Calculus and mathematical analysis Books

Book SynopsisSuitable for anyone working with problems of linear and nonlinear least squares fitting, this book includes an overview of computational methods together with their properties and advantages. It also includes topics from statistical regression analysis that help readers to understand and evaluate the computed solutions.Trade ReviewLeast Square Data fitting with Applications is a book that will be of great practical use to anyone whose work involves the analysis of data and its modeling. It offers a great deal of information that can be a s valuable in the lecture theater as in the lab or office. Mathematics TodayTable of ContentsPrefaceSymbols and AcronymsChapter 1. The Linear Data Fitting Problem1.1. Parameter estimation, data approximation1.2. Formulation of the data fitting problem1.3. Maximum likelihood estimation1.4. The residuals and their properties1.5. Robust regressionChapter 2. The Linear Least Squares Problem2.1. Linear least squares problem formulation2.2. The QR factorization and its role2.3. Permuted QR factorizationChapter 3. Analysis of Least Squares Problems3.1. The pseudoinverse3.2. The singular value decomposition3.3. Generalized singular value decomposition3.4. Condition number and column scaling3.5. Perturbation analysisChapter 4. Direct Methods for Full-Rank Problems4.1. Normal equations4.2. LU factorization4.3. QR factorization4.4. Modifying least squares problems4.5. Iterative refinement4.6. Stability and condition number estimation4.7. Comparison of the methodsChapter 5. Direct Methods for Rank-Deficient Problems5.1. Numerical rank5.2. Peters-Wilkinson LU factorization5.3. QR factorization with column permutations5.4. UTV and VSV decompositions5.5. Bidiagonalization5.6. SVD computationsChapter 6. Methods for Large-Scale Problems6.1. Iterative versus direct methods6.2. Classical stationary methods6.3. Non-stationary methods, Krylov methods6.4. Practicalities: preconditioning and stopping criteria6.5. Block methodsChapter 7. Additional Topics in Least Squares7.1. Constrained linear least squares problems7.2. Missing data problems7.3. Total least squares (TLS)7.4. Convex optimization7.5. Compressed sensingChapter 8. Nonlinear Least Squares Problems8.1. Introduction8.2. Unconstrained problems8.3. Optimality conditions for constrained problems8.4. Separable nonlinear least squares problems8.5. Multiobjective optimizationChapter 9. Algorithms for Solving Nonlinear LSQ Problems9.1. Newton's method9.2. The Gauss-Newton method9.3. The Levenberg-Marquardt method9.4. Additional considerations and software9.5. Iteratively reweighted LSQ algorithms for robust data fitting problems9.6. Variable projection algorithm9.7. Block methods for large-scale problemsChapter 10. Ill-Conditioned Problems10.1. Characterization10.2. Regularization methods10.3. Parameter selection techniques10.4. Extensions of Tikhonov regularization10.5. Ill-conditioned NLLSQ problemsChapter 11. Linear Least Squares Applications11.1. Splines in approximation11.2. Global temperatures data fitting11.3. Geological surface modelingChapter 12. Nonlinear Least Squares Applications12.1. Neural networks training12.2. Response surfaces, surrogates or proxies12.3. Optimal design of a supersonic aircraft12.4. NMR spectroscopy12.5. Piezoelectric crystal identification12.6. Travel time inversion of seismic dataAppendix A: Sensitivity AnalysisA.1. Floating-point arithmeticA.2. Stability, conditioning and accuracyAppendix B: Linear Algebra BackgroundB.1. NormsB.2. Condition numberB.3. OrthogonalityB.4. Some additional matrix propertiesAppendix C: Advanced Calculus BackgroundC.1. Convergence ratesC.2. Multivariable calculusAppendix D: StatisticsD.1. DefinitionsD.2. Hypothesis testingReferencesIndex

