Differential calculus and equations Books

Book SynopsisThis 2007 book discusses the design of reliable numerical methods to retrieve missing information in models of complex systems.Table of ContentsPreface; 1. Introduction; 2. Linear algebra; 3. Discrete-time signals and systems; 4. Random variables and signals; 5. Kalman filtering; 6. Estimation of spectra and frequency response functions; 7. Output-error parametric model estimation; 8. Prediction-error parametric model estimation; 9. Subspace model identification; 10. The system identification cycle; Notation and symbols; List of abbreviations; References; Index.

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Book SynopsisA differential equation involves an unknown function and its derivative. It is an important subject that lies in the heart of understanding calculus or analysis. Differential Equations For Dummies takes readers step-by-step through this intimidating subject.Table of ContentsIntroduction. Part I: Focusing on First Order Differential Equations. Chapter 1: Welcome to the World of Differential Equations. Chapter 2: Looking at Linear First Order Differential Equations. Chapter 3: Sorting Out Separable First Order Differential Equations. Chapter 4: Exploring Exact First Order Differential Equations and Euler’s Method. Part II: Surveying Second and Higher Order Differential Equations. Chapter 5: Examining Second Order Linear Homogeneous Differential Equations. Chapter 6: Studying Second Order Linear Nonhomogeneous Differential Equations. Chapter 7: Handling Higher Order Linear Homogeneous Differential Equations. Chapter 8: Taking On Higher Order Linear Nonhomogeneous Differential Equations. Part III: The Power Stuff: Advanced Techniques. Chapter 9: Getting Serious with Power Series and Ordinary Points. Chapter 10: Powering through Singular Points. Chapter 11: Working with Laplace Transforms. Chapter 12: Tackling Systems of First Order Linear Differential Equations. Chapter 13: Discovering Three Fail-Proof Numerical Methods. Part IV: The Part of Tens. Chapter 14: Ten Super-Helpful Online Differential Equation Tutorials. Chapter 15: Ten Really Cool Online Differential Equation Solving Tools. Index.

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Thinking about Ordinary Differential Equations | BookCurl

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Book SynopsisThe text material evolved from over 50 years of combined teaching experience it deals with a formulation and application of the finite element method. A meaningful course can be constructed from a subset of the chapters in this book for a quarter course; instructions for such use are given in the preface.Trade Review"Recommended for upper division undergraduates and above." (CHOICE, February 2008)Table of ContentsPreface xi 1 Introduction 1 1.1 Background 1 1.2 Applications of Finite elements 7 References 9 2 Direct Approach for Discrete Systems 11 2.1 Describing the Behavior of a Single Bar Element 11 2.2 Equations for a System 15 2.2.1 Equations for Assembly 18 2.2.2 Boundary Conditions and System Solution 20 2.3 Applications to Other Linear Systems 24 2.4 Two-Dimensional Truss Systems 27 2.5 Transformation Law 30 2.6 Three-Dimensional Truss Systems 35 References 36 Problems 37 3 Strong andWeak Forms for One-Dimensional Problems 41 3.1 The Strong Form in One-Dimensional Problems 42 3.1.1 The Strong Form for an Axially Loaded Elastic Bar 42 3.1.2 The Strong Form for Heat Conduction in One Dimension 44 3.1.3 Diffusion in One Dimension 46 3.2 TheWeak Form in One Dimension 47 3.3 Continuity 50 3.4 The Equivalence Between theWeak and Strong Forms 51 3.5 One-Dimensional Stress Analysis with Arbitrary Boundary Conditions 58 3.5.1 Strong Form for One-Dimensional Stress Analysis 58 3.5.2 Weak Form for One-Dimensional Stress Analysis 59 3.6 One-Dimensional Heat Conduction with Arbitrary Boundary Conditions 60 3.6.1 Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 60 3.6.2 Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 61 3.7 Two-Point Boundary Value Problem with Generalized Boundary Conditions 62 3.7.1 Strong Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions 62 3.7.2 Weak Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions 63 3.8 Advection–Diffusion 64 3.8.1 Strong Form of Advection–Diffusion Equation 65 3.8.2 Weak Form of Advection–Diffusion Equation 66 3.9 Minimum Potential Energy 67 3.10 Integrability 71 References 72 Problems 72 4 Approximation of Trial Solutions,Weight Functions and Gauss Quadrature for One-Dimensional Problems 77 4.1 Two-Node Linear Element 79 4.2 Quadratic One-Dimensional Element 81 4.3 Direct Construction of Shape Functions in One Dimension 82 4.4 Approximation of theWeight Functions 84 4.5 Global Approximation and Continuity 84 4.6 Gauss Quadrature 85 Reference 90 Problems 90 5 Finite Element Formulation for One-Dimensional Problems 93 5.1 Development of Discrete Equation: Simple Case 93 5.2 Element Matrices for Two-Node Element 97 5.3 Application to Heat Conduction and Diffusion Problems 99 5.4 Development of Discrete Equations for Arbitrary Boundary Conditions 105 5.5 Two-Point Boundary Value Problem with Generalized Boundary Conditions 111 5.6 Convergence of the FEM 113 5.6.1 Convergence by Numerical Experiments 115 5.6.2 Convergence by Analysis 118 5.7 FEM for Advection–Diffusion Equation 120 References 122 Problems 123 6 Strong andWeak Forms for Multidimensional Scalar Field Problems 131 6.1 Divergence Theorem and Green’s Formula 133 6.2 Strong Form 139 6.3 Weak Form 142 6.4 The Equivalence BetweenWeak and Strong Forms 144 6.5 Generalization to Three-Dimensional Problems 145 6.6 Strong andWeak Forms of Scalar Steady-State Advection–Diffusion in Two Dimensions 146 References 148 Problems 148 7 Approximations of Trial Solutions,Weight Functions and Gauss Quadrature for Multidimensional Problems 151 7.1 Completeness and Continuity 152 7.2 Three-Node Triangular Element 154 7.2.1 Global Approximation and Continuity 157 7.2.2 Higher Order Triangular Elements 159 7.2.3 Derivatives of Shape Functions for the Three-Node Triangular Element 160 7.3 Four-Node Rectangular Elements 161 7.4 Four-Node Quadrilateral Element 164 7.4.1 Continuity of Isoparametric Elements 166 7.4.2 Derivatives of Isoparametric Shape Functions 166 7.5 Higher Order Quadrilateral Elements 168 7.6 Triangular Coordinates 172 7.6.1 Linear Triangular Element 172 7.6.2 Isoparametric Triangular Elements 174 7.6.3 Cubic Element 175 7.6.4 Triangular Elements by Collapsing Quadrilateral Elements 176 7.7 Completeness of Isoparametric Elements 177 7.8 Gauss Quadrature in Two Dimensions 178 7.8.1 Integration Over Quadrilateral Elements 179 7.8.2 Integration Over Triangular Elements 180 7.9 Three-Dimensional Elements 181 7.9.1 Hexahedral Elements 181 7.9.2 Tetrahedral Elements 183 References 185 Problems 186 8 Finite Element Formulation for Multidimensional Scalar Field Problems 189 8.1 Finite Element Formulation for Two-Dimensional Heat Conduction Problems 189 8.2 Verification and Validation 201 8.3 Advection–Diffusion Equation 207 References 209 Problems 209 9 Finite Element Formulation for Vector Field Problems – Linear Elasticity 215 9.1 Linear Elasticity 215 9.1.1 Kinematics 217 9.1.2 Stress and Traction 219 9.1.3 Equilibrium 220 9.1.4 Constitutive Equation 222 9.2 Strong andWeak Forms 223 9.3 Finite Element Discretization 225 9.4 Three-Node Triangular Element 228 9.4.1 Element Body Force Matrix 229 9.4.2 Boundary Force Matrix 230 9.5 Generalization of Boundary Conditions 231 9.6 Discussion 239 9.7 Linear Elasticity Equations in Three Dimensions 240 Problems 241 10 Finite Element Formulation for Beams 249 10.1 Governing Equations of the Beam 249 10.1.1 Kinematics of Beam 249 10.1.2 Stress–Strain Law 252 10.1.3 Equilibrium 253 10.1.4 Boundary Conditions 254 10.2 Strong Form toWeak Form 255 10.2.1 Weak Form to Strong Form 257 10.3 Finite Element Discretization 258 10.3.1 Trial Solution andWeight Function Approximations 258 10.3.2 Discrete Equations 260 10.4 Theorem of Minimum Potential Energy 261 10.5 Remarks on Shell Elements 265 Reference 269 Problems 269 11 Commercial Finite Element Program ABAQUS Tutorials 275 11.1 Introduction 275 11.1.1 Steady-State Heat Flow Example 275 11.2 Preliminaries 275 11.3 Creating a Part 276 11.4 Creating a Material Definition 278 11.5 Defining and Assigning Section Properties 279 11.6 Assembling the Model 280 11.7 Configuring the Analysis 280 11.8 Applying a Boundary Condition and a Load to the Model 280 11.9 Meshing the Model 282 11.10 Creating and Submitting an Analysis Job 284 11.11 Viewing the Analysis Results 284 11.12 Solving the Problem Using Quadrilaterals 284 11.13 Refining the Mesh 285 11.13.1 Bending of a Short Cantilever Beam 287 11.14 Copying the Model 287 11.15 Modifying the Material Definition 287 11.16 Configuring the Analysis 287 11.17 Applying a Boundary Condition and a Load to the Model 288 11.18 Meshing the Model 289 11.19 Creating and Submitting an Analysis Job 290 11.20 Viewing the Analysis Results 290 11.20.1 Plate with a Hole in Tension 290 11.21 Creating a New Model 292 11.22 Creating a Part 292 11.23 Creating a Material Definition 293 11.24 Defining and Assigning Section Properties 294 11.25 Assembling the Model 295 11.26 Configuring the Analysis 295 11.27 Applying a Boundary Condition and a Load to the Model 295 11.28 Meshing the Model 297 11.29 Creating and Submitting an Analysis Job 298 11.30 Viewing the Analysis Results 299 11.31 Refining the Mesh 299 Appendix 303 A.1 Rotation of Coordinate System in Three Dimensions 303 A.2 Scalar Product Theorem 304 A.3 Taylor’s Formula with Remainder and the Mean Value Theorem 304 A.4 Green’s Theorem 305 A.5 Point Force (Source) 307 A.6 Static Condensation 308 A.7 Solution Methods 309 Direct Solvers 310 Iterative Solvers 310 Conditioning 311 References 312 Problem 312 Index 313

