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• Preliminaries.- Variational Problems with Fixed Boundaries.- Sufficient Conditions of Extrema of Functionals.- Problems with Variable Boundaries.- Variational Problems of Conditional Extrema.- Variational Problems in Parametric Forms.- Variational Principles.- Methods of Variational Problems.- Variational Principles in Mechanics and Their Applications.- Variational Problems of Functionals with Vector, Tensor and Hamiltonian Operators.
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This book focuses on the calculus of variations, including fundamental theories and applications. This textbook is intended for graduate and higher-level college and university students, introducing them to the basic concepts and calculation methods used in the calculus of variations. It covers the preliminaries, variational problems with fixed boundaries, sufficient conditions of extrema of functionals, problems with undetermined boundaries, variational problems of conditional extrema, variational problems in parametric forms, variational principles, direct methods for variational problems, variational principles in mechanics and their applications, and variational problems of functionals with vector, tensor and Hamiltonian operators. Many of the contributions are based on the authors' research, addressing topics such as the extension of the connotation of the Hilbert adjoint operator, definitions of the other three kinds of adjoint operators, the extremum function theorem of the complete functional, unified Euler equations in variational methods, variational theories of functionals with vectors, modulus of vectors, arbitrary order tensors, Hamiltonian operators and Hamiltonian operator strings, reconciling the Euler equations and the natural boundary conditions, and the application range of variational methods. The book is also a valuable reference resource for teachers as well as science and technology professionals. .
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• Holomorphic Functions.- Oka's First Coherence Theorem.- Sheaf Cohomology.- Holomorphically Convex Domains and Oka--Cartan's Fundamental Theorem.- Domains of Holomorphy.- Analytic Sets and Complex Spaces.- Pseudoconvex Domains and Oka's Theorem.- Cohomology of Coherent Sheaves and Kodaira's Embedding Theorem.- On Coherence.- Appendix.- References.- Index.- Symbols.
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The purpose of this book is to present the classical analytic function theory of several variables as a standard subject in a course of mathematics after learning the elementary materials (sets, general topology, algebra, one complex variable). This includes the essential parts of Grauert-Remmert's two volumes, GL227(236) (Theory of Stein spaces) and GL265 (Coherent analytic sheaves) with a lowering of the level for novice graduate students (here, Grauert's direct image theorem is limited to the case of finite maps).The core of the theory is "Oka's Coherence", found and proved by Kiyoshi Oka. It is indispensable, not only in the study of complex analysis and complex geometry, but also in a large area of modern mathematics. In this book, just after an introductory chapter on holomorphic functions (Chap. 1), we prove Oka's First Coherence Theorem for holomorphic functions in Chap. 2. This defines a unique character of the book compared with other books on this subject, in which the notion of coherence appears much later.The present book, consisting of nine chapters, gives complete treatments of the following items: Coherence of sheaves of holomorphic functions (Chap. 2); Oka-Cartan's Fundamental Theorem (Chap. 4); Coherence of ideal sheaves of complex analytic subsets (Chap. 6); Coherence of the normalization sheaves of complex spaces (Chap. 6); Grauert's Finiteness Theorem (Chaps. 7, 8); Oka's Theorem for Riemann domains (Chap. 8). The theories of sheaf cohomology and domains of holomorphy are also presented (Chaps. 3, 5). Chapter 6 deals with the theory of complex analytic subsets. Chapter 8 is devoted to the applications of formerly obtained results, proving Cartan-Serre's Theorem and Kodaira's Embedding Theorem. In Chap. 9, we discuss the historical development of "Coherence".It is difficult to find a book at this level that treats all of the above subjects in a completely self-contained manner. In the present volume, a number of classical proofs are improved and simplified, so that the contents are easily accessible for beginning graduate students.
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• Preface.- Some Introductory Material.- The Brachystochrone Problem: Johann and Jakob Bernoulli.- Isoperimetrical Problems: Jakob and Johann Bernoulli.- Shortest Lines and Geodesics.- Euler's Memoirs of 1738 and 1741.- Euler's Method us Inveniendi.- Lagrange's delta-Calculus.- Bibliography.- Index of Names.- Subject Index.