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Book SynopsisManifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.Trade ReviewFrom the reviews of the second edition:“This book could be called a prequel to the book ‘Differential forms in algebraic topology’ by R. Bott and the author. Assuming only basic background in analysis and algebra, the book offers a rather gentle introduction to smooth manifolds and differential forms offering the necessary background to understand and compute deRham cohomology. … The text also contains many exercises … for the ambitious reader.” (A. Cap, Monatshefte für Mathematik, Vol. 161 (3), October, 2010)Table of ContentsPreface to the Second Edition.- Preface to the First Edition.-Chapter 1. Eudlidean Spaces. 1. Smooth Functions on a Euclidean Space.- 2. Tangent Vectors in R(N) as Derivativations.- 3. The Exterior Algebra of Multicovectors.- 4. Differential Forms on R(N).- Chapter 2. Manifolds.- 5. Manifolds.- 6. Smooth Maps on a Manifold.- 7. Quotients.- Chapter 3. The Tangent Space.- 8. The Tangent Space.- 9. Submanifolds.- 10. Categories and Functors.- 11. The Rank of a Smooth Map.- 12. The Tangent Bundle.- 13. Bump Functions and Partitions of Unity.- 14. Vector Fields.-Chapter 4. Lie Groups and Lie Algebras.- 15. Lie Groups.- 16. Lie Algebras.- Chapter 5. Differential Forms.- 17. Differential 1-Forms.- 18. Differential k-Forms.- 19. The Exterior Derivative.- 20. The Lie Derivative and Interior Multiplication.- Chapter 6. Integration.- 21. Orientations.- 22. Manifolds with Boundary.- 23. Integration on Manifolds.- Chapter 7. De Rham Theory.- 24. De Rham Cohomology.- 25. The Long Exact Sequence in Cohomology.- 26. The Mayer –Vietoris Sequence.- 27. Homotopy Invariance.- 28. Computation of de Rham Cohomology.- 29. Proof of Homotopy Invariance.-Appendices.- A. Point-Set Topology.- B. The Inverse Function Theorem on R(N) and Related Results.- C. Existence of a Partition of Unity in General.- D. Linear Algebra.- E. Quaternions and the Symplectic Group.- Solutions to Selected Exercises.- Hints and Solutions to Selected End-of-Section Problems.- List of Symbols.- References.- Index.

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Book Synopsis1 The Spectrum of Linear Operators and Hilbert Spaces.- 2 The Geometry of a Hilbert Space and Its Subspaces.- 3 Exponential Decay of Eigenfunctions.- 4 Operators on Hilbert Spaces.- 5 Self-Adjoint Operators.- 6 Riesz Projections and Isolated Points of the Spectrum.- 7 The Essential Spectrum: Weyl's Criterion.- 8 Self-Adjointness: Part 1. The Kato Inequality.- 9 Compact Operators.- 10 Locally Compact Operators and Their Application to Schrödinger Operators.- 11 Semiclassical Analysis of Schrödinger Operators I: The Harmonic Approximation.- 12 Semiclassical Analysis of Schrödinger Operators II: The Splitting of Eigenvalues.- 13 Self-Adjointness: Part 2. The Kato-Rellich Theorem 131.- 14 Relatively Compact Operators and the Weyl Theorem.- 15 Perturbation Theory: Relatively Bounded Perturbations.- 16 Theory of Quantum Resonances I: The Aguilar-Balslev-Combes-Simon Theorem.- 17 Spectral Deformation Theory.- 18 Spectral Deformation of Schrödinger Operators.- 19 The General Theory of Spectral Stability.- 20 Theory of Quantum Resonances II: The Shape Resonance Model.- 21 Quantum Nontrapping Estimates.- 22 Theory of Quantum Resonances III: Resonance Width.- 23 Other Topics in the Theory of Quantum Resonances.- Appendix 1. Introduction to Banach Spaces.- A1.1 Linear Vector Spaces and Norms.- A1.2 Elementary Topology in Normed Vector Spaces.- A1.3 Banach Spaces.- A1.4 Compactness.- 1. Density results.- 2. The Hölder Inequality.- 3. The Minkowski Inequality.- 4. Lebesgue Dominated Convergence.- Appendix 3. Linear Operators on Banach Spaces.- A3.1 Linear Operators.- A3.2 Continuity and Boundedness of Linear Operators.- A3.3 The Graph of an Operator and Closure.- A3.4 Inverses of Linear Operators.- A3.5 Different Topologies on L(X).- Appendix 4. The Fourier Transform, SobolevSpaces, and Convolutions.- A4.1 Fourier Transform.- A4.2 Sobolev Spaces.- A4.3 Convolutions.- References.Table of Contents1 The Spectrum of Linear Operators and Hilbert Spaces.- 2 The Geometry of a Hilbert Space and Its Subspaces.- 3 Exponential Decay of Eigenfunctions.- 4 Operators on Hilbert Spaces.- 5 Self-Adjoint Operators.- 6 Riesz Projections and Isolated Points of the Spectrum.- 7 The Essential Spectrum: Weyl’s Criterion.- 8 Self-Adjointness: Part 1. The Kato Inequality.- 9 Compact Operators.- 10 Locally Compact Operators and Their Application to Schrödinger Operators.- 11 Semiclassical Analysis of Schrödinger Operators I: The Harmonic Approximation.- 12 Semiclassical Analysis of Schrödinger Operators II: The Splitting of Eigenvalues.- 13 Self-Adjointness: Part 2. The Kato-Rellich Theorem 131.- 14 Relatively Compact Operators and the Weyl Theorem.- 15 Perturbation Theory: Relatively Bounded Perturbations.- 16 Theory of Quantum Resonances I: The Aguilar-Balslev-Combes-Simon Theorem.- 17 Spectral Deformation Theory.- 18 Spectral Deformation of Schrödinger Operators.- 19 The General Theory of Spectral Stability.- 20 Theory of Quantum Resonances II: The Shape Resonance Model.- 21 Quantum Nontrapping Estimates.- 22 Theory of Quantum Resonances III: Resonance Width.- 23 Other Topics in the Theory of Quantum Resonances.- Appendix 1. Introduction to Banach Spaces.- A1.1 Linear Vector Spaces and Norms.- A1.2 Elementary Topology in Normed Vector Spaces.- A1.3 Banach Spaces.- A1.4 Compactness.- 1. Density results.- 2. The Hölder Inequality.- 3. The Minkowski Inequality.- 4. Lebesgue Dominated Convergence.- Appendix 3. Linear Operators on Banach Spaces.- A3.1 Linear Operators.- A3.2 Continuity and Boundedness of Linear Operators.- A3.3 The Graph of an Operator and Closure.- A3.4 Inverses of Linear Operators.- A3.5 Different Topologies on L(X).- Appendix 4. The Fourier Transform, Sobolev Spaces, and Convolutions.- A4.1 Fourier Transform.- A4.2 Sobolev Spaces.- A4.3 Convolutions.- References.

