Calculus and mathematical analysis Books

Book SynopsisThe series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

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Book SynopsisAn application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.Trade ReviewM.P. Do Carmo Differential Forms and Applications "This book treats differential forms and uses them to study some local and global aspects of differential geometry of surfaces. Each chapter is followed by interesting exercises. Thus, this is an ideal book for a one-semester course."—ACTA SCIENTIARUM MATHEMATICARUMTable of Contents1. Differential Forms in Rn.- 2. Line Integrals.- 3. Differentiable Manifolds.- 4. Integration on Manifolds; Stokes Theorem and Poincaré’s Lemma.- 1. Integration of Differential Forms.- 2. Stokes Theorem.- 3. Poincaré’s Lemma.- 5. Differential Geometry of Surfaces.- 1. The Structure Equations of Rn.- 2. Surfaces in R3.- 3. Intrinsic Geometry of Surfaces.- 6. The Theorem of Gauss-Bonnet and the Theorem of Morse.- 1. The Theorem of Gauss-Bonnet.- 2. The Theorem of Morse.- References.

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Book SynopsisDas Lehrbuch bietet umfangreiche Anleitungen und Übungsaufgaben zum Band „Mathematik für Physiker“ desselben Autors. Die Studienanleitungen mit Fragen und Kontrollaufgaben erleichtern Lesern das eigenständige Erarbeiten des Stoffs. In zusätzlichen Erläuterungen vertieft der Autor einzelne Themenfelder und geht auf individuelle Lernschwierigkeiten ein. Band 2 des Übungswerks enthält über 700 Aufgaben mit ausführlichen Lösungen und ist der ideale Begleiter für Bachelor-Studierende der Physik während des zweiten Semesters.Table of ContentsFunktionen mehrerer Variablen.- Partielle Ableitung, totales Differential.- Mehrfachintegrale, Parameterdarstellung.- Oberflächenintegrale.- Divergenz und Rotation.- Koordinatentransformation, Matrizen.- Lineare Gleichungssysteme.- Eigenwerte, Eigenvektoren.- Fourierreihen.- Fourier-Integrale, Fourier-Transformation.- Laplace-Transformation.- Wellengleichungen.

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Book SynopsisDieses Buch behandelt in verständlicher und klarer Sprache den klassischen Inhalt einer „Analysis 1“-Vorlesung. Das Besondere dabei ist die Zusammensetzung des Autorenteams: zwei Promotions-Studenten und ein Professor. In die Darstellung der einzelnen Themen wie Folgen, unendliche Reihen, Stetigkeit, Differential- und Integralrechnung, fließen so einerseits die Erfahrungen eines Hochschullehrers – der die Vorlesung mehrmals gehalten hat – und andererseits die Erfahrungen ehemaliger Studenten über typische Schwierigkeiten beim Übergang von der Oberstufen- zur Hochschulmathematik ein.Die mathematisch exakt formulierten Sätze und Definitionen werden durch viele Beispiele, Erklärungen sowie Anschauungen aufgelockert, die das Behandelte greifbar machen und das Verständnis erleichtern. Historische Exkurse beleuchten die Entwicklung des Gebietes, sind harmonisch in den Text eingefügt und dienen der Motivation. Zudem fördern didaktisch aufbereitete Beweise den Einstieg in die mathematische Denkweise. Am Ende eines jeden Kapitels wird schließlich das Wichtigste noch einmal übersichtlich zusammengefasst. Auf Grund der zahlreichen Aufgaben samt Lösungsvorschlag eignet sich dieses Buch nicht nur zur Vorlesungsbegleitung, sondern auch zum Selbststudium und zur Prüfungsvorbereitung.Die ZielgruppenLehramtsstudierende der Mathematik sowie Bachelorstudierende der Mathematik, Physik und Informatik, aber auch Lehrerinnen und Lehrer an Gymnasien und Schülerinnen und Schüler der gymnasialen OberstufeTrade Review“... Das mit zahlreichen Beispielen, Abbildungen und Übungsaufgaben inkl. Lösungen versehene, in Darstellung und Aufbau ungewöhnlich transparente, übersichtliche und zugängliche Werk liefert dabei eingangs elementarste Grundlagen der Logik und Mengenlehre, um anschließend die eigentlichen Inhalte zu behandeln und behutsam an die Gegenstände der Analysis heranzuführen ...” (Philipp Kastendieck, in: ekz-Informationsdienst, Heft 39, 2019)Table of ContentsElementar(st)e Logik und Mengenlehre.- Vollständige Induktion.- AngeordneteKörper.- Funktionen.- Folgen.- Unendliche Reihen.- Spezielle Funktionen.- Stetigkeit.- Gleichmäßige Stetigkeit.- Differentialrechnung.- Das (Riemann-)Integral.- Konvergenz von Funktionenfolgen.

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Book SynopsisThis guide book to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Easy to understand, and convenient to use, this guide book gives concisely the information necessary to evaluate most problems which occur in concrete applications. In the newer editions emphasis was laid on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes, namely Numerical Mathematics, Probability Theory and Statistics, as well as Information Processing. Besides many enhancements and new paragraphs, new sections on Geometric and Coordinate Transformations, Quaternions and Applications, and Lie Groups and Lie Algebras were added for the sixth edition.Trade Review“Russian scholars Bronshtein and Semendyayev created a math classic over seven decades ago. … This new Springer edition details over 1,500 entries in its table of contents, including new entries for analytical geometry, Lie groups and Lie algebra, nonlinear optimization, and computer algebra systems. … Summing Up: Recommended. All mathematics library collections.” (K. L. Swetland, Choice, Vol. 53 (11), July, 2016)Table of ContentsArithmetics.- Functions.- Geometry.- Linear Algebra.- Algebra and Discrete Mathematics.- Differentiation.- Infinite Series.- Integral Calculus.- Differential Equations.- Calculus of Variations.- Linear Integral Equations.- Functional Analysis.- Vector Analysis and Vector Fields.- Function Theory.- Integral Transformations.- Probability Theory and Mathematical Statistics.- Dynamical Systems and Chaos.- Optimization.- Numerical Analysis.- Computer Algebra Systems-Example Mathematica.

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Book SynopsisDies ist ein Buch über die Mathematik, welches insbesondere die Anforderungen des Bachelorstudiums sinnvoll bedient. Es behandelt die Analysis in mehreren Variablen sowie gewöhnliche und partielle Differenzialgleichungen. Dabei wenden wir uns an Physiker, Mathematiker sowie ambitionierte Lehramtskandidaten und Ingenieure. Das Buch fördert sowohl das Verständnis als auch das konzentrierte Lernen für Klausuren und mündliche Prüfungen. Die Autoren bringen ihre Erfahrungen aus zahlreichen erfolgreichen Vorlesungen und Übungen zum Nutzen der Studierenden ein. Auf einen Blick: Klarer Stil, klare Sprache, klare Struktur. Zahlreiche Erläuterungen. Zu jedem Thema wird gesondert ein informativer Ein- und Ausblick geliefert. Grafiken und viele Beispiele helfen beim Verstehen. Fragen zum Selbsttest unterstützen zusätzlich beim Lernen. Aufgaben mit vollständigen Lösungen dienen der Vertiefung und Vorbereitung auf Prüfungen jeglicher Art. Table of ContentsI Mehrdimensionale Analysis 1 Metrische Räume 2 Kompakte Mengen in Rn, Abbildungen und Funktionen in Rn 3 Stetige Abbildungen von Rn nach Rm 4 Differenzierbare Abbildungen von Rn nach Rm 5 Gradient, Divergenz und Rotation 6 Höhere partielle Ableitungen und der Laplace-Operator 7 Potenziale 8 Lokale Extrema und Taylor-Polynom 9 Lokale Extrema unter Nebenbedingungen 10 Kurven in Rn 11 Kurvenintegrale 12 Mehrfachintegration in R2 und R3 13 Koordinatentransformation von Integralen in R2 14 Flächen in R3, Oberächen- und Flussintegral 15 Der Satz von Gauß 16 Der Satz von Stokes Aufgaben zur mehrdimensionalen Analysis II Differenzialgleichungen 17 Grundlegendes zu Differenzialgleichungen 18 Lösungsansatz für homogene lineare Differenzialgleichungen mit konstanten Koeffzienten 19 Anfangswertprobleme I 20 Anfangswertprobleme II, inhomogene lineare Differenzialgleichungssysteme und Variation der Konstanten 21 Inhomogene lineare Differenzialgleichungssysteme und Ansatz vom Typ der rechten Seite, Wronski-Test 22 Lösungsansätze für nicht lineare Differenzialgleichungen 23 Nicht lineare Differenzialgleichungssysteme und Stabilität 24 Partielle Differenzialgleichungen: Separationsansatz 25 Wellengleichung, holomorphe und harmonische Funktionen 26 Weiteres zur Wellengleichung, Überblick 27 Fourier-Reihen 28 Variationsrechnung Aufgaben zu Differenzialgleichungen Lösungen der Selbsttests Lösungen der Aufgaben Literatur und Ausklang Index

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Book SynopsisOne SCI\'ice mathematics bas rendered the 'Et moi, ...si j'avait su comment en revcnir. je n'y serais point aile: human race. It bas put common sc:nsc back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com- puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.Table of Contents0. Preliminary Informationh.- 0.1 Probability Space.- 0.2 Random Functions and Processes.- 0.3 Conditional Probabilities.- 0.4 Independence.- 1. Sums of Independent Random Variables.- 1.1 Main Inequalities.- 1.2 Renewal Scheme.- 1.3 Random Walks. Recurrence.- 1.4 Distribution of Ladder Functions.- 2. General Processes with Independent Increments (Random Measures).- 2.1 Nonnegative Random Measures with Independent Values (r.m.i.v.).- 2.2 Random Measures with Alternating Signs.- 2.3 Stochastic Integrals and Countably Additive r.m.i.v.- 2.4 Random Linear Functional and Generalized Functions.- 3. Processes with Independent Increments. General Properties.- 3.1 Decomposition of a Process. Properties of Sample Functions.- 3.2 Stochastically Continuous Processes.- 3.3 Properties of Sample Functions.- 3.4 Locally Homogeneous Processes with Independent Increments.- 4. Homogeneous Processes.- 4.1 General Properties.- 4.2 Additive Functionals.- 4.3 Composed Poisson Process.- 4.4 Homogeneous Processes in R.- 5. Multiplicative Processes.- 5.1 Definition and General Properties.- 5.2 Multiplicative Processes in Abelian Groups.- 5.3 Stochastic Semigroups of Linear Operators in Rd.- Notes.- References.

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Book SynopsisA guide for any student of vector analysis, this text separates those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition.

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Book SynopsisA new approach to the foundations of single variable calculus, based on the introductory course taught at CaltechTrade Review"The author’s stress on repeatable techniques . . . and the real numbers treated as infinite decimals results in a distinctive excursion through familiar territory.”—Nick Lord, The Mathematical Gazette"A very useful and constructive way to teach the subject."—Dominic Thorrington, IMA“Every science and engineering student takes calculus, but few learn the subject with depth and rigor. Calculus for Cranks addresses this gap head-on, introducing fundamental concepts in analysis that are valuable for all students – not just math majors.”—Carina Curto, Professor of Mathematics, Pennsylvania State University “Nets Katz has written a calculus textbook for students who don’t like being lied to. It will be essential for those who are constantly harassing their teachers with questions beginning with ‘why’ and ‘how.’”—Deane Yang, Professor of Mathematics, New York University “Calculus for Cranks unspools like a good novel! Katz deftly weaves abstraction and computation into a single narrative, with an entertaining set of exercises along the way.”—Amie Wilkinson, Professor of Mathematics, University of Chicago “Blending formal and informal insights, Katz pulls back the curtain on calculus, revealing its foundations, especially for those who think they’ve seen it before.”—Francis Su, author of Mathematics for Human Flourishing “Calculus for Cranks is a beautiful, rigorous, intuitive, introduction to real and complex analysis starting from logical reasoning and the number system. I recommend it highly for serious students.”—Wilhelm Schlag, Professor of Mathematics, Yale University