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This monograph explores the early development of the calculus of variations in continental Europe during the Eighteenth Century by illustrating the mathematics of its founders. Closely following the original papers and correspondences of Euler, Lagrange, the Bernoullis, and others, the reader is immersed in the challenge of theory building. We see what the founders were doing, the difficulties they faced, the mistakes they made, and their triumphs. The authors guide the reader through these works with instructive commentaries and complements to the original proofs, as well as offering a modern perspective where useful. The authors begin in 1697 with Johann Bernoulli's work on the brachystochrone problem and the events leading up to it, marking the dawn of the calculus of variations. From there, they cover key advances in the theory up to the development of Lagrange's delta-calculus, including: * The isoperimetrical problems * Shortest lines and geodesics * Euler's Methodus Inveniendi and the two Additamenta Finally, the authors give the readers a sense of how vast the calculus of variations has become in centuries hence, providing some idea of what lies outside the scope of the book as well as the current state of affairs in the field. This book will be of interest to anyone studying the calculus of variations who wants a deeper intuition for the techniques and ideas that are used, as well as historians of science and mathematics interested in the development and evolution of modern calculus and analysis.
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• Calculus of Variations Historical Notes on the Calculus of Variations Some Typical Problems Some Important Dates and People
• Introduction and Preliminaries Motivating Problems Mathematical Background Function Spaces Mathematical Formulation of Problems
• The Simplest Problem in the Calculus of Variations The Mathematical Formulation of the SPCV The Fundamental Lemma of the Calculus of Variations The First Necessary Condition for a Global Minimizer Implications and Applications of the FLCV
• Necessary Conditions for Local Minima Weak and Strong Local Minimizers The Euler Necessary Condition - (I) The Legendre Necessary Condition - (III) Jacobi Necessary Condition - (IV) Weierstrass Necessary Condition - (II) Applying the Four Necessary Conditions
• Sufficient Conditions for the Simplest Problem A Field of Extremals The Hilbert Integral Fundamental Sufficient Results
• Summary for the Simplest Problem
• Extensions and Generalizations Properties of the First Variation The Free Endpoint Problem The Simplest Point to Curve Problem Vector Formulations and Higher Order Problems Problems with Constraints: Isoperimetric Problem Problems with Constraints: Finite Constraints An Introduction to Abstract Optimization Problems
• Applications Solution of the Brachistochrone Problem Classical Mechanics and Hamilton's Principle A Finite Element Method for the Heat Equation Optimal Control Optimal Control Problems An Introduction to Optimal Control Problems The Rocket Sled Problem Problems in the Calculus of Variations Time Optimal Control
• Simplest Problem in Optimal Control SPOC: Problem Formulation The Fundamental Maximum Principle Application of the Maximum Principle to Some Simple Problems
• Extensions of the Fundamental Maximum Principle A Fixed-Time Optimal Control Problem Application to Problems in the Calculus of Variations Application to the Farmer's Allocation Problem Application to a Forced Oscillator Control Problem Application to the Linear Quadratic Control Problem The Maximum Principle for a Problem of Bolza The Maximum Principle for Nonautonomous Systems Application to the Nonautonomous LQ Control Problem
• Linear Control Systems Introduction to Linear Control Systems Linear Control Systems Arising from Nonlinear Problems Linear Quadratic Optimal Control The Riccati Differential Equation for a Problem of Bolza Estimation and Observers The Time Invariant Infinite Interval Problem The Time Invariant Min-Max Controller
• Problems appear at the end of each chapter.