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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 SynopsisBasic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.Trade ReviewFrom the book reviews:“This books presents an alternative route, aiming to provide the student with an introduction not only to Riemannian geometry, but also to contact and symplectic geometry. … the book is leavened with an excellent collection of illustrative examples, and a wealth of exercises on which students can hone their skills. Each chapter also includes a short guide to further reading on the topic with a helpful brief commentary on the suggestions.” (Robert J. Low, Mathematical Reviews, May, 2014)“This book is a distinctive and ambitious effort to bring modern notions of differential geometry to undergraduates. … Mclnerney’s writing is well constructed and very clear … . Summing Up: Recommended. Upper-division undergraduates and graduate students.” (S. J. Colley, Choice, Vol. 51 (8), April, 2014)“The author does make a considerable effort to keep things as accessible as possible, with fairly detailed explanations, extensive motivational discussions and homework problems … . this book provides a different way of looking at the subject of differential geometry, one that is more modern and sophisticated than is provided by many of the standard undergraduate texts and which will certainly do a good job of preparing the student for additional work in this area down the road.” (Mark Hunacek, MAA Reviews, January, 2014)“This text provides an early and broad view of geometry to mathematical students … . Altogether, this book is easy to read because there are plenty of figures, examples and exercises which make it intuitive and perfect for undergraduate students.” (Teresa Arias-Marco, zbMATH, Vol. 1283, 2014)Table of ContentsBasic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.