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Book SynopsisFundamental Fixed-Point Principles.- 1 The Banach Fixed-Point Theorem and Iterative Methods.- 2 The Schauder Fixed-Point Theorem and Compactness.- Applications of the Fundamental Fixed-Point Principles.- 3 Ordinary Differential Equations in B-spaces.- 4 Differential Calculus and the Implicit Function Theorem.- 5 Newton's Method.- 6 Continuation with Respect to a Parameter.- 7 Positive Operators.- 8 Analytic Bifurcation Theory.- 9 Fixed Points of Multivalued Maps.- 10 Nonexpansive Operators and Iterative Methods.- 11 Condensing Maps and the BourbakiKneser Fixed-Point Theorem.- The Mapping Degree and the Fixed-Point Index.- 12 The Leray-Schauder Fixed-Point Index.- 13 Applications of the Fixed-Point Index.- 14 The Fixed-Point Index of Differentiable and Analytic Maps.- 15 Topological Bifurcation Theory.- 16 Essential Mappings and the Borsuk Antipodal Theorem.- 17 Asymptotic Fixed-Point Theorems.- References.- Additional References to the Second Printing.- List of Symbols.- List of TheoreTable of ContentsFundamental Fixed-Point Principles.- 1 The Banach Fixed-Point Theorem and Iterative Methods.- §1.1. The Banach Fixed-Point Theorem.- §1.2. Continuous Dependence on a Parameter.- §1.3. The Significance of the Banach Fixed-Point Theorem.- §1.4. Applications to Nonlinear Equations.- §1.5. Accelerated Convergence and Newton’s Method.- § 1.6. The Picard-Lindelof Theorem.- §1.7. The Main Theorem for Iterative Methods for Linear Operator Equations.- §1.8. Applications to Systems of Linear Equations.- §1.9. Applications to Linear Integral Equations.- 2 The Schauder Fixed-Point Theorem and Compactness.- §2.1. Extension Theorem.- §2.2. Retracts.- §2.3. The Brouwer Fixed-Point Theorem.- §2.4. Existence Principle for Systems of Equations.- §2.5. Compact Operators.- §2.6. The Schauder Fixed-Point Theorem.- §2.7. Peano’s Theorem.- §2.8. Integral Equations with Small Parameters.- §2.9. Systems of Integral Equations and Semilinear Differential Equations.- §2.10. A General Strategy.- §2.11. Existence Principle for Systems of Inequalities.- Applications of the Fundamental Fixed-Point Principles.- 3 Ordinary Differential Equations in B-spaces.- §3.1. Integration of Vector Functions of One Real Variable t.- §3.2. Differentiation of Vector Functions of One Real Variable t.- §3.3. Generalized Picard-Lindelöf Theorem.- §3.4. Generalized Peano Theorem.- §3.5. Gronwall’s Lemma.- §3.6. Stability of Solutions and Existence of Periodic Solutions.- §3.7. Stability Theory and Plane Vector Fields, Electrical Circuits, Limit Cycles.- §3.8. Perspectives.- 4 Differential Calculus and the Implicit Function Theorem.- §4.1. Formal Differential Calculus.- §4.2. The Derivatives of Fréchet and Gâteaux.- §4.3. Sum Rule, Chain Rule, and Product Rule.- §4.4. Partial Derivatives.- §4.5. Higher Differentials and Higher Derivatives.- §4.6. Generalized Taylor’s Theorem.- §4.7. The Implicit Function Theorem.- §4.8. Applications of the Implicit Function Theorem.- §4.9. Attracting and Repelling Fixed Points and Stability.- §4.10. Applications to Biological Equilibria.- §4.11. The Continuously Differentiable Dependence of the Solutions of Ordinary Differential Equations in B-spaces on the Initial Values and on the Parameters.- §4.12. The Generalized Frobenius Theorem and Total Differential Equations.- §4.13. Diffeomorphisms and the Local Inverse Mapping Theorem.- §4.14. Proper Maps and the Global Inverse Mapping Theorem.- §4.15. The Suijective Implicit Function Theorem.- §4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank Theorem.- §4.17. A Look at Manifolds.- §4.18. Submersions and a Look at the Sard-Smale Theorem.- §4.19. The Parametrized Sard Theorem and Constructive Fixed-Point Theory.- 5 Newton’s Method.- §5.1. A Theorem on Local Convergence.- §5.2. The Kantorovi? Semi-Local Convergence Theorem.- 6 Continuation with Respect to a Parameter.- §6.1. The Continuation Method for Linear Operators.- §6.2. B-spaces of Hölder Continuous Functions.- §6.3. Applications to Linear Partial Differential Equations.- §6.4. Functional-Analytic Interpretation of the Existence Theorem and its Generalizations.- §6.5. Applications to Semi-linear Differential Equations.- §6.6. The Implicit Function Theorem and the Continuation Method.- §6.7. Ordinary Differential Equations in B-spaces and the Continuation Method.- §6.8. The Leray—Schauder Principle.- §6.9. Applications to Quasi-linear Elliptic Differential Equations.- 7 Positive Operators.- §7.1. Ordered B-spaces.- §7.2. Monotone Increasing Operators.- §7.3. The Abstract Gronwall Lemma and its Applications to Integral Inequalities.- §7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability.- §7.5. Applications.- §7.6. Minorant Methods and Positive Eigensolutions.- §7.7. Applications.- §7.8. The Krein-Rutman Theorem and its Applications.- §7.9. Asymptotic Linear Operators.- §7.10. Main Theorem for Operators of Monotone Type.- §7.11. Application to a Heat Conduction Problem.- §7.12. Existence of Three Solutions.- §7.13. Main Theorem for Abstract Hammerstein Equations in Ordered B-spaces.- §7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation, Stability, and the Nonlinear Krein-Rutman Theorem.- §7.15. Applications to Hammerstein Integral Equations.- §7.16. Applications to Semi-linear Elliptic Boundary-Value Problems.- §7.17. Application to Elliptic Equations with Nonlinear Boundary Conditions.- §7.18. Applications to Boundary Initial-Value Problems for Parabolic Differential Equations and Stability.- 8 Analytic Bifurcation Theory.- §8.1. A Necessary Condition for Existence of a Bifurcation Point.- §8.2. Analytic Operators.- §8.3. An Analytic Majorant Method.- §8.4. Fredholm Operators.- §8.5. The Spectrum of Compact Linear Operators (Riesz—Schauder Theory).- §8.6. The Branching Equations of Ljapunov—Schmidt.- §8.7. The Main Theorem on the Generic Bifurcation From Simple Zeros.- §8.8. Applications to Eigenvalue Problems.- §8.9. Applications to Integral Equations.- §8.10. Application to Differential Equations.- §8.11. The Main Theorem on Generic Bifurcation for Multiparametric Operator Equations—The Bunch Theorem.- §8.12. Main Theorem for Regular Semi-linear Equations.- §8.13. Parameter-Induced Oscillation.- §8.14. Self-Induced Oscillations and Limit Cycles.- §8.15. Hopf Bifurcation.- §8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros.- §8.17. Stability of Bifurcation Solutions.- §8.18. Generic Point Bifurcation.- 9 Fixed Points of Multivalued Maps.- §9.1. Generalized Banach Fixed-Point Theorem.- §9.2. Upper and Lower Semi-continuity of Multivalued Maps.- §9.3. Generalized Schauder Fixed-Point Theorem.- §9.4. Variational Inequalities and the Browder Fixed-Point Theorem.- §9.5. An Extremal Principle.- §9.6. The Minimax Theorem and Saddle Points.- §9.7. Applications in Game Theory.- §9.8. Selections and the Marriage Theorem.- §9.9. Michael’s Selection Theorem.- §9.10. Application to the Generalized Peano Theorem for Differential Inclusions.- 10 Nonexpansive Operators and Iterative Methods.- §10.1. Uniformly Convex B-spaces.- §10.2. Demiclosed Operators.- §10.3. The Fixed-Point Theorem of Browder, Göhde, and Kirk.- §10.4. Demicompact Operators.- §10.5. Convergence Principles in B-spaces.- §10.6. Modified Successive Approximations.- §10.7. Application to Periodic Solutions.- 11 Condensing Maps and the Bourbaki—Kneser Fixed-Point Theorem.- §11.1. A Noncompactness Measure.- §11.2. Applications to Generalized Interval Nesting.- §11.3. Condensing Maps.- §11.4. Operators with Closed Range and an Approximation Technique for Constructing Fixed Points.- §11.5. Sadovskii’s Fixed-Point Theorem for Condensing Maps.- §11.6. Fixed-Point Theorems for Perturbed Operators.- §11.7. Application to Differential Equations in B-spaces.- §11.8. The Bourbaki-Kneser Fixed-Point Theorem.- § 11.9. The Fixed-Point Theorems of Amann and Tarski.- §11.10. Application to Interval Arithmetic.- §11.11. Application to Formal Languages.- The Mapping Degree and the Fixed-Point Index.- 12 The Leray-Schauder Fixed-Point Index.- §12.1. Intuitive Background and Basic Concepts.- §12.2. Homotopy.- §12.3. The System of Axioms.- §12.4. An Approximation Theorem.- §12.5. Existence and Uniqueness of the Fixed-Point Index in ?N.- §12.6. Proof of Theorem 12.A..- §12.7. Existence and Uniqueness of the Fixed-Point Index in B-spaces.- §12.8. Product Theorem and Reduction Theorem.- 13 Applications of the Fixed-Point Index.- §13.1. A General Fixed-Point Principle.- §13.2. A General Eigenvalue Principle.- §13.3. Existence of Multiple Solutions.- §13.4. A Continuum of Fixed Points.- §13.5. Applications to Differential Equations.- §13.6. Properties of the Mapping Degree.- §13.7. The Leray Product Theorem and Homeomorphisms.- §13.8. The Jordan-Brouwer Separation Theorem and Brouwer’s Invariance of Dimension Theorem.- §13.9. A Brief Glance at the History of Mathematics.- §13.10. Topology and Intuition.- §13.11. Generalization of the Mapping Degree.- 14 The Fixed-Point Index of Differentiable and Analytic Maps.- §14.1. The Fixed-Point Index of Classical Analytic Functions.- §14.2. The Leray—Schauder Index Theorem.- §14.3. The Fixed-Point Index of Analytic Mappings on Complex B-spaces.- §14.4. The Schauder Fixed-Point Theorem with Uniqueness.- §14.5. Solution of Analytic Operator Equations.- §14.6. The Global Continuation Principle of Leray—Schauder.- §14.7. Unbounded Solution Components.- §14.8. Applications to Systems of Equations.- §14.9. Applications to Integral Equations.- §14.10. Applications to Boundary-Value Problems.- §14.11. Applications to Integral Power Series.- 15 Topological Bifurcation Theory.- §15.1. The Index Jump Principle.- §15.2. Applications to Systems of Equations.- §15.3. Duality Between the Index Jump Principle and the Leray—Schauder Continuation Principle.- §15.4. The Geometric Heart of the Continuation Method.- §15.5. Stability Change and Bifurcation.- §15.6. Local Bifurcation.- §15.7. Global Bifurcation.- §15.8. Application to Systems of Equations.- §15.9. Application to Integral Equations.- §15.10. Application to Differential Equations.- §15.11. Application to Bifurcation at Infinity.- §15.12. Proof of the Main Theorem.- §15.13. Preventing Secondary Bifurcation.- 16 Essential Mappings and the Borsuk Antipodal Theorem.- §16.1. Intuitive Introduction.- §16.2. Essential Mappings and their Homotopy Invariance.- §16.3. The Antipodal Theorem.- §16.4. The Invariance of Domain Theorem and Global Homeomorphisms.- §16.5. The Borsuk—Ulam Theorem and its Applications.- §16.6. The Mapping Degree and Essential Maps.- §16.7. The Hopf Theorem.- §16.8. A Glance at Homotopy Theory.- 17 Asymptotic Fixed-Point Theorems.- §17.1. The Generalized Banach Fixed-Point Theorem.- §17.2. The Fixed-Point Index of Iterated Mappings.- §17.3. The Generalized Schauder Fixed-Point Theorem.- §17.4. Application to Dissipati ve Dynamical Systems.- §17.5. Perspectives.- References.- Additional References to the Second Printing.- List of Symbols.- List of Theorems.- List of the Most Important Definitions.- Schematic Overviews.- General References to the Literature.- List of Important Principles.- of the Other Parts.

