Calculus and mathematical analysis Books

Book SynopsisThis book is a very well-accepted introduction to the subject. Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy’s example of a linear equation without solutions.Trade ReviewFourth Edition F. John Partial Differential Equations "An excellent second-reading text. Should be accessible to any mathematician. Highly recommended." —THE MATHEMATICAL GAZETTE

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Book SynopsisBased on the lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Munster, this book provides a modern introduction to this subject, presenting methods used in the study of complex manifolds in the special case of complex dimension one.Trade ReviewO. Forster and B. Gilligan Lectures on Riemann Surfaces "A very attractive addition to the list in the form of a well-conceived and handsomely produced textbook based on several years' lecturing experience . . . This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces. The reviewer is inclined to think that it may well become a favorite."—MATHEMATICAL REVIEWS Table of Contents1 Covering Spaces.- §1. The Definition of Riemann Surfaces.- §2. Elementary Properties of Holomorphic Mappings.- §3. Homotopy of Curves. The Fundamental Group.- §4. Branched and Unbranched Coverings.- §5. The Universal Covering and Covering Transformations.- §6. Sheaves.- §7. Analytic Continuation.- §8. Algebraic Functions.- §9. Differential Forms.- §10. The Integration of Differential Forms.- §11. Linear Differential Equations.- 2 Compact Riemann Surfaces.- §12. Cohomology Groups.- §13. Dolbeault’s Lemma.- §14. A Finiteness Theorem.- §15. The Exact Cohomology Sequence.- §16. The Riemann-Roch Theorem.- §17. The Serre Duality Theorem.- §18. Functions and Differential Forms with Prescribed Principal Parts.- §19. Harmonic Differential Forms.- §20. Abel’s Theorem.- §21. The Jacobi Inversion Problem.- 3 Non-compact Riemann Surfaces.- §22. The Dirichlet Boundary Value Problem.- §23. Countable Topology.- §24. Weyl’s Lemma.- §25. The Runge Approximation Theorem.- §26. The Theorems of Mittag-Leffler and Weierstrass.- §27. The Riemann Mapping Theorem.- §28. Functions with Prescribed Summands of Automorphy.- §29. Line and Vector Bundles.- §30. The Triviality of Vector Bundles.- §31. The Riemann-Hilbert Problem.- A. Partitions of Unity.- B. Topological Vector Spaces.- References.- Symbol Index.- Author and Subject Index.

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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 SynopsisThis book is an introduction to hyperbolic and differential geometry that provides material in the early chapters that can serve as a textbook for a standard upper division course on hyperbolic geometry. The book goes well beyond the standard course in later chapters, and there is enough material for an honors course, or for supplementary reading.Trade Review"The book is well laid out with no shortage of diagrams and with each chapter prefaced with its own useful introduction...Also well written, it makes pleasurable reading." Proceedings of the Edinburgh Mathematical SocietyTable of ContentsPreface; Introduction; 1. Axioms for Plane Geometry; 2. Some Neutral Theorems of Plane Geometry; 3. Qualitative Description of the Hyperbolic Plane; 4. H3 and Euclidean Approximations in H2; 5. Differential Geometry of Surface; 6. Quantitative Considerations; 7. Consistency and Categoricalness of the Hyperbolic Axioms- the Classical Models; 8. Matrix Representation of the Isometry Group; 9. Differential and Hyperbolic Geometry in More Dimensions; 10. Connections with the Lorentz Group of Special Relativity; 11. Constructions by Straightedge and Compass in the Hyperbolic Plane; Index

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Book SynopsisInfinite dimensional systems is now an established area of research. Given the recent trend in systems theory and in applications towards a synthesis of time- and frequency-domain methods, there is a need for an introductory text which treats both state-space and frequency-domain aspects in an integrated fashion.Table of Contents1 Introduction.- 1.1 Motivation.- 1.2 Systems theory concepts in finite dimensions.- 1.3 Aims of this book.- 2 Semigroup Theory.- 2.1 Strongly continuous semigroups.- 2.2 Contraction and dual semigroups.- 2.3 Riesz-spectral operators.- 2.4 Delay equations.- 2.5 Invariant subspaces.- 2.6 Exercises.- 2.7 Notes and references.- 3 The Cauchy Problem.- 3.1 The abstract Cauchy problem.- 3.2 Perturbations and composite systems.- 3.3 Boundary control systems.- 3.4 Exercises.- 3.5 Notes and references.- 4 Inputs and Outputs.- 4.1 Controllability and observability.- 4.2 Tests for approximate controllability and observability.- 4.3 Input-output maps.- 4.4 Exercises.- 4.5 Notes and references.- 5 Stability, Stabilizability, and Detectability.- 5.1 Exponential stability.- 5.2 Exponential stabilizability and detectability.- 5.3 Compensator design.- 5.4 Exercises.- 5.5 Notes and references.- 6 Linear Quadratic Optimal Control.- 6.1 The problem on a finite-time interval.- 6.2 The problem on the infinite-time interval.- 6.3 Exercises.- 6.4 Notes and references.- 7 Frequency-Domain Descriptions.- 7.1 The Callier-Desoer class of scalar transfer functions.- 7.2 The multivariable extension.- 7.3 State-space interpretations.- 7.4 Exercises.- 7.5 Notes and references.- 8 Hankel Operators and the Nehari Problem.- 8.1 Frequency-domain formulation.- 8.2 Hankel operators in the time domain.- 8.3The Nehari extension problem for state linear systems.- 8.4 Exercises.- 8.5 Notes and references.- 9 Robust Finite-Dimensional Controller Synthesis.- 9.1 Closed-loop stability and coprime factorizations.- 9.2 Robust stabilization of uncertain systems.- 9.3 Robust stabilization under additive uncertainty.- 9.4 Robust stabilization under normalized left-coprime-factor uncertainty.- 9.5 Robustness in the presence of small delays.- 9.6 Exercises.- 9.7 Notes and references.- A. Mathematical Background.- A.1 Complex analysis.- A.2 Normed linear spaces.- A.2.1 General theory.- A.2.2 Hilbert spaces.- A.3 Operators on normed linear spaces.- A.3.1 General theory.- A.3.2 Operators on Hilbert spaces.- A.4 Spectral theory.- A.4.1 General spectral theory.- A.4.2 Spectral theory for compact normal operators.- A.5 Integration and differentiation theory.- A.5.1 Integration theory.- A.5.2 Differentiation theory.- A.6 Frequency-domain spaces.- A.6.1 Laplace and Fourier transforms.- A.6.2 Frequency-domain spaces.- A.6.3 The Hardy spaces.- A.7 Algebraic concepts.- A.7.1 General definitions.- A.7.2 Coprime factorizations over principal ideal domains.- A.7.3 Coprime factorizations over commutative integral domains.- References.- Notation.

