubit systems cannot be shared freely. The rules of quantum mechanics impose bounds on how multi-qubit systems can be correlated. In this paper, we utilize a concept of entangled graphs with weighted edges in order to analyse pure quantum states of multi-qubit systems. Here qubits are represented by vertexes of the graph, while the presence of bi-partite entanglement is represented by an edge between corresponding vertexes. The weight of each edge is defined to be the entanglement between the two qubits connected by the edge, as measured by the concurrence. We prove that each entangled graph with entanglement bounded by a specific value of the concurrence can be represented by a pure multi-qubit state. In addition, we present a logic network with O(N²) elementary gates that can be used for preparation of the weighted entangled graphs of N qubits.

The transfer matrix technique is much used to study one-dimensional scattering, such as electrons in semiconductor superlattices. We elaborate on the geometrical interpretation of the transfer matrix, in particular its relation to conformal mapping in the unit disc, and the analogy to Lorentz transformations on a Dirac spinor in 2 + 1 dimensions.

TBA integral equations are proposed for one-particle states in the sausage and SS models and their σ-model limits. Combined with the ground-state TBA equations the exact mass gap is computed in the O(3) and O(4) nonlinear σ-model and the results are compared to 3-loop perturbation theory and Monte Carlo data.

We propose thermodynamic Bethe ansatz (TBA) integral equations for multi-particle soliton (fermion) states in the sine–Gordon (massive Thirring) model. This is based on T-system and Y-system equations, which follow from the Bethe ansatz solution in the light-cone lattice formulation of the model. Even and odd charge sectors are treated on an equal footing, corresponding to periodic and twisted boundary conditions, respectively. The analytic properties of the Y-system functions are conjectured on the basis of the large volume solution of the system, which we find explicitly. A simple relation between the TBA Y-functions and the counting function variable of the alternative non-linear integral equation (Destri–deVega equation) description of the model is given. At the special value β² = 6π of the sine–Gordon coupling, exact expressions for energy and momentum eigenvalues of one-particle states are found.

A complete investigation of hidden symmetries present in the d-dimensional fluid dynamical model will be carried out. This will be done in the context of Wess–Zumino (WZ) extension of phase space by using the symplectic embedding formalism. As a consequence, a set of dynamically equivalent symmetries existent in fluid field theory will be discovered. Further, an interesting relation between the WZ symmetries with hidden symmetries (Bazeia D and Jackiw R 1998 Ann. Phys., NY270 246 (Preprint hep-th/9803165); Jackiw R 2000 Preprint physics/0010042) will be performed. Indeed, the global status of the symmetries will be lifted to a local one.

This book provides an overview of the theory and practice of continuous and discrete wavelet transforms. Divided into seven chapters, the first three chapters of the book are introductory, describing the various forms of the wavelet transform and their computation, while the remaining chapters are devoted to applications in fluids, engineering, medicine and miscellaneous areas. Each chapter is well introduced, with suitable examples to demonstrate key concepts. Illustrations are included where appropriate, thus adding a visual dimension to the text. A noteworthy feature is the inclusion, at the end of each chapter, of a list of further resources from the academic literature which the interested reader can consult.

The first chapter is purely an introduction to the text. The treatment of wavelet transforms begins in the second chapter, with the definition of what a wavelet is. The chapter continues by defining the continuous wavelet transform and its inverse and a description of how it may be used to interrogate signals. The continuous wavelet transform is then compared to the short-time Fourier transform. Energy and power spectra with respect to scale are also discussed and linked to their frequency counterparts. Towards the end of the chapter, the two-dimensional continuous wavelet transform is introduced. Examples of how the continuous wavelet transform is computed using the Mexican hat and Morlet wavelets are provided throughout.

The third chapter introduces the discrete wavelet transform, with its distinction from the discretized continuous wavelet transform having been made clear at the end of the second chapter. In the first half of the chapter, the logarithmic discretization of the wavelet function is described, leading to a discussion of dyadic grid scaling, frames, orthogonal and orthonormal bases, scaling functions and multiresolution representation. The fast wavelet transform is introduced and its computation is illustrated with an example using the Haar wavelet. The second half of the chapter groups together miscellaneous points about the discrete wavelet transform, including coefficient manipulation for signal denoising and smoothing, a description of Daubechies' wavelets, the properties of translation invariance and biorthogonality, the two-dimensional discrete wavelet transforms and wavelet packets.

The fourth chapter is dedicated to wavelet transform methods in the author's own specialty, fluid mechanics. Beginning with a definition of wavelet-based statistical measures for turbulence, the text proceeds to describe wavelet thresholding in the analysis of fluid flows. The remainder of the chapter describes wavelet analysis of engineering flows, in particular jets, wakes, turbulence and coherent structures, and geophysical flows, including atmospheric and oceanic processes.