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Book SynopsisFirst Part.- I The Complex Plane and Elementary Functions.- II Analytic Functions.- III Line Integrals and Harmonic Functions.- IV Complex Integration and Analyticity.- V Power Series.- VI Laurent Series and Isolated Singularities.- VII The Residue Calculus.- Second Part.- VIII The Logarithmic Integral.- IX The Schwarz Lemma and Hyperbolic Geometry.- X Harmonic Functions and the Reflection Principle.- XI Conformal Mapping.- Third Part.- XII Compact Families of Meromorphic Functions.- XIII Approximation Theorems.- XIV Some Special Functions.- XV The Dirichlet Problem.- XVI Riemann Surfaces.- Hints and Solutions for Selected Exercises.- References.- List of Symbols.Table of Contents* The Complex Plane and Elementary Functions * Analytic Functions * Line Integrals and Harmonic Functions * Complex Integration and Analyticity * Power Series * Laurent Series and Isolated Singularities * The Residue Calculus * The Logarithmic Integral * The Schwarz Lemma and Hyperbolic Geometry * Harmonic Functions and the Reflection Principle * Conformal Mapping * Compact Families of Meromorphic Functions * Approximation Theorems * Some Special Functions * The Dirichlet Problem * Riemann Surfaces

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Book SynopsisThe first part of this introduction to ergodic theory addresses measure-preserving transformations of probability spaces and covers such topics as recurrence properties and the Birkhoff ergodic theorem. The second part focuses on the ergodic theory of continuous transformations of compact metrizable spaces.Table of Contents0 Preliminaries.- §0.1 Introduction.- §0.2 Measure Spaces.- §0.3 Integration.- §0.4 Absolutely Continuous Measures and Conditional Expectations.- §0.5 Function Spaces.- §0.6 Haar Measure.- §0.7 Character Theory.- §0.8 Endomorphisms of Tori.- §0.9 Perron—Frobenius Theory.- §0.10 Topology.- 1 Measure-Preserving Transformations.- §1.1 Definition and Examples.- §1.2 Problems in Ergodic Theory.- §1.3 Associated Isometries.- §1.4 Recurrence.- §1.5 Ergodicity.- §1.6 The Ergodic Theorem.- §1.7 Mixing.- 2 Isomorphism, Conjugacy, and Spectral Isomorphism.- §2.1 Point Maps and Set Maps.- §2.2 Isomorphism of Measure-Preserving Transformations.- §2.3 Conjugacy of Measure-Preserving Transformations.- §2.4 The Isomorphism Problem.- §2.5 Spectral Isomorphism.- §2.6 Spectral Invariants.- 3 Measure-Preserving Transformations with Discrete Spectrum.- §3.1 Eigenvalues and Eigenfunctions.- §3.2 Discrete Spectrum.- §3.3 Group Rotations.- 4 Entropy.- §4.1 Partitions and Subalgebras.- §4.2 Entropy of a Partition.- §4.3 Conditional Entropy.- §4.4 Entropy of a Measure-Preserving Transformation.- §4.5 Properties of h (T, A) and h (T).- §4.6 Some Methods for Calculating h (T).- §4.7 Examples.- §4.8 How Good an Invariant is Entropy?.- §4.9 Bernoulli Automorphisms and Kolmogorov Automorphisms.- §4.10 The Pinsker ?-Algebra of a Measure-Preserving Transformation.- §4.11 Sequence Entropy.- §4.12 Non-invertible Transformations.- §4.13 Comments.- 5 Topological Dynamics.- §5.1 Examples.- §5.2 Minimality.- §5.3 The Non-wandering Set.- §5.4 Topological Transitivity.- §5.5 Topological Conjugacy and Discrete Spectrum.- §5.6 Expansive Homeomorphisms.- 6 Invariant Measures for Continuous Transformations.- §6.1 Measures on Metric Spaces.- §6.2 Invariant Measures for Continuous Transformations.- §6.3 Interpretation of Ergodicity and Mixing.- §6.4 Relation of Invariant Measures to Non-wandering Sets, Periodic Points and Topological Transitivity.- §6.5 Unique Ergodicity.- §6.6 Examples.- 7 Topological Entropy.- §7.1 Definition Using Open Covers.- §7.2 Bowen’s Definition.- §7.3 Calculation of Topological Entropy.- 8 Relationship Between Topological Entropy and Measure-Theoretic Entropy.- §8.1 The Entropy Map.- §8.2 The Variational Principle.- §8.3 Measures with Maximal Entropy.- §8.4 Entropy of Affine Transformations.- §8.5 The Distribution of Periodic Points.- §8.6 Definition of Measure-Theoretic Entropy Using the Metrics dn.- 9 Topological Pressure and Its Relationship with Invariant Measures.- §9.1 Topological Pressure.- §9.2 Properties of Pressure.- §9.3 The Variational Principle.- §9.4 Pressure Determines M(X, T).- §9.5 Equilibrium States.- 10 Applications and Other Topics.- §10.1 The Qualitative Behaviour of Diffeomorphisms.- §10.2 The Subadditive Ergodic Theorem and the Multiplicative Ergodic Theorem.- §10.3 Quasi-invariant Measures.- §10.4 Other Types of Isomorphism.- §10.5 Transformations of Intervals.- §10.6 Further Reading.- References.

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Book SynopsisIn particu­ lar, integral representations are the principal tool used to develop the global theory, in contrast to many earlier books on the subject which involved methods from commutative algebra and sheaf theory, and/or partial differ­ ential equations.Table of ContentsI Elementary Local Properties of Holomorphic Functions.- II Domains of Holomorphy and Pseudoconvexity.- III Differential Forms and Hermitian Geometry.- IV Integral Representations in ?n.- V The Levi Problem and the Solution of ?? on Strictly Pseudoconvex Domains.- VI Function Theory on Domains of Holomorphy in ?n.- VII Topics in Function Theory on Strictly Pseudoconvex Domains.- Appendix A.- Appendix B.- Appendix C.- Glossary of Symbols and Notations.

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Book SynopsisChapter 1 First-order differential equations * Chapter 2 Second-order linear differential equations * Chapter 3 Systems of differential equations * Chapter 4 Qualitative theory of differential equations * Chapter 5 Separation of variables and Fourier series * Chapter 6 Sturm -Liouville boundary value problems * Appendix A Some simple facts concerning functions of several variables * Appendix B Sequences and series * Appendix C C Programs * Answers to odd-numbered exercises * IndexTable of ContentsChapter 1 First-order differential equations * Chapter 2 Second-order linear differential equations * Chapter 3 Systems of differential equations * Chapter 4 Qualitative theory of differential equations * Chapter 5 Separation of variables and Fourier series * Chapter 6 Sturm -Liouville boundary value problems * Appendix A Some simple facts concerning functions of several variables * Appendix B Sequences and series * Appendix C C Programs * Answers to odd-numbered exercises * Index

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Book SynopsisPraise for the Second Edition This book should be an essential part of the personal library of every practicing statistician.Technometrics Thoroughly revised and updated, the new edition of Nonparametric Statistical Methods includes additional modern topics and procedures, more practical data sets, and new problems from real-life situations. The book continues to emphasize the importance of nonparametric methods as a significant branch of modern statistics and equips readers with the conceptual and technical skills necessary to select and apply the appropriate procedures for any given situation. Written by leading statisticians, Nonparametric Statistical Methods, Third Edition provides readers with crucial nonparametric techniques in a variety of settings, emphasizing the assumptions underlying the methods. The book provides an extensive array of examples that clearly illustrate how to use nonparametric approaches for handling one- or Table of ContentsPreface xiii 1. Introduction 1 1.1. Advantages of Nonparametric Methods 1 1.2. The Distribution-Free Property 2 1.3. Some Real-World Applications 3 1.4. Format and Organization 6 1.5. Computing with R 8 1.6. Historical Background 9 2. The Dichotomous Data Problem 11 Introduction 11 2.1. A Binomial Test 11 2.2. An Estimator for the Probability of Success 22 2.3. A Confidence Interval for the Probability of Success (Wilson) 24 2.4. Bayes Estimators for the Probability of Success 33 3. The One-Sample Location Problem 39 Introduction 39 Paired Replicates Analyses by Way of Signed Ranks 39 3.1. A Distribution-Free Signed Rank Test (Wilcoxon) 40 3.2. An Estimator Associated with Wilcoxon’s Signed Rank Statistic (Hodges–Lehmann) 56 3.3. A Distribution-Free Confidence Interval Based on Wilcoxon’s Signed Rank Test (Tukey) 59 Paired Replicates Analyses by Way of Signs 63 3.4. A Distribution-Free Sign Test (Fisher) 63 3.5. An Estimator Associated with the Sign Statistic (Hodges–Lehmann) 76 3.6. A Distribution-Free Confidence Interval Based on the Sign Test (Thompson, Savur) 80 One-Sample Data 84 3.7. Procedures Based on the Signed Rank Statistic 84 3.8. Procedures Based on the Sign Statistic 90 3.9. An Asymptotically Distribution-Free Test of Symmetry (Randles–Fligner–Policello–Wolfe, Davis–Quade) 94 Bivariate Data 102 3.10. A Distribution-Free Test for Bivariate Symmetry (Hollander) 102 3.11. Efficiencies of Paired Replicates and One-Sample Location Procedures 112 4. The Two-Sample Location Problem 115 Introduction 115 4.1. A Distribution-Free Rank Sum Test (Wilcoxon, Mann and Whitney) 115 4.2. An Estimator Associated with Wilcoxon’s Rank Sum Statistic (Hodges–Lehmann) 136 4.3. A Distribution-Free Confidence Interval Based on Wilcoxon’s Rank Sum Test (Moses) 142 4.4. A Robust Rank Test for the Behrens–Fisher Problem (Fligner–Policello) 145 4.5. Efficiencies of Two-Sample Location Procedures 149 5. The Two-Sample Dispersion Problem and Other Two-Sample Problems 151 Introduction 151 5.1. A Distribution-Free Rank Test for Dispersion–Medians Equal (Ansari–Bradley) 152 5.2. An Asymptotically Distribution-Free Test for Dispersion Based on the Jackknife–Medians Not Necessarily Equal (Miller) 169 5.3. A Distribution-Free Rank Test for Either Location or Dispersion (Lepage) 181 5.4. A Distribution-Free Test for General Differences in Two Populations (Kolmogorov–Smirnov) 190 5.5. Efficiencies of Two-Sample Dispersion and Broad Alternatives Procedures 200 6. The One-Way Layout 202 Introduction 202 6.1. A Distribution-Free Test for General Alternatives (Kruskal–Wallis) 204 6.2. A Distribution-Free Test for Ordered Alternatives (Jonckheere–Terpstra) 215 6.3. Distribution-Free Tests for Umbrella Alternatives (Mack–Wolfe) 225 6.3A. A Distribution-Free Test for Umbrella Alternatives, Peak Known (Mack–Wolfe) 226 6.3B. A Distribution-Free Test for Umbrella Alternatives, Peak Unknown (Mack–Wolfe) 241 6.4. A Distribution-Free Test for Treatments Versus a Control (Fligner–Wolfe) 249 Rationale For Multiple Comparison Procedures 255 6.5. Distribution-Free Two-Sided All-Treatments Multiple Comparisons Based on Pairwise Rankings–General Configuration (Dwass, Steel, and Critchlow–Fligner) 256 6.6. Distribution-Free One-Sided All-Treatments Multiple Comparisons Based on Pairwise Rankings-Ordered Treatment Effects (Hayter–Stone) 265 6.7. Distribution-Free One-Sided Treatments-Versus-Control Multiple Comparisons Based on Joint Rankings (Nemenyi, Damico–Wolfe) 271 6.8. Contrast Estimation Based on Hodges–Lehmann Two-Sample Estimators (Spjøtvoll) 278 6.9. Simultaneous Confidence Intervals for All Simple Contrasts (Critchlow–Fligner) 282 6.10. Efficiencies of One-Way Layout Procedures 287 7. The Two-Way Layout 289 Introduction 289 7.1. A Distribution-Free Test for General Alternatives in a Randomized Complete Block Design (Friedman, Kendall-Babington Smith) 292 7.2. A Distribution-Free Test for Ordered Alternatives in a Randomized Complete Block Design (Page) 304 Rationale for Multiple Comparison Procedures 315 7.3. Distribution-Free Two-Sided All-Treatments Multiple Comparisons Based on Friedman Rank Sums–General Configuration (Wilcoxon, Nemenyi, McDonald-Thompson) 316 7.4. Distribution-Free One-Sided Treatments Versus Control Multiple Comparisons Based on Friedman Rank Sums (Nemenyi, Wilcoxon-Wilcox, Miller) 322 7.5. Contrast Estimation Based on One-Sample Median Estimators (Doksum) 328 Incomplete Block Data–Two-Way Layout with Zero or One Observation Per Treatment–Block Combination 331 7.6. A Distribution-Free Test for General Alternatives in a Randomized Balanced Incomplete Block Design (BIBD) (Durbin–Skillings–Mack) 332 7.7. Asymptotically Distribution-Free Two-Sided All-Treatments Multiple Comparisons for Balanced Incomplete Block Designs (Skillings–Mack) 341 7.8. A Distribution-Free Test for General Alternatives for Data From an Arbitrary Incomplete Block Design (Skillings–Mack) 343 Replications–Two-Way Layout with at Least One Observation for Every Treatment–Block Combination 354 7.9. A Distribution-Free Test for General Alternatives in a Randomized Block Design with an Equal Number c(>1) of Replications Per Treatment–Block Combination (Mack–Skillings) 354 7.10. Asymptotically Distribution-Free Two-Sided All-Treatments Multiple Comparisons for a Two-Way Layout with an Equal Number of Replications in Each Treatment–Block Combination (Mack–Skillings) 367 Analyses Associated with Signed Ranks 370 7.11. A Test Based on Wilcoxon Signed Ranks for General Alternatives in a Randomized Complete Block Design (Doksum) 370 7.12. A Test Based on Wilcoxon Signed Ranks for Ordered Alternatives in a Randomized Complete Block Design (Hollander) 376 7.13. Approximate Two-Sided All-Treatments Multiple Comparisons Based on Signed Ranks (Nemenyi) 379 7.14. Approximate One-Sided Treatments-Versus-Control Multiple Comparisons Based on Signed Ranks (Hollander) 382 7.15. Contrast Estimation Based on the One-Sample Hodges–Lehmann Estimators (Lehmann) 386 7.16. Efficiencies of Two-Way Layout Procedures 390 8. The Independence Problem 393 Introduction 393 8.1. A Distribution-Free Test for Independence Based on Signs (Kendall) 393 8.2. An Estimator Associated with the Kendall Statistic (Kendall) 413 8.3. An Asymptotically Distribution-Free Confidence Interval Based on the Kendall Statistic (Samara-Randles, Fligner–Rust, Noether) 415 8.4. An Asymptotically Distribution-Free Confidence Interval Based on Efron’s Bootstrap 420 8.5. A Distribution-Free Test for Independence Based on Ranks (Spearman) 427 8.6. A Distribution-Free Test for Independence Against Broad Alternatives (Hoeffding) 442 8.7. Efficiencies of Independence Procedures 450 9. Regression Problems 451 Introduction 451 One Regression Line 452 9.1. A Distribution-Free Test for the Slope of the Regression Line (Theil) 452 9.2. A Slope Estimator Associated with the Theil Statistic (Theil) 458 9.3. A Distribution-Free Confidence Interval Associated with the Theil Test (Theil) 460 9.4. An Intercept Estimator Associated with the Theil Statistic and Use of the Estimated Linear Relationship for Prediction (Hettmansperger–McKean–Sheather) 463 k(≥2) Regression Lines 466 9.5. An Asymptotically Distribution-Free Test for the Parallelism of Several Regression Lines (Sen, Adichie) 466 General Multiple Linear Regression 475 9.6. Asymptotically Distribution-Free Rank-Based Tests for General Multiple Linear Regression (Jaeckel, Hettmansperger–McKean) 475 Nonparametric Regression Analysis 490 9.7. An Introduction to Non-Rank-Based Approaches to Nonparametric Regression Analysis 490 9.8. Efficiencies of Regression Procedures 494 10. Comparing Two Success Probabilities 495 Introduction 495 10.1. Approximate Tests and Confidence Intervals for the Difference between Two Success Probabilities (Pearson) 496 10.2. An Exact Test for the Difference between Two Success Probabilities (Fisher) 511 10.3. Inference for the Odds Ratio (Fisher, Cornfield) 515 10.4. Inference for k Strata of 2 × 2 Tables (Mantel and Haenszel) 522 10.5. Efficiencies 534 11. Life Distributions and Survival Analysis 535 Introduction 535 11.1. A Test of Exponentiality Versus IFR Alternatives (Epstein) 536 11.2. A Test of Exponentiality Versus NBU Alternatives (Hollander–Proschan) 545 11.3. A Test of Exponentiality Versus DMRL Alternatives (Hollander–Proschan) 555 11.4. A Test of Exponentiality Versus a Trend Change in Mean Residual Life (Guess–Hollander–Proschan) 563 11.5. A Confidence Band for the Distribution Function (Kolmogorov) 568 11.6. An Estimator of the Distribution Function When the Data are Censored (Kaplan–Meier) 578 11.7. A Two-Sample Test for Censored Data (Mantel) 594 11.8. Efficiencies 605 12. Density Estimation 609 Introduction 609 12.1. Density Functions and Histograms 609 12.2. Kernel Density Estimation 617 12.3. Bandwidth Selection 624 12.4. Other Methods 628 13. Wavelets 629 Introduction 629 13.1. Wavelet Representation of a Function 630 13.2. Wavelet Thresholding 644 13.3. Other Uses of Wavelets in Statistics 655 14. Smoothing 656 Introduction 656 14.1. Local Averaging (Friedman) 657 14.2. Local Regression (Cleveland) 662 14.3. Kernel Smoothing 667 14.4. Other Methods of Smoothing 675 15. Ranked Set Sampling 676 Introduction 676 15.1. Rationale and Historical Development 676 15.2. Collecting a Ranked Set Sample 677 15.3. Ranked Set Sampling Estimation of a Population Mean 685 15.4. Ranked Set Sample Analogs of the Mann–Whitney–Wilcoxon Two-Sample Procedures (Bohn–Wolfe) 717 15.5. Other Important Issues for Ranked Set Sampling 737 15.6. Extensions and Related Approaches 742 16. An Introduction to Bayesian Nonparametric Statistics via the Dirichlet Process 744 Introduction 744 16.1. Ferguson’s Dirichlet Process 745 16.2. A Bayes Estimator of the Distribution Function (Ferguson) 749 16.3. Rank Order Estimation (Campbell and Hollander) 752 16.4. A Bayes Estimator of the Distribution When the Data are Right-Censored (Susarla and Van Ryzin) 755 16.5. Other Bayesian Approaches 759 Bibliography 763 R Program Index 791 Author Index 799 Subject Index 809

