Nonlinear science Books

Book SynopsisThis book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book.Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bring about macroscopic phenomena that cannot be understood by considering a single part alone. The phenomenology of phase transitions is introduced by considering percolation, a simple model with a purely geometrical phase transition, thus enabling the reader to become intuitively familiar with concepts such as scale invariance and renormalisation. The Ising model is then introduced, which captures a thermodynamic phase transition from a disordered to an ordered system as the temperature is lowered in zero external field. By emphasising analogies between percolation and the Ising model, the reader's intuition of phase transitions is developed so that the underlying theoretical formalism may be appreciated fully. These equilibrium systems undergo a phase transition only if an external agent finely tunes certain external parameters to particular values.Besides fractals and phase transitions, there are many examples in Nature of the emergence of such complex behaviour in slowly driven non-equilibrium systems: earthquakes in seismic systems, avalanches in granular media and rainfall in the atmosphere. A class of non-equilibrium systems, not constrained by having to tune external parameters to obtain critical behaviour, is addressed in the framework of simple models, revealing that the repeated application of simple rules may spontaneously give rise to emergent complex behaviour not encoded in the rules themselves. The common basis of complexity and criticality is identified and applied to a range of non-equilibrium systems. Finally, the reader is invited to speculate whether self-organisation in non-equilibrium systems might be a unifying concept for disparate fields such as statistical mechanics, geophysics and atmospheric physics.Visit for animations for the models in the book (available for Windows and Linux), solutions to exercises, as well as a list with corrections.Trade Review"Personally, I enjoyed reading this book very much. The arguments are clear and draw attention to a number of useful insights ... Students will find the presentation on self-organized criticality fun to read, particularly because it deals with real phenomena, such as earthquakes, rice-pile avalanches and rainfall ... I strongly agree with these authors that undergraduates need to be exposed to issues related to complexity and criticality. Their textbook is the first that I have seen that makes developing such courses feasible."Mathematical ReviewsTable of ContentsPercolation: Percolating Phase Transition; In One and Two Dimensions, and in the Bethe Lattice; Geometric Properties of Clusters; Scaling Ansatz, Scaling Functions and Scaling Relations; Universality; Real-Space Renormalisation Group; Ising Model: Review of Thermodynamics and Statistical Mechanics; Symmetry Breaking; Ferromagnetic Phase Transition; In One and Two Dimensions, and in the Mean-Field; Landau Theory of Continuous Phase Transitions; Scaling Ansatz, Scaling Functions and Scaling Relations; Universality; Real-Space Renormalisation Group; Self-Organised Criticality: BTW Model in One and Two Dimensions, and in the Mean-Field; A Rice Pile Experiment and the Oslo Model; Earthquakes and the OFC Model; Rainfall; Self-Organised Criticality as a Unifying Principle.

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Book SynopsisThe focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved. The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, rational functions and meromorphic maps. Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hölder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.Table of Contents1. Introduction.- 2. Singular Perturbations of Classical Original Perron–Frobenius Operators on Countable Alphabet Symbol Spaces.- 3. Symbol Escape Rates and the Survivor Set K(Un).- 4. Escape Rates for Conformal GDMSs and IFSs.- 5. Applications: Escape Rates for Multimodal Mapsand One-Dimensional Complex Dynamics.

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Book SynopsisThe importance of complexity is well-captured by Hawking''s comment: Complexity is the science of the 21st century. From the movement of flocks of birds to the Internet, environmental sustainability, and market regulation, the study and understanding of complex non-linear systems has become highly influential over the last 30 years.In this Very Short Introduction, one of the leading figures in the field, John Holland, introduces the key elements and conceptual framework of complexity. From complex physical systems such as fluid flow and the difficulties of predicting weather, to complex adaptive systems such as the highly diverse and interdependent ecosystems of rainforests, he combines simple, well-known examples -- Adam Smith''s pin factory, Darwin''s comet orchid, and Simon''s ''watchmaker'' -- with an account of the approaches, involving agents and urn models, taken by complexity theory. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Table of Contents1. Complex systems ; 2. Complex physical systems ; 3. Complex adaptive systems ; 4. Agents, networks, degree, and recirculation ; 5. Specialization and diversity ; 6. Emergence ; 7. Co-evolution and the formation of niches ; 8. Putting it all together ; Further reading ; Index

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Book SynopsisA need for a deeper understanding of the convergence properties of augmented Lagrangian algorithms and of their relationship to operator-splitting methods such as alternating-methods direction and the development of more efficient algorithms prompted the authors to write this book. The volume is oriented to applications in continuum mechanics. This volume deals with the numerical simulation of the behavior of continuous media by augmented Lagrangian and operator-splitting methods (coupled to finite-element approximations). It begins with a description of the mechanical and mathematical frameworks of the considered applications as well as a general analysis of the basic numerical methods additionally used to study them. These ideas are then applied to specific classes of mechanical problems.