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Book Synopsis1 Real Numbers.- 1.1 Sets, Relations, Functions.- 1.2 Numbers.- 1.3 Infinite Sets.- 1.4 Incommensurability.- 1.5 Ordered Fields.- 1.6 Functions on R.- 1.7 Intervals in R.- 1.8 Algebraic and Transcendental Numbers.- 1.9 Existence of R.- 1.10 Exercises.- 1.11 Notes.- 2 Sequences and Series.- 2.1 Sequences.- 2.2 Continued Fractions.- 2.3 Infinite Series.- 2.4 Rearrangements of Series.- 2.5 Unordered Series.- 2.6 Exercises.- 2.7 Notes.- 3 Continuous Functions on Intervals.- 3.1 Limits and Continuity.- 3.2 Two Fundamental Theorems.- 3.3 Uniform Continuity.- 3.4 Sequences of Functions.- 3.5 The Exponential function.- 3.6 Trigonometric Functions.- 3.7 Exercises.- 3.8 Notes.- 4 Differentiation.- 4.1 Derivatives.- 4.2 Derivatives of Some Elementary Functions.- 4.3 Convex Functions.- 4.4 The Differential Calculus.- 4.5 L'Hospital's Rule.- 4.6 Higher Order Derivatives.- 4.7 Analytic Functions.- 4.8 Exercises.- 4.9 Notes.- 5 The Riemann Integral.- 5.1 Riemann Sums.- 5.2 Existence Results.- 5.3 ProTrade ReviewThis is a very good textbook presenting a modern course in analysis both at the advanced undergraduate and at the beginning graduate level. It contains 14 chapters, a bibliography, and an index. At the end of each chapter interesting exercises and historical notes are enclosed.\par From the cover: ``The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral (of a real-valued function defined on a compact interval). The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean spaces). The final part of the book deals with manifolds, differential forms, and Stokes' theorem [in the spirit of M. Spivak's: ``Calculus on manifolds'' (1965; Zbl 141.05403)] which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle''. ZENTRALBLATT MATH A. Browder Mathematical Analysis An Introduction "Everything needed is clearly defined and formulated, and there is a reasonable number of examples…. Anyone teaching a year course at this level to should seriously consider this carefully written book. In the reviewer's opinion, it would be a real pleasure to use this text with such a class."—MATHEMATICAL REVIEWSTable of Contents1 Real Functions 2 Sequences and Series 3 Continuous Functions on Intervals 4 Differentiation 5 The Riemann Integral 6 Topology 7 Function Spaces 8 Differentiable Maps 9 Measures 10 Integration 11 Manifolds 12 Multilinear Algebra 13 Differential Forms 14 Integration on Manifolds

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Book Synopsis1 Complexity and Criticality.- 2 The Discovery of Self-Organized Criticality.- 3 The Sandpile Paradigm.- 4 Real Sandpiles and Landscape Formation.- 5 Earthquakes, Starquakes, and Solar Flares.- 6 The Game of Life: Complexity Is Criticality.- 7 Is Life a Self-Organized Critical Phenomenon?.- 8 Mass Extinctions and Punctuated Equilibria in a Simple Model of Evolution.- 9 Theory of the Punctuated Equilibrium Model.- 10 The Brain.- 11 On Economics and Traffic Jams.Table of Contents1 Complexity and Criticality.- 2 The Discovery of Self-Organized Criticality.- 3 The Sandpile Paradigm.- 4 Real Sandpiles and Landscape Formation.- 5 Earthquakes, Starquakes, and Solar Flares.- 6 The “Game of Life”: Complexity Is Criticality.- 7 Is Life a Self-Organized Critical Phenomenon?.- 8 Mass Extinctions and Punctuated Equilibria in a Simple Model of Evolution.- 9 Theory of the Punctuated Equilibrium Model.- 10 The Brain.- 11 On Economics and Traffic Jams.

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Book SynopsisOne Review of Calculus.- 0 Sets and Mappings.- I Real Numbers.- II Limits and Continuous Functions.- III Differentiation.- IV Elementary Functions.- V The Elementary Real Integral.- Two Convergence.- VI Normed Vector Spaces.- VII Limits.- VIII Compactness.- IX Series.- X The Integral in One Variable.- Three Applications of the Integral.- XI Approximation with Convolutions.- XII Fourier Series.- XIII Improper Integrals.- XIV The Fourier Integral.- Four Calculus in Vector Spaces.- XV Functions on n-Space.- XVI The Winding Number and Global Potential Functions.- XVII Derivatives in Vector Spaces.- XVIII Inverse Mapping Theorem.- XIX Ordinary Differential Equations.- Five Multiple Integration.- XX Multiple Integrals.- XXI Differential Forms.Trade ReviewSecond Edition S. Lang Undergraduate Analysis "[A] fine book . . . logically self-contained . . . This material can be gone over quickly by the really well-prepared reader, for it is one of the book’s pedagogical strengths that the pattern of development later recapitulates this material as it deepens and generalizes it."—AMERICAN MATHEMATICAL SOCIETYTable of ContentsChapter 0: Sets and Mappings Chapter 1: Real Numbers Chapter 2: Limits and Continuous Functions Chapter 3: Differentiation Chapter 4: Elementary Functions Chapter 5: The Elementary Real Integral Chapter 6: Normed Vector Spaces Chapter 7: Limits Chapter 8: Compactness Chapter 9: Series Chapter 10: The Integral in One Variable Appendix: The Lebesgue Integral Chapter 11: Approximation with Convolutions Chapter 12: Fourier Series Chapter 13, Improper Integrals Chapter 14: The Fourier Integral Chapter 15: Calculus in Vector Spaces Chapter 16: The Winding Number and Global Potential Functions Chapter 17: Derivatives in Vector Spaces Chapter 18: Inverse Mapping Theorem Chapter 19: Ordinary Differential Equations Chapter 20: Multiple Integration Chapter 22: Differential Forms Appendix