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Book SynopsisDifferential Equations Workbook For Dummies is a course supplement and practice guide for students taking a course that involves the use of differential equations. This book takes readers step-by-step through this intimidating subject and features numerous practice exercises and clear, concise examples to improve problem-solving skills.Table of ContentsIntroduction 1 Part I: Tackling First Order Differential Equations 5 Chapter 1: Looking Closely at Linear First Order Differential Equations 7 Chapter 2: Surveying Separable First Order Differential Equations 29 Chapter 3: Examining Exact First Order Differential Equations 59 Part II: Finding Solutions to Second and Higher Order Differential Equations 79 Chapter 4: Working with Linear Second Order Differential Equations 81 Chapter 5: Tackling Nonhomogeneous Linear Second Order Differential Equations 105 Chapter 6: Handling Homogeneous Linear Higher Order Differential Equations 129 Chapter 7: Taking On Nonhomogeneous Linear Higher Order Differential Equations 153 Part III: The Power Stuff: Advanced Techniques 175 Chapter 8: Using Power Series to Solve Ordinary Differential Equations 177 Chapter 9: Solving Differential Equations with Series Solutions Near Singular Points 199 Chapter 10: Using Laplace Transforms to Solve Differential Equations 225 Chapter 11: Solving Systems of Linear First Order Differential Equations 249 Part IV: The Part of Tens 273 Chapter 12: Ten Common Ways of Solving Differential Equations 275 Chapter 13: Ten Real-World Applications of Differential Equations 279 Index 283

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Book SynopsisThis textbook is designed with the needs of today’s student in mind. It is the ideal textbook for a first course in elementary differential equations for future engineers and scientists, including mathematicians. This book is accessible to anyone who has a basic knowledge of precalculus algebra and differential and integral calculus. Its carefully crafted text adopts a concise, simple, no-frills approach to differential equations, which helps students acquire a solid experience in many classical solution techniques. With a lighter accent on the physical interpretation of the results, a more manageable page count than comparable texts, a highly readable style, and over 1000 exercises designed to be solved without a calculating device, this book emphasizes the understanding and practice of essential topics in a succinct yet fully rigorous fashion. Apart from several other enhancements, the second edition contains one new chapter on numerical methods of solution.The book formally splits the "pure" and "applied" parts of the contents by placing the discussion of selected mathematical models in separate chapters. At the end of most of the 246 worked examples, the author provides the commands in Mathematica® for verifying the results. The book can be used independently by the average student to learn the fundamentals of the subject, while those interested in pursuing more advanced material can regard it as an easily taken first step on the way to the next level. Additionally, practitioners who encounter differential equations in their professional work will find this text to be a convenient source of reference.Table of Contents1. Introduction.- 2. First Order Equations.- 3. Mathematical Models with First-Order Equations.- 4. Linear Second-Order Equations.- 4. Higher-Order Equations.- 5. Mathematical Models with Second-Order Equations.- 6. Higher-Order Linear Equations.- 7. Systems of Differential Equations.- 8. The Laplace Transformation.- 9. Series Solutions.- 10. Numerical Methods.- A. Algebra Techniques.- B. Calculus Techniques.- C. Table of Laplace Transforms.- D. The Greek Alphabet.- Further Reading.- Answers to Odd-Numbered Exercises.- Index.