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Introduction to the Calculus of Variations and Control with Modern Applications provides the fundamental background required to develop rigorous necessary conditions that are the starting points for theoretical and numerical approaches to modern variational calculus and control problems. The book also presents some classical sufficient conditions and discusses the importance of distinguishing between the necessary and sufficient conditions. In the first part of the text, the author develops the calculus of variations and provides complete proofs of the main results. He explains how the ideas behind the proofs are essential to the development of modern optimization and control theory. Focusing on optimal control problems, the second part shows how optimal control is a natural extension of the classical calculus of variations to more complex problems. By emphasizing the basic ideas and their mathematical development, this book gives you the foundation to use these mathematical tools to then tackle new problems. The text moves from simple to more complex problems, allowing you to see how the fundamental theory can be modified to address more difficult and advanced challenges. This approach helps you understand how to deal with future problems and applications in a realistic work environment.
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• Foreword Jean Mawhin
• Preface
• Part I. Variational Principles in Mathematical Physics: 1. Variational principles
• 2. Variational inequalities
• 3. Nonlinear eigenvalue problems
• 4. Elliptic systems of gradient type
• 5. Systems with arbitrary growth nonlinearities
• 6. Scalar field systems
• 7. Competition phenomena in Dirichlet problems
• 8. Problems to Part I
• Part II. Variational Principles in Geometry: 9. Sublinear problems on Riemannian manifolds
• 10. Asymptotically critical problems on spheres
• 11. Equations with critical exponent
• 12. Problems to Part II
• Part III. Variational Principles in Economics: 13. Mathematical preliminaries
• 14. Minimization of cost-functions on manifolds
• 15. Best approximation problems on manifolds
• 16. A variational approach to Nash equilibria
• 17. Problems to Part III
• Appendix A. Elements of convex analysis
• Appendix B. Function spaces
• Appendix C. Category and genus
• Appendix D. Clarke and Degiovanni gradients
• Appendix E. Elements of set-valued analysis
• References
• Index.
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This comprehensive introduction to the calculus of variations and its main principles also presents their real-life applications in various contexts: mathematical physics, differential geometry, and optimization in economics. Based on the authors' original work, it provides an overview of the field, with examples and exercises suitable for graduate students entering research. The method of presentation will appeal to readers with diverse backgrounds in functional analysis, differential geometry and partial differential equations. Each chapter includes detailed heuristic arguments, providing thorough motivation for the material developed later in the text. Since much of the material has a strong geometric flavor, the authors have supplemented the text with figures to illustrate the abstract concepts. Its extensive reference list and index also make this a valuable resource for researchers working in a variety of fields who are interested in partial differential equations and functional analysis.
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• Preliminaries
• Classical Methods
• Direct Methods: Existence
• Direct Methods: Regularity
• Minimal Surfaces
• Isoperimetric Inequality
• Solutions to the Exercises.
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The calculus of variations is one of the oldest subjects in mathematics, yet is very much alive and is still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology. This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist - mathematicians, physicists, engineers, students or researchers - in discovering the subject's most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions. In this new edition, the chapter on regularity has been significantly expanded and 27 new exercises have been added. The book, containing a total of 103 exercises with detailed solutions, is well designed for a course at both undergraduate and graduate levels.
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• Introduction.- Convex Analysis and the Scalar Case.- Convex Sets and Convex Functions.- Lower Semicontinuity and Existence Theorems.- The one Dimensional Case.- Quasiconvex Analysis and the Vectorial Case.- Polyconvex, Quasiconvex and Rank one Convex Functions.- Polyconvex, Quasiconvex and Rank one Convex Envelopes.- Polyconvex, Quasiconvex and Rank one Convex Sets.- Lower Semi Continuity and Existence Theorems in the Vectorial Case.- Relaxation and Non Convex Problems.- Relaxation Theorems.- Implicit Partial Differential Equations.- Existence of Minima for Non Quasiconvex Integrands.- Miscellaneous.- Function Spaces.- Singular Values.- Some Underdetermined Partial Differential Equations.- Extension of Lipschitz Functions on Banach Spaces.- Bibliography.- Index.- Notations.