The fifth chapter describes the application of wavelet methods in various branches of engineering, including machining, materials, dynamics and information engineering. Unlike previous chapters, this (and subsequent) chapters are styled more as literature reviews that describe the findings of other authors. The areas addressed in this chapter include: the monitoring of machining processes, the monitoring of rotating machinery, dynamical systems, chaotic systems, non-destructive testing, surface characterization and data compression.

The sixth chapter continues in this vein with the attention now turned to wavelets in the analysis of medical signals. Most of the chapter is devoted to the analysis of one-dimensional signals (electrocardiogram, neural waveforms, acoustic signals etc.), although there is a small section on the analysis of two-dimensional medical images.

The seventh and final chapter of the book focuses on the application of wavelets in three seemingly unrelated application areas: fractals, finance and geophysics. The treatment on wavelet methods in fractals focuses on stochastic fractals with a short section on multifractals. The treatment on finance touches on the use of wavelets by other authors in studying stock prices, commodity behaviour, market dynamics and foreign exchange rates. The treatment on geophysics covers what was omitted from the fourth chapter, namely, seismology, well logging, topographic feature analysis and the analysis of climatic data. The text concludes with an assortment of other application areas which could only be mentioned in passing.

Unlike most other publications in the subject, this book does not treat wavelet transforms in a mathematically rigorous manner but rather aims to explain the mechanics of the wavelet transform in a way that is easy to understand. Consequently, it serves as an excellent overview of the subject rather than as a reference text. Keeping the mathematics to a minimum and omitting cumbersome and detailed proofs from the text, the book is best-suited to those who are new to wavelets or who want an intuitive understanding of the subject. Such an audience may include graduate students in engineering and professionals and researchers in engineering and the applied sciences.

In the last decade decoherence has become a very popular topic mainly due to the progress in experimental techniques which allow monitoring of the process of decoherence for single microscopic or mesoscopic systems. The other motivation is the rapid development of quantum information and quantum computation theory where decoherence is the main obstacle in the implementation of bold theoretical ideas. All that makes the second improved and extended edition of this book very timely.

Despite the enormous efforts of many authors decoherence with its consequences still remains a rather controversial subject. It touches on, namely, the notoriously confusing issues of quantum measurement theory and interpretation of quantum mechanics. The existence of different points of view is reflected by the structure and content of the book. The first three authors (Joos, Zeh and Kiefer) accept the standard formalism of quantum mechanics but seem to reject orthodox Copenhagen interpretation, Giulini and Kupsch stick to both while Stamatescu discusses models which go beyond the standard quantum theory. Fortunately, most of the presented results are independent of the interpretation and the mathematical formalism is common for the (meta)physically different approaches.

After a short introduction by Joos followed by a more detailed review of the basic concepts by Zeh, chapter 3 (the longest chapter) by Joos is devoted to the environmental decoherence. Here the author considers mostly rather `down to earth' and well-motivated mechanisms of decoherence through collisions with atoms or molecules and the processes of emission, absorption and scattering of photons. The issues of decoherence induced superselection rules and localization of objects including the possible explanation of the molecular structure are discussed in details. Many other topics are also reviewed in this chapter, e.g., the so-called Zeno effect, relationships between quantum chaos and decoherence, the role of decoherence in quantum information processing and even decoherence in the brain.

The next chapter, written by Kiefer, is devoted to decoherence in quantum field theory and quantum gravity which is a much more speculative and less explored topic. Two complementary aspects are studied in this approach: decoherence of particle states by the quantum fields and decoherence of field states by the particles. Cosmological issues related to decoherence are discussed, not only within the standard Friedmann cosmology, but also using the elements of the theory of black holes, wormholes and strings.

The relations between the formalism of consistent histories defined in terms of decoherence functionals and the environmental decoherence are discussed in chapter 5, also written by Kiefer. The Feynman--Vernon influence functional for the quantum open system is presented in detail as the first example of decoherence functional. Then the general theory is outlined together with possible interpretations including cosmological aspects.

The next chapter by Giulini presents an overview of the superselection rules arising from physical symmetries and gauge transformations both for nonrelativistic quantum mechanics and quantum field theory. Critical discussion of kinematical superselection rules versus dynamical ones is illustrated by numerous examples like Galilei invariant quantum mechanics, quantum electrodynamics and quantum gravity.

The introduction to the theory of quantum open systems and its applications to decoherence models is given in chapter 7 by Kupsch. Generalized master equations, Markovian approximation and a few Hamiltonian models relevant for decoherence are discussed. Some mathematical tools, e.g., complete positivity and entropy inequalities are also presented.

The last chapter by Stamatescu is devoted to stochastic collapse models which can be interpreted either as certain representations of the dynamics of open quantum systems or as fundamental modifications of the Schr\"odinger equation.

The final part of the book consists of remarks by Zeh on related concepts and methods and seven appendices.

The broad spectrum, mathematically-friendly presentation, inclusion of the very recent developments and the extensive bibliography (about 550 references) make this book a valuable reference for all researchers, graduate and PhD students interested in the foundations of quantum mechanics, quantum open systems and quantum information. The relative independence of the chapters and numerous redundancies allow for selective reading, which is very helpful for newcomers to this field.