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Book SynopsisDifferential equations and linear algebra are two central topics in the undergraduate mathematics curriculum. This innovative textbook allows the two subjects to be developed either separately or together, illuminating the connections between two fundamental topics, and giving increased flexibility to instructors. It can be used either as a semester-long course in differential equations, or as a one-year course in differential equations, linear algebra, and applications. Beginning with the basics of differential equations, it covers first and second order equations, graphical and numerical methods, and matrix equations. The book goes on to present the fundamentals of vector spaces, followed by eigenvalues and eigenvectors, positive definiteness, integral transform methods and applications to PDEs. The exposition illuminates the natural correspondence between solution methods for systems of equations in discrete and continuous settings. The topics draw on the physical sciences, engineer

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Book SynopsisThis is a graduate-level introduction to the key ideas and theoretical foundation of the vibrant field of optimal mass transport in the Euclidean setting. Taking a pedagogical approach, it introduces concepts gradually and in an accessible way, while also remaining technically and conceptually complete.Trade Review'Francesco Maggi's book is a detailed and extremely well written explanation of the fascinating theory of Monge-Kantorovich optimal mass transfer. I especially recommend Part IV's discussion of the 'linear' cost problem and its subtle mathematical resolution.' Lawrence C. Evans, University of California, Berkeley'Over the last three decades, optimal transport has revolutionized the mathematical analysis of inequalities, differential equations, dynamical systems, and their applications to physics, economics, and computer science. By exposing the interplay between the discrete and Euclidean settings, Maggi's book makes this development uniquely accessible to advanced undergraduates and mathematical researchers with a minimum of prerequisites. It includes the first textbook accounts of the localization technique known as needle decomposition and its solution to Monge's centuries old cutting and filling problem (1781). This book will be an indispensable tool for advanced undergraduates and mathematical researchers alike.' Robert McCann, University of TorontoTable of ContentsPreface; Notation; Part I. The Kantorovich Problem: 1. An introduction to the Monge problem; 2. Discrete transport problems; 3. The Kantorovich problem; Part II. Solution of the Monge Problem with Quadratic Cost: the Brenier-McCann Theorem: 4. The Brenier theorem; 5. First order differentiability of convex functions; 6. The Brenier-McCann theorem; 7. Second order differentiability of convex functions; 8. The Monge-Ampère equation for Brenier maps; Part III. Applications to PDE and the Calculus of Variations and the Wasserstein Space: 9. Isoperimetric and Sobolev inequalities in sharp form; 10. Displacement convexity and equilibrium of gases; 11. The Wasserstein distance W2 on P2(Rn); 12. Gradient flows and the minimizing movements scheme; 13. The Fokker-Planck equation in the Wasserstein space; 14. The Euler equations and isochoric projections; 15. Action minimization, Eulerian velocities and Otto's calculus; Part IV. Solution of the Monge Problem with Linear Cost: the Sudakov Theorem: 16. Optimal transport maps on the real line; 17. Disintegration; 18. Solution to the Monge problem with linear cost; 19. An introduction to the needle decomposition method; Appendix A: Radon measures on Rn and related topics; Appendix B: Bibliographical Notes; Bibliography; Index.

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Book SynopsisThe book provides the reader with an overview of the actual state of research in ordinary and partial differential equations in the complex domain. Topics include summability and asymptotic study of both ordinary and partial differential equations, and also q-difference and differential-difference equations. This book will be of interest to researchers and students who wish to expand their knowledge of these fields.With the latest results and research developments and contributions from experts in their field, Formal and Analytic Solutions of Differential Equations provides a valuable contribution to methods, techniques, different mathematical tools, and study calculations.

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Book SynopsisThis book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the study of measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.Table of ContentsPreface.- 1. Logic and Sets.- 2. Integers.- 3. Construction of the Real Numbers.- 4. The Real Numbers.- 5. Real Sequences.- 6. Some Applications of Real Sequences.- 7. Real Series.- 8. Additional Topics in Real Series.- 9. Functions and Limits.- 10. Continuity.- 11. Function Sequences and Series.- 12. Power Series.- 13. Differentiation.- 14. Some Applications of Differentiation.- 15. Riemann and Darboux Integration.- 16. The Fundamental Theorem of Calculus.- 17. Taylor and MacLaurin Series.- 18. Introduction to Measure Theory.- 19. Lebesgue Integration.- 20. Double Integrals.- Solutions to the Exercises.- Bibliography.- Index.

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Book SynopsisThis book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.Trade Review“The book presents a new approach to … inverse problems, where the authors mainly use as an essential tool the Nambu bracket. They deduce new properties of this bracket, which plays a fundamental role in the proof of all the results and in their applications throughout the book. … The book is well written and contains new and valuable results in the development of the inverse problem in ordinary differential equations and its applications.” (Leonardo Colombo, Mathematical Reviews, January, 2017)Table of ContentsPreface.- 1.Differential Equations with Given Partial and First Integrals.- 2.Polynomial Vector Fields with Given Partial and First Integrals.- 3.16th Hilbert Problem for Algebraic Limit Cycles.- 4.Inverse Problem for Constrained Lagrangian Systems.- 5.Inverse Problem for Constrained Hamiltonian Systems.- 6.Integrability of the Constrained Rigid Body.- 7.Inverse Problem in the Vakonomic Mechanics.- Index.- Bibliography.

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Book SynopsisWritten in a straightforward and easily accessible style, this volume is suitable as a textbook for advanced undergraduate or first-year graduate students in mathematics, physical sciences, and engineering. The aim is to provide students with a strong background in the theories of Ordinary Differential Equations, Dynamical Systems and Boundary Value Problems, including regular and singular perturbations. It is also a valuable resource for researchers.This volume presents an abundance of examples in physical and biological sciences, and engineering to illustrate the applications of the theorems in the text. Readers are introduced to some important theorems in Nonlinear Analysis, for example, Brouwer fixed point theorem and fundamental theorem of algebras. A chapter on Monotone Dynamical Systems takes care of the new developments in Ordinary Differential Equations and Dynamical Systems.In this third edition, an introduction to Hamiltonian Systems is included to enhance and complete its coverage on Ordinary Differential Equations with applications in Mathematical Biology and Classical Mechanics.

