Differential calculus and equations Books

Book SynopsisFilling a gap in the literature, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions reveals important results on ordinary differential equations (ODEs) and partial differential equations (PDEs). It presents very recent results relating to the existence, boundedness, regularity, and asymptotic behavior of global solutions for differential equations and inclusions, with or without delay, subjected to nonlocal implicit initial conditions.After preliminaries on nonlinear evolution equations governed by dissipative operators, the book gives a thorough study of the existence, uniqueness, and asymptotic behavior of global bounded solutions for differential equations with delay and local initial conditions. It then focuses on two important nonlocal cases: autonomous and quasi-autonomous. The authors next discuss sufficient conditions for the existence of almost periodic solutions, describe evolution systems with delay and nonlocal initial cTrade Review"This book will be useful to researchers and graduate students interested in delay evolution equations and inclusions subjected to nonlocal initial conditions." - Sotiris K. Ntouyas (Ioannina)Table of ContentsPreliminaries. Local Initial Conditions. Nonlocal Initial Conditions: The Autonomous Case. Nonlocal Initial Conditions: The Quasi-Autonomous Case. Almost Periodic Solutions. Evolution Systems with Nonlocal Initial Conditions. Delay Evolution Inclusions. Multivalued Reaction-Diffusion Systems. Viability for Nonlocal Evolution Inclusions. Bibliography. Index.

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Book SynopsisIterative processes are the tools used to generate sequences approximating solutions of equations describing real life problems. Intended for researchers in computational sciences and as a reference book for advanced computational method in nonlinear analysis, this book is a collection of the recent results on the convergence analysis of numerical algorithms in both finite-dimensional and infinite-dimensional spaces and presents several applications and connections with fixed point theory. It contains an abundant and updated bibliography and provides comparisons between various investigations made in recent years in the field of computational nonlinear analysis.The book also provides recent advancements in the study of iterative procedures and can be used as a source to obtain the proper method to use in order to solve a problem. The book assumes a basic background in Mathematical Statistics, Linear Algebra and Numerical Analysis and may be used as a self-study referTable of ContentsHalley’s method. Newton’s method for k-Fréchet differentiable operators. Nonlinear Ill-posed quations. Sixth-order iterative methods. Local convergence and basins of attraction of a two-step Newton like method for equations with solutions of multiplicity greater than one. Extending the Kantorovich theory for solving equations. Robust convergence for inexact Newton method. Inexact Gauss-Newton-like method for least square problems. Lavrentiev Regularization Methods for Ill-posed Equations. King-Werner-type methods of order 1+sqrt(2). Generalized equations and Newton’s method. Newton’s method for generalized equations using restricted domains. Secant-like methods. King-Werner-like methods free of derivatives. Müller’s method. Generalized Newton Method with applications. Newton-secant methods with values in a cone. Gauss-Newton method with applications to convex optimization. Directional Newton methods and restricted domains. Gauss-Newton method for convex optimization. Ball Convergence for eighth order method. Expanding Kantorovich’s theorem for solving generalized equations.

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Book SynopsisThis book is designed for a systematic understanding of nuclear diffusion theory along with fuzzy/interval/stochastic uncertainty. This will serve to be a benchmark book for graduate & postgraduate students, teachers, engineers and researchers throughout the globe. In view of the recent developments in nuclear engineering, it is important to study the basic concepts of this field along with the diffusion processes for nuclear reactor design. Also, it is known that uncertainty is a must in every field of engineering and science and, in particular, with regards to nuclear-related problems. As such, one may need to understand the nuclear diffusion principles/theories corresponding with reliable and efficient techniques for the solution of such uncertain problems. Accordingly this book aims to provide a new direction for readers with basic concepts of reactor physics as well as neutron diffusion theory. On the other hand, it also includes uncertainty (in terms ofTable of Contents Basic Reactor Principles Atomic Structure Binding energy Nuclear fusion Nuclear fission Radioactivity Principles, Production, and interaction of neutrons with matter Production of neutrons Neutron reactions and radiation Inelastic and elastic scattering of neutrons Maxwell-Boltzmann distribution Neutron diffusion theory Cross section of neutron reactions Rates of neutron reactions Fission neutrons Prompt neutrons Delayed neutrons Neutron transport and diffusion equation Fundamentals of Uncertainty Probabilistic uncertainty Non-probabilistic uncertainty Interval uncertainty Fuzzy uncertainty Uncertain Neutron diffusion Uncertain factors involved in neutron diffusion theory Modeling of uncertain neutron diffusion equations One group model Analytical methods Numerical methods Finite difference method Finite element method Conclusion Uncertain One Group Model Interval arithmetic and Fuzzy Finite Element Method (FFEM) Formulation of the uncertain stiffness matrices and force vectors Bare square homogeneous reactor Multi group model Uncertain factors involved in multi group neutron diffusion theory Formulation of uncertain multi group neutron diffusion equations Uncertain Multi Group Model Fuzzy finite element for coupled differential equations Fuzzy multi group neutron diffusion equation Case study Results and discussion Conclusion Point Kinetic Diffusion Theory of point kinetic neutron diffusion equation Case study Conclusion Stochastic Point Kinetic Diffusion Stochastic point kinetic model Euler-Maruyama method Example Hybridised uncertainty in point kinetic diffusion Development of stochastic point kinetic model with fuzzy parameters Fuzzy Euler-Maruyama method Case Study Conclusion Index

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Book SynopsisAlthough the analysis of scattering for closed bodies of simple geometric shape is well developed, structures with edges, cavities, or inclusions have seemed, until now, intractable to analytical methods. This two-volume set describes a breakthrough in analytical techniques for accurately determining diffraction from classes of canonical scatterers with comprising edges and other complex cavity features. It is an authoritative account of mathematical developments over the last two decades that provides benchmarks against which solutions obtained by numerical methods can be verified.The first volume, Canonical Structures in Potential Theory, develops the mathematics, solving mixed boundary potential problems for structures with cavities and edges. The second volume, Acoustic and Electromagnetic Diffraction by Canonical Structures, examines the diffraction of acoustic and electromagnetic waves from several classes of open structures with edges or cavities. Together these volumes present an authoritative and unified treatment of potential theory and diffraction-the first complete description quantifying the scattering mechanisms in complex structures.Table of ContentsMathematical Aspects of Potential Theory. Dual or Triple Series and Integral Equations. Electrostatic Potential Theory for Open Spherical Shells and Cavities. Open Spheroidal Conducting Shells and Cavities. Charged Toroidal Shells and Cavities. Potential Theory for Conical Structures with Edges. Two-Dimensional Potential Theory. Rigorous Solution Methods for more Complicated Structures.

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Book SynopsisInverse problems arise in many disciplines and hold great importance to practical applications. However, sound new methods are needed to solve these problems. Over the past few years, Japanese and Korean mathematicians have obtained a number of very interesting and unique results in inverse problems.Inverse Problems and Related Topics compiles papers authored by some of the top researchers in Korea and Japan. It presents a number of original and useful results and offers a unique opportunity to explore the current trends of research in inverse problems in these countries. Highlighting the existence and active work of several Japanese and Korean groups, it also serves as a guide to those seeking future scientific exchange with researchers in these countries.Trade Review"The aim of this book is to fill the gap between high-school mathematics and mathematics taught at university…the reader is shown what it means to prove something rigourously…This book is easy to read for anyone with a high-school mathematics background." - European Mathematical Society NewsletterTable of ContentsA Finite Difference Model for Calderón's Boundary Inverse Problem. Inverse Problems for Equations with Memory. Parameter Estimation of Elastic Media. The Probe Method and its Applications. Recent Progress in the Inverse Conductivity Problem with Single Measurement. A Moment Method on Inverse Problems for the Heat Equation. Some Remarks on Free Boundaries of Recirculation Euler Flows with Constant Vorticity. Algorithms for the Identification of Spatially Varying/Invariant Stiffness and Dampings in Flexible Beams. Numerical Solutions of the Cauchy Problem in Potential and Elastostatics. Inverse Source Problems in the Helmholtz Equations. A Numerical Method for a Magnetostatic Inverse Problem using the Edge Element. Exact Controllability Method and Multidimensional Linear Inverse Problems. Impedance Computed Tomo-Electrocardiography. An Inverse Problem for Free Channel Scattering. Surface Impedance Tensor and Boundary Value Problem. Aysmptotics for the Spectral and Weyl Functions of the Operator-Value Sturm-Liouville Problem. Exact Controllability Method and Multidimensional Linear Inverse Problems

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Book SynopsisThe theory of holomorphic functions of several complex variables emerged from the attempt to generalize the theory in one variable to the multidimensional situation. Research in this area has led to the discovery of many sophisticated facts, structures, ideas, relations, and applications. This deepening of knowledge, however, has also revealed more and more paradoxical differences between the structures of the two theories. The authors of this Research Note were driven by the quest to construct a theory in several complex variables that has the same structure as the one-variable theory. That is, they sought a reproducing kernel for the whole class that is universal and from same class. Integral Theorems for Functions and Differential Forms in Cm documents their success. Their highly original approach allowed them to obtain new results and refine some well-known results from the classical theory of several complex variables. The 'hyperholomorphic" theory they developed proved to be a kind of direct sum of function theories for two Dirac-type operators of Clifford analysis considered in the same domain.In addition to new results and methods, this work presents a first-look at a brand new setting, based upon the natural language of differential forms, for complex analysis. Integral Theorems for Functions and Differential Forms in Cm reveals a deep link between the fields of several complex variables theory and Clifford analysis. It will have a strong influence on researchers in both areas, and undoubtedly will change the general viewpoint on the methods and ideas of several complex variables theory.Trade Review"…the book will be interesting to specialists in complex analysis and its applications".- Mathematical Reviews, 2003a"This well-written book is a valuable contribution to the broad field of interactions between complex analysis and partial differential equations...Moreover, the book can be used for individual studies, because fundamental concepts and important theorems are explained in detail."-Mathematical Reviews, Issue 94aTable of ContentsIntroduction. Differential Forms. Differential Forms with Co-Efficients in 2x2 Matrices. Hyperholomorphic Functions and Differential Forms in Cm. Cauchy's Theorem. Morera's Theorem. Cauchy's Integral Representation. Hyperholomorphic D-problem. Complex Hodge-Dolbeault System. Relations with Clifford Analysis.