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The present monograph is the successor of "Direct methods in the calculus of variations" which was published in the "Applied Mathematical Sciences" series and is currently out of print. Although the core and the structure of the present book is similar to the old one, it is much more than a revised version. Fifteen years have passed since the publication of the "Direct methods in the calculus of variations" book and since the subject is a very active one, almost half of the book presently consists of new material. The perspective has also slightly changed, which is reflected in the change of the title. Indeed a new subject, "quasiconvex analysis" has now been developed. The present monograph, which is essentially a reference book on the subject of quasiconvex analysis, can be used, as was the earlier book for an advanced course on the calculus of variations.
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• The First Variation.- Some Generalizations.- Isoperimetric Problems.- Applications to Eigenvalue Problems.- Holonomic and Nonholonomic Constraints.- Problems with Variable Endpoints.- The Hamiltonian Formulation.- Noether's Theorem.- The Second Variation.
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Suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering, this introduction to the calculus of variations focuses on variational problems involving one independent variable. It also discusses more advanced topics such as the inverse problem, eigenvalue problems, and Noether's theorem. The text includes numerous examples along with problems to help students consolidate the material.
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• Preliminaries
• Classical Methods
• Direct Methods
• Regularity
• Minimal Surfaces
• Isoperimetric Inequality
• Solutions to the Exercises.
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The book, containing more than seventy exercises with detailed solutions, is well designed for a course both at the undergraduate and graduate levels.
(source: Nielsen Book Data)
The calculus of variations is one of the oldest subjects in mathematics, yet is very much alive and is still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology. This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist -- mathematicians, physicists, engineers, students or researchers -- in discovering the subjects most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions. The book, containing more than seventy exercises with detailed solutions, is well designed for a course both at the undergraduate and graduate levels.
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• Calculus of variations: What does "variations" mean? by F. Jones How many equilibria are there? An introduction to Morse theory by R. Forman Aye, there's the rub. An inquiry into why a plucked string comes to rest by S. J. Cox Proof of the double bubble conjecture by F. Morgan Minimal surfaces, flat cone spheres and moduli spaces of staircases by M. Wolf Hold that light! Modeling of traffic flow by differential equations by B. L. Keyfitz.
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The calculus of variations is a beautiful subject with a rich history and with origins in the minimization problems of calculus. Although it is now at the core of many modern mathematical fields, it does not have a well-defined place in most undergraduate mathematics curricula. This volume should nevertheless give the undergraduate reader a sense of its great character and importance. Interesting functionals, such as area or energy, often give rise to problems for which the most natural solution occurs by differentiating a one-parameter family of variations of some function.The critical points of the functional are related to the solutions of the associated Euler-Lagrange equation. These differential equations are at the heart of the calculus of variations and its applications to other subjects. Some of the topics addressed in this book are Morse theory, wave mechanics, minimal surfaces, soap bubbles, and modeling traffic flow. All are readily accessible to advanced undergraduates. This book is derived from a workshop sponsored by Rice University. It is suitable for advanced undergraduates, graduate students and research mathematicians interested in the calculus of variations and its applications to other subjects.
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• Semi-Classical Theory
• Integrable Functions
• Sobolev Spaces
• Semicontinuity
• Quasi-Convex Functionals
• Quasi-Minima
• Regularity of Quasi-Minima
• First Derivatives
• Partial Regularity
• Higher Derivatives.
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This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory.
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• Part I. One-Dimensional Variational Problems: 1. The classical theory
• 2. Geodesic curves
• 3. Saddle point constructions
• 4. The theory of Hamilton and Jacobi
• 5. Dynamic optimization
• Part II. Multiple Integrals in the Calculus of Variations: 6. Lebesgue integration theory
• 7. Banach spaces
• 8. Lp and Sobolev spaces
• 9. The direct methods
• 10. Nonconvex functionals: relaxation
• 11. G-convergence
• 12. BV-functionals and G-convergence: the example of Modica and Mortola
• Appendix A. The coarea formula
• Appendix B. The distance function from smooth hypersurfaces
• 13. Bifurcation theory
• 14. The Palais-Smale condition and unstable critical points of variational problems.