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Book SynopsisThe emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, facilitates a deeper understanding of the subject. Among a big number of CAS, we choose the two systems, Maple and Mathematica, that are used worldwide by students, research mathematicians, scientists, and engineers. As in the our previous books, we propose the idea to use in parallel both systems, Maple and Mathematica, since in many research problems frequently it is required to compare independent results obtained by using different computer algebra systems, Maple and/or Mathematica, at all stages of the solution process. One of the main points (related to CAS) is based on the implementation of a whole solution method (e.g. starting from an analytical derivation of exact governing equations, constructing discretizations and analytical formulas of a numerical method, performing numerical procedure, obtaining various visualizations, and comparing the numerical solution obtained with other types of solutions considered in the book, e.g. with asymptotic solution).Trade ReviewFrom the reviews:“The authors consider the problem of constructing closed-form and approximate solutions to nonlinear partial differential equations with the help of computer algebra systems. … The book will be useful for readers who want to try modern methods for solving nonlinear partial differential equations on concrete examples without bothering too much about the mathematics behind the methods. Thus it is mainly of interest for applied scientists. Mathematicians may use it in connection with more theoretical works; some references are given throughout the book.” (Werner M. Seiler, Zentralblatt MATH, Vol. 1233, 2012)Table of Contents1 Introduction 1.1 Basic Concepts 2 Algebraic Approach 2.1 Point Transformations 2.2 Contact Transformations 2.3 Transformations Relating Differential Equations 2.4 Linearizing and Bilinearizing Transformations 2.5 Reductions of Nonlinear PDEs 2.6 Separation of Variables 2.7 Transformation Groups 2.8 Nonlinear Systems 3 Geometric-Qualitative Approach 3.1 Method of Characteristics 3.2 Generalized Method of Characteristics 3.3 Qualitative Analysis 4 General Analytical Approach. Integrability 4.1 Painlevé Test and Integrability 4.2 Complete Integrability. Evolution Equations 4.3 Nonlinear Systems. Integrability Conditions 5 Approximate Analytical Approach 5.1 Adomian Decomposition Method 5.2 Asymptotic Expansions. Perturbation Methods 6 Numerical Approach 6.1 Embedded Numerical Methods 6.2 Finite DifferenceMethods 7 Analytical-Numerical Approach 7.1 Method of Lines 7.2 Spectral Collocation Method; A Brief Description of Maple A.1 Introduction A.2 Basic Concepts A.3 Maple Language B Brief Description of Mathematica B.1 Introduction B.2 Basic Concepts B.3 Mathematica Language; References, Index

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Book SynopsisBased on notes written during the author's many years of teaching, Analysis in Euclidean Space mainly covers Differentiation and Integration theory in several real variables, but also an array of closely related areas including measure theory, differential geometry, classical theory of curves, geometric measure theory, integral geometry, and others.With several original results, new approaches and an emphasis on concepts and rigorous proofs, the book is suitable for undergraduate students, particularly in mathematics and physics, who are interested in acquiring a solid footing in analysis and expanding their background. There are many examples and exercises inserted in the text for the student to work through independently.Analysis in Euclidean Space comprises 21 chapters, each with an introduction summarizing its contents, and an additional chapter containing miscellaneous exercises. Lecturers may use the varied chapters of this book for different undergraduate courses in analysis. The only prerequisites are a basic course in linear algebra and a standard first-year calculus course in differentiation and integration. As the book progresses, the difficulty increases such that some of the later sections may be appropriate for graduate study.

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Book SynopsisThis textbook uses insight from differential equations to analyse fundamental subjects of modern theoretical physics, including classical and quantum mechanics, thermodynamics, electromagnetism, superconductivity, gravitational physics, and quantum field theories.Table of ContentsPreface Notation and Convention 1: Hamiltonian Systems and Applications 2: Schrödinger Equation and Quantum Mechanics 3: Maxwell Equations, Dirac Monopole, and Gauge Fields 4: Special Relativity 5: Abelian Gauge Field Equations 6: Dirac Equations 7: GinzburgDSLandau Equations for Superconductivity 8: Magnetic Vortices in Abelian Higgs Theory 9: Non-Abelian Gauge Field Equations 10: Einstein Equations and Related Topics 11: Charged Vortices and ChernDSSimons Equations 12: Skyrme Model and Related Topics 13: Strings and Branes 14: BornDSInfeld Theory of Electromagnetism 15: Canonical Quantization of Fields Appendices Bibliography Index

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Book SynopsisThis volume presents a systematic and mathematically rigorous exposition of methods for studying linear partial differential equations. It focuses on quantization of the corresponding objects (states, observables and canonical transformations) in the phase space. The quantization of all three types of classical objects is carried out in a unified way with the use of a special integral transform. This book covers recent as well as established results, treated within the framework of a universal approach. It also includes applications and provides a useful reference text for graduate and research-level readers.Table of ContentsSemiclassical Quantization. Quantization and Microlocalization. Quantization by the Wave Packet Transform. Maslov's Canonical Operator and Hormander's Oscillatory Integrals. Topological Aspects of Quantization Conditions. The Schrodinger Equation. The Maxwell Equations. Equations with Trapping Hamiltonians. Quantization by the Method of Ordered Operators (Noncommutative Analysis). Noncommutative Analysis: Main Ideas, Definitions, and Theorems. Exactly Soluble Commutation Relations. Operator Algebras on Singular Manifolds. The High-Frequency Asymptotics in the Problem of Wave Propagation in Plasma. Appendices.

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Book SynopsisThis highly useful text for students and professionals working in the applied sciences shows how to formulate and solve partial differential equations. Realistic, practical coverage of diffusion-type problems, hyperbolic-type problems, elliptic-type problems and numerical and approximate methods. Suggestions for further reading. Solution guide available upon request. 1982 edition.