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Book SynopsisThis book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs. The authors concentrate on three types of absorbing sets: attractors, exponential attractors, and inertial manifolds. They present the fundamental properties of these sets, and then proceed to show the existence of some of these sets for a number of dynamical systems generated by well-known physical models. In particular, they consider in full detail two particular PDEEs: a semilinear version of the heat equation and a corresponding version of the dissipative wave equation. These examples illustrate the most important features of the theory of semiflows and provide a sort of template that can be applied to the analysis of other models.The material builds in a careful, gradual progression, developing the background needed by newcomers to the field, and culminating in a more detailed presentation of the main topics than found in most sources. The authors' approach to and treatment of the subject builds the foundation for more advanced references and research on global attractors, exponential attractors, and inertial manifolds.Table of ContentsDynamical Processes. Attractors of Semiflows. Attractors for Semilinear Evolution Equations. Exponential Attractors. Inertial Manifolds. Examples. A Non-Existence Result for Inertial Manifolds. Appendix: Selected Results from Analysis. Bibliography. Index. Nomenclature

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Book SynopsisWith contributions from some of the leading authorities in the field, the work in Differential Equations: Inverse and Direct Problems stimulates the preparation of new research results and offers exciting possibilities not only in the future of mathematics but also in physics, engineering, superconductivity in special materials, and other scientific fields.Exploring the hypotheses and numerical approaches that relate to pure and applied mathematics, this collection of research papers and surveys extends the theories and methods of differential equations. The book begins with discussions on Banach spaces, linear and nonlinear theory of semigroups, integrodifferential equations, the physical interpretation of general Wentzell boundary conditions, and unconditional martingale difference (UMD) spaces. It then proceeds to deal with models in superconductivity, hyperbolic partial differential equations (PDEs), blowup of solutions, reaction-diffusion equation with memory, and Navier-Stokes equations. The volume concludes with analyses on Fourier-Laplace multipliers, gradient estimates for Dirichlet parabolic problems, a nonlinear system of PDEs, and the complex Ginzburg-Landau equation.By combining direct and inverse problems into one book, this compilation is a useful reference for those working in the world of pure or applied mathematics.Trade Review"…Almost all of the fourteen contributions contain original results; they do not just survey or explain results already published elsewhere. They cover a wide scope of up-to-date topics from the field of differential equations. … The book will be an interesting and stimulating read for research workers in the field."-EMS Newsletter, June 2007Table of ContentsDegenerate first order identification problems in Banach spaces. A non-isothermal dynamical Ginzburg-Landau model of superconductivity. Some global in time results for integrodifferential parabolic inverse problems. Fourth order ordinary differential operators with general Wentzell boundary conditions. Study of elliptic differential equations in UMD spaces. Degenerate integrodifferential equations of parabolic type. Exponential attractors for semiconductor equations. Convergence to stationary states of solutions to the semilinear equa-tion of viscoelasticity. Asymptotic behavior of a phase field system with dynamic boundary conditions. The power potential and nonexistence of positive solutions. The Model-Problem associated to the Stefan Problem with Surface Tension: an Approach via Fourier-Laplace Multipliers. Identification problems for nonautonomous degenerate integrodifferential equations of parabolic type with Dirichlet boundary conditions. Existence results for a phase transition model based on microscopic movements. Strong L2-wellposedness in the complex Ginzburg-Landau equation.

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Book SynopsisProviding a basic tool for studying nonlinear problems, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications focuses on the principal spectral theory for general time-dependent and random parabolic equations and systems. The text contains many new results and considers existing results from a fresh perspective.Taking a clear, unified, and self-contained approach, the authors first develop the abstract general theory in the framework of weak solutions, before turning to cases of random and nonautonomous equations. They prove that time dependence and randomness do not reduce the principal spectrum and Lyapunov exponents of nonautonomous and random parabolic equations. The book also addresses classical Faber–Krahn inequalities for elliptic and time-periodic problems and extends the linear theory for scalar nonautonomous and random parabolic equations to cooperative systems. The final chapter presents applications to Kolmogorov systems of parabolic equations. By thoroughly explaining the spectral theory for nonautonomous and random linear parabolic equations, this resource reveals the importance of the theory in examining nonlinear problems.Trade Review... a clear and interesting account of the principal spectral theory for general time-dependent and random linear parabolic equations and systems. It contains many new results and puts some already known results in a new framework. ... -Krystyna Twardowksa, Mathematical Reviews, Issue 2010gTable of ContentsIntroduction. Fundamental Properties in the General Setting. Spectral Theory in the General Setting. Spectral Theory in Nonautonomous and Random Cases. Influence of Spatial-Temporal Variations and the Shape of Domain. Cooperative Systems of Parabolic Equations. Applications to Kolmogorov Systems of Parabolic Equations. References. Index.

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Book SynopsisPart II of the Selected Works of Ivan Georgievich Petrowsky, contains his major papers on second order Partial differential equations, systems of ordinary. Differential equations, the theory, of Probability, the theory of functions, and the calculus of variations. Many of the articles contained in this book have Profoundly, influenced the development of modern mathematics. Of exceptional value is the article on the equation of diffusion with growing quantity of the substance. This work has found extensive application in biology, genetics, economics and other branches of natural science. Also of great importance is Petrowsky's work on a Problem which still remains unsolved - that of the number of limit cycles for ordinary differential equations with rational right-hand sides.Table of ContentsForeword, Editor’s Preface, Ivan Georgievich Petrowsky, PETROWSKY’S ARTICLES ON PARTIAL DIFFERENTIAL EQUATIONS, PETROWSKY’S ARTICLES ON ORDINARY DIFFERENTIAL EQUATIONS, PETROWSKY’S ARTICLES ON THE THEORY OF PROBABILITY AND OTHER PROBLEMS OF ANALYSIS, APPENDIX: COMMENTARIES, Index

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Book SynopsisThe monograph is devoted to one of the most important trends in contemporary mathematical physics, the investigation of evolution equations of many-particle systems of statistical mechanics. The book systematizes rigorous results obtained in this field in recent years, and it presents contemporary methods for the investigation of evolution equations of infinite-particle systems. The book is intended for experts in statistical physics, mathematical physics, and probability theory and for students of universities specialized in mathematics and physics.Trade Review"This book may be useful for advanced graduate students and for scientists who are interested in mathematical problems of the statistical mechanics and rare ed gasesow.?"Oleg A. Sinkevich in: Zentralblatt Math 1/2010<

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Book SynopsisThis textbook is for the standard, one-semester, junior-senior course that often goes by the title "Elementary Partial Differential Equations" or "Boundary Value Problems". The audience consists of students in mathematics, engineering, and the sciences. The topics include derivations of some of the standard models of mathematical physics and methods for solving those equations on unbounded and bounded domains, and applications of PDE's to biology. The text differs from other texts in its brevity; yet it provides coverage of the main topics usually studied in the standard course, as well as an introduction to using computer algebra packages to solve and understand partial differential equations.For the 3rd edition the section on numerical methods has been considerably expanded to reflect their central role in PDE's. A treatment of the finite element method has been included and the code for numerical calculations is now written for MATLAB. Nonetheless the brevity of the text has been maintained. To further aid the reader in mastering the material and using the book, the clarity of the exercises has been improved, more routine exercises have been included, and the entire text has been visually reformatted to improve readability.Trade Review“The aim of this book is to provide the reader with basic ideas encountered in partial differential equations. … The mathematical content is highly motivated by physical problems and the emphasis is on motivation, methods, concepts and interpretation rather than formal theory. The textbook is a valuable material for undergraduate science and engineering students.” (Marius Ghergu, zbMATH 1310.35001, 2015)Table of ContentsPreface to the Third Edition.- To the Students.- 1: The Physical Origins of Partial Differential Equations.- 1.1 PDE Models.- 1.2 Conservation Laws.- 1.3 Diffusion.- 1.4 Diffusion and Randomness.- 1.5 Vibrations and Acoustics.- 1.6 Quantum Mechanics*.- 1.7 Heat Conduction in Higher Dimensions.- 1.8 Laplace’s Equation.- 1.9 Classification of PDEs.- 2. Partial Differential Equations on Unbounded Domains.- 2.1 Cauchy Problem for the Heat Equation.- 2.2 Cauchy Problem for the Wave Equation.- 2.3 Well-Posed Problems.- 2.4 Semi-Infinite Domains.- 2.5 Sources and Duhamel’s Principle.- 2.6 Laplace Transforms.- 2.7 Fourier Transforms.- 3. Orthogonal Expansions.- 3.1 The Fourier Method.- 3.2 Orthogonal Expansions.- 3.3 Classical Fourier Series.-4. Partial Differential Equations on Bounded Domains.- 4.1 Overview of Separation of Variables.- 4.2 Sturm–Liouville Problems - 4.3 Generalization and Singular Problems.- 4.4 Laplace's Equation.- 4.5 Cooling of a Sphere.- 4.6 Diffusion inb a Disk.- 4.7 Sources on Bounded Domains.- 4.8 Poisson's Equation*.-5. Applications in the Life Sciences.-5.1 Age-Structured Models.- 5.2 Traveling Waves Fronts.- 5.3 Equilibria and Stability.- References.- Appendix A. Ordinary Differential Equations.- Index.