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Book SynopsisChaos exists in systems all around us. Even the simplest system of cause and effect can be subject to chaos, denying us accurate predictions of its behaviour, and sometimes giving rise to astonishing structures of large-scale order. Our growing understanding of Chaos Theory is having fascinating applications in the real world - from technology to global warming, politics, human behaviour, and even gambling on the stock market.Leonard Smith shows that we all have an intuitive understanding of chaotic systems. He uses accessible maths and physics (replacing complex equations with simple examples like pendulums, railway lines, and tossing coins) to explain the theory, and points to numerous examples in philosophy and literature (Edgar Allen Poe, Chang-Tzu, Arthur Conan Doyle) that illuminate the problems. The beauty of fractal patterns and their relation to chaos, as well as the history of chaos, and its uses in the real world and implications for the philosophy of science are all discussed in this Very Short Introduction.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Trade ReviewLeonard Smith's Chaos (part of the Oxford Very Short Introduction series) will give you the clearest (but not too painful idea) of the maths involved... There's a lot packed into this little book, and for such a technical exploration it's surprisingly readble and enjoyable - I really wanted to keep turning the pages. Smith also has some excellent words of wisdom about common misunderstandings of chaos theory... One of the best books so far in this useful and informative series. * popularscience.co.uk *Table of ContentsPreface ; 1. The Emergence of Chaos ; 2. Exponential Growth, Nonlinearity, Common Sense ; 3. Chaos in Context: Determinism Randomness and Noise ; 4. Chaos in Mathematical Models ; 5. Fractals, Strange Attractors, and Dimension(s) ; 6. Quantifying the Dynamics of Uncertainty ; 7. Real numbers, Real Observations and Computers ; 8. Sorry, Wrong Number: Statistics and Chaos ; 9. Predictability: Does Chaos Constrain Our Forecasts? ; 10. Applied Chaos: Can We See Through Our Models? ; 11. Philosophy in Chaos ; Glossary ; Further Reading

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Book SynopsisOffering a comprehensive treatment of adhesive particle flows, this book adopts a particle-level approach oriented toward directly simulating the various fluid, electric field, collision, and adhesion forces and torques acting on the particles, within the framework of a discrete-element model. It is ideal for professionals and graduate students working in engineering and atmospheric and condensed matter physics, materials science, environmental science, and other disciplines where particulate flows have a significant role. The presentation is applicable to a wide range of flow fields, including aerosols, colloids, fluidized beds, and granular flows. It describes both physical models of the various forces and torques on the particles as well as practical aspects necessary for efficient implementation of these models in a computational framework.Table of Contents1. Introduction; 2. Modeling viewpoints and approaches; 3. Contact mechanics without adhesion; 4. Adhesion forces; 5. Fluid forces on particles; 6. Particle dispersion in turbulent flows; 7. Ellipsoidal particles; 8. Particle interactions with electric and magnetic fields; 9. Nanoscale particle dynamics; 10. Computer implementation and data analysis; 11. Applications.

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Book SynopsisNonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.Trade Review'This book has been written by a well-established researcher in the field. His expertise is evidenced by the deft exposition of relatively challenging material. In that regard, one of the very useful functions of this book is its provision of a number of background mathematical techniques in Hamiltonians systems, symplectic geometry, Noether theory and Lagrangian field theory.' K. Alan Shore, Contemporary Physics'The book is clearly written, and only the most basic knowledge of Hamiltonian and Lagrangian theories is required.' Wen-Xiu Ma, MathSciNetTable of Contents1. Introduction; 2. Hamiltonian ODEs and relative equilibria; 3. Modulation of relative equilibria; 4. Revised modulation near a singularity; 5. Introduction to Whitham Modulation Theory – the Lagrangian viewpoint; 6. From Lagrangians to Multisymplectic PDEs; 7. Whitham Modulation Theory – the multisymplectic viewpoint; 8. Phase modulation and the KdV equation; 9. Classical view of KdV in shallow water; 10. Phase modulation of uniform flows and KdV; 11. Generic Whitham Modulation Theory in 2+1; 12. Phase modulation in 2+1 and the KP equation; 13. Shallow water hydrodynamics and KP; 14. Modulation of three-dimensional water waves; 15. Modulation and planforms; 16. Validity of Lagrangian-based modulation equations; 17. Non-conservative PDEs and modulation; 18. Phase modulation – extensions and generalizations; Appendix A. Supporting calculations – 4th and 5th order terms; Appendix B. Derivatives of a family of relative equilibria; Appendix C. Bk and the spectral problem; Appendix D. Reducing dispersive conservation laws to KdV; Appendix E. Advanced topics in multisymplecticity; References; Index.

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Book SynopsisThis book provides research summaries from a number of different focuses in Mathematics, and compiles biographical sketches of top professionals in this important field.

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Book SynopsisThis brief presents a suite of computationally efficient methods for bounding trajectories of dynamical systems with multi-dimensional intervals, or ‘boxes’. It explains the importance of bounding trajectories for evaluating the robustness of systems in the face of parametric uncertainty, and for verification or control synthesis problems with respect to safety and reachability properties. The methods presented make use of: interval analysis; monotonicity theory; contraction theory; and data-driven techniques that sample trajectories. The methods are implemented in an accompanying open-source Toolbox for Interval Reachability Analysis. This brief provides a tutorial description of each method, focusing on the requirements and trade-offs relevant to the user, requiring only basic background on dynamical systems. The second part of the brief describes applications of interval reachability analysis. This makes the brief of interest to a wide range of academic researchers, graduate students, and practising engineers in the field of control and verification. Trade Review“The motivation of this book is to provide to the readers tutorial presentations of several approaches for interval reachability analysis, without requiring any previous knowledge and experience of reachability analysis. Two parts, Part I and Part II, are used for this purpose. Part I describes six main methods for interval reachability analysis and in Part II several applications are presented.” (Takashi Amemiya, Mathematical Reviews, October, 2022)Table of ContentsChapter 1. Introduction.- Part 1: Reachability Methods.- Chapter 2. Interval Analysis.- Chapter 3. Monotonicity.- Chapter 4. Mixed-Monotonicity.- Chapter 5. Sampled-Data Mixed-Monotonicity.- Chapter 6. Growth Bounds.- Chapter 7. Sampling-Based Methods.- Part 2: Applications.- Chapter 8. Safety and Reachability Verification.- Chapter 9. Interval Volume as a Robustness Measure.- Chapter 10. Abstraction-Based Control Synthesis.