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Book SynopsisThis new edition continues to serve as a comprehensive guide to modern and classical methods of statistical computing. The book is comprised of four main parts spanning the field: Optimization Integration and Simulation Bootstrapping Density Estimation and Smoothing Within these sections,each chapter includes a comprehensive introduction and step-by-step implementation summaries to accompany the explanations of key methods. The new edition includes updated coverage and existing topics as well as new topics such as adaptive MCMC and bootstrapping for correlated data. The book website now includes comprehensive R code for the entire book. There are extensive exercises, real examples, and helpful insights about how to use the methods in practice.Table of ContentsPREFACE xv ACKNOWLEDGMENTS xvii 1 REVIEW 1 1.1 Mathematical Notation 1 1.2 Taylor’s Theorem and Mathematical Limit Theory 2 1.3 Statistical Notation and Probability Distributions 4 1.4 Likelihood Inference 9 1.5 Bayesian Inference 11 1.6 Statistical Limit Theory 13 1.7 Markov Chains 14 1.8 Computing 17 PART I OPTIMIZATION 2 OPTIMIZATION AND SOLVING NONLINEAR EQUATIONS 21 2.1 Univariate Problems 22 2.2 Multivariate Problems 34 Problems 54 3 COMBINATORIAL OPTIMIZATION 59 3.1 Hard Problems and NP-Completeness 59 3.2 Local Search 65 3.3 Simulated Annealing 68 3.4 Genetic Algorithms 75 3.5 Tabu Algorithms 85 Problems 92 4 EM OPTIMIZATION METHODS 97 4.1 Missing Data, Marginalization, and Notation 97 4.2 The EM Algorithm 98 4.3 EM Variants 111 Problems 121 PART II INTEGRATION AND SIMULATION 5 NUMERICAL INTEGRATION 129 5.1 Newton–Côtes Quadrature 129 5.2 Romberg Integration 139 5.3 Gaussian Quadrature 142 5.4 Frequently Encountered Problems 146 Problems 148 6 SIMULATION AND MONTE CARLO INTEGRATION 151 6.1 Introduction to the Monte Carlo Method 151 6.2 Exact Simulation 152 6.3 Approximate Simulation 163 6.4 Variance Reduction Techniques 180 Problems 195 7 MARKOV CHAIN MONTE CARLO 201 7.1 Metropolis–Hastings Algorithm 202 7.2 Gibbs Sampling 209 7.3 Implementation 218 Problems 230 8 ADVANCED TOPICS IN MCMC 237 8.1 Adaptive MCMC 237 8.2 Reversible Jump MCMC 250 8.3 Auxiliary Variable Methods 256 8.4 Other Metropolis–Hastings Algorithms 260 8.5 Perfect Sampling 264 8.6 Markov Chain Maximum Likelihood 268 8.7 Example: MCMC for Markov Random Fields 269 Problems 279 PART III BOOTSTRAPPING 9 BOOTSTRAPPING 287 9.1 The Bootstrap Principle 287 9.2 Basic Methods 288 9.3 Bootstrap Inference 292 9.4 Reducing Monte Carlo Error 302 9.5 Bootstrapping Dependent Data 303 9.6 Bootstrap Performance 315 9.7 Other Uses of the Bootstrap 316 9.8 Permutation Tests 317 Problems 319 PART IV DENSITY ESTIMATION AND SMOOTHING 10 NONPARAMETRIC DENSITY ESTIMATION 325 10.1 Measures of Performance 326 10.2 Kernel Density Estimation 327 10.3 Nonkernel Methods 341 10.4 Multivariate Methods 345 Problems 359 11 BIVARIATE SMOOTHING 363 11.1 Predictor–Response Data 363 11.2 Linear Smoothers 365 11.3 Comparison of Linear Smoothers 377 11.4 Nonlinear Smoothers 379 11.5 Confidence Bands 384 11.6 General Bivariate Data 388 Problems 389 12 MULTIVARIATE SMOOTHING 393 12.1 Predictor–Response Data 393 12.2 General Multivariate Data 413 Problems 416 DATA ACKNOWLEDGMENTS 421 REFERENCES 423 INDEX 457

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Book SynopsisIntroducing several important and useful methods for analyzing planar transmission line structures, this text discusses such topics as the theory and applications of Green's functions, the conformal mapping method, spectral domain methods, variational methods.Trade Review"...this book introduces the most commonly used techniques for analyzing microwave planar transmission live structures." (SciTech Book News, Vol. 25, No. 2, June 2001) "All important fundamental concepts and principles are covered as far as is possible with in a text of reasonable size...addresses student of electromagnetic theory...also...the engineer who is need of knowledge and practical, easy-to-apply formulas for the various line systems." (Measurement Science & Technology, Vol. 12, No. 10, October 2001) "...covers the analysis methods...from basics to advanced levels. All important fundamental concepts and principles are covered as far as is possible within a text of reasonable size." (Measurement Science & Technology, Vol. 12, No. 10, October 2001)Table of ContentsFundamentals of Electromagnetic Theory. Green's Function. Planar Transmission Lines. Conformal Mapping. Variational Methods. Spectral-Domain Method. Mode-Matching Method. Index.

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Book SynopsisContains 36 lectures solely on Fourier analysis and the FFT. Time and frequency domains, representation of waveforms in terms of complex exponentials and sinusoids, convolution, impulse response and the frequency transfer function, modulation and demodulation are among the topics covered. The text is linked to a complete FFT system on the accompanying disk where almost all of the exercises can be either carried out or verified. End-of-chapter exercises have been carefully constructed to serve as a development and consolidation of concepts discussed in the text.Table of ContentsCONTINUOUS FOURIER ANALYSIS. Background. Fourier Series for Periodic Functions. The Fourier Integral. Fourier Transforms of Some Important Functions. The Method of Successive Differentiation. Frequency-Domain Analysis. Time-Domain Analysis. The Properties. The Sampling Theorems. DISCRETE FOURIER ANALYSIS. The Discrete Fourier Transform. Inside the Fast Fourier Transform. The Discrete Fourier Transform as an Estimator. The Errors in Fast Fourier Transform Estimation. The Four Kinds of Convolution. Emulating Dirac Deltas and Differentiation on the Fast FourierTransform. THE USER'S MANUAL FOR THE ACCOMPANYING DISKS. Appendices. Answers to the Exercises. Index.