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Book SynopsisMethod of Variation of Parameters for Dynamic Systems presents a systematic and unified theory of the development of the theory of the method of variation of parameters, its unification with Lyapunov's method and typical applications of these methods. No other attempt has been made to bring all the available literature into one volume. This book is a clear exposition of this important topic in control theory, which is not covered by any other text. Such an exposition finally enables the comparison and contrast of the theory and the applications, thus facilitating further development in this fascinating field.Table of Contents1. Ordinary Differential Equations 2. Integrodifferential Equations 3. Differential Equations with Delay 4. Difference Equations 5. Stochastic Differential Equations 6. Abstract Differential Equations 7. Impulsive Differential Equations Equations 3. Differential Equations with Delay 4. Difference Equations 5. Abstract Differential Equations 6. Impulsive Differential Equations 7. Stochastic Differential Equations

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Book SynopsisThis volume comprises selected papers presented at the Volterra Centennial Symposium and is dedicated to Volterra and the contribution of his work to the study of systems - an important concept in modern engineering. Vito Volterra began his study of integral equations at the end of the nineteenth century and this was a significant development in the theory of integral equations and nonlinear functional analysis. Volterra series are of interest and use in pure and applied mathematics and engineering.Table of Contents1. Retrospective of Vito Volterra and His Influence on Nonlinear Systems Theory 2. Volterra Integral Equations at Wisconsin 3. Stability and Asymptotic Behaviour of Solutions of Equations with Aftereffect 4. Generalized Halay Inequalities for Volterra Functional Differential Equations and Discretized Versions 5. Stochastic Convolutions with Kernels Arising in Volterra Equations 6. An Example of Lp-Regularity for Hyperbolic Integrodifferential Equations 7. The Present Status of UAS for Volterra and Delay Equations 8. Myopic Maps and Volterra Series Approximations 9. State Space Theory for Abstract Volterra Operators

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Book SynopsisThe principles of symmetry and self-symmetry are expressed in fractals, the subject of study in dimension theory. This book introduces an area of research which has recently appeared in the interface between dimension theory and the theory of dynamical systems, focusing on invariant fractals.

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Book SynopsisLinear Quadratic Differential Games is an assessment of the state of the art in its field and modern book on linear-quadratic game theory, one of the most commonly used tools for modelling and analysing strategic decision making problems in economics and management.Table of ContentsPreface. Notation and symbols. 1 Introduction. 1.1 Historical perspective. 1.2 How to use this book. 1.3 Outline of this book. 1.4 Notes and references. 2 Linear algebra. 2.1 Basic concepts in linear algebra. 2.2 Eigenvalues and eigenvectors. 2.3 Complex eigenvalues. 2.4 Cayley–Hamilton theorem. 2.5 Invariant subspaces and Jordan canonical form. 2.6 Semi-definite matrices. 2.7 Algebraic Riccati equations. 2.8 Notes and references. 2.9 Exercises. 2.10 Appendix. 3 Dynamical systems. 3.1 Description of linear dynamical systems. 3.2 Existence–uniqueness results for differential equations. 3.2.1 General case. 3.2.2 Control theoretic extensions. 3.3 Stability theory: general case. 3.4 Stability theory of planar systems. 3.5 Geometric concepts. 3.6 Performance specifications. 3.7 Examples of differential games. 3.8 Information, commitment and strategies. 3.9 Notes and references. 3.10 Exercises. 3.11 Appendix. 4 Optimization techniques. 4.1 Optimization of functions. 4.2 The Euler–Lagrange equation. 4.3 Pontryagin’s maximum principle. 4.4 Dynamic programming principle. 4.5 Solving optimal control problems. 4.6 Notes and references. 4.7 Exercises. 4.8 Appendix. 5 Regular linear quadratic optimal control. 5.1 Problem statement. 5.2 Finite-planning horizon. 5.3 Riccati differential equations. 5.4 Infinite-planning horizon. 5.5 Convergence results. 5.6 Notes and references. 5.7 Exercises. 5.8 Appendix. 6 Cooperative games. 6.1 Pareto solutions. 6.2 Bargaining concepts. 6.3 Nash bargaining solution. 6.4 Numerical solution. 6.5 Notes and references. 6.6 Exercises. 6.7 Appendix. 7 Non-cooperative open-loop information games. 7.1 Introduction. 7.2 Finite-planning horizon. 7.3 Open-loop Nash algebraic Riccati equations. 7.4 Infinite-planning horizon. 7.5 Computational aspects and illustrative examples. 7.6 Convergence results. 7.7 Scalar case. 7.8 Economics examples. 7.8.1 A simple government debt stabilization game. 7.8.2 A game on dynamic duopolistic competition. 7.9 Notes and references. 7.10 Exercises. 7.11 Appendix. 8 Non-cooperative feedback information games. 8.1 Introduction. 8.2 Finite-planning horizon. 8.3 Infinite-planning horizon. 8.4 Two-player scalar case. 8.5 Computational aspects. 8.5.1 Preliminaries. 8.5.2 A scalar numerical algorithm: the two-player case. 8.5.3 The N-player scalar case. 8.6 Convergence results for the two-player scalar case. 8.7 Notes and references. 8.8 Exercises. 8.9 Appendix. 9 Uncertain non-cooperative feedback information games. 9.1 Stochastic approach. 9.2 Deterministic approach: introduction. 9.3 The one-player case. 9.4 The one-player scalar case. 9.5 The two-player case. 9.6 A fishery management game. 9.7 A scalar numerical algorithm. 9.8 Stochastic interpretation. 9.9 Notes and references. 9.10 Exercises. 9.11 Appendix. References. Index.

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Book SynopsisOur understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.Table of ContentsChapter 1/Where PDEs Come From 1.1* What is a Partial Differential Equation? 1 1.2* First-Order Linear Equations 6 1.3* Flows, Vibrations, and Diffusions 10 1.4* Initial and Boundary Conditions 20 1.5 Well-Posed Problems 25 1.6 Types of Second-Order Equations 28 Chapter 2/Waves and Diffusions 2.1* The Wave Equation 33 2.2* Causality and Energy 39 2.3* The Diffusion Equation 42 2.4* Diffusion on the Whole Line 46 2.5* Comparison of Waves and Diffusions 54 Chapter 3/Reflections and Sources 3.1 Diffusion on the Half-Line 57 3.2 Reflections of Waves 61 3.3 Diffusion with a Source 67 3.4 Waves with a Source 71 3.5 Diffusion Revisited 80 Chapter 4/Boundary Problems 4.1* Separation of Variables, The Dirichlet Condition 84 4.2* The Neumann Condition 89 4.3* The Robin Condition 92 Chapter 5/Fourier Series 5.1* The Coefficients 104 5.2* Even, Odd, Periodic, and Complex Functions 113 5.3* Orthogonality and General Fourier Series 118 5.4* Completeness 124 5.5 Completeness and the Gibbs Phenomenon 136 5.6 Inhomogeneous Boundary Conditions 147 Chapter 6/Harmonic Functions 6.1* Laplace’s Equation 152 6.2* Rectangles and Cubes 161 6.3* Poisson’s Formula 165 6.4 Circles, Wedges, and Annuli 172 Chapter 7/Green’s Identities and Green’s Functions 7.1 Green’s First Identity 178 7.2 Green’s Second Identity 185 7.3 Green’s Functions 188 7.4 Half-Space and Sphere 191 Chapter 8/Computation of Solutions 8.1 Opportunities and Dangers 199 8.2 Approximations of Diffusions 203 8.3 Approximations of Waves 211 8.4 Approximations of Laplace’s Equation 218 8.5 Finite Element Method 222 Chapter 9/Waves in Space 9.1 Energy and Causality 228 9.2 The Wave Equation in Space-Time 234 9.3 Rays, Singularities, and Sources 242 9.4 The Diffusion and Schrodinger Equations 248 ¨ 9.5 The Hydrogen Atom 254 Chapter 10/Boundaries in the Plane and in Space 10.1 Fourier’s Method, Revisited 258 10.2 Vibrations of a Drumhead 264 10.3 Solid Vibrations in a Ball 270 10.4 Nodes 278 10.5 Bessel Functions 282 10.6 Legendre Functions 289 10.7 Angular Momentum in Quantum Mechanics 294 Chapter 11/General Eigenvalue Problems 11.1 The Eigenvalues Are Minima of the Potential Energy 299 11.2 Computation of Eigenvalues 304 11.3 Completeness 310 11.4 Symmetric Differential Operators 314 11.5 Completeness and Separation of Variables 318 11.6 Asymptotics of the Eigenvalues 322 Chapter 12/Distributions and Transforms 12.1 Distributions 331 12.2 Green’s Functions, Revisited 338 12.3 Fourier Transforms 343 12.4 Source Functions 349 12.5 Laplace Transform Techniques 353 Chapter 13/PDE Problems from Physics 13.1 Electromagnetism 358 13.2 Fluids and Acoustics 361 13.3 Scattering 366 13.4 Continuous Spectrum 370 13.5 Equations of Elementary Particles 373 Chapter 14/Nonlinear PDEs 14.1 Shock Waves 380 14.2 Solitons 390 14.3 Calculus of Variations 397 14.4 Bifurcation Theory 401 14.5 Water Waves 406 Appendix A.1 Continuous and Differentiable Functions 414 A.2 Infinite Series of Functions 418 A.3 Differentiation and Integration 420 A.4 Differential Equations 423 A.5 The Gamma Function 425 References 427 Answers and Hints to Selected Exercises 431 Index 446

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Book SynopsisThis book introduces the reader to the fundamental principles, definitions, and results of dynamical systems and chaos. Rather than relegating chaos to the last chapter in the book, as is usually the case, this work treats chaos as an integral part of dynamical systems theory.Trade Review"From the preface: 'The purpose of this book is to present the fundamental ideas on discrete dynamical systems and chaos at the level of those undergraduates...who have completed the standard calculus sequence, with the inclusion of functions of several variables and linear algebra.'" (Mathematical Reviews, Issue 2001k)Table of ContentsDiscrete Dynamical Systems. One-Dimensional Dynamical Systems. R¯q, Matrices, and Functions. Discrete Linear Dynamical Systems. Nonlinear Dynamical Systems. Chaotic Behavior. Analysis of Four Dynamical Systems. Appendices. Index.