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Book SynopsisSymmetry is the key to solving differential equations. There are many well-known techniques for obtaining exact solutions, but most of them are special cases of a few powerful symmetry methods. Furthermore, these methods can be applied to differential equations of an unfamiliar type; they do not rely on special 'tricks'. Instead, a given differential equation is forced to reveal its symmetries, which are then used to construct exact solutions. This book is a straightforward introduction to the subject, and is aimed at applied mathematicians, physicists, and engineers. The presentation is informal, using many worked examples to illustrate the main symmetry methods. It is written at a level suitable for postgraduates and advanced undergraduates, and is designed to enable the reader to master the main techniques quickly and easily. The book contains methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.Trade Review'Hydon's book stands out as perhaps the best introductory level text currently available … Hydon's book is extremely well-written, and a welcome addition to the literature on Lie's methods. The author has clearly devoted a lot of effort to pedagogical details, and the exposition is designed to effortlessly bring the beginning student up to speed in basic applications of the method.' Peter Olver, ZAMM'I really enjoyed reading this book and I am planning on using some parts for one of my next courses.' Monatshefte für Mathematik'… a nice introduction to symmetry methods for ordinary and partial differential equations written with passion by a specialist … after a few pages it becomes clear that the book is written in a lucid and precise manner.' Zentralblatt MATH'This new book by Peter Hydon … is eminently suitable for advanced undergraduates and beginning postgraduate students … Overall I thoroughly recommend this book and believe that it will be a useful textbook for introducing students to symmetry methods for differential equations.' Journal of Nonlinear Mathematical Physics'Throughout the text numerous examples are worked out in detail and the exercises have been well chosen. this is the most readable text on this material I have seen and I would recommend the book for self-study (as an introduction).' MathSciNet'It is very suitable for, and is specifically aimed at, postgraduate courses in the field. it is the more enjoyable for being written with infectious enthusiasm and there is a good selection of examples.' Mathematical GazetteTable of Contents1. Introduction to symmetries; 1.1. Symmetries of planar objects; 1.2. Symmetries of the simplest ODE; 1.3. The symmetry condition for first-order ODEs; 1.4. Lie symmetries solve first-order ODEs; 2. Lie symmetries of first order ODEs; 2.1. The action of Lie symmetries on the plane; 2.2. Canonical coordinates; 2.3. How to solve ODEs with Lie symmetries; 2.4. The linearized symmetry condition; 2.5. Symmetries and standard methods; 2.6. The infinitesimal generator; 3. How to find Lie point symmetries of ODEs; 3.1 The symmetry condition. 3.2. The determining equations for Lie point symmetries; 3.3. Linear ODEs; 3.4. Justification of the symmetry condition; 4. How to use a one-parameter Lie group; 4.1. Reduction of order using canonical coordinates; 4.2. Variational symmetries; 4.3. Invariant solutions; 5. Lie symmetries with several parameters; 5.1. Differential invariants and reduction of order; 5.2. The Lie algebra of point symmetry generators; 5.3. Stepwise integration of ODEs; 6. Solution of ODEs with multi-parameter Lie groups; 6.1 The basic method: exploiting solvability; 6.2. New symmetries obtained during reduction; 6.3. Integration of third-order ODEs with sl(2); 7. Techniques based on first integrals; 7.1. First integrals derived from symmetries; 7.2. Contact symmetries and dynamical symmetries; 7.3. Integrating factors; 7.4. Systems of ODEs; 8. How to obtain Lie point symmetries of PDEs; 8.1. Scalar PDEs with two dependent variables; 8.2. The linearized symmetry condition for general PDEs; 8.3. Finding symmetries by computer algebra; 9. Methods for obtaining exact solutions of PDEs; 9.1. Group-invariant solutions; 9.2. New solutions from known ones; 9.3. Nonclassical symmetries; 10. Classification of invariant solutions; 10.1. Equivalence of invariant solutions; 10.2. How to classify symmetry generators; 10.3. Optimal systems of invariant solutions; 11. Discrete symmetries; 11.1. Some uses of discrete symmetries; 11.2. How to obtain discrete symmetries from Lie symmetries; 11.3. Classification of discrete symmetries; 11.4. Examples.

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Book SynopsisNonlinear dynamics has been successful in explaining complicated phenomena in well-defined low-dimensional systems. Now it is time to focus on real-life problems that are high-dimensional or ill-defined, for example, due to delay, spatial extent, stochasticity, or the limited nature of available data. How can one understand the dynamics of such systems? Written by international experts, Nonlinear Dynamics and Chaos: Where Do We Go from Here? assesses what the future holds for dynamics and chaos. The chapters address one or more of the broad and interconnected main themes: neural and biological systems, spatially extended systems, and experimentation in the physical sciences. The contributors offer suggestions as to what they see as the way forward, often in the form of open questions for future research.Trade Review"This handsome volume is the proceedings of a conference held in Bristol in 2001, which had the aim of charting new directions for the exploration of nonlinear dynamical systems. The editors must be commended for their work: the individual chapters have been given a clean, uniform style that reflects a serious effort to present the volume as a unified book rather than a recollection of articles, with several cross-references between the chapters. The book is also remarkably free of typographical errors. I heartily recommend this collection to students looking for some direction (as long as they don't think this is all of nonlinear dynamics!)." -UK Nonlinear News, May 2003 "This timely and important book is a record of papers presented at a conference in Bristol and is very well edited, and produced … The very richness of this book, in both theory and real-world applications, makes it difficult to summarize and even more difficult to put down." -Nonlinear Dynamics, Psychology and Life Sciences "The book is written by authors who are champions of their field. All researchers in nonlinear dynamics should have access to this book. It is a valuable resource of references and it contains a lot of ideas and open problems in various fields. One might think of it as a catalogue of problems in nonlinear dynamics. The introduction of the book is a 'must-read.' It presents the nature and the philosophy of the book (and the symposium). Reading the introduction, the editors clearly have done a great job of managing each of the invited lecturers to translate the philosophy of the symposium into their lectures … my impression is that all authors did a good job presenting the excitement of their research and addressing the interesting questions. This book in general is a valuable addition to the literature of the theory and practice of nonlinear dynamics and chaos." -Theo Tuwankotta, Institute of Technology,ITB, Bandung, IndonesiaTable of ContentsPreface. Bifurcation and Degenerate Decomposition in Multiple Time Scale Dynamical Systems. Many-body quantum mechanics. Unfolding Complexity: Hereditory Dynamical Systems-New Bifurcation Schemes and High Dimensional Chaos. Creating stability out of instability.Signal or Noise? A nonlinear dynamics approach to spatiotemporal communication. Outstanding problems in the theory of pattern formation. Is Chaos relevant to Fluid Mechanics?. Time-Reversed Acoustics and Chaos.Reduction methods applied to nonlocally coupled oscillator systems. A prime number of prime questions about vortex dynamcis in nonlinear media. Spontaneous pattern formation in primary visual cortex. Models for Pattern Formation in Development. Spatiotemporal nonlinear dynamics: a new beginning. Author index.