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Book Synopsis1 Euclidean spaces.- 1.1 The real number system.- 1.2 Euclidean En.- 1.3 Elementary geometry of En.- 1.4 Basic topological notions in En.- *1.5 Convex sets.- 2 Elementary topology of En.- 2.1 Functions.- 2.2 Limits and continuity of transformations.- 2.3 Sequences in En.- 2.4 Bolzano-Weierstrass theorem.- 2.5 Relative neighborhoods, continuous transformations.- 2.6 Topological spaces.- 2.7 Connectedness.- 2.8 Compactness.- 2.9 Metric spaces.- 2.10 Spaces of continuous functions.- *2.11 Noneuclidean norms on En.- 3 Differentiation of real-valued functions.- 3.1 Directional and partial derivatives.- 3.2 Linear functions.- **3.3 Difierentiable functions.- 3.4 Functions of class C(q).- 3.5 Relative extrema.- *3.6 Convex and concave functions.- 4 Vector-valued functions of several variables.- 4.1 Linear transformations.- 4.2 Affine transformations.- 4.3 Differentiable transformations.- 4.4 Composition.- 4.5 The inverse function theorem.- 4.6 The implicit function theorem.- 4.7 Manifolds.- 4Table of Contents1 Euclidean spaces.- 1.1 The real number system.- 1.2 Euclidean En.- 1.3 Elementary geometry of En.- 1.4 Basic topological notions in En.- *1.5 Convex sets.- 2 Elementary topology of En.- 2.1 Functions.- 2.2 Limits and continuity of transformations.- 2.3 Sequences in En.- 2.4 Bolzano-Weierstrass theorem.- 2.5 Relative neighborhoods, continuous transformations.- 2.6 Topological spaces.- 2.7 Connectedness.- 2.8 Compactness.- 2.9 Metric spaces.- 2.10 Spaces of continuous functions.- *2.11 Noneuclidean norms on En.- 3 Differentiation of real-valued functions.- 3.1 Directional and partial derivatives.- 3.2 Linear functions.- **3.3 Difierentiable functions.- 3.4 Functions of class C(q).- 3.5 Relative extrema.- *3.6 Convex and concave functions.- 4 Vector-valued functions of several variables.- 4.1 Linear transformations.- 4.2 Affine transformations.- 4.3 Differentiable transformations.- 4.4 Composition.- 4.5 The inverse function theorem.- 4.6 The implicit function theorem.- 4.7 Manifolds.- 4.8 The multiplier rule.- 5 Integration.- 5.1 Intervals.- 5.2 Measure.- 5.3 Integrals over En.- 5.4 Integrals over bounded sets.- 5.5 Iterated integrals.- 5.6 Integrals of continuous functions.- 5.7 Change of measure under affine transformations.- 5.8 Transformation of integrals.- 5.9 Coordinate systems in En.- 5.10 Measurable sets and functions; further properties.- 5.11 Integrals: general definition, convergence theorems.- 5.12 Differentiation under the integral sign.- 5.13 Lp-spaces.- 6 Curves and line integrals.- 6.1 Derivatives.- 6.2 Curves in En.- 6.3 Differential 1-forms.- 6.4 Line integrals.- *6.5 Gradient method.- *6.6 Integrating factors; thermal systems.- 7 Exterior algebra and differential calculus.- 7.1 Covectors and differential forms of degree 2.- 7.2 Alternating multilinear functions.- 7.3 Multicovectors.- 7.4 Differential forms.- 7.5 Multivectors.- 7.6 Induced linear transformations.- 7.7 Transformation law for differential forms.- 7.8 The adjoint and codifferential.- *7.9 Special results for n = 3.- *7.10 Integrating factors (continued).- 8 Integration on manifolds.- 8.1 Regular transformations.- 8.2 Coordinate systems on manifolds.- 8.3 Measure and integration on manifolds.- 8.4 The divergence theorem.- *8.5 Fluid flow.- 8.6 Orientations.- 8.7 Integrals of r-forms.- 8.8 Stokes’s formula.- 8.9 Regular transformations on submanifolds.- 8.10 Closed and exact differential forms.- 8.11 Motion of a particle.- 8.12 Motion of several particles.- Axioms for a vector space.- Mean value theorem; Taylor’s theorem.- Review of Riemann integration.- Monotone functions.- References.- Answers to problems.

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Book SynopsisIntended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups.Trade ReviewFrom the book reviews:“This textbook provides a comprehensive and consistent introduction to measure and integration theory. … The book can be recommended to anyone having basic knowledge of calculus and point-set topology. It is very self-contained, and can thus serve as an excellent reference book as well.” (Ville Suomala, Mathematical Reviews, July, 2014)“In this second edition, Cohn has updated his excellent introduction to measure theory … and has made this great textbook even better. Those readers unfamiliar with Cohn’s style will discover that his writing is lucid. … this is a wonderful text to learn measure theory from and I strongly recommend it.” (Tushar Das, MAA Reviews, June, 2014)Table of Contents1. Measures.- Algebras and sigma-algebras.- Measures.- Outer measures.- Lebesgue measure.- Completeness and regularity.- Dynkin classes.- 2. Functions and Integrals.- Measurable functions.- Properties that hold almost everywhere.- The integral.- Limit theorems.- The Riemann integral.- Measurable functions again, complex-valued functions, and image measures.- 3. Convergence.- Modes of Convergence.- Normed spaces.- Definition of L^p and L^p.- Properties of L^p and L-p.- Dual spaces.- 4. Signed and Complex Measures.- Signed and complex measures.- Absolute continuity.- Singularity.- Functions of bounded variation.- The duals of the L^p spaces.- 5. Product Measures.- Constructions.- Fubini’s theorem.- Applications.- 6. Differentiation.- Change of variable in R^d.- Differentiation of measures.- Differentiation of functions.- 7. Measures on Locally Compact Spaces.- Locally compact spaces.- The Riesz representation theorem.- Signed and complex measures; duality.- Additional properties of regular measures.- The µ^*-measurable sets and the dual of L^1.- Products of locally compact spaces.- 8. Polish Spaces and Analytic Sets.- Polish spaces.- Analytic sets.- The separation theorem and its consequences.- The measurability of analytic sets.- Cross sections.- Standard, analytic, Lusin, and Souslin spaces.- 9. Haar Measure.- Topological groups.- The existence and uniqueness of Haar measure.- The algebras L^1 (G) and M (G).- Appendices.- A. Notation and set theory.- B. Algebra.- C. Calculus and topology in R^d.- D. Topological spaces and metric spaces.- E. The Bochner integral.- F Liftings.- G The Banach-Tarski paradox.- H The Henstock-Kurzweil and McShane integralsBibliography.- Index of notation.- Index.