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Book SynopsisMany ecological phenomena may be modelled using apparently random processes involving space (and possibly time). Such phenomena are classified as spatial in their nature and include all aspects of pollution. This book addresses the problem of modelling spatial effects in ecology and population dynamics using reaction-diffusion models. * Rapidly expanding area of research for biologists and applied mathematicians * Provides a unified and coherent account of methods developed to study spatial ecology via reaction-diffusion models * Provides the reader with the tools needed to construct and interpret models * Offers specific applications of both the models and the methods * Authors have played a dominant role in the field for years Essential reading for graduate students and researchers working with spatial modelling from mathematics, statistics, ecology, geography and biology.Trade Review"…particularly attractive and useful for graduate students and other researchers who are interested in studying applications of reaction-diffusion theory to spatial ecology." (Mathematical Reviews, Issue 2007a) "…I would recommend this book to anyone who wants a well supported journey into the modern theory of partial differential equations and dynamic systems…" (The Mathematical Gazette, March 2005)Table of ContentsPreface. Series Preface. 1 Introduction. 1.1 Introductory Remarks. 1.2 Nonspatial Models for a Single Species. 1.3 Nonspatial Models For Interacting Species. 1.4 Spatial Models: A General Overview. 1.5 Reaction-Diffusion Models. 1.6 Mathematical Background. 2 Linear Growth Models for a Single Species: Averaging Spatial Effects Via Eigenvalues. 2.1 Eigenvalues, Persistence, and Scaling in Simple Models. 2.2 Variational Formulations of Eigenvalues: Accounting for Heterogeneity. 2.3 Effects of Fragmentation and Advection/Taxis in Simple Linear Models. 2.4 Graphical Analysis in One Space Dimension. 2.5 Eigenvalues and Positivity. 2.6 Connections with Other Topics and Models. Appendix. 3 Density Dependent Single-Species Models. 3.1 The Importance of Equilibria in Single Species Models. 3.2 Equilibria and Stability: Sub- and Supersolutions. 3.3 Equilibria and Scaling: One Space Dimension. 3.4 Continuation and Bifurcation of Equilibria. 3.5 Applications and Properties of Single Species Models. 3.6 More General Single Species Models. Appendix. 4 Permanence. 4.1 Introduction. 4.2 Definition of Permanence. 4.3 Techniques for Establishing Permanence. 4.4 Invasibility Implies Coexistence. 4.5 Permanence in Reaction-Diffusion Models for Predation. 4.6 Ecological Permanence and Equilibria. Appendix. 5 Beyond Permanence: More Persistence Theory. 5.1 Introduction. 5.2 Compressivity. 5.3 Practical Persistence. 5.4 Bounding Transient Orbits. 5.5 Persistence in Nonautonomous Systems. 5.6 Conditional Persistence. 5.7 Extinction Results. Appendix. 6 Spatial Heterogeneity in Reaction-Diffusion Models. 6.1 Introduction. 6.2 Spatial Heterogeneity within the Habitat Patch. 6.3 Edge Mediated Effects. 6.4 Estimates and Consequences. Appendix. 7 Nonmonotone Systems. 7.1 Introduction. 7.2 Predator Mediated Coexistence. 7.3 Three Species Competition. 7.4 Three Trophic Level Models. Appendix. References. Index.

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Book SynopsisThe Schwarz function originates in classical complex analysis and potential theory. Here the author presents the advantages favoring a mode of treatment which unites the subject with modern theory of distributions and partial differential equations thus bridging the gap between two-dimensional geometric and multi-dimensional analysts. Examines the Schwarz function and its relationship to recent investigations regarding inverse problems of Newtonian gravitation, free boundaries, Hele-Shaw flows and the propagation of singularities for holomorphic p.d.e.Table of ContentsThe Schwarz Principle of Reflection. The Logarithmic Potential, Balayage, and Quadrature Domains. Examples of ``Quadrature Identities''. Quadrature Domains: Basic Properties, 1. Quadrature Domains: Basic Properties, 2. Schwarzian Reflection, Revisited. Projectors from L? (dOmega) to H? (dOmega). The Friedrichs Operator. Concluding Remarks. Bibliography. Index.

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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 SynopsisNumerical Methods for Ordinary Differential Systems The Initial Value Problem J.D. Lambert Professor of Numerical Analysis University of Dundee Scotland In 1973 the author published a book entitled Computational Methods in Ordinary Differential Equations.Table of ContentsBackground Material. Introduction to Numerical Methods. Linear Multistep Methods. Predictor-Corrector Methods. Runge-Kutta Methods. Stiffness: Linear Stability Theory. Stiffness: Nonlinear Stability Theory. References. Index.

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Book SynopsisThis book systematically studies upwind methods for initial value problems for scalar conservation laws in one- and multidimensions. The mathematical theory of convergence theory and of a priori error estimates is presented in detail for structure (finite difference methods) as well as for unstructured grids (finite volume methods).Table of ContentsInitial Value Problems for Scalar Conservation Laws in 1-D. Initial Value Problems for Scalar Conservation Laws in 2-D. Initial Value Problems for Systems in 1-D. Initial Value Problems for Systems of Conservation Laws in 2-D. Initial Boundary Value Problems for Conservation Laws. Convection-Dominated Problems. List of Figures. References. Index.

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Book SynopsisPresents many of the main developments in the study of real submanifolds in complex space, providing background material for researchers and advanced graduate students. This work addresses topics such as the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds.Table of ContentsPrefaceCh. IHypersurfaces and Generic Submanifolds in C[superscript N]3Ch. IIAbstract and Embedded CR Structures35Ch. IIIVector Fields: Commutators, Orbits, and Homogeneity62Ch. IVCoordinates for Generic Submanifolds94Ch. VRings of Power Series and Polynomial Equations119Ch. VIGeometry of Analytic Discs156Ch. VIIBoundary Values of Holomorphic Functions in Wedges184Ch. VIIIHolomorphic Extension of CR Functions205Ch. IXHolomorphic Extension of Mappings of Hypersurfaces241Ch. XSegre Sets281Ch. XINondegeneracy Conditions for Manifolds315Ch. XIIHolomorphic Mappings of Submanifolds349Ch. XIIIMappings of Real-algebraic Subvarieties379References390Index401

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Book SynopsisServes as a text for mathematics students at the intermediate graduate level. This book aims to acquaint readers with the fundamental classical results of partial differential equations and to guide them into some aspects of the modern theory to the point where they will be equipped to read advanced treatises and research papers.Trade Review"The first edition of Folland's text on PDEs used to be my favorite source for a course on DPEs. The new edition includes many more exercises and offers a new chapter on pseudodifferential operators. ... This text book is a pleasant compromise between the modern abstract theory and the concrete classical examples and applications."--Monatshefte fur MathematikTable of Contents* Local Existence Theory * The Laplace Operator * Layer Potentials * The Heat Operator * The Wave Operator * The L2 Theory of Derivatives * Elliptic Boundary Value Problems * Pseudodifferential Operators

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Book SynopsisWhen John Nash won the Nobel prize in economics in 1994, many people were surprised to learn that he was alive. This book presents Nash's contributions not only to game theory, for which he received the Nobel, but to mathematics - from Riemannian geometry and partial differential equations - in which he commands greater acclaim among academics.Trade Review"If you want to see a sugary Hollywood depiction of John Nash's life, go to the cinema. Afterwards, if you are curious about his insights, pick up a new book that explains his work and reprints his most famous papers. It is just as amazing as his personal story."--Chris Giles, Financial Times "One of the most beautifully designed economics books I have ever seen and at a low price... Why are we so intrigued by the story of John Nash? We are curious to understand a person who proves theorems we are unable to fathom. We imagine the voices from another world he has heard. We ask where he was for 30 years during which he walked among us but wasn't here. We are frightened and we are attracted by this combination of 'crazy' and 'genius', an invitation for visiting the edge of our own minds."--Ariel Rubinstein, The Times Higher Education Supplement "Any mathematician who read A Beautiful Mind ... had to be looking for the appendices--the ones explaining what Nash actually did to earn his formidable reputation within the mathematical community. Well, here they are, in a beautifully produced volume... Kuhn, Nasar, and the other contributors have performed a most welcome service by collaborating to bring together the pieces missing from A Beautiful Mind... The mathematical community is eternally in their debt."--SIAM News "The book is written in a pleasant and informal style, addressed to a large audience."--P.T. Moranu, MathematicaTable of ContentsPREFACE by Harold W. Kuhn vii INTRODUCTION by Sylvia Nasar xi Chapter 1: Press Release--The Royal Swedish Academy of Sciences 1 Chapter 2: Autobiography 5 Photo Essay 13 Editor's introduction to Chapter 3 29 Chapter 3: The Game of Hex by John Milnor 31 Editor's Introduction to Chapter 4 35 Chapter 4: The bargaining problem 37 Editor's Introduction to Chapters 5, 6, and 7 47 Chapter 5: Equilibrium Points in n-Person games 49 Chapter 6: Non-Cooperative Games Facsimile of Ph.D. Thesis 51 Chapter 7: Non-Cooperative Games 85 Chapter 8: Two-Person Coooperative Games 99 Editor's Introduction to Chapter 9 115 Chapter 9: Parallel Control 117 Chapter 10: real Algebraic Manifolds 127 Chapter 11: The Imbedding problem for Riemannian Manifolds 151 Chapter 12: Continuity of Solutions of Parabolic and Elliptic Equations 211 AFTERWORD 241 SOURCES 243

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Book SynopsisExplores developments in the theory of planar quasiconformal mappings with a focus on the interactions with partial differential equations and nonlinear analysis. This book presents a modern approach to the classical theory and features applications across a spectrum of mathematics such as dynamical systems and singular integral operators.Trade Review"The nature of the writing is impressive, and any library owning this volume, and other volumes of he series, will be a rich library indeed. This book can work out well as a text for further study at higher graduate level and beyond. For many a mathematician, it works well as a collection of enjoyable chapters; and most importantly, it can comfortably serve well as a reference resource and study material. They will be grateful to the publishers and the authors, for the volume includes a wealth of interesting and useful information on many important topics in the subject... In short, a scintillating volume, full of detailed and thought-provoking contributions. Readers who bring to this book a reasonably strong background of the topics treated in the volume and an open mind will be well rewarded."--Current Engineering PracticeTable of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. xv*Chapter 1. Introduction, pg. 1*Chapter 2. A Background In Conformal Geometry, pg. 12*Chapter 3. The Foundations Of Quasiconformal Mappings, pg. 48*Chapter 4. Complex Potentials, pg. 92*Chapter 5. The Measurable Riemann Mapping Theorem: The Existence Theory Of Quasiconformal Mappings, pg. 161*Chapter 6. Parameterizing General Linear Elliptic Systems, pg. 195*Chapter 7. The Concept Of Ellipticity, pg. 210*Chapter 8. Solving General Nonlinear First-Order Elliptic Systems, pg. 235*Chapter 9. Nonlinear Riemann Mapping Theorems, pg. 259*Chapter 10. Conformal Deformations And Beltrami Systems, pg. 275*Chapter 11. A Quasilinear Cauchy Problem, pg. 289*Chapter 12. Holomorphic Motions, pg. 293*Chapter 13. Higher Integrability, pg. 316*Chapter 14. Lp-Theory Of Beltrami Operators, pg. 362*Chapter 15. Schauder Estimates For Beltrami Operators, pg. 389*Chapter 16. Applications To Partial Differential Equations, pg. 403*Chapter 17. PDEs Not Of Divergence Type: Pucci'S Conjecture, pg. 472*Chapter 18. Quasiconformal Methods In Impedance Tomography: Calderon's Problem, pg. 490*Chapter 19. Integral Estimates For The Jacobian, pg. 514*Chapter 20. Solving The Beltrami Equation: Degenerate Elliptic Case, pg. 527*Chapter 21. Aspects Of The Calculus Of Variations, pg. 586*Appendix: Elements Of Sobolev Theory And Function Spaces, pg. 624*Basic Notation, pg. 643*Bibliography, pg. 647*Index, pg. 671