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Book SynopsisExamines a nonlinear system of parabolic partial differential equations (PDE) arising in mathematical biology and statistical mechanics. This book describes the mathematical and physical principles: derivation of a series of equations, biological modeling based on biased random walks, and quantized blowup mechanism based on several PDE techniques.Trade ReviewFrom the reviews: "This book is primarily concerned with systems of partial differential equations of elliptic-parabolic type. Such systems are special cases of systems of pdes that occur naturally in biochemistry, biophysics, molecular biology, and in chemical thermodynamics and statistical mechanics—in short wherever chemical kinetics and diffusive processes are present in a system…. The book itself is well organized and readable although very technical. It is a book about the mathematics of systems and not biology or thermodynamics…. [It] certainly belongs on the bookshelf of specialists in nonlinear pde/math biology." —Bulletin of the AMS “This book is introductory in the sense of being accessible to students not only of mathematics, but also of the physical and the engineering sciences once they have mastered the introductory mathematical courses. …Overall, the presentation is quite detailed regarding the needs of the practitioner with many examples to engineering, earth sciences, etc. (among others elasticity, vibrations, heat transfer, fluid flow; also eigenvalue problems), and special but important items not so often covered in other texts, e.g., how to cope with the specific difficulties arising in polar coordinates. Both numerous exercises and codes in Ansys, Fortran, Mathematica® and MATLAB direct the reader towards experimentation of his own.” (Monatshefte fur Mathematik) "This book examines a nonlinear system of parabolic partial differential equations (PDEs) arising in mathematical biology and statistical mechanics. … Free Energy and Self-Interacting Particles is suitable for researchers and graduate students of mathematics and applied mathematics who are interested in non-linear PDEs in stochastic processes, cellular automata, variational methods, and their applications to natural sciences. It is also suitable for researchers in other fields such as physics, chemistry, biology, and engineering." (Christopher Mayer, Simulation News Europe, Vol. 16 (1), 2006)Table of Contents* Dedication * Preface * Summary * Background * Fundamental Theorem * Trudinger–Moser Inequality * The Green's Function * Equilibrium States * Blowup Analysis for Stationary Solutions * Multiple Existence * Dynamical Equivalence * Formation of Collapses * Finiteness of Blowup Points * Concentration Lemma * Weak Solution * Hyperparabolicity * Quantized Blowup Mechanism * Theory of Dual Variation * References * Index

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Book SynopsisThis text serves as an exploration of the beautiful topic of mathematical biology through the lens of discrete and differential equations. Intended for students who have completed differential and integral calculus, Mathematical Biology: Discrete and Differential Equations allows students to explore topics such as bifurcation diagrams, nullclines, discrete dynamics, and SIR models for disease spread, which are often reserved for more advanced undergraduate or graduate courses. These exciting topics are sprinkled throughout the book alongside the more typical first- and second-order linear differential equations and systems of linear differential equations.This class-tested text is written in a conversational, welcoming voice, which should help invite students along as they discover the magic of mathematical biology and both discrete and differential equations. A focus is placed on examples with solutions written out step by step, including co

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Book SynopsisPartial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners' course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor. This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The teaching-by-examples approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book's level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses. Highlights: Table of Contents Introduction Basic definitions Examples First-order equations Linear first-order equations General solution Initial condition Quasilinear first-order equations Characteristic curves Examples Second-order equations Classification of second-order equations Canonical forms Hyperbolic equations Elliptic equations Parabolic equations The Sturm-Liouville Problem General consideration Examples of Sturm-Liouville Problems One-Dimensional Hyperbolic Equations Wave Equation Boundary and Initial Conditions Longitudinal Vibrations of a Rod and Electrical Oscillations Rod oscillations: Equations and boundary conditions Electrical Oscillations in a Circuit Traveling Waves: D'Alembert Method Cauchy problem for nonhomogeneous wave equation D'Alembert's formula The Green's function Well-posedness of the Cauchy problem Finite intervals: The Fourier Method for Homogeneous Equations The Fourier Method for Nonhomogeneous Equations The Laplace Transform Method: simple cases Equations with Nonhomogeneous Boundary Conditions The Consistency Conditions and Generalized Solutions Energy in the Harmonics Dispersion of waves Cauchy problem in an infinite region Propagation of a wave train One-Dimensional Parabolic Equations Heat Conduction and Diffusion: Boundary Value Problems Heat conduction Diffusion equation One-dimensional parabolic equations and initial and boundary conditions The Fourier Method for Homogeneous Equations Nonhomogeneous Equations The Green's function and Duhamel's principle The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions Large time behavior of solutions Maximum principle The heat equation in an infinite region Elliptic equations Elliptic differential equations and related physical problems Harmonic functions Boundary conditions Example of an ill-posed problem Well-posed boundary value problems Maximum principle and its consequences Laplace equation in polar coordinates Laplace equation and interior BVP for circular domain Laplace equation and exterior BVP for circular domain Poisson equation: general notes and a simple case Poisson Integral Application of Bessel functions for the solution of Poisson equations in a circle Three-dimensional Laplace equation for a cylinder Three-dimensional Laplace equation for a ball Axisymmetric case Non-axisymmetric case BVP for Laplace Equation in a Rectangular Domain The Poisson Equation with Homogeneous Boundary Conditions Green's function for Poisson equations Homogeneous boundary conditions Nonhomogeneous boundary conditions Some other important equations Helmholtz equation Schrӧdinger equation Two Dimensional Hyperbolic Equations Derivation of the Equations of Motion Boundary and Initial Conditions Oscillations of a Rectangular Membrane The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions Small Transverse Oscillations of a Circular Membrane The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions Axisymmetric Oscillations of a Membrane The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions Forced Axisymmetric Oscillations The Fourier Method for Equations with Nonhomogeneous Boundary Conditions Two-Dimensional Parabolic Equations Heat Conduction within a Finite Rectangular Domain The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange) The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary conditions Heat Conduction within a Circular Domain The Fourier Method for the Homogeneous Heat Equation The Fourier Method for the Nonhomogeneous Heat Equation Heat conduction in an Infinite Medium Heat Conduction in a Semi-Infinite Medium Nonlinear equations Burgers equation Kink solution Symmetries of the Burgers equation General solution of the Cauchy problem. Interaction of kinks Korteweg-de Vries equation Symmetry properties of the KdV equation Cnoidal waves Solitons Bilinear formulation of the KdV equation Hirota's method Multisoliton solutions Nonlinear Schrӧdinger equation Symmetry properties of NSE Solitary waves Appendix A. Fourier Series, Fourier and Laplace Transforms Appendix B. Bessel and Legendre Functions Appendix C. Sturm-Liouville problem and auxiliary functions for one and two dimensions Appendix D. D1. The Sturm-Liouville problem for a circle D2. The Sturm-Liouville problem for the rectangle Appendix E. E1. The Laplace and Poisson equations for a rectangular domain with nonhomogeneous boundary conditions. E2. The heat conduction equations with nonhomogeneous boundary conditions.