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Book SynopsisImplicit Functions and Solution MappingsTrade Review“The book represents the state of the art of the modern theory of inverse and implicit functions and provides an important source for studies of numerical methods and applications in this area. It can be warmly recommended to all specialists and advanced students working in optimization, analysis, numerical mathematics, and other mathematical fields, as well as to all those who apply variational analysis in engineering, physics, operations research, economics, finance, and more.” (Diethard Klatte, SIAM Review, Vol. 57 (2), June, 2015)“The book commences with a helpful context-setting preface followed by six chapters. Each chapter starts with a useful preamble and concludes with a careful and instructive commentary, while a good set of references, a notation guide and a somewhat brief index complete this study. … I unreservedly recommended this book to all practitioners and graduate students interested in modern optimization theory or control theory or to those just engaged by beautiful analysis cleanly described.” (Jonathan Michael Browein, IEEE Control Systems Magazine, February, 2012).“This book is devoted to the theory of inverse and implicit functions and some of its modifications for solution mappings in variational problems. … The book is targeted to a broad audience of researchers, teachers and graduate students. It can be used as well as a textbook as a reference book on the topic. Undoubtedly, it will be used by mathematicians dealing with functional and numerical analysis, optimization, adjacent branches and also by specialists in mechanics, physics, engineering, economics and so on.” (Peter Zabreiko, Zentralblatt MATH, Vol. 1178, 2010).“The present monograph will be a most welcome and valuable addition. … This book will save much time and effort, both for those doing research in variational analysis and for students learning the field. This important contribution fills a gap in the existing literature.” (Stephen M. Robinson, Mathematical Reviews, Issue 2010).Table of ContentsIntroduction and equation-solving background.- Solution mappings for variational problems.- Set-valued analysis of solution mappings.- Regularity properties through generalized derivatives.- Metric regularity in infinite dimensions.- Applications in numerical variational analysis.

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Book SynopsisPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.Trade Review“The list of main topics covered is quite standard: sequences, series, limits, continuity, differentiation, Riemann integration, uniform convergence … . This is a well-written textbook with an abundance of worked examples and exercises that is intended for a first course in analysis with modest ambitions.” (Brian S. Thomson, Mathematical Reviews, March, 2016)“The authors … introduce sequences and series at the beginning and build the fundamental concepts of analysis from them. … it achieves the same goal of introducing students to mathematical rigor and basic concepts and results in real analysis. … Summing Up: Recommended. Upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 53 (5), January, 2016)“This textbook is based on the central idea that concepts such as continuity, differentiation and integration are approached via the concepts of sequences and series. … Most of the sections are followed by exercises. The textbook is recommended for a first course in mathematical analysis.” (Sorin Gheorghe Gal, zbMATH, Vol. 1325.26002, 2016)Table of ContentsPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.

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Book SynopsisPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.Trade Review“The list of main topics covered is quite standard: sequences, series, limits, continuity, differentiation, Riemann integration, uniform convergence … . This is a well-written textbook with an abundance of worked examples and exercises that is intended for a first course in analysis with modest ambitions.” (Brian S. Thomson, Mathematical Reviews, March, 2016)“The authors … introduce sequences and series at the beginning and build the fundamental concepts of analysis from them. … it achieves the same goal of introducing students to mathematical rigor and basic concepts and results in real analysis. … Summing Up: Recommended. Upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 53 (5), January, 2016)“This textbook is based on the central idea that concepts such as continuity, differentiation and integration are approached via the concepts of sequences and series. … Most of the sections are followed by exercises. The textbook is recommended for a first course in mathematical analysis.” (Sorin Gheorghe Gal, zbMATH, Vol. 1325.26002, 2016)Table of ContentsPreface.- 1. Introduction.- 2. Sequences.- 3. Series.- 4. Limits of Functions.- 5. Continuity.- 6. Differentiability.- 7. The Riemann Integral.- 8. Taylor Polynomials and Taylor Series.- 9. The Fixed Point Problem.- 10. Sequences of Functions.- Bibliography.- Index.