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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 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 Synopsis1 Examples.- 1.1. Introduction.- 1.2. Iteration of Möbius Transformations.- 1.3. Iteration of z ? z2.- 1.4. Tchebychev Polynomials.- 1.5. Iteration of z ? z2 ? 1.- 1.6. Iteration of z ? z2 + c.- 1.7. Iteration of z ? z + 1/z.- 1.8. Iteration of z ? 2z ? 1/z.- 1.9. Newton's Approximation.- 1.10. General Remarks.- 2 Rational Maps.- 2.1. The Extended Complex Plane.- 2.2. Rational Maps.- 2.3. The Lipschitz Condition.- 2.4. Conjugacy.- 2.5. Valency.- 2.6. Fixed Points.- 2.7. Critical Points.- 2.8. A Topology on the Rational Functions.- 3 The Fatou and Julia Sets.- 3.1. The Fatou and Julia Sets.- 3.2. Completely Invariant Sets.- 3.3. Normal Families and Equicontinuity.- Appendix I. The Hyperbolic Metric.- 4 Properties of the Julia Set.- 4.1. Exceptional Points.- 4.2. Properties of the Julia Set.- 4.3. Rational Maps with Empty Fatou Set.- Appendix II. Elliptic Functions.- 5 The Structure of the Fatou Set.- 5.1. The Topology of the Sphere.- 5.2. Completely Invariant Components of the Fatou SetTrade ReviewA.F. Beardon Iteration of Rational Functions Complex Analytic Dynamical Systems "This book makes available a comprehensive, detailed, and organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The material covered extends from the original memoirs of Fatou and Julia to the recent and important results and methods of Sullivan and Shishikura. Many of the details of the proofs have not occurred in print before."—ZENTRALBLATT MATHTable of Contents1 Examples.- 1.1. Introduction.- 1.2. Iteration of Möbius Transformations.- 1.3. Iteration of z ? z2.- 1.4. Tchebychev Polynomials.- 1.5. Iteration of z ? z2 ? 1.- 1.6. Iteration of z ? z2 + c.- 1.7. Iteration of z ? z + 1/z.- 1.8. Iteration of z ? 2z ? 1/z.- 1.9. Newton’s Approximation.- 1.10. General Remarks.- 2 Rational Maps.- 2.1. The Extended Complex Plane.- 2.2. Rational Maps.- 2.3. The Lipschitz Condition.- 2.4. Conjugacy.- 2.5. Valency.- 2.6. Fixed Points.- 2.7. Critical Points.- 2.8. A Topology on the Rational Functions.- 3 The Fatou and Julia Sets.- 3.1. The Fatou and Julia Sets.- 3.2. Completely Invariant Sets.- 3.3. Normal Families and Equicontinuity.- Appendix I. The Hyperbolic Metric.- 4 Properties of the Julia Set.- 4.1. Exceptional Points.- 4.2. Properties of the Julia Set.- 4.3. Rational Maps with Empty Fatou Set.- Appendix II. Elliptic Functions.- 5 The Structure of the Fatou Set.- 5.1. The Topology of the Sphere.- 5.2. Completely Invariant Components of the Fatou Set.- 5.3. The Euler Characteristic.- 5.4. The Riemann-Hurwitz Formula for Covering Maps.- 5.5. Maps Between Components of the Fatou Set.- 5.6. The Number of Components of the Fatou Set.- 5.7. Components of the Julia Set.- 6 Periodic Points.- 6.1. The Classification of Periodic Points.- 6.2. The Existence of Periodic Points.- 6.3. (Super) Attracting Cycles.- 6.4. Repelling Cycles.- 6.5. Rationally Indifferent Cycles.- 6.6. Irrationally Indifferent Cycles in F.- 6.7. Irrationally Indifferent Cycles in J.- 6.8. The Proof of the Existence of Periodic Points.- 6.9. The Julia Set and Periodic Points.- 6.10. Local Conjugacy.- Appendix III. Infinite Products.- Appendix IV. The Universal Covering Surface.- 7 Forward Invariant Components.- 7.1. The Five Possibilities.- 7.2. Limit Functions.- 7.3. Parabolic Domains.- 7.4. Siegel Discs and Herman Rings.- 7.5. Connectivity of Invariant Components.- 8 The No Wandering Domains Theorem.- 8.1. The No Wandering Domains Theorem.- 8.2. A Preliminary Result.- 8.3. Conformal Structures.- 8.4. Quasiconformal Conjugates of Rational Maps.- 8.5. Boundary Values of Conjugate Maps.- 8.6. The Proof of Theorem 8.1.2.- 9 Critical Points.- 9.1. Introductory Remarks.- 9.2. The Normality of Inverse Maps.- 9.3. Critical Points and Periodic Domains.- 9.4. Applications.- 9.5. The Fatou Set of a Polynomial.- 9.6. The Number of Non-Repelling Cycles.- 9.7. Expanding Maps.- 9.8. Julia Sets as Cantor Sets.- 9.9. Julia Sets as Jordan Curves.- 9.10. The Mandelbrot Set.- 10 Hausdorff Dimension.- 10.1. Hausdorff Dimension.- 10.2. Computing Dimensions.- 10.3. The Dimension of Julia Sets.- 11 Examples.- 11.1. Smooth Julia Sets.- 11.2. Dendrites.- 11.3. Components of F of Infinite Connectivity.- 11.4. F with Infinitely Connected and Simply Connected Components.- 11.5. J with Infinitely Many Non-Degenerate Components.- 11.6. F of Infinite Connectivity with Critical Points in J.- 11.7. A Finitely Connected Component of F.- 11.8. J Is a Cantor Set of Circles.- 11.9. The Function (z ? 2)2/z2.- References.- Index of Examples.

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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 SynopsisBasic Properties of Harmonic Functions.- Bounded Harmonic Functions.- Positive Harmonic Functions.- The Kelvin Transform.- Harmonic Polynomials.- Harmonic Hardy Spaces.- Harmonic Functions on Half-Spaces.- Harmonic Bergman Spaces.- The Decomposition Theorem.- Annular Regions.- The Dirichlet Problem and Boundary Behavior.Trade ReviewFrom the reviews of the second edition: "There are several major changes in this second edition … . Many exercises have been added and several photographs of mathematicians related to harmonic functions are included. The book is a nice introduction to the fundamental notions of potential theory." (European Mathematical Society Newsletter, June, 2002) "We warmly recommend this textbook to graduate students interested in Harmonic Function Theory and/or related areas. We are sure that the reader will be able to appreciate the lively and illuminating discussions in this book, and therefore, will certainly gain a better understanding of the subject." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 67, 2001) "This is a new edition of a nice textbook … on harmonic functions in Euclidean spaces, suitable for a beginning graduate level course. … New exercises are added and numerous minor improvements throughout the text are made." (Alexander Yu. Rashkovsky, Zentralblatt MATH, Vol. 959, 2001)Table of Contents* Basic Properties of Harmonic Functions * Bounded Harmonic Functions * Positive Harmonic Functions * The Kelvin Transform * Harmonic Polynomials * Harmonic Hardy Spaces * Harmonic Functions on Half-Spaces * Harmonic Bergman Spaces * The Decomposition Theorem * Annular Regions * The Dirichlet Problem and Boundary Behavior * Volume, Surface Area, and Integration on Spheres * Harmonic Function Theory and Mathematica * References * Symbol Index * Index

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Book SynopsisThe first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level.Trade ReviewFrom the reviews of the second edition:“In the second edition of this text, Koblitz presents a wide-ranging introduction to the theory of p-adic numbers and functions. … there are some really nice exercises that allow the reader to explore the material. … And with the exercises, the book would make a good textbook for a graduate course, provided the students have a decent background in analysis and number theory.” (Donald L. Vestal, The Mathematical Association of America, April, 2011)Table of ContentsI p-adic numbers.- 1. Basic concepts.- 2. Metrics on the rational numbers.- Exercises.- 3. Review of building up the complex numbers.- 4. The field of p-adic numbers.- 5. Arithmetic in ?p.- Exercises.- II p-adic interpolation of the Riemann zeta-function.- 1. A formula for ?(2k).- 2. p-adic interpolation of the function f(s) = as.- Exercises.- 3. p-adic distributions.- Exercises.- 4. Bernoulli distributions.- 5. Measures and integration.- Exercises.- 6. The p-adic ?-function as a Mellin-Mazur transform.- 7. A brief survey (no proofs).- Exercises.- III Building up ?.- 1. Finite fields.- Exercises.- 2. Extension of norms.- Exercises.- 3. The algebraic closure of ?p.- 4. ?.- Exercises.- IV p-adic power series.- 1. Elementary functions.- Exercises.- 2. The logarithm, gamma and Artin-Hasse exponential functions.- Exercises.- 3. Newton polygons for polynomials.- 4. Newton polygons for power series.- Exercises.- V Rationality of the zeta-function of a set of equations over a finite field.- 1. Hypersurfaces and their zeta-functions.- Exercises.- 2. Characters and their lifting.- 3. A linear map on the vector space of power series.- 4. p-adic analytic expression for the zeta-function.- Exercises.- 5. The end of the proof.- Answers and Hints for the Exercises.