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Book SynopsisThis engaging graduate-level introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. Explanatory pictures, detailed proofs, exercises and helpful remarks make it suitable for self-study and also a useful reference for researchers.Trade Review'The book is a clear exposition of the theory of sets of finite perimeter, that introduces this topic in a very elegant and original way, and shows some deep and important results and applications … Although most of the results contained in this book are classical, some of them appear in this volume for the first time in book form, and even the more classical topics which one may find in several other books are presented here with a strong touch of originality which makes this book pretty unique … I strongly recommend this excellent book to every researcher or graduate student in the field of calculus of variations and geometric measure theory.' Alessio Figalli, Canadian Mathematical Society Notes'The first aim of the book is to provide an introduction for beginners to the theory of sets of finite perimeter, presenting results concerning the existence, symmetry, regularity and structure of singularities in some variational problems involving length and area … The secondary aim … is to provide a multi-leveled introduction to the study of other variational problems … an interested reader is able to enter with relative ease several parts of geometric measure theory and to apply some tools from this theory in the study of other problems from mathematics … This is a well-written book by a specialist in the field … It provides generous guidance to the reader [and] is recommended … not only to beginners who can find an up-to-date source in the field but also to specialists … It is an invitation to understand and to approach some deep and difficult problems from mathematics and physics.' Vasile Oproiu, Zentralblatt MATHTable of ContentsPart I. Radon Measures on Rn: 1. Outer measures; 2. Borel and Radon measures; 3. Hausdorff measures; 4. Radon measures and continuous functions; 5. Differentiation of Radon measures; 6. Two further applications of differentiation theory; 7. Lipschitz functions; 8. Area formula; 9. Gauss–Green theorem; 10. Rectifiable sets and blow-ups of Radon measures; 11. Tangential differentiability and the area formula; Part II. Sets of Finite Perimeter: 12. Sets of finite perimeter and the Direct Method; 13. The coarea formula and the approximation theorem; 14. The Euclidean isoperimetric problem; 15. Reduced boundary and De Giorgi's structure theorem; 16. Federer's theorem and comparison sets; 17. First and second variation of perimeter; 18. Slicing boundaries of sets of finite perimeter; 19. Equilibrium shapes of liquids and sessile drops; 20. Anisotropic surface energies; Part III. Regularity Theory and Analysis of Singularities: 21. (Λ, r0)-perimeter minimizers; 22. Excess and the height bound; 23. The Lipschitz approximation theorem; 24. The reverse Poincaré inequality; 25. Harmonic approximation and excess improvement; 26. Iteration, partial regularity, and singular sets; 27. Higher regularity theorems; 28. Analysis of singularities; Part IV. Minimizing Clusters: 29. Existence of minimizing clusters; 30. Regularity of minimizing clusters; References; Index.

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Book SynopsisTable of ContentsP. What is Number Theory? 1. The Integers. Numbers and Sequences. Sums and Products. Mathematical Induction. The Fibonacci Numbers. 2. Integer Representations and Operations. Representations of Integers. Computer Operations with Integers. Complexity of Integer Operations. 3. Primes and Greatest Common Divisors. Prime Numbers. The Distribution of Primes. Greatest Common Divisors. The Euclidean Algorithm. The Fundemental Theorem of Arithmetic. Factorization Methods and Fermat Numbers. Linear Diophantine Equations. 4. Congruences. Introduction to Congruences. Linear Congrences. The Chinese Remainder Theorem. Solving Polynomial Congruences. Systems of Linear Congruences. Factoring Using the Pollard Rho Method. 5. Applications of Congruences. Divisibility Tests. The perpetual Calendar. Round Robin Tournaments. Hashing Functions. Check Digits. 6. Some Special Congruences. Wilson's Theorem and Fermat's Little Theorem. Pseudoprimes. Euler's Theorem. 7. Multiplicative Functions. The Euler Phi-Function. The Sum and Number of Divisors. Perfect Numbers and Mersenne Primes. Mobius Inversion. Partitions. 8. Cryptology. Character Ciphers. Block and Stream Ciphers. Exponentiation Ciphers. Knapsack Ciphers. Cryptographic Protocols and Applications. 9. Primitive Roots. The Order of an Integer and Primitive Roots. Primitive Roots for Primes. The Existence of Primitive Roots. Index Arithmetic. Primality Tests Using Orders of Integers and Primitive Roots. Universal Exponents. 10. Applications of Primitive Roots and the Order of an Integer. Pseudorandom Numbers. The EIGamal Cryptosystem. An Application to the Splicing of Telephone Cables. 11. Quadratic Residues. Quadratic Residues and nonresidues. The Law of Quadratic Reciprocity. The Jacobi Symbol. Euler Pseudoprimes. Zero-Knowledge Proofs. 12. Decimal Fractions and Continued. Decimal Fractions. Finite Continued Fractions. Infinite Continued Fractions. Periodic Continued Fractions. Factoring Using Continued Fractions. 13. Some Nonlinear Diophantine Equations. Pythagorean Triples. Fermat's Last Theorem. Sums of Squares. Pell's Equation. Congruent Numbers. 14. The Gaussian Integers. Gaussian Primes. Unique Factorization of Gaussian Integers. Gaussian Integers and Sums of Squares.

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Book SynopsisDiscovering Evolution Equations with Applications: Volume 1-Deterministic Equations provides an engaging, accessible account of core theoretical results of evolution equations in a way that gradually builds intuition and culminates in exploring active research. It gives nonspecialists, even those with minimal prior exposure to analysis, the foundation to understand what evolution equations are and how to work with them in various areas of practice.After presenting the essentials of analysis, the book discusses homogenous finite-dimensional ordinary differential equations. Subsequent chapters then focus on linear homogenous abstract, nonhomogenous linear, semi-linear, functional, Sobolev-type, neutral, delay, and nonlinear evolution equations. The final two chapters explore research topics, including nonlocal evolution equations. For each class of equations, the author develops a core of theoretical results concerning the existence and uniqueness of solTable of ContentsA Basic Analysis Toolbox. Homogenous Linear Evolution Equations in RN. Abstract Homogenous Linear Evolution Equations. Nonhomogenous Linear Evolution Equations. Semi-Linear Evolution Equations. Functional Evolution Equations. Implicit Evolution Equations. Delay Evolution Equations. Nonlinear Evolution Equations. Nonlocal Evolution Equations. Beyond Volume 1. Bibliography. Index.

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Book SynopsisMost existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations. The text begins with hands-on introductions to the essentials of real and stochastic analysis. It then develops the theory for homogenous one-dimensional stochastic ordinary differential equations (ODEs) and extends the theory to systems of homogenous linear stochastic ODEs. The next several chapters focus on abstract homogenous linear, nonhomogenous linear, and semi-linear stochastic evolution equations. The author also addresses the case in wTable of ContentsA Basic Analysis Toolbox. The Bare-Bone Essentials of Probability Theory. Linear Homogenous Stochastic Evolution Equations in R. Homogenous Linear Stochastic Evolution Equations in RN. Abstract Homogenous Linear Stochastic Evolution Equations. Nonhomogenous Linear Stochastic Evolution Equations. Semi-Linear Stochastic Evolution Equations. Functional Stochastic Evolution Equations. Sobolev-Type Stochastic Evolution Equations. Beyond Volume 2. Bibliography. Index.