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Book SynopsisThe 5th edition of this classic textbook covers the central concepts of practical optimization techniques, with an emphasis on methods that are both state-of-the-art and popular. One major insight is the connection between the purely analytical character of an optimization problem and the behavior of algorithms used to solve that problem. End-of-chapter exercises are provided for all chapters. The material is organized into three separate parts. Part I offers a self-contained introduction to linear programming. The presentation in this part is fairly conventional, covering the main elements of the underlying theory of linear programming, many of the most effective numerical algorithms, and many of its important special applications. Part II, which is independent of Part I, covers the theory of unconstrained optimization, including both derivations of the appropriate optimality conditions and an introduction to basic algorithms. This part of the book explores the general properties of algorithms and defines various notions of convergence. In turn, Part III extends the concepts developed in the second part to constrained optimization problems. Except for a few isolated sections, this part is also independent of Part I. As such, Parts II and III can easily be used without reading Part I and, in fact, the book has been used in this way at many universities. New to this edition are popular topics in data science and machine learning, such as the Markov Decision Process, Farkas’ lemma, convergence speed analysis, duality theories and applications, various first-order methods, stochastic gradient method, mirror-descent method, Frank-Wolf method, ALM/ADMM method, interior trust-region method for non-convex optimization, distributionally robust optimization, online linear programming, semidefinite programming for sensor-network localization, and infeasibility detection for nonlinear optimization.Table of Contents

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Book SynopsisThe 5th edition of this classic textbook covers the central concepts of practical optimization techniques, with an emphasis on methods that are both state-of-the-art and popular. One major insight is the connection between the purely analytical character of an optimization problem and the behavior of algorithms used to solve that problem. End-of-chapter exercises are provided for all chapters. The material is organized into three separate parts. Part I offers a self-contained introduction to linear programming. The presentation in this part is fairly conventional, covering the main elements of the underlying theory of linear programming, many of the most effective numerical algorithms, and many of its important special applications. Part II, which is independent of Part I, covers the theory of unconstrained optimization, including both derivations of the appropriate optimality conditions and an introduction to basic algorithms. This part of the book explores the general properties of algorithms and defines various notions of convergence. In turn, Part III extends the concepts developed in the second part to constrained optimization problems. Except for a few isolated sections, this part is also independent of Part I. As such, Parts II and III can easily be used without reading Part I and, in fact, the book has been used in this way at many universities. New to this edition are popular topics in data science and machine learning, such as the Markov Decision Process, Farkas’ lemma, convergence speed analysis, duality theories and applications, various first-order methods, stochastic gradient method, mirror-descent method, Frank-Wolf method, ALM/ADMM method, interior trust-region method for non-convex optimization, distributionally robust optimization, online linear programming, semidefinite programming for sensor-network localization, and infeasibility detection for nonlinear optimization.Table of Contents

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Book SynopsisThe goal of this book is to provide a mathematical perspective on some key elements of the so-called deep neural networks (DNNs). Much of the interest in deep learning has focused on the implementation of DNN-based algorithms. Our hope is that this compact textbook will offer a complementary point of view that emphasizes the underlying mathematical ideas. We believe that a more foundational perspective will help to answer important questions that have only received empirical answers so far. The material is based on a one-semester course Introduction to Mathematics of Deep Learning" for senior undergraduate mathematics majors and first year graduate students in mathematics. Our goal is to introduce basic concepts from deep learning in a rigorous mathematical fashion, e.g introduce mathematical definitions of deep neural networks (DNNs), loss functions, the backpropagation algorithm, etc. We attempt to identify for each concept the simplest setting that minimizes technicalities but still contains the key mathematics.

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Book SynopsisNonlinear dynamics and chaos involves the study of apparently random happenings within a system or process. The subject has wide applications within mathematics, engineering, physics and other physical sciences.This second edition covers the latest research conducted in this area.Trade Review"... much more extensive than before." (The Mathematical Review, March 2004) "The fully updated second edition provides a self-contained introduction to the theory and applications of nonlinear dynamics and chaos." (International Journal of Environmental Analytical Chemistry, Vol.84, No.14 – 15, 10 – 20 December 2004)Table of ContentsPreface. Preface to the First Edition. Acknowledgements from the First Edition. Introduction PART I: BASIC CONCEPTS OF NONLINEAR DYNAMICS An overview of nonlinear phenomena Point attractors in autonomous systems Limit cycles in autonomous systems Periodic attractors in driven oscillators Chaotic attractors in forced oscillators Stability and bifurcations of equilibria and cycles PART II ITERATED MAPS AS DYNAMICAL SYSTEMS Stability and bifurcation of maps Chaotic behaviour of one-and two-dimensional maps PART III FLOWS, OUTSTRUCTURES AND CHAOS The Geometry of Recurrence The Lorenz system Rosslers band Geometry of bifurcations PART IV APPLICATIONS IN THE PHYSICAL SCIENCES Subharmonic resonances of an offshore structure Chaotic motions of an impacting system Escape from a potential well Appendix. Illustrated Glossary. Bibliography. Online Resource. Index.