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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 SynopsisThe term "weakly differentiable functions" in the title refers to those inte­ n grable functions defined on an open subset of R whose partial derivatives in the sense of distributions are either LP functions or (signed) measures with finite total variation.Table of Contents1 Preliminaries.- 1.1 Notation.- Inner product of vectors.- Support of a function.- Boundary of a set.- Distance from a point to a set.- Characteristic function of a set.- Multi-indices.- Partial derivative operators.- Function spaces—continuous, Hölder continuous, Hölder continuous derivatives.- 1.2 Measures on Rn.- Lebesgue measurable sets.- Lebesgue measurability of Borel sets.- Suslin sets.- 1.3 Covering Theorems.- Hausdorff maximal principle.- General covering theorem.- Vitali covering theorem.- Covering lemma, with n-balls whose radii vary in Lipschitzian way.- Besicovitch covering lemma.- Besicovitch differentiation theorem.- 1.4 Hausdorff Measure.- Equivalence of Hausdorff and Lebesgue measures.- Hausdorff dimension.- 1.5 Lp-Spaces.- Integration of a function via its distribution function.- Young’s inequality.- Hölder’s and Jensen’s inequality.- 1.6 Regularization.- Lp-spaces and regularization.- 1.7 Distributions.- Functions and measures, as distributions.- Positive distributions.- Distributions determined by their local behavior.- Convolution of distributions.- Differentiation of distributions.- 1.8 Lorentz Spaces.- Non-increasing rearrangement of a function.- Elementary properties of rearranged functions.- Lorentz spaces.- O’Neil’s inequality, for rearranged functions.- Equivalence of Lp-norm and (p, p)-norm.- Hardy’s inequality.- Inclusion relations of Lorentz spaces.- Exercises.- Historical Notes.- 2 Sobolev Spaces and Their Basic Properties.- 2.1 Weak Derivatives.- Sobolev spaces.- Absolute continuity on lines.- Lp-norm of difference quotients.- Truncation of Sobolev functions.- Composition of Sobolev functions.- 2.2 Change of Variables for Sobolev Functions.- Rademacher’s theorem.- Bi-Lipschitzian change of variables.- 2.3 Approximation of Sobolev Functions by Smooth Functions.- Partition of unity.- Smooth functions are dense in Wk,p.- 2.4 Sobolev Inequalities.- Sobolev’s inequality.- 2.5 The Rellich-Kondrachov Compactness Theorem.- Extension domains.- 2.6 Bessel Potentials and Capacity.- Riesz and Bessel kernels.- Bessel potentials.- Bessel capacity.- Basic properties of Bessel capacity.- Capacitability of Suslin sets.- Minimax theorem and alternate formulation of Bessel capacity.- Metric properties of Bessel capacity.- 2.7 The Best Constant in the Sobolev Inequality.- Co-area formula.- Sobolev’s inequality and isoperimetric inequality.- 2.8 Alternate Proofs of the Fundamental Inequalities.- Hardy-Littlewood-Wiener maximal theorem.- Sobolev’s inequality for Riesz potentials.- 2.9 Limiting Cases of the Sobolev Inequality.- The case kp=n by infinite series.- The best constant in the case kp = n.- An L?-bound in the limiting case.- 2.10 Lorentz Spaces, A Slight Improvement.- Young’s inequality in the context of Lorentz spaces.- Sobolev’s inequality in Lorentz spaces.- The limiting case.- Exercises.- Historical Notes.- 3 Pointwise Behavior of Sobolev Functions.- 3.1 Limits of Integral Averages of Sobolev Functions.- Limiting values of integral averages except for capacity null set.- 3.2 Densities of Measures.- 3.3 Lebesgue Points for Sobolev Functions.- Existence of Lebesgue points except for capacity null set.- Approximate continuity.- Fine continuity everywhere except for capacity null set.- 3.4 LP-Derivatives for Sobolev Functions.- Existence of Taylor expansions Lp.- 3.5 Properties of Lp-Derivatives.- The Spaces TktkTk,ptk,p.- The implication of a function being in Tk,pat all points of a closed set.- 3.6 An Lp-Version of the Whitney Extension Theorem.- Existence of a C? function comparable to the.- distance function to a closed set.- The Whitney extension theorem for functions in Tk,p and tk,p.- 3.7 An Observation on Differentiation.- 3.8 Rademacher’s Theorem in the Lp-Context.- A function in Tk,peverywhere implies it is in tk,palmost everywhere.- 3.9 The Implications of Pointwise Differentiability.- Comparison of Lp-derivatives and distributional derivatives.- If u ? tk,p(x)for everyxand if the.- LP-derivatives are in Lpthen u ? Wk,p.- 3.10 A Lusin-Type Approximation for Sobolev Functions.- Integral averages of Sobolev functions are uniformly close to their limits on the complement of sets of small capacity.- Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity.- 3.11 The Main Approximation.- Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity and are close in norm.- Exercises.- Historical Notes.- 4 Poincaré Inequalities—A Unified Approach.- 4.1 Inequalities in a General Setting.- An abstract version of the Poincaré inequality.- 4.2 Applications to Sobolev Spaces.- An interpolation inequality.- 4.3 The Dual of WM,p(?).- The representation of (W0M,p(?) )*.- 4.4 Some Measures in (W0M,p(?))*.- Poincaré inequalities derived from the abstract version by identifying Lebesgue and Hausdorff measure with elements in (WM,p(?))*.- The trace of Sobolev functions on the boundary of Lipschitz domains.- Poincaré inequalities involving the trace of a Sobolev function.- 4.5 Poincaré Inequalities.- Inequalities involving the capacity of the set on which a function vanishes.- 4.6 Another Version of Poincaré’s Inequality.- An inequality involving dependence on the set on which the function vanishes, not merely on its capacity.- 4.7 More Measures in (WM,p(?))*.- Sobolev’s inequality for Riesz potentials involving measures other than Lebesgue measure.- Characterization of measures in (WM,p(?))*.- 4.8 Other Inequalities Involving Measures in (WM,p)*.- Inequalities involving the restriction of Hausdorff measure to lower dimensional manifolds.- 4.9 The Case p= 1.- Inequalities involving the L1-norm of the gradient.- Exercises.- Historical Notes.- 5 Functions of Bounded Variation.- 5.1 Definitions.- Definition of BV functions.- The total variation measure ? Du?.- 5.2 Elementary Properties of BV Functions.- Lower semicontinuity of the total variation measure.- A condition ensuring continuity of the total variation measure.- 5.3 Regularization of BV Functions.- Regularization does not increase the BV norm.- Approximation of BV functions by smooth functions Compactness in L1of the unit ball in BV.- 5.4 Sets of Finite Perimeter.- Definition of sets of finite perimeter.- The perimeter of domains with smooth boundaries.- Isoperimetric and relative isoperimetric inequality for sets of finite perimeter.- 5.5 The Generalized Exterior Normal.- A preliminary version of the Gauss-Green theorem.- Density results at points of the reduced boundary.- 5.6 Tangential Properties of the Reduced Boundary and the Measure-Theoretic Normal.- Blow-up at a point of the reduced boundary.- The measure-theoretic normal.- The reduced boundary is contained in the measure-theoretic boundary.- A lower bound for the density of ?DXE?.- Hausdorff measure restricted to the reduced boundary is bounded above by ?DXE?.- 5.7 Rectifiability of the Reduced Boundary.- Countably (n — 1)-rectifiable sets.- Countable (n — 1)-rectifiability of the measure-theoretic boundary.- 5.8 The Gauss-Green Theorem.- The equivalence of the restriction of Hausdorff measure to the measure-theoretic boundary and ?DXE?.- The Gauss-Green theorem for sets of finite perimeter.- 5.9 Pointwise Behavior of BV Functions.- Upper and lower approximate limits.- The Boxing inequality.- The set of approximate jump discontinuities.- 5.10 The Trace of a BV Function.- The bounded extension of BV functions.- Trace of a BV function defined in terms of the upper and lower approximate limits of the extended function.- The integrability of the trace over the.- measure-theoretic boundary.- 5.11 Sobolev-Type Inequalities for BV Functions.- Inequalities involving elements in (BV(?))*.- 5.12 Inequalities Involving Capacity.- Characterization of measure in (BV(?))*.- Poincaré inequality for BV functions.- 5.13 Generalizations to the Case p> 1.- 5.14 Trace Defined in Terms of Integral Averages.- Exercises.- Historical Notes.- List of Symbols.