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Book SynopsisBased on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. It shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic.Trade Review"The book is very well written... I would definitely recommend it to anybody who wants to learn spectral geometry."--Leonid Friedlander, Mathematical ReviewsTable of ContentsPreface ix 1A review: The Laplacian and the d'Alembertian 1 1.1 The Laplacian 1 1.2 Fundamental solutions of the d'Alembertian 6 2Geodesics and the Hadamard parametrix 16 2.1 Laplace-Beltrami operators 16 2.2 Some elliptic regularity estimates 20 2.3 Geodesics and normal coordinates|a brief review 24 2.4 The Hadamard parametrix 31 3The sharp Weyl formula 39 3.1 Eigenfunction expansions 39 3.2 Sup-norm estimates for eigenfunctions and spectral clusters 48 3.3 Spectral asymptotics: The sharp Weyl formula 53 3.4 Sharpness: Spherical harmonics 55 3.5 Improved results: The torus 58 3.6 Further improvements: Manifolds with nonpositive curvature 65 4Stationary phase and microlocal analysis 71 4.1 The method of stationary phase 71 4.2 Pseudodifferential operators 86 4.3 Propagation of singularities and Egorov's theorem 103 4.4 The Friedrichs quantization 111 5Improved spectral asymptotics and periodic geodesics 120 5.1 Periodic geodesics and trace regularity 120 5.2 Trace estimates 123 5.3 The Duistermaat-Guillemin theorem 132 5.4 Geodesic loops and improved sup-norm estimates 136 6Classical and quantum ergodicity 141 6.1 Classical ergodicity 141 6.2 Quantum ergodicity 153 Appendix 165 A.1 The Fourier transform and the spaces S( n) and S'( n)) 165 A.2 The spaces D'(OMEGA) and E'(OMEGA) 169 A.3 Homogeneous distributions 173 A.4 Pullbacks of distributions 176 A.5 Convolution of distributions 179 Notes 183 Bibliography 185 Index 191 Symbol Glossary 193

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Book Synopsis

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Book SynopsisTable of ContentsPreface xiii Conventions and Notations xv Leitfaden xvii Dramatis Personae xix Introduction and Overview 1 A Differential Field with No Escape 1 Strategy and Main Results 10 Organization 21 The Next Volume 24 Future Challenges 25 A Historical Note on Transseries 26 1 Some Commutative Algebra 29 1.1 The Zariski Topology and Noetherianity 29 1.2 Rings and Modules of Finite Length 36 1.3 Integral Extensions and Integrally Closed Domains 39 1.4 Local Rings 43 1.5 Krull's Principal Ideal Theorem 50 1.6 Regular Local Rings 52 1.7 Modules and Derivations 55 1.8 Differentials 59 1.9 Derivations on Field Extensions 67 2 Valued Abelian Groups 70 2.1 Ordered Sets 70 2.2 Valued Abelian Groups 73 2.3 Valued Vector Spaces 89 2.4 Ordered Abelian Groups 98 3 Valued Fields 110 3.1 Valuations on Fields 110 3.2 Pseudoconvergence in Valued Fields 126 3.3 Henselian Valued Fields 136 3.4 Decomposing Valuations 157 3.5 Valued Ordered Fields 171 3.6 Some Model Theory of Valued Fields 179 3.7 The Newton Tree of a Polynomial over a Valued Field 186 4 Differential Polynomials 199 4.1 Differential Fields and Differential Polynomials 199 4.2 Decompositions of Differential Polynomials 209 4.3 Operations on Differential Polynomials 214 4.4 Valued Differential Fields and Continuity 221 4.5 The Gaussian Valuation 227 4.6 Differential Rings 231 4.7 Differentially Closed Fields 237 5 Linear Differential Polynomials 241 5.1 Linear Differential Operators 241 5.2 Second-Order Linear Differential Operators 258 5.3 Diagonalization of Matrices 264 5.4 Systems of Linear Differential Equations 270 5.5 Differential Modules 276 5.6 Linear Differential Operators in the Presence of a Valuation 285 5.7 Compositional Conjugation 290 5.8 The Riccati Transform 298 5.9 Johnson's Theorem 303 6 Valued Differential Fields 310 6.1 Asymptotic Behavior of vP 311 6.2 Algebraic Extensions 314 6.3 Residue Extensions 316 6.4 The Valuation Induced on the Value Group 320 6.5 Asymptotic Couples 322 6.6 Dominant Part 325 6.7 The Equalizer Theorem 329 6.8 Evaluation at Pseudocauchy Sequences 334 6.9 Constructing Canonical Immediate Extensions 335 7 Differential-Henselian Fields 340 7.1 Preliminaries on Differential-Henselianity 341 7.2 Maximality and Differential-Henselianity 345 7.3 Differential-Hensel Configurations 351 7.4 Maximal Immediate Extensions in the Monotone Case 353 7.5 The Case of Few Constants 356 7.6 Differential-Henselianity in Several Variables 359 8 Differential-Henselian Fields with Many Constants 365 8.1 Angular Components 367 8.2 Equivalence over Substructures 369 8.3 Relative Quantifier Elimination 374 8.4 A Model Companion 377 9 Asymptotic Fields and Asymptotic Couples 378 9.1 Asymptotic Fields and Their Asymptotic Couples 379 9.2 H-Asymptotic Couples 387 9.3 Application to Differential Polynomials 398 9.4 Basic Facts about Asymptotic Fields 402 9.5 Algebraic Extensions of Asymptotic Fields 409 9.6 Immediate Extensions of Asymptotic Fields 413 9.7 Differential Polynomials of Order One 416 9.8 Extending H-Asymptotic Couples 421 9.9 Closed H-Asymptotic Couples 425 10 H-Fields 433 10.1 Pre-Differential-Valued Fields 433 10.2 Adjoining Integrals 439 10.3 The Differential-Valued Hull 443 10.4 Adjoining Exponential Integrals 445 10.5 H-Fields and Pre-H-Fields 451 10.6 Liouville Closed H-Fields 460 10.7 Miscellaneous Facts about Asymptotic Fields 468 11 Eventual Quantities, Immediate Extensions, and Special Cuts 474 11.1 Eventual Behavior 474 11.2 Newton Degree and Newton Multiplicity 482 11.3 Using Newton Multiplicity and Newton Weight 487 11.4 Constructing Immediate Extensions 492 11.5 Special Cuts in H-Asymptotic Fields 499 11.6 The Property of l-Freeness 505 11.7 Behavior of the Function ! 511 11.8 Some Special Definable Sets 519 12 Triangular Automorphisms 532 12.1 Filtered Modules and Algebras 532 12.2 Triangular Linear Maps 541 12.3 The Lie Algebra of an Algebraic Unitriangular Group 545 12.4 Derivations on the Ring of Column-Finite Matrices 548 12.5 Iteration Matrices 552 12.6 Riordan Matrices 563 12.7 Derivations on Polynomial Rings 568 12.8 Application to Differential Polynomials 579 13 The Newton Polynomial 585 13.1 Revisiting the Dominant Part 586 13.2 Elementary Properties of the Newton Polynomial 593 13.3 The Shape of the Newton Polynomial 598 13.4 Realizing Cuts in the Value Group 606 13.5 Eventual Equalizers 610 13.6 Further Consequences of w-Freeness 615 13.7 Further Consequences of l-Freeness 622 13.8 Asymptotic Equations 628 13.9 Some Special H-Fields 635 14 Newtonian Differential Fields 640 14.1 Relation to Differential-Henselianity 641 14.2 Cases of Low Complexity 645 14.3 Solving Quasilinear Equations 651 14.4 Unravelers 657 14.5 Newtonization 665 15 Newtonianity of Directed Unions 671 15.1 Finitely Many Exceptional Values 671 15.2 Integration and the Extension K(x) 672 15.3 Approximating Zeros of Differential Polynomials 673 15.4 Proof of Newtonianity 676 16 Quantifier Elimination 678 16.1 Extensions Controlled by Asymptotic Couples 680 16.2 Model Completeness 685 16.3 LW-Cuts and LW-Fields 688 16.4 Embedding Pre-LW-Fields into w-Free LW-Fields 697 16.5 The Language of LW-Fields 701 16.6 Elimination of Quantifiers with Applications 704 A Transseries 712 B Basic Model Theory 724 B.1 Structures and Their Definable Sets 724 B.2 Languages 729 B.3 Variables and Terms 734 B.4 Formulas 738 B.5 Elementary Equivalence and Elementary Substructures 744 B.6 Models and the Compactness Theorem 749 B.7 Ultraproducts and Proof of the Compactness Theorem 755 B.8 Some Uses of Compactness 759 B.9 Types and Saturated Structures 763 B.10 Model Completeness 767 B.11 Quantifier Elimination 771 B.12 Application to Algebraically Closed and Real Closed Fields 776 B.13 Structures without the Independence Property 782 Bibliography 787 List of Symbols 817 Index 833

