This book is a new volume of a series designed to introduce the curious reader to anything from ancient Egypt and Indian philosophy to conceptual art and cosmology. Very handy in pocket size, Chaos promises an introduction to fundamental concepts of nonlinear science by using mathematics that is `no more complicated than X=2.' Anyone who ever tried to give a popular science account of research knows that this is a more challenging task than writing an ordinary research article. Lenny Smith brilliantly succeeds to explain in words, in pictures and by using intuitive models the essence of mathematical dynamical systems theory and time series analysis as it applies to the modern world. In a more technical part he introduces the basic terms of nonlinear theory by means of simple mappings. He masterly embeds this analysis into the social, historical and cultural context by using numerous examples, from poems and paintings over chess and rabbits to Olbers' paradox, card games and `phynance'. Fundamental problems of the modelling of nonlinear systems like the weather, sun spots or golf balls falling through an array of nails are discussed from the point of view of mathematics, physics and statistics by touching upon philosophical issues. At variance with Laplace's demon, Smith's 21st century demon makes `real world' observations only with limited precision. This poses a severe problem to predictions derived from complex chaotic models, where small variations of initial conditions typically yield totally different outcomes. As Smith argues, this difficulty has direct implications on decision-making in everyday modern life. However, it also asks for an inherently probabilistic theory, which somewhat reminds us of what we are used to in the microworld. There is little to criticise in this nice little book except that some figures are of poor quality thus not really reflecting the beauty of fractals and other wonderful objects in this field. I feel that occasionally the book is also getting a bit too intricate for the complete layman, and experts may not agree on all details of the more conceptual discussions. Altogether I thoroughly enjoyed reading this book. It was a happy companion while travelling and a nice bedtime literature. It is furthermore an excellent reminder of the `big picture' underlying nonlinear science as it applies to the real world. I will gladly recommend this book as background literature for students in my introductory course on dynamical systems. However, the book will be of interest to anyone who is looking for a very short account on fundamental problems and principles in modern nonlinear science.

The hallmark of a good book of problems is that it allows you to become acquainted with an unfamiliar topic quickly and efficiently. The Quantum Mechanics Solver fits this description admirably. The book contains 27 problems based mainly on recent experimental developments, including neutrino oscillations, tests of Bell's inequality, Bose–Einstein condensates, and laser cooling and trapping of atoms, to name a few. Unlike many collections, in which problems are designed around a particular mathematical method, here each problem is devoted to a small group of phenomena or experiments. Most problems contain experimental data from the literature, and readers are asked to estimate parameters from the data, or compare theory to experiment, or both. Standard techniques (e.g., degenerate perturbation theory, addition of angular momentum, asymptotics of special functions) are introduced only as they are needed. The style is closer to a non-specialist seminar rather than an undergraduate lecture. The physical models are kept simple; the emphasis is on cultivating conceptual and qualitative understanding (although in many of the problems, the simple models fit the data quite well). Some less familiar theoretical techniques are introduced, e.g. a variational method for lower (not upper) bounds on ground-state energies for many-body systems with two-body interactions, which is then used to derive a surprisingly accurate relation between baryon and meson masses. The exposition is succinct but clear; the solutions can be read as worked examples if you don't want to do the problems yourself. Many problems have additional discussion on limitations and extensions of the theory, or further applications outside physics (e.g., the accuracy of GPS positioning in connection with atomic clocks; proton and ion tumor therapies in connection with the Bethe–Bloch formula for charged particles in solids). The problems use mainly non-relativistic quantum mechanics and are organised into three sections: Elementary Particles, Nuclei and Atoms; Quantum Entanglement and Measurement; and Complex Systems. The coverage is not comprehensive; there is little on scattering theory, for example, and some areas of recent interest, such as topological aspects of quantum mechanics and semiclassics, are not included. The problems are based on examination questions given at the Ecole Polytechnique in the last 15 years. The book is accessible to undergraduates, but working physicists should find it a delight.

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrodinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equations. Here, a novel analytical approach reveals that the evolution of small-amplitude Helmholtz–Kerr dark solitons is also governed by a KdV equation. This broadens the class of nonlinear systems that are known to possess KdV soliton solutions, and provides a framework for perturbative analyses when propagation angles are not negligibly small. The derivation of this KdV equation involves an element that appears new to weakly nonlinear analyses, since transformations are required to preserve the rotational symmetry inherent to Helmholtz-type equations.

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional probabilities. The beta-binomial distribution is derived by a strong symmetric assumption on the conditional correlations. Our derivation clarifies the 'correlation' structure of the beta-binomial distribution. It is also possible to study the correlation structures of other probability distributions of exchangeable (homogeneous) correlated Bernoulli random variables. We study some distribution functions and discuss their behaviours in terms of their correlation structures.

The standard relation between the field momentum and the force is generalized for the system with a field singularity: in addition to the regular force, there appears the singular one. This approach is applied to the description of the gyroscopic dynamics of the classical field with topological defects. The collective-variable Lagrangian description is considered for gyroscopical systems taking into account singularities. Using this method, we describe the dynamics of two-dimensional magnetic solitons. We establish a relation between the gyroscopic force and the singular one. An effective Lagrangian description is discussed for the magnetic soliton dynamics.

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup Γ0(2) of the modular group is constructed. These nonlinear equations are analogues of the well-known Ramanujan equations, as well as the Chazy and Darboux–Halphen equations associated with the modular group. The general solutions of these equations can be realized in terms of the Schwarz triangle function S(0, 0, 1/2; z).

In this paper we consider vector fields in that are invariant under a suitable symmetry and that possess a 'generalized heteroclinic loop' formed by two singular points (e+ and e−) and their invariant manifolds: one of dimension 2 (a sphere minus the points e+ and e−) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e−). In particular, we analyse the dynamics of the vector field near the heteroclinic loop by means of a convenient Poincare map, and we prove the existence of infinitely many symmetric periodic orbits near . We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in , and the second one is the charged rhomboidal four-body problem.

Harmonic maps from or one-connected domain into and U(m) are treated. The GBDT version of the B?cklund?Darboux transformation is applied to the case of the harmonic maps and a new and simple algebraic procedure to construct new harmonic maps from the initial ones is given, using some methods from system theory. A new general formula on the GBDT transformations of the Sym?Tafel immersions is derived. A new class of the unitary harmonic maps with asymptotics along one line essentially different from the asymptotics in all other directions, similar in certain ways to line solutions, is obtained explicitly and studied.

Loop models have been widely studied in physics and mathematics, in problems ranging from polymers to topological quantum computation to Schramm-Loewner evolution. I present new loop models which have critical points described by conformal field theories. Examples include both fully-packed and dilute loop models with critical points described by the superconformal minimal models and the SU(2)_2 WZW models. The dilute loop models are generalized to include SU(2)_k models as well.

The generalized vector is defined on an n-dimensional manifold. The interior product and Lie derivative acting on generalized p-forms, −1 ≤ p ≤ n are introduced. The generalized commutator of two generalized vectors is defined. Adding a correction term to Cartan's formula, the generalized Lie derivative's action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application.