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Book SynopsisThe standard topics for a one-term undergraduate real analysis course are covered in this book. In addition, examples are given that show the ways in which real analysis is used in ordinary and partial differential equations, probability theory, numerical analysis, and number theory.Table of ContentsPreface Chapter 1 Preliminaries 1 The Real Numbers 1 Sets and Functions 6 Cardinality 15 Methods of Proof 20 Chapter 2 Sequences 27 Convergence 27 Limit Theorems 35 Two-state Markov Chains 40 Cauchy Sequences 44 Supremum and Infimum 52 The Bolzano-Weierstrass Theorem 55 The Quadratic Map 60 Projects 68 Chapter 3 The Riemann Integral 73 Continuity 73 Continuous Functions on Closed Intervals 80 The Riemann Integral 87 Numerical Methods 95 Discontinuities 103 Improper Integrals 113 Projects 119 Chapter 4 Differentiation 121 Differentiable Functions 121 The Fundamental Theorem of Calculus 129 Taylor’s Theorem 134 Newton’s Method 140 Inverse Functions 147 Functions of Two Variables 151 Projects 159 Chapter 5 Sequences of Functions 163 Pointwise and Uniform Convergence 163 Limit Theorems 169 The Supremum Norm 175 Integral Equations 183 The Calculus of Variations 188 Metric Spaces 196 The Contraction Mapping Principle 203 Normed Linear Spaces 210 Projects 219 Chapter 6 Series of Functions 223 Lim sup and Lim inf 223 Series of Real Constants 228 The Weierstrass M-test 238 Power Series 245 Complex Numbers 252 Infinite Products and Prime Numbers 260 Projects 270 Chapter 7 Differential Equations 273 Local Existence 273 Global Existence 283 The Error Estimate for Euler’s Method 289 Projects 296 Chapter 8 Complex Analysis 299 Analytic Functions 299 Integration on Paths 305 Cauchy's Theorem 312 Projects 320 Chapter 9 Fourier Series 323 The Heat Equation 323 Definitions and Examples 331 Pointwise Convergence 337 Mean-square Convergence 345 Projects 355 Chapter 10 Probability Theory 359 Discrete Random Variables 359 Coding Theory 368 Continuous Random Variables 376 The Variation Metric 386 Projects 398 Bibliography 403 Symbol Index 406 Index 409

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Book SynopsisEngineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems.Table of ContentsLinear Spaces. Hilbert Space. Least-Squares Estimation. Dual Spaces. Linear Operators and Adjoints. Optimization of Functionals. Global Theory of Constrained Optimization. Local Theory of Constrained Optimization. Iterative Methods of Optimization. Indexes.

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Book SynopsisThe authors are pioneering a new approach to classifying existing chemometric techniques for data analysis in one and two dimensions, using a practical applications approach to illustrating chemical examples and problems. Written in a simple, balanced, applications-based style, the book will appeal to both theorists and non-mathematicians.Trade Review"Statisticians, biochemists, engineers, and health researchers will benefit a lot from this wonderful book." (Journal of Statistical Computation and Simulation, November 2005) "...quite useful for persons who apply signal processing methods in chemistry." (Technometrics, May 2005) "…my overall impression of the text is favorable…I would recommend this book to chemists who are interested in using wavelets in their research and to faculty…" (Journal of the American Chemical Society, February 23, 2005) "I recommend this book to chemists who are interested in using wavelets in their research and to faculty who would like to teach graduate students about signal processing..." (Analytical Chemistry, February 1, 2005) "The presentation of information makes it easy for reader to find the relevant information. The text is well-written and understandable." (E-STREAMS, October 2004)Table of ContentsPreface xiii Chapter 1 Introduction 1 1.1. Modern Analytical Chemistry 1 1.1.1. Developments in Modern Chemistry 1 1.1.2. Modern Analytical Chemistry 2 1.1.3. Multidimensional Dataset 3 1.2. Chemometrics 5 1.2.1. Introduction to Chemometrics 5 1.2.2. Instrumental Response and Data Processing 8 1.2.3. White, Black, and Gray Systems 9 1.3. Chemometrics-Based Signal Processing Techniques 10 1.3.1. Common Methods for Processing Chemical Data 10 1.3.2. Wavelets in Chemistry 11 1.4. Resources Available on Chemometrics and Wavelet Transform 12 1.4.1. Books 12 1.4.2. Online Resources 14 1.4.3. Mathematics Software 15 Chapter 2 One-dimensional Signal Processing Techniques in Chemistry 23 2.1. Digital Smoothing and Filtering Methods 23 2.1.1. Moving-Window Average Smoothing Method 24 2.1.2. Savitsky-Golay Filter 25 2.1.3. Kalman Filtering 32 2.1.4. Spline Smoothing 36 2.2. Transformation Methods of Analytical Signals 39 2.2.1. Physical Meaning of the Convolution Algorithm 39 2.2.2. Multichannel Advantage in Spectroscopy and Hadamard Transformation 41 2.2.3. Fourier Transformation 44 2.2.3.1. Discrete Fourier Transformation and Spectral Multiplex Advantage 45 2.2.3.2. Fast Fourier Transformation 48 2.2.3.3. Fourier Transformation as Applied to Smooth Analytical Signals 50 2.2.3.4. Fourier Transformation as Applied to Convolution and Deconvolution 52 2.3. Numerical Differentiation 54 2.3.1. Simple Difference Method 54 2.3.2. Moving-Window Polynomial Least-Squares Fitting Method 55 2.4. Data Compression 57 2.4.1. Data Compression Based on B-Spline Curve Fitting 57 2.4.2. Data Compression Based on Fourier Transformation 64 2.4.3. Data Compression Based on Principal-Component Analysis 64 Chapter 3 Two-dimensional Signal Processing Techniques in Chemistry 69 3.1. General Features of Two-Dimensional Data 69 3.2. Some Basic Concepts for Two-Dimensional Data from Hyphenated Instrumentation 70 3.2.1. Chemical Rank and Principal-Component Analysis (PCA) 71 3.2.2. Zero-Component Regions and Estimation of Noise Level and Background 75 3.3. Double-Centering Technique for Background Correction 77 3.4. Congruence Analysis and Least-Squares Fitting 78 3.5. Differentiation Methods for Two-Dimensional Data 80 3.6 Resolution Methods for Two-Dimensional Data 81 3.6.1. Local Principal-Component Analysis and Rankmap 83 3.6.2. Self-Modeling Curve Resolution and Evolving Resolution Methods 85 3.6.2.1. Evolving Factor Analysis (EFA) 88 3.6.2.2. Window Factor Analysis (WFA) 90 3.6.2.3. Heuristic Evolving Latent Projections (HELP) 94 Chapter 4 Fundamentals of Wavelet Transform 99 4.1. Introduction to Wavelet Transform and Wavelet Packet Transform 100 4.1.1. A Simple Example: Haar Wavelet 103 4.1.2. Multiresolution Signal Decomposition 108 4.1.3. Basic Properties of Wavelet Function 112 4.2. Wavelet Function Examples 113 4.2.1. Meyer Wavelet 113 4.2.2. B-Spline (Battle--Lemarié) Wavelets 114 4.2.3. Daubechies Wavelets 116 4.2.4. Coiflet Functions 117 4.3. Fast Wavelet Algorithm and Packet Algorithm 118 4.3.1. Fast Wavelet Transform 119 4.3.2. Inverse Fast Wavelet Transform 122 4.3.3. Finite Discrete Signal Handling with Wavelet Transform 125 4.3.4. Packet Wavelet Transform 132 4.4. Biorthogonal Wavelet Transform 134 4.4.1. Multiresolution Signal Decomposition of Biorthogonal Wavelet 134 4.4.2. Biorthogonal Spline Wavelets 136 4.4.3. A Computing Example 137 4.5. Two-Dimensional Wavelet Transform 140 4.5.1. Multidimensional Wavelet Analysis 140 4.5.2. Implementation of Two-Dimensional Wavelet Transform 141 Chapter 5 Application of Wavelet Transform In Chemistry 147 5.1. Data Compression 148 5.1.1. Principle and Algorithm 149 5.1.2. Data Compression Using Wavelet Packet Transform 155 5.1.3. Best-Basis Selection and Criteria for Coefficient Selection 158 5.2. Data Denoising and Smoothing 166 5.2.1. Denoising 167 5.2.2. Smoothing 173 5.2.3. Denoising and Smoothing Using Wavelet Packet Transform 179 5.2.4. Comparison between Wavelet Transform and Conventional Methods 182 5.3. Baseline/Background Removal 183 5.3.1. Principle and Algorithm 184 5.3.2. Background Removal 185 5.3.3. Baseline Correction 191 5.3.4. Background Removal Using Continuous Wavelet Transform 191 5.3.5. Background Removal of Two-Dimensional Signals 196 5.4. Resolution Enhancement 199 5.4.1. Numerical Differentiation Using Discrete Wavelet Transform 200 5.4.2. Numerical Differentiation Using Continuous Wavelet Transform 205 5.4.3. Comparison between Wavelet Transform and other Numerical Differentiation Methods 210 5.4.4. Resolution Enhancement 212 5.4.5. Resolution Enhancement by Using Wavelet Packet Transform 220 5.4.6. Comparison between Wavelet Transform and Fast Fourier Transform for Resolution Enhancement 221 5.5. Combined Techniques 225 5.5.1. Combined Method for Regression and Calibration 225 5.5.2. Combined Method for Classification and Pattern Recognition 227 5.5.3. Combined Method of Wavelet Transform and Chemical Factor Analysis 228 5.5.4. Wavelet Neural Network 230 5.6. An Overview of the Applications in Chemistry 232 5.6.1. Flow Injection Analysis 233 5.6.2. Chromatography and Capillary Electrophoresis 234 5.6.3. Spectroscopy 238 5.6.4. Electrochemistry 244 5.6.5. Mass Spectrometry 246 5.6.6. Chemical Physics and Quantum Chemistry 248 5.6.7. Conclusion 249 Appendix Vector and Matrix Operations and Elementary MATLAB 257 A.1. Elementary Knowledge in Linear Algebra 257 A.1.1. Vectors and Matrices in Analytical Chemistry 257 A.1.2. Column and Row Vectors 259 A.1.3. Addition and Subtraction of Vectors 259 A.1.4. Vector Direction and Length 260 A.1.5. Scalar Multiplication of Vectors 261 A.1.6. Inner and Outer Products between Vectors 262 A.1.7. The Matrix and Its Operations 263 A.1.8. Matrix Addition and Subtraction 264 A.1.9. Matrix Multiplication 264 A.1.10. Zero Matrix and Identity Matrix 264 A.1.11. Transpose of a Matrix 265 A.1.12. Determinant of a Matrix 265 A.1.13. Inverse of a Matrix 266 A.1.14. Orthogonal Matrix 266 A.1.15. Trace of a Square Matrix 267 A.1.16. Rank of a Matrix 268 A.1.17. Eigenvalues and Eigenvectors of a Matrix 268 A.1.18. Singular-Value Decomposition 269 A.1.19. Generalized Inverse 270 A.1.20. Derivative of a Matrix 271 A.1.21. Derivative of a Function with Vector as Variable 271 A.2. Elementary Knowledge of MATLAB 273 A.2.1. Matrix Construction 275 A.2.2. Matrix Manipulation 275 A.2.3. Basic Mathematical Functions 276 A.2.4. Methods for Generating Vectors and Matrices 278 A.2.5. Matrix Subscript System 280 A.2.6. Matrix Decomposition 286 A.2.6.1. Singular-Value Decomposition (SVD) 286 A.2.6.2. Eigenvalues and Eigenvectors (eig) 287 A.2.7. Graphic Functions 288 Index 293

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Book SynopsisA timely book on a topic that has witnessed a surge of interest over the last decade, owing in part to several novel applications, most notably in data compression and computational molecular biology. It describes methods employed in average case analysis of algorithms, combining both analytical and probabilistic tools in a single volume.Trade Review"Surveying the major techniques of average case analysis, this graduate textbook presents both analytical methods used for well-structured algorithms and probabilistic methods used for more structurally complex algorithms." (SciTech Book News, Vol. 25, No. 3, September 2001) "...contains a comprehensive treatment on probabilistic, combinatorial, and analytical techniques and methods...treatment is clear, rigorous, self-contained, with many examples and exercises." (Zentralblatt MATH Vol. 968, 2001/18) "This well-organized book...is certainly useful...It is a valuable source for a deeper and more precise understanding of the behaviors of algorithms on sequences." (Mathematical Reviews, 2002f)Table of ContentsForeword. Preface. Acknowledgments. PROBLEMS ON WORDS. Data Structures and Algorithms on Words. Probabilistic and Analytical Models. PROBABILISTIC AND COMBINATORIAL TECHNIQUES. Inclusion-Exclusion Principle. The First and Second Moment Methods. Subadditive Ergodic Theorem and Large Deviations. Elements of Information Theory. ANALYTIC TECHNIQUES. Generating Functions. Complex Asymptotic Methods. Mellin Transform and Its Applications. Analytic Poissonization and Depoissonization. Bibliography. Index.