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Book SynopsisTrade Review“Innovative, mathematically exact, and very well written. Garoche is a rare resource, and his book will enrich the knowledge of both the computer-science and control-systems communities.”—Eric Feron, Georgia Institute of Technology "This book makes a timely contribution at the crossroads of formal computer science, optimization, and control. It should be of interest to computer scientists and control engineers."—Didier Henrion, LAAS-CNRS Toulouse and Czech Technical University in Prague“A pleasure to read. Garoche’s excellent and timely book presents state-of-the-art methods building on convex optimization to perform static analysis for control systems and software.”—Taylor Johnson, Vanderbilt University

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Book SynopsisTrade Review“This remarkable book studies thermodynamics within the framework of dynamical systems theory. A major contribution by any standard, it is a gem in the tiara of books being written by one of the most prolific, deep-thinking, and insightful researchers working today.”—Frank Lewis, University of Texas, Arlington“Haddad develops an original mathematical framework for thermodynamics deeply rooted in modern systems theory, threading postulates and analyses of a science that has evolved from the seemingly mundane quest for efficiency in steam engines to the flow of time and the workings of the cosmos and life itself. He succeeds in presenting an all-encompassing treatise, from the early works of Carnot and Clausius to the insights of relativity and the conundrum of the time arrow, in a lucid exposition that systematically details a rigorous base for future generations of scientists and theorists.”—Tryphon Georgiou, University of California, Irvine"By applying ideas and techniques from compartmental systems theory, Haddad’s treatise places thermodynamics on a solid foundation for the twenty-first century."—Dennis Bernstein, University of Michigan"This effective blend of thermodynamics and the theory of dynamical systems provides a unified, coherent, and mathematically accurate framework that is currently missing in the literature. This is a significant contribution to several fields spanning dynamical systems, mathematics, physics, chemistry, and more. It will provide the underlying foundation for additional research and conceptual understanding of physical phenomena."—Kyriakos G. Vamvoudakis, Georgia Institute of Technology

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Book SynopsisTrade Review"This book . . . . is a major contribution to the state of the art in MFGs which is a must read for researchers in the field. . . . . The authors use the book format (and not a more compact paper format) to explain all their steps carefully. Because of its structured approach, it could be used as a textbook for an advanced course on the subject."---Adhemar Bultheel, European Mathematical Society

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Book SynopsisThis book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population.Trade Review"This book . . . . is a major contribution to the state of the art in MFGs which is a must read for researchers in the field. . . . . The authors use the book format (and not a more compact paper format) to explain all their steps carefully. Because of its structured approach, it could be used as a textbook for an advanced course on the subject."---Adhemar Bultheel, European Mathematical Society

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Book SynopsisMathematical Notes, 29 Originally published in 1983. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperTable of Contents*FrontMatter, pg. i*Table of Contents, pg. 1*Introduction, pg. 3*Chapter 0. Preliminaries, pg. 8*Chapter 1. The Main Theorem, pg. 11*Chapter 2. Geometric Spectral Analysis, pg. 32*Chapter 3. SeU-Adjointness, pg. 41*Chapter 4. L2 Exponenttal Decay Applications to eigenfunctions of N-body Schrodmger Operators, pg. 52*Chapter 5. Pointwise Exponential Bounds, pg. 83*Appendix 1. Approximallon of Metrics and Completeness, pg. 99*Appendix 2. Proof of Lemma 1.2, pg. 102*Appenduc 3. Proof of Lemma 2.2, pg. 104*Appendix 4. Proof of Lemma 5.7, pg. 110*Bibliographical Comments, pg. 112*References, pg. 115

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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 SynopsisOrdinary Differential Equations covers the fundamentals of the theory of ordinary differential equations (ODEs), including an extensive discussion of the integration of differential inequalities, on which this theory relies heavily. In addition to these results, the text illustrates techniques involving simple topological arguments, fixed point theorems, and basic facts of functional analysis. Unlike many texts, which supply only the standard simplified theorems, Ordinary Differential Equations presents the basic theory of ODEs in a general way, making it a valuable reference. This SIAM reissue of the 1982 second edition covers invariant manifolds, perturbations, and dichotomies, making the text relevant to current studies of geometrical theory of differential equations and dynamical systems.

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Book SynopsisDifferent linear mathematical methods can be used to solve these models analytically, such as the Inverse Scattering Transformation (IST), Adomian Decomposition Method, Variational Iteration Method (VIM), Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM).Table of ContentsNonlinear Water Waves and Nonlinear Evolution Equations with ApplicationsInverse Scattering Transform and the Theory of SolitonsKorteweg-de Vries Equation (KdV), Different Analytical Methods for Solving theKorteweg-de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of theSemi-analytical Methods for Solving the KdV and mKdV EquationsKorteweg-de Vries Equation (KdV), Some Numerical Methods for Solving theNonlinear Internal WavesPartial Differential Equations that Lead to SolitonsShallow Water Waves and Solitary WavesSoliton PerturbationSolitons and CompactonsSolitons: Historical and Physical IntroductionSolitons InteractionsSolitons, Introduction toSolitons, Tsunamis and Oceanographical Applications ofWater Waves and the Korteweg-de Vries EquationSoliton Solutions for Some Nonlinear Water Wave Dynamical ModelsAnalytical Soliton Solutions for Some Nonlinear Dynamical Water Waves ModelsSoliton Propagation in Solids: Advances and ApplicationsApplications of lump and interaction soliton solutions to the model of liquid crystals and nerve fibersPeriodic cross-kink, rogue-waves, and lump interaction soliton solutions with kink and periodic waves for fractional Bogoyavlenskii equationDouble Tchebyshev spectral tau algorithm for solving KdV equation, with soliton application

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Book SynopsisPresents a self-contained introduction to continuum mechanics that illustrates how many of the important partial differential equations of applied mathematics arise from continuum modeling principles Written as an accessible introduction, Continuum Mechanics: The Birthplace of Mathematical Models provides a comprehensive foundation for mathematical models used in fluid mechanics, solid mechanics, and heat transfer. The book features derivations of commonly used differential equations based on the fundamental continuum mechanical concepts encountered in various fields, such as engineering, physics, and geophysics. The book begins with geometric, algebraic, and analytical foundations before introducing topics in kinematics. The book then addresses balance laws, constitutive relations, and constitutive theory. Finally, the book presents an approach to multiconstituent continua based on mixture theory to illustrate how phenomena, such as diffusion and porous-Table of ContentsPreface v 1 Geometric Setting 1 1.1 Vectors and Euclidean Point Space 2 1.1.1 Vectors 2 1.1.2 Euclidean Point Space 6 1.1.3 Summary 8 1.2 Tensors 8 1.2.1 First-Order Tensors and Vectors 8 1.2.2 Second-Order Tensors 11 1.2.3 Cross Products, Triple Products, and Determinants 15 1.2.4 Orthogonal Tensors 20 1.2.5 Invariants of a Tensor 21 1.2.6 Derivatives of Tensor-Valued Functions 24 1.2.7 Summary 27 2 Kinematics I: The Calculus of Motion 29 2.1 Bodies, Motions, and Deformations 29 2.1.1 Deformation 32 2.1.2 Examples of Motions 33 2.1.3 Summary 36 2.2 Derivatives of Motion 36 2.2.1 Time Derivatives 37 2.2.2 Derivatives with Respect to Position 38 2.2.3 The Deformation Gradient 40 2.2.4 Summary 42 2.3 Pathlines, Streamlines, and Streaklines 43 2.3.1 Three Types of Arc 43 2.3.2 An Example 45 2.3.3 Summary 49 2.4 Integrals Under Motion 49 2.4.1 Arc, Surface, and Volume Integrals 49 2.4.2 Reynolds Transport Theorem 55 2.4.3 Summary 57 3 Kinematics II: Strain and its Rates 59 3.1 Strain 59 3.1.1 Symmetric Tensors 60 3.1.2 Polar Decomposition and the Deformation Gradient 64 3.1.3 Examples 66 3.1.4 Cauchy–Green and Strain Tensors 68 3.1.5 Strain Invariants 70 3.1.6 Summary 71 3.2 Infinitesimal Strain 72 3.2.1 The Infinitesimal Strain Tensor 72 3.2.2 Summary 75 3.3 Strain Rates 75 3.3.1 Stretching and Spin Tensors 76 3.3.2 Skew Tensors, Spin, and Vorticity 79 3.3.3 Summary 84 3.4 Vorticity and Circulation 84 3.4.1 Circulation 84 3.4.2 Summary 88 3.5 Observer Transformations 89 3.5.1 Changes in Frame of Reference 89 3.5.2 Summary 95 4 Balance Laws 97 4.1 Mass Balance 98 4.1.1 Local Forms of Mass Balance 99 4.1.2 Summary 102 4.2 Momentum Balance 102 4.2.1 Analysis of Stress 104 4.2.2 Inertial Frames of Reference 110 4.2.3 Momentum Balance in Referential Coordinates 113 4.2.4 Summary 114 4.3 Angular Momentum Balance 115 4.3.1 Symmetry of the Stress Tensor 117 4.3.2 Summary 118 4.4 Energy Balance 119 4.4.1 Thermal Energy Balance 122 4.4.2 Summary 124 4.5 Entropy Inequality 124 4.5.1 Motivation 125 4.5.2 Clausius–Duhem Inequality 126 4.5.3 Summary 127 4.6 Jump Conditions 127 4.6.1 Singular Surfaces 129 4.6.2 Localization 132 4.6.3 Summary 135 5 Constitutive Relations: Examples of Mathematical Models 137 5.1 Heat Transfer 138 5.1.1 Properties of the Heat Equation 140 5.1.2 Summary 142 5.2 Potential Theory 143 5.2.1 Motivation 143 5.2.2 Boundary Conditions 144 5.2.3 Uniqueness of Solutions to the Poisson Equation 146 5.2.4 Maximum Principle 147 5.2.5 Mean Value Property 150 5.2.6 Summary 151 5.3 Fluid Mechanics 152 5.3.1 Ideal Fluids 152 5.3.2 An Ideal Fluid in a Rotating Frame of Reference 154 5.3.3 Acoustics 155 5.3.4 Incompressible Newtonian Fluids 158 5.3.5 Stokes Flow 159 5.3.6 Summary 163 5.4 Solid Mechanics 164 5.4.1 Static Displacements 164 5.4.2 Elastic Waves 167 5.4.3 Summary 170 6 Constitutive Theory 173 6.1 Conceptual Setting 174 6.1.1 The Need to Close the System 174 6.1.2 Summary 176 6.2 Determinism and Equipresence 177 6.2.1 Determinism 177 6.2.2 Equipresence 177 6.2.3 Summary 178 6.3 Objectivity 179 6.3.1 Reducing Functional Dependencies 180 6.3.2 Summary 182 6.4 SYMMETRY 183 6.4.1 Changes in Reference Configuration 183 6.4.2 Symmetry Groups 186 6.4.3 Classification of Materials 189 6.4.4 Implications for Thermoviscous Fluids 193 6.4.5 Summary 193 6.5 Admissibility 194 6.5.1 Implications of the Entropy Inequality 195 6.5.2 Analysis of Equilibrium 197 6.5.3 Linear, Isotropic, Thermoelastic Solids 199 6.5.4 Summary 202 7 Multiconstituent Continua 203 7.1 Constituents 204 7.1.1 Configurations and Motions 204 7.1.2 Volume Fractions and Densities 206 7.1.3 Summary 208 7.2 Multiconstituent Balance Laws 209 7.2.1 Multiconstituent Mass Balance 210 7.2.2 Multiconstituent Momentum Balance 212 7.2.3 Multiconstituent Angular Momentum Balance 214 7.2.4 Multiconstituent Energy Balance 215 7.2.5 Multiconstituent Entropy Inequality 216 7.2.6 Isothermal, Nonreacting Multiphase Mixtures 217 7.2.7 Summary 219 7.3 Fluid Flow in a Porous Solid 220 7.3.1 Modeling Assumptions for Porous Media 221 7.3.2 Balance Laws for the Fluid and Solid Phases 223 7.3.3 Equilibrium Constraints 225 7.3.4 Linear Extensions From Equilibrium 226 7.3.5 Commentary 228 7.3.6 Potential Formulation of Darcy’s Law 229 7.3.7 Summary 233 7.4 Diffusion in a Binary Fluid Mixture 234 7.4.1 Modeling Assumptions for Binary Diffusion 235 7.4.2 Balance Laws for the Two Species 235 7.4.3 Constitutive Relationships for Diffusion 236 7.4.4 Modeling Solute Transport 239 7.4.5 Summary 242 A Guide to Notation 243 A.1 General Conventions 243 A.2 Letters Reserved for Dedicated Uses 244 A.3 Special Symbols 245 B Vector Integral Theorems 247 B.1 Stokes’s Theorem 248 B.2 The Divergence Theorem 249 B.3 The Change-of-variables Theorem 252 C Hints and Solutions to Exercises 253 References 265 Index 269