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Book SynopsisI Asymptotic Solutions of Evolution Problems.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VIII Bifurcation of Periodic Solutions in the General Case.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf's Type) in the Autonomous Case.- XII Stability and Bifurcation in Conservative Systems.Table of ContentsI Asymptotic Solutions of Evolution Problems.- I.1 One-Dimensional, Two-Dimensional n-Dimensional, and Infinite-Dimensional Interpretations of (I.1).- I.2 Forced Solutions; Steady Forcing and T-Periodic Forcing; Autonomous and Nonautonomous Problems.- I.3 Reduction to Local Form.- I.4 Asymptotic Solutions.- I.5 Asymptotic Solutions and Bifurcating Solutions.- I.6 Bifurcating Solutions and the Linear Theory of Stability.- I.7 Notation for the Functional Expansion of F(t µ,U).- Notes.- II Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension.- II.1 The Implicit Function Theorem.- II.2 Classification of Points on Solution Curves.- 1I.3 The Characteristic Quadratic. Double Points, Cusp Points, and Conjugate Points.- II.4 Double-Point Bifurcation and the Implicit Function Theorem.- II.5 Cusp-Point Bifurcation.- II.6 Triple-Point Bifurcation.- II.7 Conditional Stability Theorem.- II.8 The Factorization Theorem in One Dimension.- II.9 Equivalence of Strict Loss of Stability and Double-Point Bifurcation.- II.10 Exchange of Stability at a Double Point.- II.1 1 Exchange of Stability at a Double Point for Problems Reduced to Local Form.- II.12 Exchange of Stability at a Cusp Point.- II.13 Exchange of Stability at a Triple Point.- II.14 Global Properties of Stability of Isolated Solutions.- III Imperfection Theory and Isolated Solutions Which Perturb Bifurcation.- III.1 The Structure of Problems Which Break Double-Point Bifurcation.- III.2 The Implicit Function Theorem and the Saddle Surface Breaking Bifurcation.- III.3 Examples of Isolated Solutions Which Break Bifurcation.- III.4 Iterative Procedures for Finding Solutions.- III.5 Stability of Solutions Which Break Bifurcation.- III.6 Isolas.- Exercise.- Notes.- IV Stability of Steady Solutions of Evolution Equations in Two Dimensions and nDimensions.- IV.1 Eigenvalues and Eigenvectors of an n x n Matrix.- IV.2 Algebraic and Geometric Multiplicity—The Riesz Index.- IV.3 The Adjoint Eigenvalue Problem.- IV.4 Eigenvalues and Eigenvectors of a 2 x 2 Matrix.- 4.1 Eigenvalues.- 4.2 Eigenvectors.- 4.3 Algebraically Simple Eigenvalues.- 4.4 Algebraically Double Eigenvalues.- 4.4.1 Riesz Index 1.- 4.4.2 Riesz Index 2.- IV.5 The Spectral Problem and Stability of the Solution u = 0 in ?n.- IV.6 Nodes, Saddles, and Foci.- IV.7 Criticality and Strict Loss of Stability.- Appendix IV.I Biorthogonality for Generalized Eigenvectors.- Appendix IV.2 Projections.- V Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions.- V.1 The Form of Steady Bifurcating Solutions and Their Stability.- V.2 Necessary Conditions for the Bifurcation of Steady Solutions.- V.3 Bifurcation at a Simple Eigenvalue.- V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue.- V.5 Bifurcation at a Double Eigenvalue of Index Two.- V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue of Index Two.- V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a Double Eigenvalue of Index One (Semi-Simple).- V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi-Simple Double Eigenvalue.- V.9 Examples of Stability Analysis at a Double Semi-Simple (Index-One) Eigenvalue.- V.10 Saddle-Node Bifurcation.- Appendix V.1 Implicit Function Theorem for a System of Two Equations in Two Unknown Functions of One Variable.- Exercises.- VI Methods of Projection for General Problems of Bifurcation into Steady Solutions.- VI.1 The Evolution Equation and the Spectral Problem.- VI.2 Construction of Steady Bifurcating Solutions as Power Series in the Amplitude.- VI.3 ?1 and ?1 in Projection.- VI.4 Stability of the Bifurcating Solution.- VI.5 The Extra Little Part for ?1 in Projection.- V1.6 Projections of Higher-Dimensional Problems.- VI.7 The Spectral Problem for the Stability of u = 0.- VI.8 The Spectral Problem and the Laplace Transform.- VI.9 Projections into ?1.- VI.10 The Method of Projection for Isolated Solutions Which Perturb Bifurcation at a Simple Eigenvalue (Imperfection Theory).- VI.1 1 The Method of Projection at a Double Eigenvalue of Index Two.- VI.12 The Method of Projection at a Double Semi-Simple Eigenvalue.- VI.13 Examples of the Method of Projection.- VI.14 Symmetry and Pitchfork Bifurcation.- VII Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions.- VII.1 The Structure of the Two-Dimensional Problem Governing Hopf Bifurcation.- VII.2 Amplitude Equation for Hopf Bifurcation.- VII.3 Series Solution.- VII.4 Equations Governing the Taylor Coefficients.- VII.5 Solvability Conditions (the Fredholm Alternative).- VII.6 Floquet Theory.- 6.1 Floquet Theory in ?1.- 6.2 Floquet Theory in ?2 and ?n.- VII.7 Equations Governing the Stability of the Periodic Solutions.- VII.8 The Factorization Theorem.- VII.9 Interpretation of the Stability Result.- Example.- VIII Bifurcation of Periodic Solutions in the General Case.- VIII.1 Eigenprojections of the Spectral Problem.- VIII.2 Equations Governing the Projection and the Complementary Projection.- VIII.3 The Series Solution Using the Fredholm Alternative.- VIII.4 Stability of the Hopf Bifurcation in the General Case.- VIII.5 Systems with Rotational Symmetry.- Examples.- Notes.- IX Subharmonic Bifurcation of Forced T-Periodic Solutions.- Notation.- IX.1 Definition of the Problem of Subharmonic Bifurcation.- IX.2 Spectral Problems and the Eigenvalues ?( µ).- IX.3 Biorthogonality.- IX.4 Criticality.- IX.S The Fredholm Alternative for J( µ) —?( µ)and a Formula Expressing the Strict Crossing (IX.20).- IX.6 Spectral Assumptions.- IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality.- IX.8 The Operator $$\mathbb{J}$$ and its Eigenvectors.- IX.9 The Adjoint Operator $${{\mathbb{J}}^{*}}$$ Biorthogonality, Strict Crossing, and the Fredholm Alternative for $$\mathbb{J}$$.- IX.10 The Amplitude ?and the Biorthogonal Decomposition of Bifurcating Subharmonic Solutions.- IX.11 The Equations Governing the Derivatives of Bifurcating Subharmonic Solutions with Respect to ?at ? =0.- IX.12 Bifurcation and Stability of T-Periodic and 2 T-Periodic Solutions.- IX.13 Bifurcation and Stability of n T-Periodic Solutions with n >2.- IX.14 Bifurcation and Stability of 3T-Periodic Solutions.- IX.15 Bifurcation of 4 T-Periodic Solutions.- IX.16 Stability of 4 T-Periodic Solutions.- IX.17 Nonexistence of Higher-Order Subharmonic Solutions and Weak Resonance.- IX.18 Summary of Results About Subharmonic Bifurcation.- IX.19 Imperfection Theory with a Periodic Imperfection.- Exercises.- IX.20 Saddle-Node Bifurcation of T-Periodic Solutions.- IX.21 General Remarks About Subharmonic Bifurcations.- X Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions.- X.1 Decomposition of the Solution and Amplitude Equation.- Exercise.- X.2 Derivation of the Amplitude Equation.- X.3 The Normal Equations in Polar Coordinates.- X.4 The Torus and Trajectories on the Torus in the Irrational Case.- X.5 The Torus and Trajectories on the Torus When ?0T/2? Is a Rational Point of Higher Order (n?5).- X.6 The Form of the Torus in the Case n =5.- X.7 Trajectories on the Torus When n =5.- X.8 The Form of the Torus When n >5.- X.9 Trajectories on the Torus When n?5.- X.10 Asymptotically Quasi-Periodic Solutions.- X.11 Stability of the Bifurcated Torus.- X.12 Subharmonic Solutions on the Torus.- X.13 Stability of Subharmonic Solutions on the Torus.- X.14 Frequency Locking.- Appendix X.1 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Irrational Points Using the Method of Two Times, Power Series, and the Fredholm Alternative.- Appendix X.2 Direct Computation of Asymptotically Quasi-Periodic Solutions Which Bifurcate at Rational Points of Higher Order Using the Method of Two Times.- Exercise.- Notes.- XI Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case.- Notation.- XI.1 Spectral Problems.- XI.2 Criticality and Rational Points.- XI.3 Spectral Assumptions About J0.- XI.4 Spectral Assumptions About $$\mathbb{J}$$ in the Rational Case.- XI.5 Strict Loss of Stability at a Simple Eigenvalue of J0.- XI.6 Strict Loss of Stability at a Double Semi-Simple Eigenvalue of J0.- XI.7 Strict Loss of Stability at a Double Eigenvalue of Index Two.- XI.8 Formulation of the Problem of Subharmonic Bifurcation of Periodic Solutions of Autonomous Problems.- XI.9 The Amplitude of the Bifurcating Solution.- XI.10 Power-Series Solutions of the Bifurcation Problem.- XI.11 Subharmonic Bifurcation When n =2.- XI.12 Subharmonic Bifurcation When n >2.- XI.13 Subharmonic Bifurcation When n = 1in the Semi-Simple Case.- XI.14 “Subharmonic” Bifurcation When n =1 in the Case When Zero is an Index-Two Double Eigenvalue of Jo.- XI.15 Stability of Subharmonic Solutions.- XI.16 Summary of Results About Subharmonic Bifurcation in the Autonomous Case.- XI.17 Amplitude Equations.- XI.18 Amplitude Equations for the Cases n?3 or ?0/?0Irrational.- XI.19 Bifurcating Tori. Asymptotically Quasi-Periodic Solutions.- XI.20 Period Doubling n =2.- XI.21 Pitchfork Bifurcation of Periodic Orbits in the Presence of Symmetry n = 1.- Exercises.- XI.22 Rotationally Symmetric Problems.- Exercise.- XII Stability and Bifurcation in Conservative Systems.- XII.1 The Rolling Ball.- XII.2 Euler Buckling.- Exercises.- XII.3 Some Remarks About Spectral Problems for Conservative Systems.- XII.4 Stability and Bifurcation of Rigid Rotation of Two Immiscible Liquids.- Steady Rigid Rotation of Two Fluids.