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Book SynopsisAn accessible, clearly organized survey of the basic topics of measure theory for students and researchers in mathematics, statistics, and physics In order to fully understand and appreciate advanced probability, analysis, and advanced mathematical statistics, a rudimentary knowledge of measure theory and like subjects must first be obtained.Trade Review"…an excellent read…I was impressed with the wealth of information and the amount of flawless detail." (Journal of the American Statistical Association, March 2006) “…contains many really good exercises…the style is clear and the notation appropriate…” (Zentralbaltt MATH, May 2005)Table of ContentsPreface. Acknowledgments. 1. Set Systems. 2. Measures. 3. Extensions of Measures. 4. Lebesgue Measure. 5. Measurable Functions. 6. The Lebesgue Integral. 7. Integrals Relative to Lebesgue Measure. 8. The Lp Spaces. 9. The Radon–Nikodym Theorem. 10. Products of Two Measure Spaces. 11. Arbitrary Products of Measure Spaces. References. Index.

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Book SynopsisRegression analysis is the study of the dependence of a response variable on one or more predictor variables. It is among the most widely used methods in statistics. In recent years, several new ways to approach regression have been presented.Trade Review"...with its up-to-date discussion of regression graphics at a very accessible level, Applied Regression Including Computing and Graphics is a must for everyone working in the area of regression analysis. I strongly recommend it as a text..." (Journal of the American Statistical Association, September 2001) "...a must for everyone working in the area of regression analysis. I strongly recommend it as a text..." (Journal of the American Statistical Association, September 2001)Table of ContentsLooking Forward and Back. Introduction to Regression. Introduction to Smoothing. Bivariate Distributions. Two-Dimensional Plots. TOOLS. Simple Linear Regression. Introduction to Multiple Linear Regression. Three-Dimensional Plots. Weights and Lack-of-Fit. Understanding Coefficients. Relating Mean Functions. Factors and Interactions. Response Transformations. Diagnostics I: Curvature and Nonconstant Variance. Diagnostics II: Influence and Outliers. Predictor Transformations. Model Assessment. REGRESSION GRAPHICS. Visualizing Regression. Visualizing Regression with Many Predictors. Graphical Regression. LOGISTIC REGRESSION AND GENERALIZED LINEAR MODELS. Binomial Regression. Graphical and Diagnostic Methods for Logistic Regression. Generalized Linear Models. Appendix. References. Indexes.

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Book SynopsisDiscusses techniques that have important applications to wireless engineering.Table of ContentsPreface xv 1 Notations and Mathematical Preliminaries 1 1.1 Notations and Abbreviations 1 1.2 Mathematical Preliminaries 2 1.2.1 Functions and Integration 2 1.2.2 The Fourier Transform 4 1.2.3 Regularity 4 1.2.4 Linear Spaces 7 1.2.5 Functional Spaces 8 1.2.6 Sobolev Spaces 10 1.2.7 Bases in Hilbert Space H 11 1.2.8 Linear Operators 12 Bibliography 14 2 Intuitive Introduction to Wavelets 15 2.1 Technical History and Background 15 2.1.1 Historical Development 15 2.1.2 When Do Wavelets Work? 16 2.1.3 A Wave Is a Wave but What Is a Wavelet? 17 2.2 What Can Wavelets Do in Electromagnetics and Device Modeling? 18 2.2.1 Potential Benefits of Using Wavelets 18 2.2.2 Limitations and Future Direction of Wavelets 19 2.3 The Haar Wavelets and Multiresolution Analysis 20 2.4 How Do Wavelets Work? 23 Bibliography 28 3 Basic Orthogonal Wavelet Theory 30 3.1 Multiresolution Analysis 30 3.2 Construction of Scalets 3.2.1 Franklin Scalet 32 3.2.2 Battle-Lemarie Scalets 39 3.2.3 Preliminary Properties of Scalets 40 3.3 Wavelet ^ ( r ) 42 3.4 Franklin Wavelet 48 3.5 Properties of Scalets (p(co) 51 3.6 Daubechies Wavelets 56 3.7 Coifman Wavelets (Coiflets) 64 3.8 Constructing Wavelets by Recursion and Iteration 69 3.8.1 Construction of Scalets 69 3.8.2 Construction of Wavelets 74 3.9 Meyer Wavelets 75 3.9.1 Basic Properties of Meyer Wavelets 75 3.9.2 Meyer Wavelet Family 83 3.9.3 Other Examples of Meyer Wavelets 92 3.10 Mallat's Decomposition and Reconstruction 92 3.10.1 Reconstruction 92 3.10.2 Decomposition 93 3.11 Problems 95 3.11.1 Exercise 1 95 3.11.2 Exercise 2 95 3.11.3 Exercise 3 97 3.11.4 Exercise 4 97 Bibliography 98 4 Wavelets in Boundary Integral Equations 100 4.1 Wavelets in Electromagnetics 100 4.2 Linear Operators 102 4.3 Method of Moments (MoM) 103 4.4 Functional Expansion of a Given Function 107 4.5 Operator Expansion: Nonstandard Form 110 4.5.1 Operator Expansion in Haar Wavelets 111 4.5.2 Operator Expansion in General Wavelet Systems 113 4.5.3 Numerical Example 114 4.6 Periodic Wavelets 120 4.6.1 Construction of Periodic Wavelets 120 4.6.2 Properties of Periodic Wavelets 123 4.6.3 Expansion of a Function in Periodic Wavelets 127 4.7 Application of Periodic Wavelets: 2D Scattering 128 4.8 Fast Wavelet Transform (FWT) 133 4.8.1 Discretization of Operation Equations 133 4.8.2 Fast Algorithm 134 4.8.3 Matrix Sparsification Using FWT 135 4.9 Applications of the FWT 140 4.9.1 Formulation 140 4.9.2 Circuit Parameters 141 4.9.3 Integral Equations and Wavelet Expansion 143 4.9.4 Numerical Results 144 4.10 Intervallic Coifman Wavelets 144 4.10.1 Intervallic Scalets 145 4.10.2 Intervallic Wavelets on [0, 1] 154 4.11 Lifting Scheme and Lazy Wavelets 156 4.11.1 Lazy Wavelets 156 4.11.2 Lifting Scheme Algorithm 157 4.11.3 Cascade Algorithm 159 4.12 Green's Scalets and Sampling Series 159 4.12.1 Ordinary Differential Equations (ODEs) 160 4.12.2 Partial Differential Equations (PDEs) 166 4.13 Appendix: Derivation of Intervallic Wavelets on [0, 1] 172 4.14 Problems 185 4.14.1 Exercise 5 185 4.14.2 Exercise 6 185 4.14.3 Exercise 7 185 4.14.4 Exercise 8 186 4.14.5 Project 1 187 Bibliography 187 5 Sampling Biorthogonal Time Domain Method (SBTD) 189 5.1 Basis FDTD Formulation 189 5.2 Stability Analysis for the FDTD 194 5.3 FDTD as Maxwell's Equations with Haar Expansion 198 5.4 FDTD with Battle-Lemarie Wavelets 201 5.5 Positive Sampling and Biorthogonal Testing Functions 205 5.6 Sampling Biorthogonal Time Domain Method 215 5.6.1 SBTD versus MRTD 215 5.6.2 Formulation 215 5.7 Stability Conditions for Wavelet-Based Methods 219 5.7.1 Dispersion Relation and Stability Analysis 219 5.7.2 Stability Analysis for the SBTD 222 5.8 Convergence Analysis and Numerical Dispersion 223 5.8.1 Numerical Dispersion 223 5.8.2 Convergence Analysis 225 5.9 Numerical Examples 228 5.10 Appendix: Operator Form of the MRTD 233 5.11 Problems 236 5.11.1 Exercise 9 236 5.11.2 Exercise 10 237 5.11.3 Project 2 237 Bibliography 238 6 Canonical Multiwavelets 240 6.1 Vector-Matrix Dilation Equation 240 6.2 Time Domain Approach 242 6.3 Construction of Multiscalets 245 6.4 Orthogonal Multiwavelets yjr(t) 255 6.5 Intervallic Multiwavelets xj/(t) 258 6.6 Multiwavelet Expansion 261 6.7 Intervallic Dual Multiwavelets \j/(t) 264 6.8 Working Examples 269 6.9 Multiscalet-Based ID Finite Element Method (FEM) 276 6.10 Multiscalet-Based Edge Element Method 280 6.11 Spurious Modes 285 6.12 Appendix 287 6.13 Problems 296 6.13.1 Exercise 11 296 Bibliography 297 7 Wavelets in Scattering and Radiation 299 7.1 Scattering from a 2D Groove 299 7.1.1 Method of Moments (MoM) Formulation 300 7.1.2 Coiflet-Based MoM 304 7.1.3 Bi-CGSTAB Algorithm 305 7.1.4 Numerical Results 305 7.2 2D and 3D Scattering Using Intervallic Coiflets 309 7.2.1 Intervallic Scalets on [0,1] 309 7.2.2 Expansion in Coifman Intervallic Wavelets 312 7.2.3 Numerical Integration and Error Estimate 313 7.2.4 Fast Construction of Impedance Matrix 317 7.2.5 Conducting Cylinders, TM Case 319 7.2.6 Conducting Cylinders with Thin Magnetic Coating 322 7.2.7 Perfect Electrically Conducting (PEC) Spheroids 324 7.3 Scattering and Radiation of Curved Thin Wires 329 7.3.1 Integral Equation for Curved Thin-Wire Scatterers and Antennae 330 7.3.2 Numerical Examples 331 7.4 Smooth Local Cosine (SLC) Method 340 7.4.1 Construction of Smooth Local Cosine Basis 341 7.4.2 Formulation of 2D Scattering Problems 344 7.4.3 SLC-Based Galerkin Procedure and Numerical Results 347 7.4.4 Application of the SLC to Thin-Wire Scatterers and Antennas 355 7.5 Microstrip Antenna Arrays 357 7.5.1 Impedance Matched Source 358 7.5.2 Far-Zone Fields and Antenna Patterns 360 Bibliography 363 8 Wavelets in Rough Surface Scattering 366 8.1 Scattering of EM Waves from Randomly Rough Surfaces 366 8.2 Generation of Random Surfaces 368 8.2.1 Autocorrelation Method 370 8.2.2 Spectral Domain Method 373 8.3 2D Rough Surface Scattering 376 8.3.1 Moment Method Formulation of 2D Scattering 376 8.3.2 Wavelet-Based Galerkin Method for 2D Scattering 380 8.3.3 Numerical Results of 2D Scattering 381 8.4 3D Rough Surface Scattering 387 8.4.1 Tapered Wave of Incidence 388 8.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets 391 8.4.3 Numerical Results of 3D Scattering 394 Bibliography 399 9 Wavelets in Packaging, Interconnects, and EMC 401 9.1 Quasi-static Spatial Formulation 402 9.1.1 What Is Quasi-static? 402 9.1.2 Formulation 403 9.1.3 Orthogonal Wavelets in L2([0, 1]) 406 9.1.4 Boundary Element Method and Wavelet Expansion 408 9.1.5 Numerical Examples 412 9.2 Spatial Domain Layered Green's Functions 415 9.2.1 Formulation 417 9.2.2 Prony's Method 423 9.2.3 Implementation of the Coifman Wavelets 424 9.2.4 Numerical Examples 426 9.3 Skin-Effect Resistance and Total Inductance 429 9.3.1 Formulation 431 9.3.2 Moment Method Solution of Coupled Integral Equations 433 9.3.3 Circuit Parameter Extraction 435 9.3.4 Wavelet Implementation 437 9.3.5 Measurement and Simulation Results 438 9.4 Spectral Domain Green's Function-Based Full-Wave Analysis 440 9.4.1 Basic Formulation 440 9.4.2 Wavelet Expansion and Matrix Equation 444 9.4.3 Evaluation of Sommerfeld-Type Integrals 447 9.4.4 Numerical Results and Sparsity of Impedance Matrix 451 9.4.5 Further Improvements 455 9.5 Full-Wave Edge Element Method for 3D Lossy Structures 455 9.5.1 Formulation of Asymmetric Functionals with Truncation Conditions 456 9.5.2 Edge Element Procedure 460 9.5.3 Excess Capacitance and Inductance 464 9.5.4 Numerical Examples 466 Bibliography 469 10 Wavelets in Nonlinear Semiconductor Devices 474 10.1 Physical Models and Computational Efforts 474 10.2 An Interpolating Subdivision Scheme 476 10.3 The Sparse Point Representation (SPR) 478 10.4 Interpolation Wavelets in the FDM 479 10.4.1 ID Example of the SPR Application 480 10.4.2 2D Example of the SPR Application 481 10.5 The Drift-Diffusion Model 484 10.5.1 Scaling 486 10.5.2 Discretization 487 10.5.3 Transient Solution 489 10.5.4 Grid Adaptation and Interpolating Wavelets 490 10.5.5 Numerical Results 492 10.6 Multiwavelet Based Drift-Diffusion Model 498 10.6.1 Precision and Stability versus Reynolds 499 10.6.2 MWFEM-Based ID Simulation 502 10.7 The Boltzmann Transport Equation (BTE) Model 504 10.7.1 Why BTE? 505 10.7.2 Spherical Harmonic Expansion of the BTE 505 10.7.3 Arbitrary Order Expansion and Galerkin's Procedure 509 10.7.4 The Coupled Boltzmann-Poisson System 515 10.7.5 Numerical Results 517 Bibliography 524 Index 527

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Book SynopsisA comprehensive solutions manual for students using the Vector Calculus text This book gives a comprehensive and thorough introduction to ideas and major results of the theory of functions of several variables and of modern vector calculus in two and three dimensions. Clear and easy-to-follow writing style, carefully crafted examples, wide spectrum of applications and numerous illustrations, diagrams, and graphs invite students to use the textbook actively, helping them to both enforce their understanding of the material and to brush up on necessary technical and computational skills. The Student Solutions Manual to Accompany Vector Calculus also pays particular attention to material that some students find challenging, such as the chain rule, Implicit Function Theorem, parametrizations, or the Change of Variables Theorem.