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Book SynopsisSince the bestselling first edition was published, there has been a lot of new research conducted in the area of nonlinear dynamics and chaos. This revised edition provides new material, including a glossary and bibliography, as well as a generous supplement of new figures and illustrations.Trade Review"... much more extensive than before." (The Mathematical Review, March 2004) "The fully updated second edition provides a self-contained introduction to the theory and applications of nonlinear dynamics and chaos." (International Journal of Environmental Analytical Chemistry, Vol.84, No.14 – 15, 10 – 20 December 2004)Table of ContentsPreface vi Preface to the First Edition xv Acknowledgements from the First Edition xxi 1 Introduction 1 1.1 Historical background 1 1.2 Chaotic dynamics in Duffing's oscillator 3 1.3 Attractors and bifurcations 8 Part I Basic Concepts of Nonlinear Dynamics 2 An overview of nonlinear phenomena 15 2.1 Undamped, unforced linear oscillator 15 2.2 Undamped, unforced nonlinear oscillator 17 2.3 Damped, unforced linear oscillator 18 2.4 Damped, unforced nonlinear oscillator 20 2.5 Forced linear oscillator 21 2.6 Forced nonlinear oscillator: periodic attractors 22 2.7 Forced nonlinear oscillator: chaotic attractor 24 3 Point attractors in autonomous systems 26 3.1 The linear oscillator 26 3.2 Nonlinear pendulum oscillations 34 3.3 Evolving ecological systems 41 3.4 Competing point attractors 45 3.5 Attractors of a spinning satellite 47 4 Limit cycles in autonomous systems 50 4.1 The single attractor 50 4.2 Limit cycle in a neural system 51 4.3 Bifurcations of a chemical oscillator 55 4.4 Multiple limit cycles in aeroelastic galloping 58 4.5 Topology of two-dimensional phase space 61 5 Periodic attractors in driven oscillators 62 5.1 The Poincare map 62 5.2 Linear resonance 64 5.3 Nonlinear resonance 66 5.4 The smoothed variational equation 71 5.5 Variational equation for subharmonics 72 5.6 Basins ofattraction by mapping techniques 73 5.7 Resonance ofa self-exciting system 76 5.8 The ABC ofnonlinear dynamics 79 6 Chaotic attractors in forced oscillators 80 6.1 Relaxation oscillations and heartbeat 80 6.2 The Birkhoff±Shaw chaotic attractor 82 6.3 Systems with nonlinear restoring force 93 7 Stability and bifurcations of equilibria and cycles 106 7.1 Liapunov stability and structural stability 106 7.2 Centre manifold theorem 109 7.3 Local bifurcations of equilibrium paths 111 7.4 Local bifurcations of cycles 123 7.5 Basin changes at local bifurcations 126 7.6 Prediction ofincipient instability 128 Part II Iterated Maps as Dynamical Systems 8 Stability and bifurcation of maps 135 8.1 Introduction 135 8.2 Stability of one-dimensional maps 138 8.3 Bifurcations of one-dimensional maps 139 8.4 Stability of two-dimensional maps 149 8.5 Bifurcations of two-dimensional maps 156 8.6 Basin changes at local bifurcations of limit cycles 158 9 Chaotic behaviour of one- and two-dimensional maps 161 9.1 General outline 161 9.2 Theory for one-dimensional maps 164 9.3 Bifurcations to chaos 167 9.4 Bifurcation diagram of one-dimensional maps 170 9.5 He non map 174 Part III Flows, Outstructures, and Chaos 10 The geometry of recurrence 183 10.1 Finite-dimensional dynamical systems 183 10.2 Types ofrecurrent behaviour 187 10.3 Hyperbolic stability types for equilibria 195 10.4 Hyperbolic stability types for limit cycles 200 10.5 Implications ofhyperbolic structure 205 11 The Lorenz system 207 11.1 A model ofthermal convection 207 11.2 First convective instability 209 11.3 The chaotic attractor ofLorenz 214 11.4 Geometry ofa transition to chaos 222 1 2 RoÈssler's band 229 12.1 The simply folded band in an autonomous system 229 12.2 Return map and bifurcations 233 12.3 Smale's horseshoe map 238 12.4 Transverse homoclinic trajectories 243 12.5 Spatial chaos and localized buckling 246 13 Geometry of bifurcations 249 13.1 Local bifurcations 249 13.2 Global bifurcations in the phase plane 258 13.3 Bifurcations of chaotic attractors 266 Part IV Applications in the Physical Sciences 14 Subharmonic resonances of an offshore structure 285 14.1 Basic equation and non-dimensional form 286 14.2 Analytical solution for each domain 288 14.3 Digital computer program 289 14.4 Resonance response curves 290 14.5 Effect of damping 294 14.6 Computed phase projections 296 14.7 Multiple solutions and domains ofattraction 298 15 Chaotic motions of an impacting system 302 15.1 Resonance response curve 302 15.2 Application to moored vessels 306 15.3 Period-doubling and chaotic solutions 306 16 Escape from a potential well 313 16.1 Introduction 313 16.2 Analytical formulation 314 16.3 Overview ofthe steady-state response 319 16.4 The two-band chaotic attractor 324 16.5 Resonance ofthe steady states 328 16.6 Transients and basins ofattraction 333 16.7 Homoclinic phenomena 340 16.8 Heteroclinic phenomena 346 16.9 Indeterminate bifurcations 352 Appendix 359 Illustrated Glossary 369 Bibliography 402 Online Resources 428 Index 429