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Book SynopsisReal Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions. All the concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy.Trade ReviewVol. 85 (504), 2001) "The book is a clear and structured introduction to real analysis. ... Fully worked out examples and exercises with solutions extend and illustrate the theory. Written in an easy-to-read style, combining informality and precision, the book is ideal for self-study or as a course textbook for first- and second-year undergraduates." (I. Rasa, Zentralblatt MATH, Vol. 969, 2001)Table of Contents1. Introductory Ideas.- 1.1 Foreword for the Student: Is Analysis Necessary?.- 1.2 The Concept of Number.- 1.3 The Language of Set Theory.- 1.4 Real Numbers.- 1.5 Induction.- 1.6 Inequalities.- 2. Sequences and Series.- 2.1 Sequences.- 2.2 Sums, Products and Quotients.- 2.3 Monotonie Sequences.- 2.4 Cauchy Sequences.- 2.5 Series.- 2.6 The Comparison Test.- 2.7 Series of Positive and Negative Terms.- 3. Functions and Continuity.- 3.1 Functions, Graphs.- 3.2 Sums, Products, Compositions; Polynomial and Rational Functions.- 3.3 Circular Functions.- 3.4 Limits.- 3.5 Continuity.- 3.6 Uniform Continuity.- 3.7 Inverse Functions.- 4. Differentiation.- 4.1 The Derivative.- 4.2 The Mean Value Theorems.- 4.3 Inverse Functions.- 4.4 Higher Derivatives.- 4.5 Taylor’s Theorem.- 5. Integration.- 5.1 The Riemann Integral.- 5.2 Classes of Integrable Functions.- 5.3 Properties of Integrals.- 5.4 The Fundamental Theorem.- 5.5 Techniques of Integration.- 5.6 Improper Integrals of the First Kind.- 5.7 Improper Integrals of the Second Kind.- 6. The Logarithmic and Exponential Functions.- 6.1 A Function Defined by an Integral.- 6.2 The Inverse Function.- 6.3 Further Properties of the Exponential and Logarithmic Functions.- Sequences and Series of Functions.- 7.1 Uniform Convergence.- 7.2 Uniform Convergence of Series.- 7.3 Power Series.- 8. The Circular Functions.- 8.1 Definitions and Elementary Properties.- 8.2 Length.- 9. Miscellaneous Examples.- 9.1 Wallis’s Formula.- 9.2 Stirling’s Formula.- 9.3 A Continuous, Nowhere Differentiable Function.- Solutions to Exercises.- The Greek Alphabet.