When the editorial office of Journal of Physics A: Mathematical and General of the Institute of Physics Publishing asked me to review a book on nonlinear dynamics I experienced an undeniable apprehension. Indeed, the domain is a rapidly expanding one and writing a book aiming at a certain degree of completeness looks like an almost impossible task. My uneasiness abated somewhat when I saw the names of the authors, two well-known specialists of the nonlinear domain, but it was only when I held the book in my hands that I felt really reassured.

The book is not just a review of the recent (and less so) findings on nonlinear systems. It is also a textbook. The authors set out to provide a detailed, step by step, introduction to the domain of nonlinearity and its various subdomains: chaos, integrability and pattern formation (although this last topic is treated with far less detail than the other two). The public they have in mind is obviously that of university students, graduate or undergraduate, who are interested in nonlinear phenomena. I suspect that a non-negligible portion of readers will be people who have to teach topics which figure among those included in the book: they will find this monograph an excellent companion to their course.

The book is written in a pedagogical way, with a profusion of examples, detailed explanations and clear diagrams. The point of view is that of a physicist, which to my eyes is a major advantage. The mathematical formulation remains simple and perfectly intelligible. Thus the reader is not bogged down by fancy mathematical formalism, which would have discouraged the less experienced ones. A host of exercises accompanies every chapter. This will give the novice the occasion to develop his/her problem-solving skills and acquire competence in the use of nonlinear techniques. Some exercises are quite straightforward, like `verify the relation 14.81'. Others are less so, such as `prepare a write-up on a) frequency-locking and b) devil's staircase'. I do not quite grasp the usefulness of such project-like exercises. Projects must be assigned by the person who indeed teaches the course.

There are things that I really like a lot in this book. For instance, the section on `chaos in nonlinear electronic circuits' is particularly interesting. It offers a simple and rather inexpensive way to visualize chaos in the laboratory. The closing section of the book devoted to technological applications of nonlinear dynamics is also quite useful. The fact that the treatment remains rather elementary, based on review articles and monographs rather than research articles, adds to the intelligibility of the chapter, which will certainly prove stimulating to many a student.

Of course, not everything can be perfect, and a 600-page book is bound to have some weak points. I find the treatment of quantum chaos rather sketchy and that of chaotic scattering even more so. Also, while the authors are aware of the importance of complex time in integrability, they do not attempt an explanation of the fundamental puzzle: `why, while the physical time is par excellence real, do we need a complex time in order to study the long-time behaviour of dynamical systems?'. Also the book devotes just four pages to integrable discrete systems. Given the tremendous development of this domain over the past decade, this short presentation is not doing justice to the subject. (However as the present reviewer is editing Springer Lecture Notes in Physics on precisely `Integrable Discrete Systems', to appear in early 2004, he would be the last one to complain about the absence of more details on the matter in the present book.)

To sum it up, the monograph of Lakshmanan and Rajasekar is a book written by physicists and for physicists. It will be of interest to both the experienced practitioner and to the uninitiated. Its main quality resides in its thorough, pedagogical approach to the matter. Moreover the relaxed, not too formal, style makes for easy reading. Given that I am writing this review just a few days before Christmas I cannot help thinking that this book could be a nice present for a physicist.

In these two volumes the author provides a comprehensive survey of the various mathematically-based models used in the research literature to predict the mechanical, thermal and electrical properties of hetereogeneous materials, i.e., materials containing two or more phases such as fibre-reinforced polymers, cast iron and porous ceramic kiln furniture. Volume I covers linear properties such as linear dielectric constant, effective electrical conductivity and elastic moduli, while Volume II covers nonlinear properties, fracture and atomistic and multiscale modelling. Where appropriate, particular attention is paid to the use of fractal geometry and percolation theory in describing the structure and properties of these materials.

The books are advanced level texts reflecting the research interests of the author which will be of significant interest to research scientists working at the forefront of the areas covered by the books. Others working more generally in the field\newpage\noindent of materials science interested in comparing predictions of properties with experimental results may well find the mathematical level quite daunting initially, as it is apparent that the author assumes a level of mathematics consistent with that taught in final year undergraduate and graduate theoretical physics courses. However, for such readers it is well worth persevering because of the in-depth coverage to which the various models are subjected, and also because of the extensive reference lists at the back of both volumes which direct readers to the various source references in the scientific literature. Thus, for the wider materials science scientific community the two volumes will be a valuable library resource.

While I would have liked to see more comparison with experimental data on both ideal and 'real' heterogeneous materials than is provided by the author and a discussion of how to model strong nonlinear current--voltage behaviour in systems such as zinc oxide varistors, my overall impression of the books is that together they are an impressive tour de force and provide a valuable summary of the state of knowledge of the mathematical modelling of heterogeneous materials as we begin the 21st century.