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Book SynopsisA new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world's leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics. In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numTable of ContentsForeword xiii Preface to the first edition xv Preface to the second edition xix Preface to the third edition xxi 1 Differential and Difference Equations 1 10 Differential Equation Problems 1 100 Introduction to differential equations 1 101 The Kepler problem 4 102 A problem arising from the method of lines 7 103 The simple pendulum 11 104 A chemical kinetics problem 14 105 The Van der Pol equation and limit cycles 16 106 The Lotka–Volterra problem and periodic orbits 18 107 The Euler equations of rigid body rotation 20 11 Differential Equation Theory 22 110 Existence and uniqueness of solutions 22 111 Linear systems of differential equations 24 112 Stiff differential equations 26 12 Further Evolutionary Problems 28 120 Many-body gravitational problems 28 121 Delay problems and discontinuous solutions 30 122 Problems evolving on a sphere 33 123 Further Hamiltonian problems 35 124 Further differential-algebraic problems 36 13 Difference Equation Problems 38 130 Introduction to difference equations 38 131 A linear problem 39 132 The Fibonacci difference equation 40 133 Three quadratic problems 40 134 Iterative solutions of a polynomial equation 41 135 The arithmetic-geometric mean 43 14 Difference Equation Theory 44 140 Linear difference equations 44 141 Constant coefficients 45 142 Powers of matrices 46 15 Location of Polynomial Zeros 50 150 Introduction 50 151 Left half-plane results 50 152 Unit disc results 52 Concluding remarks 53 2 Numerical Differential Equation Methods 55 20 The Euler Method 55 200 Introduction to the Euler method 55 201 Some numerical experiments 58 202 Calculations with stepsize control 61 203 Calculations with mildly stiff problems 65 204 Calculations with the implicit Euler method 68 21 Analysis of the Euler Method 70 210 Formulation of the Euler method 70 211 Local truncation error 71 212 Global truncation error 72 213 Convergence of the Euler method 73 214 Order of convergence 74 215 Asymptotic error formula 78 216 Stability characteristics 79 217 Local truncation error estimation 84 218 Rounding error 85 22 Generalizations of the Euler Method 90 220 Introduction 90 221 More computations in a step 90 222 Greater dependence on previous values 92 223 Use of higher derivatives 92 224 Multistep–multistage–multiderivative methods 94 225 Implicit methods 95 226 Local error estimates 96 23 Runge–Kutta Methods 97 230 Historical introduction 97 231 Second order methods 98 232 The coefficient tableau 98 233 Third order methods 99 234 Introduction to order conditions 100 235 Fourth order methods 101 236 Higher orders 103 237 Implicit Runge–Kutta methods 103 238 Stability characteristics 104 239 Numerical examples 108 24 Linear MultistepMethods 111 240 Historical introduction 111 241 Adams methods 111 242 General form of linear multistep methods 113 243 Consistency, stability and convergence 113 244 Predictor–corrector Adams methods 115 245 The Milne device 117 246 Starting methods 118 247 Numerical examples 119 25 Taylor Series Methods 120 250 Introduction to Taylor series methods 120 251 Manipulation of power series 121 252 An example of a Taylor series solution 122 253 Other methods using higher derivatives 123 254 The use of f derivatives 126 255 Further numerical examples 126 26 MultivalueMulitistage Methods 128 260 Historical introduction 128 261 Pseudo Runge–Kutta methods 128 262 Two-step Runge–Kutta methods 129 263 Generalized linear multistep methods 130 264 General linear methods 131 265 Numerical examples 133 27 Introduction to Implementation 135 270 Choice of method 135 271 Variable stepsize 136 272 Interpolation 138 273 Experiments with the Kepler problem 138 274 Experiments with a discontinuous problem 139 Concluding remarks 142 3 Runge–KuttaMethods 143 30 Preliminaries 143 300 Trees and rooted trees 143 301 Trees, forests and notations for trees 146 302 Centrality and centres 147 303 Enumeration of trees and unrooted trees 150 304 Functions on trees 153 305 Some combinatorial questions 155 306 Labelled trees and directed graphs 156 307 Differentiation 159 308 Taylor’s theorem 161 31 Order Conditions 163 310 Elementary differentials 163 311 The Taylor expansion of the exact solution 166 312 Elementary weights 168 313 The Taylor expansion of the approximate solution 171 314 Independence of the elementary differentials 174 315 Conditions for order 174 316 Order conditions for scalar problems 175 317 Independence of elementary weights 178 318 Local truncation error 180 319 Global truncation error 181 32 Low Order ExplicitMethods 185 320 Methods of orders less than 4 185 321 Simplifying assumptions 186 322 Methods of order 4 189 323 New methods from old 195 324 Order barriers 200 325 Methods of order 5 204 326 Methods of order 6 206 327 Methods of order greater than 6 209 33 Runge–Kutta Methods with Error Estimates 211 330 Introduction 211 331 Richardson error estimates 211 332 Methods with built-in estimates 214 333 A class of error-estimating methods 215 334 The methods of Fehlberg 221 335 The methods of Verner 223 336 The methods of Dormand and Prince 223 34 Implicit Runge–Kutta Methods 226 340 Introduction 226 341 Solvability of implicit equations 227 342 Methods based on Gaussian quadrature 228 343 Reflected methods 233 344 Methods based on Radau and Lobatto quadrature 236 35 Stability of Implicit Runge–Kutta Methods 243 350 A-stability, A(α)-stability and L-stability 243 351 Criteria for A-stability 244 352 Padé approximations to the exponential function 245 353 A-stability of Gauss and related methods 252 354 Order stars 253 355 Order arrows and the Ehle barrier 256 356 AN-stability 259 357 Non-linear stability 262 358 BN-stability of collocation methods 265 359 The V and W transformations 267 36 Implementable Implicit Runge–Kutta Methods 272 360 Implementation of implicit Runge–Kutta methods 272 361 Diagonally implicit Runge–Kutta methods 273 362 The importance of high stage order 274 363 Singly implicit methods 278 364 Generalizations of singly implicit methods 283 365 Effective order and DESIRE methods 285 37 Implementation Issues 288 370 Introduction 288 371 Optimal sequences 288 372 Acceptance and rejection of steps 290 373 Error per step versus error per unit step 291 374 Control-theoretic considerations 292 375 Solving the implicit equations 293 38 Algebraic Properties of Runge–Kutta Methods 296 380 Motivation 296 381 Equivalence classes of Runge–Kutta methods 297 382 The group of Runge–Kutta tableaux 299 383 The Runge–Kutta group 302 384 A homomorphism between two groups 308 385 A generalization of G1 309 386 Some special elements of G 311 387 Some subgroups and quotient groups 314 388 An algebraic interpretation of effective order 316 39 Symplectic Runge–Kutta Methods 323 390 Maintaining quadratic invariants 323 391 Hamiltonian mechanics and symplectic maps 324 392 Applications to variational problems 325 393 Examples of symplectic methods 326 394 Order conditions 327 395 Experiments with symplectic methods 328 4 Linear Multistep Methods 333 40 Preliminaries 333 400 Fundamentals 333 401 Starting methods 334 402 Convergence 335 403 Stability 336 404 Consistency 336 405 Necessity of conditions for convergence 338 406 Sufficiency of conditions for convergence 339 41 The Order of Linear Multistep Methods 344 410 Criteria for order 344 411 Derivation of methods 346 412 Backward difference methods 347 42 Errors and Error Growth 348 420 Introduction 348 421 Further remarks on error growth 350 422 The underlying one-step method 352 423 Weakly stable methods 354 424 Variable stepsize 355 43 Stability Characteristics 357 430 Introduction 357 431 Stability regions 359 432 Examples of the boundary locus method 360 433 An example of the Schur criterion 363 434 Stability of predictor–corrector methods 364 44 Order and Stability Barriers 367 440 Survey of barrier results 367 441 Maximum order for a convergent k-step method 368 442 Order stars for linear multistep methods 371 443 Order arrows for linear multistep methods 373 45 One-leg Methods and G-stability 375 450 The one-leg counterpart to a linear multistep method 375 451 The concept of G-stability 376 452 Transformations relating one-leg and linear multistep methods 379 453 Effective order interpretation 380 454 Concluding remarks on G-stability 380 46 Implementation Issues 381 460 Survey of implementation considerations 381 461 Representation of data 382 462 Variable stepsize for Nordsieck methods 385 463 Local error estimation 386 Concluding remarks 387 5 General Linear Methods 389 50 RepresentingMethods in General Linear Form 389 500 Multivalue–multistage methods 389 501 Transformations of methods 391 502 Runge–Kutta methods as general linear methods 392 503 Linear multistep methods as general linear methods 393 504 Some known unconventional methods 396 505 Some recently discovered general linear methods 398 51 Consistency, Stability and Convergence 400 510 Definitions of consistency and stability 400 511 Covariance of methods 401 512 Definition of convergence 403 513 The necessity of stability 404 514 The necessity of consistency 404 515 Stability and consistency imply convergence 406 52 The Stability of General Linear Methods 412 520 Introduction 412 521 Methods with maximal stability order 413 522 Outline proof of the Butcher–Chipman conjecture 417 523 Non-linear stability 419 524 Reducible linear multistep methods and G-stability 422 53 The Order of General Linear Methods 423 530 Possible definitions of order 423 531 Local and global truncation errors 425 532 Algebraic analysis of order 426 533 An example of the algebraic approach to order 428 534 The underlying one-step method 429 54 Methods with Runge–Kutta stability 431 540 Design criteria for general linear methods 431 541 The types of DIMSIM methods 432 542 Runge–Kutta stability 435 543 Almost Runge–Kutta methods 438 544 Third order, three-stage ARK methods 441 545 Fourth order, four-stage ARK methods 443 546 A fifth order, five-stage method 446 547 ARK methods for stiff problems 446 55 Methods with Inherent Runge–Kutta Stability 448 550 Doubly companion matrices 448 551 Inherent Runge–Kutta stability 450 552 Conditions for zero spectral radius 452 553 Derivation of methods with IRK stability 454 554 Methods with property F 457 555 Some non-stiff methods 458 556 Some stiff methods 459 557 Scale and modify for stability 460 558 Scale and modify for error estimation 462 56 G-symplectic methods 464 560 Introduction 464 561 The control of parasitism 467 562 Order conditions 471 563 Two fourth order methods 474 564 Starters and finishers for sample methods 476 565 Simulations 480 566 Cohesiveness 481 567 The role of symmetry 487 568 Efficient starting 492 Concluding remarks 497 References 499 Index 509