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Book SynopsisThis text covers key mathematical principles and algorithms for nonlinear filters used in image processing. It offers insight into the underlying mathematical and filter design methodologies needed to construct and use nonlinear filters in a variety of applications.Table of ContentsPreface. Logical Image Operators (E. Dougherty & J. Barrera). Computational Gray-Scale Operators (E. Dougherty & J. Barrera). Translation-Invariant Set Operators (E. Dougherty). Granulometric Filters (E. Dougherty & Y. Chen) Easy Recipes for Morphological Filters (H. Heijmans). Introduction to Connected Operators (H. Heijmans). Representation and Optimization of Stack Filters (J. Astola & P. Kuosmanen). Invariant Signals of Median and Stack Filters (J. Astola & P. Kuosmanen). Binary Polynomial Transforms and Logical Correlation (K. Egiazarian, et al.). Applications of Binary Polynomial Transforms (K. Egiazarian, et al.). Random Sets in View of Image Filtering Applications (I. Molchanov). Index.

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Book SynopsisPresents a systematic and unified study of geometric nonlinear functional analysis. This book presents a study of uniformly continuous and Lipschitz functions between Banach spaces, which leads naturally also to the classification of Banach spaces and of their important subsets (mainly spheres) in the uniform and Lipschitz categories.Table of ContentsIntroduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classification of Banach spaces Nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local isometries Perturbations of global isometries Twisted sums Group structure on Banach spaces Appendices Bibliography Index.

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Book SynopsisIntroduces the notion of topological degree and develops its basic properties. This book uses these properties in the discussion of bifurcation theory (the possible branching of solutions as parameters vary), including the proof of Rabinowitz's global bifurcation theorem. It is suitable as a graduate level textbook and a supplementary course text.Trade Reviewextremely stimulating Zentralblatt fur MathematikTable of ContentsTopological approach: Finite dimensions Topological degree in Banach space Bifurcation theory Further topological methods Monotone operators and the min-max theorem Generalized implicit function theorems Bibliography.

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Book SynopsisProvides an introduction to the applications, theory, and algorithms of linear and nonlinear optimization. The emphasis is on practical aspects - discussing modern algorithms, as well as the influence of theory on the interpretation of solutions or on the design of software. The book includes several examples of realistic optimization models that address important applications. The succinct style of this second edition is punctuated with numerous real-life examples and exercises, and the authors include accessible explanations of topics that are not often mentioned in textbooks, such as duality in nonlinear optimization, primal-dual methods for nonlinear optimization, filter methods, and applications such as support-vector machines. The book is designed to be flexible. It has a modular structure, and uses consistent notation and terminology throughout. It can be used in many different ways, in many different courses, and at many different levels of sophistication.

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Book SynopsisThis book addresses modern nonlinear programming concepts and algorithms, especially as they apply to challenging applications in chemical process engineering. It relates the material to real-world problem classes in process optimisation, thus bridging the gap between the mathematical material and the practical uses. Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes shows readers which methods are best suited for specific applications, how large-scale problems should be formulated and what features of these problems should be emphasised, and how existing NLP methods can be extended to exploit specific structures of large-scale optimisation models. The book serves a dual function: it will be useful to chemical engineers who wish to understand and use nonlinear programming; it will also be of interest to experts in mathematical optimisation who want to understand process engineering problems and develop better approaches to solving them.

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Book SynopsisWeb-like waves, often observed on the surface of shallow water, are examples of nonlinear waves. They are generated by nonlinear interactions among several obliquely propagating solitary waves, also known as solitons. In this book, modern mathematical tools—algebraic geometry, algebraic combinatorics, and representation theory, among others—are used to analyze these two-dimensional wave patterns. The author’s primary goal is to explain some details of the classification problem of the soliton solutions of the KP equation (or KP solitons) and their applications to shallow water waves.This book is intended for researchers and graduate students.

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Book SynopsisInverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers—a subset of transmission eigenvalues—using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles.This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.