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Book SynopsisWith applications in pattern recognition, data compression and numerical analysis, the wavelet transform is a key area of modern mathematics that brings new approaches to the analysis and synthesis of signals. This book presents the central issues and emphasizes comparison, assessment and how to combine method and application. It reviews different approaches to guide researchers to appropriate classes of techniques.Table of ContentsPreface ix Notation xi Introduction xv 1 The Continuous Wavelet Transform 1 1.1. Definition and Elementary Properties 1 1.2 Affine Operators 10 1.3 Filter Properties of the Wavelet Transform 12 1.4 Approximation Properties 22 1.5 Decay Behaviour 32 1.6 Group-Theoretical Foundations and Generalizations 36 1.7 Extension of the One-Dimensional Wavelet Transform to Sobolev Spaces 59 Exercises 69 2 The Discrete Wavelet Transform 73 2.1 Wavelet Frames 73 2.2 Multiscale Analysis 97 2.3 Fast Wavelet Transform 121 2.4 One-Dimensional Orthogonal Wavelets 131 2.5 Two-Dimensional Orthogonal Wavelets 203 Exercises 226 3 Applications of the Wavelet Transform 231 3.1 Wavelet Analysis of One-Dimensional Signals 231 3.2 Quality Control of Texture 235 3.3 Data Compression in Digital Image Processing 239 3.4 Regularization of Inverse Problems 251 3.5 Wavelet – Galerkin Methods for Two-Point boundary Value Problems 259 3.6 Schwarz Iterations Based on Wavelet Decompositions 278 3.7 An Outlook on Two-Dimensional Boundary Value Problems 300 Exercises 306 Appendix The Fourier Transform 309 References 313 Index 321

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Book SynopsisThe theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and, in particular, to elliptic modular forms with emphasis on their number-theoretical aspects.Table of Contents* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

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Book SynopsisAn elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Developing, with many examples, the elementary theory of elliptic curves, this book goes on to the subject of modular forms and the first connections with elliptic curves.

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Book SynopsisAddressing the topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. It includes fixed point theorems and applications to functional equations and optimization theory.Trade Review"The book is intended as a textbook on real analysis for graduate students in economics. It is largely graduate level mathematics, and the students should have a solid undergraduate real analysis background... The author's writing style is ... in general quite attractive. The book should be quite successful for its intended purpose."--Gerald A. Heuer, Zentralblatt MATH "Important and commendable, this indispensable resource should be highly prized by all concerned with courses on mathematics for economists and by graduate students working on economic theory. Rarely do books meet such high aspirations and carry out their aims, yet this one certainly does. Well written in an engaging style and impressively researched in the requirements of graduate students of economics and finance, Real Analysis with Economic Applications is sure to become the definitive work for its intended audience. Real Analysis with Economic Applications with its large number of economics applications and variety of exercises represents the single most important mathematical source for students of economics applications and it will be the book, for a long time to come, to which they will turn with confidence, as well as pleasure, in all questions of economic applications."--Current Engineering PracticeTable of ContentsPreface xvii Prerequisites xxvii Basic Conventions xxix Part I: SET THEORY 1 Chapter A: Preliminaries of Real Analysis 3 A.1 Elements of Set Theory 4 A.1.1 Sets 4 A.1.2 Relations 9 A.1.3 Equivalence Relations 11 A.1.4 Order Relations 14 A.1.5 Functions 20 A.1.6 Sequences, Vectors, and Matrices 27 A.1.7* A Glimpse of Advanced Set Theory: The Axiom of Choice 29 A.2 Real Numbers 33 A.2.1 Ordered Fields 33 A.2.2 Natural Numbers, Integers, and Rationals 37 A.2.3 Real Numbers 39 A.2.4 Intervals and R 44 A.3 Real Sequences 46 A.3.1 Convergent Sequences 46 A.3.2 Monotonic Sequences 50 A.3.3 Subsequential Limits 53 A.3.4 Infinite Series 56 A.3.5 Rearrangement of Infinite Series 59 A.3.6 Infinite Products 61 A.4 Real Functions 62 A.4.1 Basic Definitions 62 A.4.2 Limits, Continuity, and Differentiation 64 A.4.3 Riemann Integration 69 A.4.4 Exponential, Logarithmic, and Trigonometric Functions 74 A.4.5 Concave and Convex Functions 77 A.4.6 Quasiconcave and Quasiconvex Functions 80 Chapter B: Countability 82 B.1 Countable and Uncountable Sets 82 B.2 Losets and Q 90 B.3 Some More Advanced Set Theory 93 B.3.1 The Cardinality Ordering 93 B.3.2* The Well-Ordering Principle 98 B.4 Application: Ordinal Utility Theory 99 B.4.1 Preference Relations 100 B.4.2 Utility Representation of Complete Preference Relations 102 B.4.3* Utility Representation of Incomplete Preference Relations 107 Part II: ANALYSIS ON METRIC SPACES 115 Chapter C: Metric Spaces 117 C.1 Basic Notions 118 C.1.1 Metric Spaces: Definition and Examples 119 C.1.2 Open and Closed Sets 127 C.1.3 Convergent Sequences 132 C.1.4 Sequential Characterization of Closed Sets 134 C.1.5 Equivalence of Metrics 136 C.2 Connectedness and Separability 138 C.2.1 Connected Metric Spaces 138 C.2.2 Separable Metric Spaces 140 C.2.3 Applications to Utility Theory 145 C.3 Compactness 147 C.3.1 Basic Definitions and the Heine-Borel Theorem 148 C.3.2 Compactness as a Finite Structure 151 C.3.3 Closed and Bounded Sets 154 C.4 Sequential Compactness 157 C.5 Completeness 161 C.5.1 Cauchy Sequences 161 C.5.2 Complete Metric Spaces: Definition and Examples 163 C.5.3 Completeness versus Closedness 167 C.5.4 Completeness versus Compactness 171 C.6 Fixed Point Theory I 172 C.6.1 Contractions 172 C.6.2 The Banach Fixed Point Theorem 175 C.6.3* Generalizations of the Banach Fixed Point Theorem 179 C.7 Applications to Functional Equations 183 C.7.1 Solutions of Functional Equations 183 C.7.2 Picard's Existence Theorems 187 C.8 Products of Metric Spaces 192 C.8.1 Finite Products 192 C.8.2 Countably Infinite Products 193 Chapter D: Continuity I 200 D.1 Continuity of Functions 201 D.1.1 Definitions and Examples 201 D.1.2 Uniform Continuity 208 D.1.3 Other Continuity Concepts 210 D.1.4* Remarks on the Differentiability of Real Functions 212 D.1.5 A Fundamental Characterization of Continuity 213 D.1.6 Homeomorphisms 216 D.2 Continuity and Connectedness 218 D.3 Continuity and Compactness 222 D.3.1 Continuous Image of a Compact Set 222 D.3.2 The Local-to-Global Method 223 D.3.3 Weierstrass' Theorem 225 D.4 Semicontinuity 229 D.5 Applications 237 D.5.1* Caristi's Fixed Point Theorem 238 D.5.2 Continuous Representation of a Preference Relation 239 D.5.3* Cauchy's Functional Equations: Additivity on Rn 242 D.5.4* Representation of Additive Preferences 247 D.6 CB(T) and Uniform Convergence 249 D.6.1 The Basic Metric Structure of CB(T) 249 D.6.2 Uniform Convergence 250 D.6.3* The Stone-Weierstrass Theorem and Separability of C(T) 257 D.6.4* The Arzela-Ascoli Theorem 262 D.7* Extension of Continuous Functions 266 D.8 Fixed Point Theory II 272 D.8.1 The Fixed Point Property 273 D.8.2 Retracts 274 D.8.3 The Brouwer Fixed Point Theorem 277 D.8.4 Applications 280 Chapter E: Continuity II 283 E.1 Correspondences 284 E.2 Continuity of Correspondences 287 E.2.1 Upper Hemicontinuity 287 E.2.2 The Closed Graph Property 294 E.2.3 Lower Hemicontinuity 297 E.2.4 Continuous Correspondences 300 E.2.5* The Hausdorff Metric and Continuity 302 E.3 The Maximum Theorem 306 E.4 Application: Stationary Dynamic Programming 311 E.4.1 The Standard Dynamic Programming Problem 312 E.4.2 The Principle of Optimality 315 E.4.3 Existence and Uniqueness of an Optimal Solution 320 E.4.4 Application: The Optimal Growth Model 324 E.5 Fixed Point Theory III 330 E.5.1 Kakutani's Fixed Point Theorem 331 E.5.2* Michael's Selection Theorem 333 E.5.3* Proof of Kakutani's Fixed Point Theorem 339 E.5.4* Contractive Correspondences 341 E.6 Application: The Nash Equilibrium 343 E.6.1 Strategic Games 343 E.6.2 The Nash Equilibrium 346 E.6.3* Remarks on the Equilibria of Discontinuous Games 351 Part III: ANALYSIS ON LINEAR SPACES 355 Chapter F: Linear Spaces 357 F.1 Linear Spaces 358 F.1.1 Abelian Groups 358 F.1.2 Linear Spaces: Definition and Examples 360 F.1.3 Linear Subspaces, Affine Manifolds, and Hyperplanes 364 F.1.4 Span and Affine Hull of a Set 368 F.1.5 Linear and Affine Independence 370 F.1.6 Bases and Dimension 375 F.2 Linear Operators and Functionals 382 F.2.1 Definitions and Examples 382 F.2.2 Linear and Affine Functions 386 F.2.3 Linear Isomorphisms 389 F.2.4 Hyperplanes, Revisited 392 F.3 Application: Expected Utility Theory 395 F.3.1 The Expected Utility Theorem 395 F.3.2 Utility Theory under Uncertainty 403 F.4* Application: Capacities and the Shapley Value 409 F.4.1 Capacities and Coalitional Games 410 F.4.2 The Linear Space of Capacities 412 F.4.3 The Shapley Value 415 Chapter G: Convexity 422 G.1 Convex Sets 423 G.1.1 Basic Definitions and Examples 423 G.1.2 Convex Cones 428 G.1.3 Ordered Linear Spaces 432 G.1.4 Algebraic and Relative Interior of a Set 436 G.1.5 Algebraic Closure of a Set 447 G.1.6 Finitely Generated Cones 450 G.2 Separation and Extension in Linear Spaces 454 G.2.1 Extension of Linear Functionals 455 G.2.2 Extension of Positive Linear Functionals 460 G.2.3 Separation of Convex Sets by Hyperplanes 462 G.2.4 The External Characterization of Algebraically Closed and Convex Sets 471 G.2.5 Supporting Hyperplanes 473 G.2.6* Superlinear Maps 476 G.3 Reflections on Rn 480 G.3.1 Separation in Rn 480 G.3.2 Support in Rn 486 G.3.3 The Cauchy-Schwarz Inequality 488 G.3.4 Best Approximation from a Convex Set in Rn 489 G.3.5 Orthogonal Complements 492 G.3.6 Extension of Positive Linear Functionals, Revisited 496 Chapter H: Economic Applications 498 H.1 Applications to Expected Utility Theory 499 H.1.1 The Expected Multi-Utility Theorem 499 H.1.2* Knightian Uncertainty 505 H.1.3* The Gilboa-Schmeidler Multi-Prior Model 509 H.2 Applications to Welfare Economics 521 H.2.1 The Second Fundamental Theorem of Welfare Economics 521 H.2.2 Characterization of Pareto Optima 525 H.2.3* Harsanyi's Utilitarianism Theorem 526 H.3 An Application to Information Theory 528 H.4 Applications to Financial Economics 535 H.4.1 Viability and Arbitrage-Free Price Functionals 535 H.4.2 The No-Arbitrage Theorem 539 H.5 Applications to Cooperative Games 542 H.5.1 The Nash Bargaining Solution 542 H.5.2* Coalitional Games without Side Payments 546 Part IV: ANALYSIS ON METRIC/NORMED LINEAR SPACES 551 Chapter I: Metric Linear Spaces 553 I.1 Metric Linear Spaces 554 I.2 Continuous Linear Operators and Functionals 561 I.2.1 Examples of (Dis-)Continuous Linear Operators 561 I.2.2 Continuity of Positive Linear Functionals 567 I.2.3 Closed versus Dense Hyperplanes 569 I.2.4 Digression: On the Continuity of Concave Functions 573 I.3 Finite-Dimensional Metric Linear Spaces 577 I.4* Compact Sets in Metric Linear Spaces 582 I.5 Convex Analysis in Metric Linear Spaces 587 I.5.1 Closure and Interior of a Convex Set 587 I.5.2 Interior versus Algebraic Interior of a Convex Set 590 I.5.3 Extension of Positive Linear Functionals, Revisited 594 I.5.4 Separation by Closed Hyperplanes 594 I.5.5* Interior versus Algebraic Interior of a Closed and Convex Set 597 Chapter J: Normed Linear Spaces 601 J.1 Normed Linear Spaces 602 J.1.1 A Geometric Motivation 602 J.1.2 Normed Linear Spaces 605 J.1.3 Examples of Normed Linear Spaces 607 J.1.4 Metric versus Normed Linear Spaces 611 J.1.5 Digression: The Lipschitz Continuity of Concave Maps 614 J.2 Banach Spaces 616 J.2.1 Definition and Examples 616 J.2.2 Infinite Series in Banach Spaces 618 J.2.3* On the "Size" of Banach Spaces 620 J.3 Fixed Point Theory IV 623 J.3.1 The Glicksberg-Fan Fixed Point Theorem 623 J.3.2 Application: Existence of the Nash Equilibrium, Revisited 625 J.3.3* The Schauder Fixed Point Theorems 626 J.3.4* Some Consequences of Schauder's Theorems 630 J.3.5* Applications to Functional Equations 634 J.4 Bounded Linear Operators and Functionals 638 J.4.1 Definitions and Examples 638 J.4.2 Linear Homeomorphisms, Revisited 642 J.4.3 The Operator Norm 644 J.4.4 Dual Spaces 648 J.4.5* Discontinuous Linear Functionals, Revisited 649 J.5 Convex Analysis in Normed Linear Spaces 650 J.5.1 Separation by Closed Hyperplanes, Revisited 650 J.5.2* Best Approximation from a Convex Set 652 J.5.3 Extreme Points 654 J.6 Extension in Normed Linear Spaces 661 J.6.1 Extension of Continuous Linear Functionals 661 J.6.2* Infinite-Dimensional Normed Linear Spaces 663 J.7* The Uniform Boundedness Principle 665 Chapter K: Differential Calculus 670 K.1 Frechet Differentiation 671 K.1.1 Limits of Functions and Tangency 671 K.1.2 What Is a Derivative? 672 K.1.3 The Frechet Derivative 675 K.1.4 Examples 679 K.1.5 Rules of Differentiation 686 K.1.6 The Second Frechet Derivative of a Real Function 690 K.1.7 Differentiation on Relatively Open Sets 694 K.2 Generalizations of the Mean Value Theorem 698 K.2.1 The Generalized Mean Value Theorem 698 K.2.2* The Mean Value Inequality 701 K.3 Frechet Differentiation and Concave Maps 704 K.3.1 Remarks on the Differentiability of Concave Maps 704 K.3.2 Frechet Differentiable Concave Maps 706 K.4 Optimization 712 K.4.1 Local Extrema of Real Maps 712 K.4.2 Optimization of Concave Maps 716 K.5 Calculus of Variations 718 K.5.1 Finite-Horizon Variational Problems 718 K.5.2 The Euler-Lagrange Equation 721 K.5.3* More on the Sufficiency of the Euler-Lagrange Equation 733 K.5.4 Infinite-Horizon Variational Problems 736 K.5.5 Application: The Optimal Investment Problem 738 K.5.6 Application: The Optimal Growth Problem 740 K.5.7* Application: The Poincare-Wirtinger Inequality 743 Hints for Selected Exercises 747 References 777 Glossary of Selected Symbols 789 Index 793