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Book SynopsisGENERALIZED ORDINARY DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES AND APPLICATIONS Explore a unified view of differential equations through the use of the generalized ODE from leading academics in mathematics Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations, including Measure Functional Differential Equations (measure FDEs). It presents a uniform collection of qualitative results of Generalized ODEs and offers readers an introduction to several theories, including ordinary differential equations, impulsive differential equations, functional differential equations, dynamical equations on time scales, and more. Throughout the book, the focus is on qualitative theory and on corresponding results for other types of differential equations, as well as the connection between Generalized OrdinarTable of ContentsList of Contributors xi Foreword xiii Preface xvii 1 Preliminaries 1Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, and Eduard Toon 1.1 Regulated Functions 2 1.1.1 Basic Properties 2 1.1.2 Equiregulated Sets 7 1.1.3 Uniform Convergence 9 1.1.4 Relatively Compact Sets 11 1.2 Functions of Bounded B-Variation 14 1.3 Kurzweil and Henstock Vector Integrals 19 1.3.1 Definitions 20 1.3.2 Basic Properties 25 1.3.3 Integration by Parts and Substitution Formulas 29 1.3.4 The Fundamental Theorem of Calculus 36 1.3.5 A Convergence Theorem 44 Appendix 1.A: The McShane Integral 44 2 The Kurzweil Integral 53Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Jaqueline G. Mesquita 2.1 The Main Background 54 2.1.1 Definition and Compatibility 54 2.1.2 Special Integrals 56 2.2 Basic Properties 57 2.3 Notes on Kapitza Pendulum 67 3 Measure Functional Differential Equations 71Everaldo M. Bonotto, Márcia Federson, Miguel V. S. Frasson, Rogelio Grau, and Jaqueline G. Mesquita 3.1 Measure FDEs 74 3.2 Impulsive Measure FDEs 76 3.3 Functional Dynamic Equations on Time Scales 86 3.3.1 Fundamentals of Time Scales 87 3.3.2 The Perron Δ-integral 89 3.3.3 Perron Δ-integrals and Perron–Stieltjes integrals 90 3.3.4 MDEs and Dynamic Equations on Time Scales 98 3.3.5 Relations with Measure FDEs 99 3.3.6 Impulsive Functional Dynamic Equations on Time Scales 104 3.4 Averaging Methods 106 3.4.1 Periodic Averaging 107 3.4.2 Nonperiodic Averaging 118 3.5 Continuous Dependence on Time Scales 135 4 Generalized Ordinary Differential Equations 145Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita 4.1 Fundamental Properties 146 4.2 Relations with Measure Differential Equations 153 4.3 Relations with Measure FDEs 160 5 Basic Properties of Solutions 173Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 5.1 Local Existence and Uniqueness of Solutions 174 5.1.1 Applications to Other Equations 178 5.2 Prolongation and Maximal Solutions 181 5.2.1 Applications to MDEs 191 5.2.2 Applications to Dynamic Equations on Time Scales 197 6 Linear Generalized Ordinary Differential Equations 205Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Miguel V. S. Frasson 6.1 The Fundamental Operator 207 6.2 A Variation-of-Constants Formula 209 6.3 Linear Measure FDEs 216 6.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs 220 7 Continuous Dependence on Parameters 225Suzete M. Afonso, Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita 7.1 Basic Theory for Generalized ODEs 226 7.2 Applications to Measure FDEs 236 8 StabilityTheory 241Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 8.1 Variational Stability for Generalized ODEs 244 8.1.1 Direct Method of Lyapunov 246 8.1.2 Converse Lyapunov Theorems 247 8.2 Lyapunov Stability for Generalized ODEs 256 8.2.1 Direct Method of Lyapunov 257 8.3 Lyapunov Stability for MDEs 261 8.3.1 Direct Method of Lyapunov 263 8.4 Lyapunov Stability for Dynamic Equations on Time Scales 265 8.4.1 Direct Method of Lyapunov 267 8.5 Regular Stability for Generalized ODEs 272 8.5.1 Direct Method of Lyapunov 275 8.5.2 Converse Lyapunov Theorem 282 9 Periodicity 295Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Maria Carolina Mesquita 9.1 Periodic Solutions and Floquet’s Theorem 297 9.1.1 Linear Differential Systems with Impulses 303 9.2 (θ,T)-Periodic Solutions 307 9.2.1 An Application to MDEs 313 10 Averaging Principles 317Márcia Federson and Jaqueline G. Mesquita 10.1 Periodic Averaging Principles 320 10.1.1 An Application to IDEs 325 10.2 Nonperiodic Averaging Principles 330 11 Boundedness of Solutions 341Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 11.1 Bounded Solutions and Lyapunov Functionals 342 11.2 An Application to MDEs 352 11.2.1 An Example 356 12 Control Theory 361Fernanda Andrade da Silva, Márcia Federson, and Eduard Toon 12.1 Controllability and Observability 362 12.2 Applications to ODEs 365 13 Dichotomies 369Everaldo M. Bonotto and Márcia Federson 13.1 Basic Theory for Generalized ODEs 370 13.2 Boundedness and Dichotomies 381 13.3 Applications to MDEs 391 13.4 Applications to IDEs 400 14 Topological Dynamics 407Suzete M. Afonso, Marielle Ap. Silva, Everaldo M. Bonotto, and Márcia Federson 14.1 The Compactness of the Class F0(Ω,h) 408 14.2 Existence of a Local Semidynamical System 411 14.3 Existence of an Impulsive Semidynamical System 418 14.4 LaSalle’s Invariance Principle 423 14.5 Recursive Properties 425 15 Applications to Functional Differential Equations of Neutral Type 429Fernando G. Andrade, Miguel V. S. Frasson, and Patricia H. Tacuri 15.1 Drops of History 429 15.2 FDEs of Neutral Type with Finite Delay 435 References 455 List of Symbols 471 Index 473

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Book SynopsisMany physical problems that are usually solved by differential equation methods can be solved more effectively by integral equation methods.Trade ReviewA nice introductory text... Presents the basics of linear integral equations theory in a very comprehensive way... [The] richness of examples and applications makes the book extremely useful for teachers and also researchers. —Applications of Mathematics (Review of the Second Edition) This second edition of this highly useful book continues the emphasis on applications and presents a variety of techniques with extensive examples...The book is ideal as a text for a beginning graduate course. Its excellent treatment of boundary value problems and an up-to-date bibliography make the book equally useful for researchers in many applied fields.—MathSciNet ​(Review of the Second Edition)Table of ContentsIntroduction.- Integral Equations with Separable Kernels.- Method Of Successive Approximations.- Classical Fredholm Theory.- Applications of Ordinary Differential Equations.- Applications of Partial Differential Equations.- Symmetric Kernels.- Singular Integral Equations.- Integral Transformation Methods.- Applications to Mixed Boundary Value Problems.- Integral Equations Perturbation Methods.- Appendix.- Bibliography.- Index.​

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