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Book SynopsisThe book is designed to serve as a textbook for courses offered to upper-undergraduate students enrolled in physics. The first edition of this book was published in 2014. As there is a demand for the next edition, it is quite natural to take note of the several advances that have occurred in the subject over the past five years and to decide which of these are appropriate for inclusion at the textbook level, given the fundamental nature and the significance of the subject area. This is the prime motivation for bringing out a revised second edition. Among the newer mechanisms and materials, the book introduces the super-continuum generation, which arises from an excellent interplay of the various mechanisms of optical nonlinearity. The topics covered in this book are quantum mechanics of nonlinear interaction of matter and radiation, formalism and phenomenology of nonlinear wave mixing processes, optical phase conjugation and applications, self-focusing and self-phase modulation and their role in pulse modification, nonlinear absorption mechanisms, and optical limiting applications, photonic switching and bi-stability, and physical mechanisms leading to a nonlinear response in a variety of materials. This book has emerged from an attempt to address the requirement of presenting the subject at the college level. This textbook includes rigorous features such as the elucidation of relevant basic principles of physics; a clear exposition of the ideas involved at an appropriate level; coverage of the physical mechanisms of non-linearity; updates on physical mechanisms and emerging photonic materials and emphasis on the experimental study of nonlinear interactions. The detailed coverage and pedagogical tools make this an ideal textbook for students and researchers enrolled in physics and related courses.Table of Contents1 From Optics to Photonics 1.1 The Charm and Challenge of Modern Optics 1.2 The Nature of Optical Non-linearity 1.3 Overcoming the Materials Bottleneck 1.4 The Expanding Frontiers 1.5 Problems and Prospects 1.6 Explorations 2 A Phenomenological View of Nonlinear Optics 2.1 Optics in the Nonlinear World 2.1.1 Introduction 2.1.2 First Order Susceptibility 2.1.3 Second Order Susceptibility 2.1.4 Third Order Susceptibility 2.2 Time Domain Response 2.2.1 First Order Polarization- Time Domain Response 2.2.2 Second Order Polarization - Time Domain Response 2.3 Frequency Domain Response 2.3.1 First Order Susceptibility 2.3.2 Second Order Susceptibility 2.3.3 General Order (n) Susceptibility 2.4 The nth order polarization 2.5 Monochromatic Waves 2.6 Calculation of the Factor K 2.6.1 Optical Rectification 2.6.2 Second Harmonic Generation 2.6.3 Pockels Effect 2.6.4 Frequency Mixing : Sum and Difference Frequency generation 2.6.5 Third Harmonic Generation 2.6.6 Nondegenerate Four Wave Mixing 2.7 Explorations 3. Symmetry and Susceptibility 3.1 Introduction 3.2 Crystal Symmetry and Susceptibility Tensors 3.2.1 Neumann Principle 3.2.2 Symmetry of Second Order Susceptibility 3.2.3 Second Harmonic Generation 3.2.4 Kleinmann Symmetry 3.2.5 Symmetry of Third Order Susceptibility 3.3 The Dielectric Permittivity Tensor 3.4 The Refractive Index Ellipsoid 3.5 Explorations 4 Calculation of Non-linear Susceptibilities 4.1 Introduction 4.1.1 Physical Quantities in Quantum Physics 4.1.2 The Projection Operator 4.2 The Equation of Motion 4.3 Ensembles of Particles 4.4 Time-dependent Perturbation 4.5 Dipolar Interaction 4.6 First Order Density Matrix 4.7 Second Order Density Matrix 4.8 Third order Density Matrix 4.9 Double Integrals in the Expressions for the Density Matrix 4.10 Second Harmonic Susceptibility 4.11 Relaxation Effects 4.12 Applications to Color Centers 4.12.1 Third Order Susceptibility 4.12.3 Second Order Susceptibility 4.13. Explorations 5 Nonlinear Wave Mixing Processes 5.1 Introduction 5.2 Elements of Electromagnetism 5.3 Travelling Electromagnetic Waves in Free Space 5.3.1 Energy Density in the Travelling Wave 5.4 Propagation of Electromagnetic Waves in Linear Materials 5.5 Propagation of Electromagnetic Waves in Nonlinear Materials 5.5.1 The Wave Equation 5.5.2 Energy Transfer Rate 5.6 Three Wave Mixing 5.6.1 An Approximation 5.7 Second Harmonic Generation 5.7.1 Phase Matching Schemes 5.7.2 Accurate Treatment of Second Harmonic Generation 5.8 Explorations 6 Optical Phase Conjugation and Bi-stability 6.1 Optical Phase Conjugation 6.1.1 Phase Conjugation as Time reversal 6.1.2 Phase Conjugation through Four-wave-mixing 6.1.3 Practical Realization 6.1.4 The Peculiar Properties of the Phase Conjugate Beam 6.1.5 The Grating Picture 6.1.6 Applications of Phase Conjugation 6.2 Optical Bi-stability and Photonic Switching 6.2.1 Refractive Index at High Intensities : An Overview 6.2.2. Fabry-Perot Etalon 6.2.2 Photonic Switching in a Nonlinear Fabry-Perot Etalon 6.3 Explorations 7 Self Focusing, Phase Modulation and Pulse Shaping 7.1 Self Focusing of Light 7.1.1 The Concept of Self Focusing 7.1.2 Self Trapping and Spatial Solitons 7.1.3 The z-Scan Experiment 7.1.4 Analysis of the z-scan trace 7.1.5 Measurement of Nonlinear Optical Absorption 7.1.6 Mechanisms of Nonlinear Absorption 7.2 Self Phase Modulation 7.3 Pulse Shaping and Optical Soliton Propagation 7.3.1 Solitary Waves and Optical Solitons 7.4 Explorations 8 Materials and Mechanisms 8.1 Introduction 8.2 Mechanisms of Non-linearity 8.2.1 Anharmonicity of Potential 8.2.2 Thermal Mechanism 8.2.3 Orientational Mechanism 8.2.4 Inelastic Photon Scattering 8.2.5 Photorefractivity 8.2.6 Saturable Absorption 8.2.7 Band Gap Distortion (Franz-Keldysh Effect) 8.2.8 Band filling Mechanism 8.2.9 Non-parabolicity of Bands 8.2.10 Delocalization of Electrons 8.3 A Perspective on Newer Materials and Mechanisms 8.3.1 Low Dimensional Materials 8.3.2 Photonic Bandgap Materials 8.3.3 Slowing of light and the effect on non-linearity 8.3.4 Super-continuum Generation 8.4 Explorations 9. Basics of Multi-photon Microscopy 9.1 Introduction 9.2. Techniques for Bio-imaging with High resolution 9.2.1 Fluorescence Microscopy 9.2.2 Confocal Scanning Microscopy 9.3. Multi-photon Microscopy with IR Laser Sources 9.3.1 Principles and Experimental Techniques 9.3.2 Fluorescent Labels in Microscopy 9.4 Use of Multiphoton Absorption in Quantum Dots 9.5 Outlook 9.6 Explorations