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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 SynopsisOne Basic Theory.- I Complex Numbers and Functions.- II Power Series.- III Cauchy's Theorem, First Part.- IV Winding Numbers and Cauchy's Theorem.- V Applications of Cauchy's integral Formula.- VI Calculus of Residues.- VII Conformal Mappings.- VIII Harmonic Functions.- Two Geometric Function Theory.- IX Schwarz Reflection.- X The Riemann Mapping Theorem.- XI Analytic Continuation Along Curves.- Three Various Analytic Topics.- XII Applications of the Maximum Modulus Principle and Jensen's Formula.- XIII Entire and Meromorphic Functions.- XIV Elliptic Functions.- XV The Gamma and Zeta Functions.- XVI The Prime Number Theorem.- 1. Summation by Parts and Non-Absolute Convergence.- 2. Difference Equations.- 3. Analytic Differential Equations.- 4. Fixed Points of a Fractional Linear Transformation.- 6. Cauchy's Theorem for Locally Integrable Vector Fields.- 7. More on Cauchy-Riemann.Trade Review"The very understandable style of explanation, which is typical for this author, makes the book valuable for both students and teachers."EMS Newsletter, Vol. 37, Sept. 2000 Fourth Edition S. Lang Complex Analysis "A highly recommendable book for a two semester course on complex analysis." —ZENTRALBLATTMATHTable of ContentsI: BASIC THEORY. 1: Complex Numbers and Functions. 2: Power Series. 3: Cauchy's Theorem, First Part. 4: Winding Numbers and Cauchy's Theorem. 5: Applications of Cauchy's Integral Formula. 6: Calculus of Residues. 7: Conformal Mappings. 8: Harmonic Functions. II: GEOMETRIC FUNCTION THEORY. 9: Schwarz Reflection. 10: The Riemann Mapping Theorem. 11: Analytic Continuation Along Curves. III: VARIOUS ANALYTIC TOPICS. 12: Applications of the Maximum Modulus Principle and Jensen's Formula. 13: Entire and Meromorphic Functions. 14: Elliptic Functions. 15: The Gamma and Zeta Functions. 16: The Prime Number Theorem.