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Book SynopsisThis book serves as an introduction to the use of mathematics in describing collective phenomena in physics and biology. Derived from a course of innovative lectures, the book shows students early in their studies how many of the topics they have encountered – partial differential equations, differential equations, Fourier series, and linear algebra – are useful in constructing, analysing and interpreting phenomena present in the real world. Throughout, ideas are developed using worked examples and exercises with solution. The text does not assume a strong background in physics. Trade ReviewFrom the reviews: "This textbook … is designed for undergraduates who have followed a first ‘mathematical methods’ course and who are ready to study in more depth the mathematics underlying phenomena … . Each chapter includes a number of worked examples and a dozen or so exercises, with solutions collected at the end of the book. The book is aimed at students of applied mathematics, physics and engineering. The emphasis on techniques and the frequent references to applications make it particularly suitable for this audience." (S.C. Russen, The Mathematical Gazette, Vol. 89 (516), 2005) "Fields, Flows, and Waves, is an introduction to continuum models based on the author’s lectures … . Ample illustrations and worked examples come with the exposition, and there are several exercises with varying degrees of difficulty; detailed solutions are included at the end of the text. … I warmly recommend this book. It reads well and is in an attractive, concise format. … It makes one yearn for a course in the curriculum where this material could be regularly taught." (SIAM Review, Vol. 46 (3), 2004) "This book is an introduction to the mathematical methods in classical fields theory. It is designed for the second-year undergraduate in physics, mathematics and engineering. … The presentation is excellent, numerous examples of increasing difficulties are considered with details. … The book ends with solutions to the exercises a short bibliography and an index. In conclusion, I warmly recommend this book to any students in physics because it’s well written … interesting and very useful." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004) "This book … is a first introduction to the mathematical description of fields, flows and waves. It shows students, early in their studies, how many of the topics they have encountered are useful … . Designed for second-year undergraduate students in mathematics, mathematical physics, and engineering, it presumes only a limited familiarity with several variable calculus and vector fields. … The ideas are developed through worked examples, and a range of exercises (with solutions) is provided to test understanding." (Läenseignement Mathematique, Vol. 49 (3-4), 2003) "This is another excellent readable book in the Springer Undergraduate Mathematics Series (SUMS). It is a refreshingly modern approach to Continuum Mechanics … . Indeed Professor Parker has written this book so that it might be used directly as an elementary course … . This is a carefully written, well structured book which contains a wealth of examples complete with solutions. … a carefully structured book from which a modern undergraduate applied mathematics course may be taught directly." (Sean McKee, Journal of Fluid Mechanics, Vol.504, 2004) "Introductory books … often struggle with the balance between the motivating physical problems and the formal mathematical structures. As the title suggests, Parker … manages to keep the more technical mathematical structure in clear view. Particularly impressive is how carefully the author leads readers … . the book has a completeness that makes it attractive as a self-contained resource as well as a textbook. … complete solutions (not just answers) to all of the exercises makes the book particularly effective for independent study of this material. Summing Up: Highly recommended." (J. Feroe, CHOICE, December, 2003) "The book is well-written and illustrated by interesting figures which make the text easy to read and attractive. Of course undergraduate students in physics and maybe in mathematics will surely benefit of a lecture and practice of this book. Each of the ten chapters indeed contains some lists of significant exercises. The more or less detailed solutions of these exercises are gathered at the end of the book." (Alain Brillard, Zentralblatt MATH, 2003) "Continuum models ignoring the substructures of fluids are useful and widely applied for the description of fields, flows, and waves in different research works. This book gives a first introduction to the mathematical methods necessary for the solution of the resulting equations. … Each chapter contains some examples and exercises. … The results of the exercises are listed at the end of the book. … the book is a useful introduction in this important branch of knowledge." (Bernd Platzer, www.zamm-journal.org, 2004) "David Parker’s book Fields, Flows and Waves: An Introduction to Continuum Models … is a fine addition to the Springer Undergraduate Mathematics Series. … For the subjects considered, the author provides masterly compact accounts of the physical phenomena … and solves interesting problems. Parker takes particular care to examine the physical implications of the mathematical solutions … . An adequate selection of student exercises is included, with solutions … . could be used to enrich an advanced undergraduate or beginning graduate course on continuum mechanics." (James Casey, Physics today, October, 2004)Table of Contents1. The Continuum Description.- 1.1 Densities and Fluxes.- 1.2 Conservation and Balance Laws in One Dimension.- 1.3 Heat Flow.- 1.4 Steady Radial Flow in Two Dimensions.- 1.5 Steady Radial Flow in Three Dimensions.- 2. Unsteady Heat Flow.- 2.1 Thermal Energy.- 2.1.1 Heat Balance in One-dimensional Problems.- 2.1.2 Some Special Solutions of Equation (2.3).- 2.2 Effects of Heat Supply.- 2.3 Unsteady, Spherically Symmetric Heat Flow.- 3. Fields and Potentials.- 3.1 Gradient of a Scalar.- 3.1.1 Some Applications.- 3.2 Gravitational Potential.- 3.2.1 Special Properties of the Function ?=r?1.- 3.3 Continuous Distributions of Mass.- 3.4 Electrostatics.- 3.4.1 Gauss’s Law of Flux.- 3.4.2 Charge-free Regions.- 3.4.3 Surface Charge Density.- 4. Laplace’s Equation and Poisson’s Equation.- 4.1 The Ubiquitous Laplacian.- 4.2 Separable Solutions.- 4.3 Poisson’s Equation.- 4.4 Dipole Solutions.- 4.4.1 Uses of Dipole Solutions to ?2?=0.- 4.4.2 Spherical Inclusions.- 5. Motion of an Elastic String.- 5.1 Tension and Extension; Kinematics and Dynamics.- 5.1.1 Dynamics.- 5.2 Planar Motions.- 5.2.1 Small Transverse Motions.- 5.2.2 Longitudinal Motions.- 5.3 Properties of the Wave Equation.- 5.3.1 Standing Waves.- 5.3.2 Superposition of Standing Waves.- 5.4 D’Alembert’s Solution, Travelling Waves and Wave Reflections.- 5.4.1 Wave Reflections.- 5.5 Other One-dimensional Waves.- 5.5.1 Acoustic Vibrations in a’ lUbe.- 5.5.2 Telegraphy and High-voltage Transmission.- 6. Fluid Flow.- 6.1 Kinematics and Streamlines.- 6.1.1 Some Important Examples of Steady Flow.- 6.2 Volume Flux and Mass Flux.- 6.2.1 Incompressible Fluids.- 6.2.2 Mass Conservation.- 6.3 Two-dimensional Flows of Incompressible Fluids.- 6.3.1 The Continuity Equation.- 6.3.2 Irrotational Flows and the Velocity Potential.- 6.3.3 The Stream Function.- 6.4 Pressure in a Fluid.- 6.4.1 Resultant Force.- 6.4.2 Hydrostatics and Archimedes’ Principle.- 6.4.3 Momentum Density and Momentum Flux.- 6.5 Bernoulli’s Equation.- 6.5.1 The Material (Advected) Derivative.- 6.5.2 Bernoulli’s Equation and Dynamic Pressure.- 6.5.3 The Principle of Aerodynamic Lift.- 6.6 Three-dimensional, Incompressible Flows.- 6.6.1 The Continuity Equation.- 6.6.2 Irrotational Flows, the Velocity Potential and Laplace’s Equation.- 7. Elastic Deformations.- 7.1 The Kinematics of Deformation.- 7.1.1 Deformation Gradient.- 7.1.2 Stretch and Rotation.- 7.2 Polar Decomposition.- 7.3 Stress.- 7.3.1 Traction Vectors.- 7.3.2 Components of Stress.- 7.3.3 Traction on a General Surface.- 7.4 Isotropic Linear Elasticity.- 7.4.1 The Constitutive Law.- 7.4.2 Stretching, Shear and Torsion.- 8. Vibrations and Waves.- 8.1 Wave Reflection and Refraction.- 8.1.1 Use of the Complex Exponential.- 8.1.2 Plane Waves.- 8.1.3 Reflection at a Rigid Wall.- 8.1.4 Refraction at an Interface.- 8.1.5 Total Internal Reflection.- 8.2 Guided Waves.- 8.2.1 Acoustic Waves in a Layer.- 8.2.2 Waveguides and Dispersion.- 8.3 Love Waves in Elasticity.- 8.4 Elastic Plane Waves.- 8.4.1 Elastic Shear Waves.- 8.4.2 Dilatational Waves.- 9. Electromagnetic VVaves and Light.- 9.1 Physical Background.- 9.1.1 The Origin of Maxwell’s Equations.- 9.1.2 Plane Electromagnetic Waves.- 9.1.3 Reflection and Refraction of Electromagnetic Waves.- 9.2 Waveguides.- 9.2.1 Rectangular Waveguides.- 9.2.2 Circular Cylindrical Waveguides.- 9.2.3 An Introduction to Fibre Optics.- 10. Chemical and Biological Models.- 10.1 Diffusion of Chemical Species.- 10.1.1 Fick’s Law of Diffusion.- 10.1.2 Self-similar Solutions.- 10.1.3 Travelling Wavefronts.- 10.2 Population Biology.- 10.2.1 Growth and Dispersal.- 10.2.2 Fisher’s Equation and Self-limitation.- 10.2.3 Population-dependent Dispersivity.- 10.2.4 Competing Species.- 10.2.5 Diffusive Instability.- 10.3 Biological Waves.- 10.3.1 The Logistic Wavefront.- 10.3.2 Travelling Pulses and Spiral Waves.- Solutions.