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Book Synopsis

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Book Synopsis

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Book SynopsisDas Buch wendet sich an Leser, die - über die rein computergraphische Darstellung hinaus - an einer analytischen Untersuchung von chaotischen und nichtchaotischen Differenzen- und Differentialgleichungssystemen interessiert sind. Breiter Raum wird der Durchrechnung von Beispielen gegeben. Dargestellt werden zunächst qualitative Methoden als auch solche, die das Auffinden von Attraktoren, Bifurkationen etc. und deren Klassifikation in Abhängigkeit von den Systemparametern gestatten. Der letzte Teil schließlich widmet sich der quantitativen Beschreibung chaotischer Systeme. Dazu werden zuerst die Begriffe Chaos und Fraktal exakt definiert und dann die verschiedenen fraktalen Dimensionen, Lyapunov-Exponenten, Entropien etc. eingeführt und durch Beispiele begründet.Table of Contents1 Einleitung.- 2 Diskrete Systeme.- 2.1 Fixpunkte.- 2.2 Lineare und nichtlineare Abbildungen.- 2.3 Abbildungen mit chaotischem Verhalten.- 2.3.1 Die Bernoulli-Abbildung.- 2.3.2 Die logistische Parabel.- 2.3.3 Die Hénon-Abbildung.- 2.4 Die Poincaré-Abbildung.- Anhang A (Verallgemeinerte Eigenvektoren und Jordan-Formen).- Aufgaben.- 3 Kontinuierliche dynamische Systeme.- 3.1 Definitionen, Existenz- und Eindeutigkeitssätze.- 3.2 Eigenschaften der Lösungen von gewöhnlichen Differentialgleichungen.- 3.2.1 Stabilität von Lösungen.- 3.2.2 Asymptotik.- 3.3 Fixpunkte.- 3.3.1 Stabilität von Fixpunkten.- 3.3.2 Struktur von Lösungen in kleinen Umgebungen von Fixpunkten.- 3.3.3 Klassifikation von Fixpunkten.- 3.3.4 Pendelschwingungen.- 3.4 Hamilton-Systeme.- 3.5 Zentrale Mannigfaltigkeiten.- 3.5.1 Parameterabhängige zentrale Mannigfaltigkeiten.- 3.6 Normalformen.- Aufgaben.- 4 Bifurkationen.- 4.1 Äquivalente und konjugierte dynamische Systeme, strukturelle Stabilität.- 4.2 Verzweigungs-Grundtypen.- 4.3 Die Sattel-Knoten-Bifurkation.- 4.4 Die transkritische Verzweigung.- 4.5 Die Pitchfork-Bifurkation.- 4.6 Die Hopf-Bifurkation.- 4.7 Methode der Projektionen.- 4.8 Stabilität periodischer Lösungen.- Anhang A (Fredholm-Alternative).- Anhang B (Hopf-Bifurkationen in kontinuierlichen Systemen).- Aufgaben.- 5 Asymptotische Methoden.- 5.1 Die Mittelwert-Methode.- 5.2 Beispiele.- 5.3 Schwach nichtlineare Oszillatoren.- 5.4 Die Viel variablen-Methode.- Aufgaben.- 6 Homokline Bifurkationen.- 6.1 Die Standardabbildung.- 6.2 Sattelpunkte flächenerhaltender Abbildungen.- 6.3 Elliptische Fixpunkte flächenerhaltender Abbildungen und KAM-Kurven.- 6.4 Winkel- und Wirkungsvariable.- 6.5 Schwach gestörte Hamilton-Systeme.- 6.6 Das Melnikov-Kriterium.- 6.6.1 Homokline Koordinaten.- 6.6.2 Abstand zwischen stabilen und instabilen Mannigfaltigkeiten gestörter Systeme.- 6.6.3 Definition der Melnikov-Funktion.- 6.7 Verallgemeinerungen des Melnikov-Kriteriums.- 6.7.1 Heterokline Bifurkationen.- 6.7.2 Melnikov-Kriterium für eine Klasse von Hamilton-Systemen mit zwei Freiheitsgraden.- 6.8 Das Shilnikov-Phänomen.- Aufgaben.- 7 Bifurkationen mit höherer Ko-Dimension.- 7.1 Verallgemeinerung der Grundtypen von Bifurkationen eindimensionaler Systeme.- 7.1.1 Eindimensionale Systeme mit kubischen Nichtlinearitäten.- 7.1.2 Eindimensionale Systeme mit quartären Nichtlinearitäten.- 7.2 Die Ko-Dimension dynamischer Systeme.- 7.2.1 Eindimensionale Systeme.- 7.2.2 Ebene Systeme.- 7.2.2.1 Zweidimensionale Potential-Systeme.- 7.2.2.2 Allgemeine zweidimensionale Systeme.- 7.3 Dynamik von Bifurkationen mit Ko-Dimension Zwei.- 7.3.1 Ein doppelter Eigenwert.- 7.3.2 Zwei Paare rein imaginärer Eigenwerte.- Anhang A Versale Entfaltung von Matrizen.- Aufgaben.- Quantitative Methoden der Beschreibung nichtlinearer und chaotischer Systeme.- 8.1 Der (Phasen-)Fluß autonomer Vektorfelder.- 8.2 Nicht-autonome dynamische Systeme.- 8.3 Zur Begriffsbildung bei chaotischen Systemen.- 8.4 Der Lyapunov-Exponent.- 8.4.1 Lyapunov-Exponenten für diskrete, eindimensionale Systeme.- 8.4.2 Lyapunov-Exponenten mehrdimensionaler Systeme.- 8.4.3 Numerische Bestimmung der Lyapunov-Exponenten.- 8.4.4 Lyapunov-Exponenten und Attraktorvolumen.- 8.5 Die Autokorrelationsfunktion.- 8.5.1 Die Autokorrelationsfunktion diskreter Systeme.- 8.5.2 Die Autokorrelationsfunktion kontinuierlicher Systeme.- 8.6 Das Leistungsspektrum.- 8.6.1 Das Leistungsspektrum diskreter Systeme.- 8.6.2 Das Leistungsspektrum kontinuierlicher Systeme.- 8.7 Fraktale Strukturen und Dimensionen.- 8.7.1 Selbstähnlichkeit und Selbstaffinität.- 8.7.2 Fraktale, Hausdorff-Dimension.- 8.7.2.1 Zufallsfraktale.- 8.7.2.2 Multi-Fraktale.- 8.7.3 Selbstähnlichkeits-Dimension.- 8.7.4 Box-Dimension.- 8.7.5 Die informationsdimension.- 8.7.6 Korrelationsdimension.- 8.7.7 Lyapunov-Dimension.- 8.7.8 Die Rényi-Dimension.- 8.7.9 Die Kolmogorov-Entropie.- 8.8 Rekonstruktion eines Attraktors aus einer Zeitreihe.- Aufgaben.- Literatur.- Sachwortverzeichnis.