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This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues like Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The only prerequisites are basic results from calculus of one and several variables. After having studied this book, the reader will be well-equipped to read research papers in the calculus of variations.
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• Part I: General Measure Theory.- Integer Rectifiable Currents.- Cartesian Maps.- Cartesian Currents in Euclidean Spaces.- Cartesian Currents in Riemannian Manifolds.- Part II: Regular Variational Integrals.- Finite Elasticity and Weak Diffeomorphisms.- The Dirichlet Integral in Sobolev Spaces.- The Dirichlet Energy for Maps into S2.- Regular and Non Regular Integrals.- The Non Parametric Area Functional.
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This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configurations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph.
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• Introduction
• 1. Classical problems and indirect methods
• 2. Absolutely continuous functions and Sobolev spaces
• 3. Semicontinuity and existence results
• 4. Regularity of minimizers
• 5. Some applications
• 6. Scholia.
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One-dimensional variational problems have been somewhat neglected in the literature on calculus of variations, as authors usually treat minimal problems for multiple integrals which lead to partial differential equations and are considerably more difficult to handle. One-dimensional problems are connected with ordinary differential equations, and hence need many fewer technical prerequisites, but they exhibit the same kind of phenomena and surprises as variational problems for multiple integrals. This book provides an modern introduction to this subject, placing special emphasis on direct methods. It combines the efforts of a distinguished team of authors who are all renowned mathematicians and expositors. Since there are fewer technical details graduate students who want an overview of the modern approach to variational problems will be able to concentrate on the underlying theory and hence gain a good grounding in the subject. Except for results from the theory of measure and integration and from the theory of convex functions, the text develops all mathematical tools needed, including the basic results on one-dimensional Sobolev spaces, absolutely continuous functions, and functions of bounded variation.
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• Max and Min.- Convexity.- Cones and Cosmic Closure.- Set Convergence.- Set-Valued Mappings.- Variational Geometry.- Epigraphical Limits.- Subderivatives and Subgradients.- Lipschitzian Properties.- Subdifferential Calculus.- Dualization.- Monotone Mappings.- Second-Order Theory.- Measurability.- References.- Index of Statements.- Index of Notation.- Index of Topics.
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From its origins in the minimization of integral functionals, the notion of 'variations' has evolved greatly in connection with applications in optimization, equilibrium, and control. It refers not only to constrained movement away from a point, but also to modes of perturbation and approximation that are best describable by 'set convergence', variational convergence of functions and the like. This book develops a unified framework and, in finite dimension, provides a detailed exposition of variational geometry and subdifferential calculus in their current forms beyond classical and convex analysis. Also covered are set-convergence, set-valued mappings, epi-convergence, duality, maximal monotone mappings, second-order subderivatives, measurable selections and normal integrands. The changes in the 3rd printing mainly concern various typographical corrections and reference omissions that came to light in the previous printings. Many of these reached the authors' notice through their own re-reading, that of their students and a number of colleagues mentioned in the Preface. The authors also included a few telling examples as well as improved a couple of statements with slightly weaker assumptions, or provided a converse.
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• 1. The Lagrangian formalism
• 2. The Hamiltonian formalism.

This is the first part of a treatise covering calculus of variations in breadth and depth, paying special attention to the historical origins, partly in applications, for example from geometrical optics, of parts of the theory. A variety of aids to the reader are provided, including an introduction to each chapter, section and subsection and historical footnotes. Later volumes deal with direct methods and regularity theory.
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This long-awaited book by two of the foremost researchers and writers in the field is the first part of a treatise that covers the subject in breadth and depth, paying special attention to the historical origins, partly in applications, e. g. from geometrical optics, of parts of the theory. A variety of aids to the reader is provided: besides the very detailed table of contents, an introduction to each chapter, section and subsection, an overview of the relevant literature (in Vol. 2) plus the references in the Scholia to each chapter, in the (historical) footnotes, and in the bibliography, and finally an index of the examples used throughout the book. Both individually and collectively these volumes have already become standard references.
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