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Book SynopsisPresents the analytic foundations to the theory of the hypoelliptic Laplacian. This book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. It gives the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus.Table of Contents*Frontmatter, pg. i*Contents, pg. v*Introduction, pg. 1*Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles, pg. 11*Chapter 2. The hypoelliptic Laplacian on the cotangent bundle, pg. 25*Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel, pg. 44*Chapter 4. Hypoelliptic Laplacians and odd Chern forms, pg. 62*Chapter 5. The limit as t --> + and b --> 0 of the superconnection forms, pg. 98*Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics, pg. 113*Chapter 7. The hypoelliptic torsion forms of a vector bundle, pg. 131*Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula, pg. 162*Chapter 9. A comparison formula for the Ray-Singer metrics, pg. 171*Chapter 10. The harmonic forms for b --> 0 and the formal Hodge theorem, pg. 173*Chapter 11. A proof of equation (8.4.6), pg. 182*Chapter 12. A proof of equation (8.4.8), pg. 190*Chapter 13. A proof of equation (8.4.7), pg. 194*Chapter 14. The integration by parts formula, pg. 214*Chapter 15. The hypoelliptic estimates, pg. 224*Chapter 16. Harmonic oscillator and the J0 function, pg. 247*Chapter 17. The limit of A'2phib,+-H as b --> 0, pg. 264*Bibliography, pg. 353*Subject Index, pg. 359*Index of Notation, pg. 361

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Book SynopsisThe hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula.Table of Contents*FrontMatter, pg. i*Contents, pg. vii*Acknowledgments, pg. xi*Introduction, pg. 1*Chapter One. Clifford and Heisenberg algebras, pg. 12*Chapter Two. The hypoelliptic Laplacian on X = G/K, pg. 22*Chapter Three. The displacement function and the return map, pg. 48*Chapter Four. Elliptic and hypoelliptic orbital integrals, pg. 76*Chapter Five. Evaluation of supertraces for a model operator, pg. 92*Chapter Six. A formula for semisimple orbital integrals, pg. 113*Chapter Seven. An application to local index theory, pg. 120*Chapter Eight. The case where [k (gamma); p0] = 0, pg. 138*Chapter Nine. A proof of the main identity, pg. 142*Chapter Ten. The action functional and the harmonic oscillator, pg. 161*Chapter Eleven. The analysis of the hypoelliptic Laplacian, pg. 187*Chapter Twelve. Rough estimates on the scalar heat kernel, pg. 212*Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b, pg. 248*Chapter Fourteen. The heat kernel qXb;t for bounded b, pg. 262*Chapter Fifteen. The heat kernel qXb;t for b large, pg. 290*Bibliography, pg. 317*Subject Index, pg. 323*Index of Notation, pg. 325

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Book SynopsisHybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. This title unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems.Trade Review"The book is carefully written and contains many examples. It will be a good resource for both researchers already familiar with hybrid systems and those starting from scratch."--Daniel Liberzon, Mathematical Reviews Clippings "The book presents a clean and self-contained exposition of hybrid systems, starting from the elementary definitions, continuing with the basic tools and finishing with more recent contributions in the literature."--Marco Castrillon Lopez, European Mathematical SocietyTable of ContentsPreface ix Chapter 1: Introduction 1 1.1 The modeling framework 1 1.2 Examples in science and engineering 2 1.3 Control system examples 7 1.4 Connections to other modeling frameworks 15 1.5 Notes 22 Chapter 2 The solution concept 25 2.1 Data of a hybrid system 25 2.2 Hybrid time domains and hybrid arcs 26 2.3 Solutions and their basic properties 29 2.4 Generators for classes of switching signals 35 2.5 Notes 41 Chapter 3 Uniform asymptotic stability, an initial treatment 43 3.1 Uniform global pre-asymptotic stability 43 3.2 Lyapunov functions 50 3.3 Relaxed Lyapunov conditions 60 3.4 Stability from containment 64 3.5 Equivalent characterizations 68 3.6 Notes 71 Chapter 4 Perturbations and generalized solutions 73 4.1 Differential and difference equations 73 4.2 Systems with state perturbations 76 4.3 Generalized solutions 79 4.4 Measurement noise in feedback control 84 4.5 Krasovskii solutions are Hermes solutions 88 4.6 Notes 94 Chapter 5 Preliminaries from set-valued analysis 97 5.1 Set convergence 97 5.2 Set-valued mappings 101 5.3 Graphical convergence of hybrid arcs 107 5.4 Differential inclusions 111 5.5 Notes 115 Chapter 6 Well-posed hybrid systems and their properties 117 6.1 Nominally well-posed hybrid systems 117 6.2 Basic assumptions on the data 120 6.3 Consequences of nominal well-posedness 125 6.4 Well-posed hybrid systems 132 6.5 Consequences of well-posedness 134 6.6 Notes 137 Chapter 7 Asymptotic stability, an in-depth treatment 139 7.1 Pre-asymptotic stability for nominally well-posed systems 141 7.2 Robustness concepts 148 7.3 Well-posed systems 151 7.4 Robustness corollaries 153 7.5 Smooth Lyapunov functions 156 7.6 Proof of robustness implies smooth Lyapunov functions 161 7.7 Notes 167 Chapter 8 Invariance principles 169 8.1 Invariance and omega-limits 169 8.2 Invariance principles involving Lyapunov-like functions 170 8.3 Stability analysis using invariance principles 176 8.4 Meagre-limsup invariance principles 178 8.5 Invariance principles for switching systems 181 8.6 Notes 184 Chapter 9 Conical approximation and asymptotic stability 185 9.1 Homogeneous hybrid systems 185 9.2 Homogeneity and perturbations 189 9.3 Conical approximation and stability 192 9.4 Notes 196 Appendix: List of Symbols 199 Bibliography 201 Index 211

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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 SynopsisTrade Review"The authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts."--Gerd Faltings, Zentralblatt MATHTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Foreword, pg. ix*Chapter I. Representations of the fundamental group and the torsor of deformations. An overview, pg. 1*Chapter II. Representations of the fundamental group and the torsor of deformations. Local study, pg. 27*Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects, pg. 179*Chapter IV. Cohomology of Higgs isocrystals, pg. 307*Chapter V. Almost etale coverings, pg. 449*Chapter VI. Covanishing topos and generalizations, pg. 485*Facsimile : A p-adic Simpson correspondence, pg. 577*Bibliography, pg. 595*Indexes, pg. 599

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Book SynopsisTrade Review"This well-written book prepares readers to take a real analysis course by carefully defining and proving all concepts one needs for this type of course. . . . Throughout the book, the style is incredibly reader friendly, and the author's enthusiasm for the subject is very clear." * Choice *"I can imagine this book proving useful to a motivated student who is finding the transition into analysis challenging through traditional textbooks."---Dominic Yeo, Mathematical GazetteTable of ContentsPreliminaries 1 1 Introduction 3 2 Basic Math and Logic* 6 3 Set Theory* 14 Real Numbers 25 4 Least Upper Bounds* 27 5 The Real Field* 35 6 Complex Numbers and Euclidean Spaces 46 Topology 59 7 Bijections 61 8 Countability 68 9 Topological Definitions* 79 10 Closed and Open Sets* 90 11 Compact Sets* 98 12 The Heine-Borel Theorem* 108 13 Perfect and Connected Sets 117 Sequences 127 14 Convergence* 129 15 Limits and Subsequences* 138 16 Cauchy and Monotonic Sequences* 148 17 Subsequential Limits 157 18 Special Sequences 166 19 Series* 174 20 Conclusion 183 Acknowledgments 187 Bibliography 189 Index 191

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