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Book SynopsisThis book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number oTrade Review'… they have been remarkably successful in presenting a clear and concise introductory summary of a large body of work on the statistical properties of stock prices.' Burton Malkiel, Journal of Economic Literature'Clearly and concisely written, this book provides an excellent introduction to the problem of understanding the empirical statistical properties of prices.' Doyne Farmer, Prediction Company, Santa Fe and the Santa Fe Institute'I feel the book is a useful introduction to the empirical aspects of econophysics.' Blake LeBaron, Nature'The authors are leading researchers in the field, and were well-regarded statistical physicists before that … the book seems aimed the other way, at physicists interested in economics, and for them it would make a good introduction to finance. The writing is clear and friendly, the production values high and the guides to further reading excellent. They will find it well worth their time and money.' Cosma Shalizi, Institute of PhysicsTable of ContentsPreface; 1. Introduction; 2. Efficient market hypothesis; 3. Random walk; 4. Lévy stochastic processes and limit theorems; 5. Scales in financial data; 6. Stationarity and time correlation; 7. Time correlation in financial time series; 8. Stochastic models of price dynamics; 9. Scaling and its breakdown; 10. ARCH and GARCH processes; 11. Financial markets and turbulence; 12. Correlation and anti-correlation between stocks; 13. Taxonomy of a stock portfolio; 14. Options in idealized markets; 15. Options in real markets; Appendix A: notation guide; Appendix B: martingales; References; Index.

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Book SynopsisSolitons are waves with exceptional stability properties which appear in many areas of physics. The basic properties of solitons are introduced here using examples from macroscopic physics (e.g. blood pressure pulses and fibre optical communications). The book then presents the main theoretical methods before discussing applications from solid state or atomic physics such as dislocations, excitations in spin chains, conducting polymers, ferroelectrics and BoseâEinstein condensates. Examples are also taken from biological physics and include energy transfer in proteins and DNA fluctuations. Throughout the book the authors emphasise a fresh approach to modelling nonlinearities in physics. Instead of a perturbative approach, nonlinearities are treated intrinsically and the analysis based on the soliton equations introduced in this book. Based on the authors' graduate course, this textbook gives an instructive view of the physics of solitons for students with a basic knowledge of general pTable of ContentsList of Portraits; Preface; Part I. Different Classes of Solitons: Introduction; 1. Nontopological solitons: the Korteweg-de Vries equation; 2. Topological soltitons: sine-Gordon equation; 3. Envelope solitons and nonlinear localisation: the nonlinear Schrödinger equation; 4. The modelling process: ion acoustic waves in a plasma; Part II. Mathematical Methods for the Study of Solitons: Introduction; 5. Linearisation around the soliton solution; 6. Collective coordinate method; 7. The inverse-scattering transform; Part III. Examples in Solid State and Atomic Physics: Introduction; 8. The Ferm–Pasta–Ulam problem; 9. A simple model for dislocations in crystals; 10. Ferroelectric domain walls; 11. Incommensurate phases; 12. Solitons in magnetic systems; 13. Solitons in Conducting polymers; 14. Solitons in Bose–Einstein condensates; Part IV. Nonlinear Excitations in Biological Molecules: Introduction; 15. Energy localisation and transfer in proteins; 16. Nonlinear dynamics and statistical physics of DNA; Conclusion: Physical solitons: do they exist?; Part V. Appendices: A. Derivation of the KdV equation for surface hydrodynamic waves; B. Mechanics of a continuous medium; C. Coherent states of an harmonic oscillator; References; Index.

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Book SynopsisUnderstanding the behaviour of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the fTrade Review'The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians … this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.' Sylvie Pic, zbMATHTable of ContentsPreface; Prologue; Part I. Microhydrodynamics: 1. Basic concepts in viscous flow; 2. One sphere in Stokes flow; 3. Sophisticated techniques; 4. Particle pair interactions; Interlude: from the microscopic to the macroscopic; 5. Statistical and stochastic concepts; Part II. Toward a Description of Macroscopic Phenomena in Suspensions: 6. Sedimentation; 7. Shear flow; 8. Beyond Stokes flow: finite inertia; Epilogue; References; Index.

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