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Book SynopsisFifteen years ago, most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely, yet today the situation has completely changed.Trade Review“Each chapter ends with a section called Notes and another called Exercises and Further Results. … It would be quite suitable for graduate students in the field.” (Lou Zengjian, zbMATH 0955.32003, 2022)Table of Contents1 The Bergman Spaces.- 1.1 Bergman Spaces.- 1.2 Some Lp Estimates.- 1.3 The Bloch Space.- 1.4 Duality of Bergman Spaces.- 1.5 Notes.- 1.6 Exercises and Further Results.- 2 The Berezin Transform.- 2.1 Algebraic Properties.- 2.2 Harmonic Functions.- 2.3 Carleson-Type Measures.- 2.4 BMO in the Bergman Metric.- 2.5 A Lipschitz Estimate.- 2.6 Notes.- 2.7 Exercises and Further Results.- 3 Ap -Inner Functions.- 3.1 Ap? -Inner Functions.- 3.2 An Extremal Problem.- 3.3 The Biharmonic Green function.- 3.4 The Expansive Multiplier Property.- 3.5 Contractive Zero Divisors in Ap.- 3.6 An Inner-Outer Factorization Theorem for Ap.- 3.7 Approximation of Subinner Functions.- 3.8 Notes.- 3.9 Exercises and Further Results.- 4 Zero Sets.- 4.1 Some Consequences of Jensen’s Formula.- 4.2 Notions of Density.- 4.3 The Growth Spaces A-? and A-?.- 4.4 A-? Zero Sets, Necessary Conditions.- 4.5 A-? Zero Sets, a Sufficient Condition.- 4.6 Zero Sets for AP?.- 4.7 The Bergman-Nevanlinna Class.- 4.8 Notes.- 4.9 Exercises and Further Results.- 5 Interpolation and Sampling.- 5.1 Interpolation Sequences for AT-?.- 5.2 Sampling Sets for A-?.- 5.3 Interpolation and Sampling in Ap?.- 5.4 Hyperbolic Lattices.- 5.5 Notes.- 5.6 Exercises and Further Results.- 6 Invariant Subspaces.- 6.1 Invariant Subspaces of Higher Index.- 6.2 Inner Spaces in A2?.- 6.3 A Beurling-Type Theorem.- 6.4 Notes.- 6.5 Exercises and Further Results.- 7 Cyclicity.- 7.1 Cyclic Vectors as Outer functions.- 7.2 Cyclicity in Ap Versus in A-?.- 7.3 Premeasures for Functions in A-?.- 7.4 Cyclicity in A-?.- 7.5 Notes.- 7.6 Exercises and Further Results.- 8 Invertible Noncyclic Functions.- 8.1 An Estimate for Harmonic Functions.- 8.2 The Building Blocks.- 8.3 The Basic Iteration Scheme.- 8.4 The Mushroom Forest.- 8.5 Finishing the Construction.- 8.6 Two Applications.- 8.7 Notes.- 8.8 Exercises and Further Results.- 9 Logarithmically Subharmonic Weights.- 9.1 Reproducing Kernels.- 9.2 Green Functions with Smooth Weights.- 9.3 Green Functions with General Weights.- 9.4 An Application.- 9.5 Notes.- 9.6 Exercises and Further Results.- References.

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Book SynopsisI Fundamentals.- 1 Ordinary Differential Equations.- 2 Difference Equations.- II Local Analysis.- 3 Approximate Solution of Linear Differential Equations.- 4 Approximate Solution of Nonlinear Differential Equations.- 5 Approximate Solution of Difference Equations.- 6 Asymptotic Expansion of Integrals.- III Perturbation Methods.- 7 Perturbation Series.- 8 Summation of Series.- IV Global Analysis.- 9 Boundary Layer Theory.- 10 WKB Theory.- 11 Multiple-Scale Analysis.Trade Review"This book is a reprint of the original published by McGraw-Hill \ref [MR0538168 (80d:00030)]. The only changes are the addition of the Roman numeral I to the title and the provision of a subtitle, "Asymptotic methods and perturbation theory". This latter improvement is much needed, as the original title suggested that this was a teaching book for undergraduate scientists and engineers. It is not, but is an excellent introduction to asymptotic and perturbation methods for master's degree students or beginning research students. Certain parts of it could be used for a course in asymptotics for final year undergraduates in applied mathematics or mathematical physics. This is a book that has stood the test of time and I cannot but endorse the remarks of the original reviewer. It is written in a fresh and lively style and the many graphs and tables, comparing the results of exact and approximate methods, were in advance of its time. I have owned a copy of the original for over twenty years, using it on a regular basis, and, after the original had gone out of print, lending it to my research students. Springer-Verlag has done a great service to users of, and researchers in, asymptotics and perturbation theory by reprinting this classic." (A.D. Wood, Mathematical Reviews) Table of ContentsI Preface. 1 Ordinary Differential Equations. 2 Difference Equations. 3 Approximate Solution of Linear Differential Equations. 4 Approximate Solution of Nonlinear Equations. 5 Approximate Solution of Difference Equations. 6 Asymptotic Expansion of Integrals. 7 Perturbation Series. 8 Summation of Series. 9 Boundary Layer Theory. 10 WKB Theory. 11 Multiple Scales Analysis. Appendix, References, Index

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Book SynopsisDuring the 90s robust control theory has seen major advances and achieved a new maturity, centered around the notion of convexity.Trade ReviewFrom the reviews"Because progress in LMI robust control theory has been explosive, only books published in the past 3 or 4 years can hope to adequatetely document the phenomenon. The textbook of Dullerud and Paganini rises admirably to the challenge, starting from the basics of linear algebra and system theory and leading the reader through the key 1990s breakthroughs in LMI robust control theory. To keep things simple, the authors relegate the issue of robustness against nonlinear uncertainties to the citations, focusing attention squarely on the linear case. (...)The book would make an excellent text for a two-semester or two-quarter course for first year graduate students beginning with no prior knowledge of state-space methods. Alternatively, for control students who already have a state-space background."IEEE Transactions on Automatics Control, Vol. 46, No. 9, September 2001Table of Contents0 Introduction.- 1 Preliminaries in Finite Dimensional Space.- 2 State Space System Theory.- 3 Linear Analysis.- 4 Model Realizations and Reduction.- 5 Stabilizing Controllers.- 6 H2 Optimal Control.- 7 H? Synthesis.- 8 Uncertain Systems.- 9 Feedback Control of Uncertain Systems.- 10 Further Topics: Analysis.- 11 Further Topics: Synthesis.- A Some Basic Measure Theory.- A.1 Sets of zero measure.- A.2 Terminology.- Notes and references.- B Proofs of Strict Separation.- Notes and references.- Notes and references.- Notation.- References.

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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 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 SynopsisFactor analysis, a mathematical technique for studying matrices of data, has long been used in the behavioural sciences. This new edition of a work on its application to chemical problems has been thoroughly revised and includes an added chapter on special methods of factor analysis.Table of ContentsMain Steps; Mathematical Formulation of Target Factor Analysis; Effects of Experimental Error on Target Factor Analysis; Numerical Examples of Target Factor Analysis; Special Methods of Factor Analysis; Component Analysis; Nuclear Magnetic Resonance; Chromatography; Additional Applications; Appendices; Bibliography; Author Index; Subject Index.

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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 SynopsisThe purpose of this book is to bridge the gap between differential geometry of Euclidean space of three dimensions and the more advanced work on differential geometry of generalised space.Table of Contents1. Some Preliminaries; 2. Coordinates, Vectors , Tensors; 3. Riemannian Metric; 4. Christoffel's Three-Index Symbols. Covariant Differentiation; 5. Curvature of a Curve. Geodeics, Parallelism of Vectors; 6. Congruences and Orthogonal Ennuples; 7. Riemann Symbols. Curvature of a Riemannian Space; 8. Hypersurfaces; 9. Hypersurfaces in Euclidean Space. Spaces of Constant Curvature; 10. Subspaces of a Riemannian Space.

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Book SynopsisIn this book, Professor Copson gives a rigorous account of the theory of partial differential equations of the first orderTable of ContentsPreface; 1. Partial differential equations of the first order; 2. Characteristics of equations of the second order; 3. Boundary value and initial value problems; 4. Equations of hyperbolic type; 5. Reimann's method; 6. The equation of wave motions; 7. Marcel Riesz's method; 8. Potential theory in the plane; 9. Subharmonic functions and the problem of Dirichlet; 10. Equations of elliptic type in the plane; 11. Equations of elliptic type in space; 12. The equation of heat; Appendix; Books for further reading; Index.

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Book SynopsisThe theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to Trade Review'A very valuable addition to the growing field of random graphs, providing a systematic coverage of these novel models.' Michael Krivelevich, Mathematical Reviews'The book is written in a friendly, chatty style, making it easy to read; I very much like that. In summary, Random Graph Dynamics is a nice contribution to the area of random graphs and a source of valuable insights.' Malwina J. Luczak, Journal of the American Statistical AssociationTable of Contents1. Overview; 2. Erdos–Renyi random graphs; 3. Fixed degree distributions; 4. Power laws; 5. Small worlds; 6. Random walks; 7. CHKNS model.