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Book SynopsisThis textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations.Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory.Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject. Trade Review“Instructors teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. Researchers in partial differential equations may find this work useful as a summary of analytic theories published in this volume.” (Vicenţiu D. Rădulescu, zbMATH 1461.35001, 2021)Table of ContentsPart I Tools and Problems.- 1 Elements of functional analysis and distributions.- 2 Statements of the problems of Chapter 1.- 3 Functional spaces.- 4 Statements of the problems of Chapter 3.- 5 Microlocal analysis.- 6 Statements of the problems of Chapter 5.- 7 The classical equations.- 8 Statements of the problems of Chapter 7.- Part II Solutions of the Problems. A Classical results. Index.

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Book SynopsisThis graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering. This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field’s growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published. Trade Review“This monograph is a thorough and insightful introduction to the mathematics of inverse problems and a solid improvement of the previous editions, which were used to educate many researchers in the field over the last two and a half decades. As such, the volume is already a classic and can be recommended without reservations to any reader interested in both the foundations and specific examples of inverse problems relevant to modern engineering and sciences.” (Alexander Mamonov, SIAM Review, Vol. 65 (2), 2023)Table of ContentsIntroduction and Basic Concepts.- Regularization Theory for Equations of the First Kind.- Regularization by Discretization.- Nonlinear Inverse Problems.- Inverse Eigenvalue Problems.- An Inverse Problem in Electrical Impedance Tomography.- An Inverse Scattering Problem.- Basic Facts from Functional Analysis.- References.- Index.

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Book SynopsisThis textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series.The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required.The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading.The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.Trade Review“The text contains a large number of problems and exercises, which should make it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and to test the understanding of concepts. … Many additional problems are proposed as homework tasks at the end of each chapter.” (Hüseyin Çakallı, zbMATH 1523.40001, 2023)Table of ContentsSequences of numbers.- Series of numbers.- Sequences of functions.- Series of functions.- Power series.

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