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Book SynopsisThe theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approachTrade Review' … this is a very valuable new reference for the subject, incorporating many modern results and perspectives that are not present in earlier texts on this topic … this book would serve as an excellent foundation with which to begin studying other aspects of random matrix theory.' Terence Tao, Mathematical Reviews'… the book aims to introduce some of the modern techniques of random matrix theory in a comprehensive and rigorous way. It has a broad range of topics and most of them are fairly accessible. The focus is on introducing and explaining the main techniques, rather than obtaining the most general results. Additional references are given for the reader who wants to continue the study of a certain topic. The writing style is careful and the book is mostly self-contained with complete proofs. This is an excellent new contribution to random matrix theory.' Journal of Approximation TheoryTable of ContentsPreface; 1. Introduction; 2. Real and complex Wigner matrices; 3. Hermite polynomials, spacings, and limit distributions for the Gaussian ensembles; 4. Some generalities; 5. Free probability; Appendices; Bibliography; General conventions; Glossary; Index.

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Book SynopsisFor the second edition of this very successful text, Professor Binmore has written two chapters on analysis in vector spaces. The discussion extends to the notion of the derivative of a vector function as a matrix and the use of second derivatives in classifying stationary points.Table of ContentsPreface to the first edition; Preface to the second edition; 1. Real numbers; 2. Continuum property; 3. Natural numbers; 4. Convergent sequences; 5. Subsequences; 6. Series; 7. Functions; 8. Limits of functions; 9. Continuity; 10. Differentiation; 11. Mean value theorems; 12. Monotone functions; 13. Integration; 14. Exponential and logarithm; 15. Power series; 16. Trigonometric functions; 17. The gamma function; 18. Vectors; 19. Vector derivatives; 20. Appendix; Solutions to exercises; Further problems; Suggested further reading; Notation; Index.

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Book SynopsisThis textbook is an introduction to the theory of Hilbert space and its applications. The notion of Hilbert space is central in functional analysis and is used in numerous branches of pure and applied mathematics. Some basic familiarity with real analysis, linear algebra and metric spaces is assumed, but otherwise the book is self-contained.Trade Review' … the author's style is a delight. Each topic is carefully motivated and succinctly presented, and the exposition is enthusiastic and limpid … Young has done a really fine job in presenting a subject of great mathematical elegance as well as genuine utility, and I recommend it heartily.' The Times Higher Education SupplementTable of ContentsForeword; Introduction; 1. Inner product spaces; 2. Normed spaces; 3. Hilbert and Banach spaces; 4. Orthogonal expansions; 5. Classical Fourier series; 6. Dual spaces; 7. Linear operators; 8. Compact operators; 9. Sturm-Liouville systems; 10. Green's functions; 11. Eigenfunction expansions; 12. Positive operators and contractions; 13. Hardy spaces; 14. Interlude: complex analysis and operators in engineering; 15. Approximation by analytic functions; 16. Approximation by meromorphic functions; Appendix; References; Answers to selected problems; Afterword; Index of notation; Subject index.

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Book SynopsisThis classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and exhaustively both the statement and proof of all the standard inequalities of analysis. The authors were well known for their powers of exposition and were able here to make the subject accessible to a wide audience of mathematicians.Trade Review'In retrospect one sees that 'Hardy, Littlewood and Pólya' has been one of the most important books in analysis in the last few decades. It had an impact on the trend of research and is still influencing it. In looking through the book now one realises how little one would like to change the existing text.' A. Zygmund, Bulletin of the AMSTable of Contents1. Introduction; 2. Elementary mean values; 3. Mean values with an arbitrary function and the theory of convex functions; 4. Various applications of the calculus; 5. Infinite series; 6. Integrals; 7. Some applications of the calculus of variations; 8. Some theorems concerning bilinear and multilinear forms; 9. Hilbert's inequality and its analogues and extensions; 10. Rearrangements; Appendices; Bibliography.

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Book SynopsisFirst published in 1941, this book, by one of the foremost geometers of his day, rapidly became a classic. In its original form the book constituted a section of Hodge's essay for which the Adam's prize of 1936 was awarded, but the author substantially revised and rewrote it.Table of Contents1. Reimannian Manifolds; 2. Integrals and their periods; 3. Harmonic Integrals; 4. Applications to algebraic varieties; 5. Applications to the theory of continuous groups.

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Book SynopsisA textbook presenting the theory and underlying techniques of perturbation methods in a manner suitable for senior undergraduates from a broad range of disciplines.Trade Review'A nice and readable introduction.' Monatshefte für MathematikTable of ContentsPreface; 1. Algebraic equations; 2. Asymptotic expansions; 3. Integrals; 4. Regular problems in PDEs; 5. Matched asymptotic expansions; 6. Method of strained coordinates; 7. Method of multiple scales; 8. Improved convergence; Bibliography; Index.

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Book SynopsisMathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education the traditional development of analysis, often rather divorced from the calculus which they learnt at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus at school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate the new ideas are related to school topics and are used to extend the reader's understanding of those topics. A first course in analysis at colleTrade Review"Bryant's style is extremely leisurely, copiously illustrated, often intuitively appealing, chatty and unintimidating, in contrast to other treatments of similar material..." ChoiceTable of ContentsPreface; 1. Firm foundations; 2. Gradually getting there; 3. A functional approach; 4. Calculus at last!; 5. An integrated conclusion; Solutions to exercises; Index.

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Book SynopsisThis text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. It is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning.Trade Review"The book is well written.... This book provides a very readable introduction to heat kernal methods and it can be strongly recommended for graduate students of mathematics looking for a thorough introduction to the topic." Friedbert PrÜfer, Mathematical ReviewsTable of ContentsIntroduction; 1. The Laplacian on a Riemannian manifold; 2. Elements of differential geometry; 3. The construction of the heat kernel; 4. The heat equation approach to the Atiyah-Singer index theorem; 5. Zeta functions of Laplacians; Bibliography; Index.

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Book SynopsisThis 1994 book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. There are nearly 200 exercises, making the book ideal for both classroom use and self-study.Trade Review"...Darling's exegesis is clear and easy to understand, and his frequent use of examples is beneficial to the reader. There are many exercises that serve to reinforce the concepts." D.P. Turner, Choice"...easy on the eyes; some nice exercises..." American Mathematical Monthly"The exposition is clear and, in the American textbook style, has many exercises, both theoretical and computational. In summary, this text provides a worthwhile elementary introduction to anyone who wants to understand the basic mathematical ingredients of Differential Geometry and its interactions with Physics." F.E. Burstall, Contemporary Physics"...a good introduction to differential geometry and its applications to physics by using the calculus of differential forms...Nearly 200 exercises and many examples will help the reader's understanding...this book can be recommended as a good textbook for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering." Akira Asada, Mathematical ReviewsTable of ContentsPreface; 1. Exterior algebra; 2. Exterior calculus on Euclidean space; 3. Submanifolds of Euclidean spaces; 4. Surface theory using moving frames; 5. Differential manifolds; 6. Vector bundles; 7. Frame fields, forms and metrics; 8. Integration on oriented manifolds; 9. Connections on vector bundles; 10. Applications to gauge field theory; Bibliography; Index.

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