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Book SynopsisIntegral Transforms and Their Applications, Third Edition covers advanced mathematical methods for many applications in science and engineering. The book is suitable as a textbook for senior undergraduate and first-year graduate students and as a reference for professionals in mathematics, engineering, and applied sciences. It presents a systematic development of the underlying theory as well as a modern approach to Fourier, Laplace, Hankel, Mellin, Radon, Gabor, wavelet, and Z transforms and their applications. New to the Third Edition New material on the historical development of classical and modern integral transforms New sections on Fourier transforms of generalized functions, the Poisson summation formula, the Gibbs phenomenon, and the Heisenberg uncertainty principle Revised material on Laplace transforms and double Laplace transforms and their applications New examples of applications in mechanical Table of ContentsIntegral Transforms. Fourier Transforms and Their Applications. Laplace Transforms and Their Basic Properties. Applications of Laplace Transforms. Fractional Calculus and Its Applications. Applications of Integral Transforms to Fractional Differential Equations. Hankel Transforms and Their Applications. Mellin Transforms and Their Applications. Hilbert and Stieltjes Transforms. Finite Fourier Sine and Cosine Transforms. Finite Laplace Transforms. Z Transforms. Finite Hankel Transforms. Legendre Transforms. Jacobi and Gegenbauer Transforms. Laguerre Transforms. Hermite Transforms. The Radon Transform and Its Application. Wavelets and Wavelet Transforms. Appendices.

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Book SynopsisDynamical systems are a principal tool in the modeling, prediction, and control of a wide range of complex phenomena. As the need for improved accuracy leads to larger and more complex dynamical systems, direct simulation often becomes the only available strategy for accurate prediction or control, inevitably creating a considerable burden on computational resources. This is the main context where one considers model reduction, seeking to replace large systems of coupled differential and algebraic equations that constitute high fidelity system models with substantially fewer equations that are crafted to control the loss of fidelity that order reduction may induce in the system response. Interpolatory methods are among the most widely used model reduction techniques, and Interpolatory Methods for Model Reduction is the first comprehensive analysis of this approach available in a single, extensive resource. It introduces state-of-the-art methods reflecting significant developments over the past two decades, covering both classical projection frameworks for model reduction and data-driven, nonintrusive frameworks.This textbook is appropriate for a wide audience of engineers and other scientists working in the general areas of large-scale dynamical systems and data-driven modeling of dynamics.

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Book SynopsisBringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems. Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics, and dissipation-induced instabilities are treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. Each chapter contains mechanical and physical examples, and the combination of advanced material and more tutorial elements makes this book attractive for both experts and non-specialists keen to expand their knowledge on modern methods and trends in stability theory. Contents 1. Surprising Instabilities of Simple Elastic Structures, Davide Bigoni, Diego Misseroni, Giovanni Noselli and Daniele Zaccaria. 2. WKB Solutions Near an Unstable Equilibrium and Applications, Jean-François Bony, Setsuro Fujiié, Thierry Ramond and Maher Zerzeri, partially supported by French ANR project NOSEVOL. 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems, Richard Cushman, Johnathan Robbins and Dimitrii Sadovskii. 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities, Olivier Doaré. 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field, Sergey Yu. Dobrokhotov and Anatoly Yu. Anikin. 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials, Nir Dror and Boris A. Malomed. 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation, Yasuhide Fukumoto, Makoto Hirota and Youichi Mie. 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance, Igor Hoveijn and Oleg N. Kirillov. 9. Index Theorems for Polynomial Pencils, Richard Kollár and Radomír Bosák. 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches, Paolo Luzzatto-Fegiz and Charles H.K. Williamson. 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows, Sherwin A. Maslowe. 12. Continuum Hamiltonian Hopf Bifurcation I, Philip J. Morrison and George I. Hagstrom. 13. Continuum Hamiltonian Hopf Bifurcation II, George I. Hagstrom and Philip J. Morrison. 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model, Philip J. Morrison, Emanuele Tassi and Cesare Tronci. 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators, Francis Nier. 16. Stability Optimization for Polynomials and Matrices, Michael L. Overton. 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations, Dmitry E. Pelinovsky. 18. Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities, Zensho Yoshida and Philip J. Morrison. About the Authors Oleg N. Kirillov has been a Research Fellow at the Magneto-Hydrodynamics Division of the Helmholtz-Zentrum Dresden-Rossendorf in Germany since 2011. His research interests include non-conservative stability problems of structural mechanics and physics, perturbation theory of non-self-adjoint boundary eigenvalue problems, magnetohydrodynamics, friction-induced oscillations, dissipation-induced instabilities and non-Hermitian problems of optics and microwave physics. Since 2013 he has served as an Associate Editor for the journal Frontiers in Mathematical Physics. Dmitry E. Pelinovsky has been Professor at McMaster University in Canada since 2000. His research profile includes work with nonlinear partial differential equations, discrete dynamical systems, spectral theory, integrable systems, and numerical analysis. He served as the guest editor of the special issue of the journals Chaos in 2005 and Applicable Analysis in 2010. He is an Associate Editor of the journal Communications in Nonlinear Science and Numerical Simulations. This book is devoted to the problems of spectral analysis, stability and bifurcations arising from the nonlinear partial differential equations of modern physics. Leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics present state-of-the-art approaches to a wide spectrum of new challenging stability problems. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics and dissipation-induced instabilities will be treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. All chapters contain mechanical and physical examples and combine both tutorial and advanced sections, making them attractive both to experts in the field and non-specialists interested in knowing more about modern methods and trends in stability theory.Table of ContentsPreface xiii Chapter 1. Surprising Instabilities of Simple Elastic Structures 1 Davide BIGONI, Diego MISSERONI, Giovanni NOSELLI and Daniele ZACCARIA Chapter 2. WKB Solutions Near an Unstable Equilibrium and Applications 15 Jean-François BONY, Setsuro FUJIIÉ, Thierry RAMOND and Maher ZERZERI Chapter 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems 41 Richard CUSHMAN, Johnathan M. ROBBINS and Dimitrii SADOVSKII Chapter 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities 67 Olivier DOARÉ Chapter 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field 85 Sergey Y. DOBROKHOTOV and Anatoly Y. ANIKIN Chapter 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials 111 Nir DROR and Boris A. MALOMED Chapter 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation 139 Yasuhide FUKUMOTO, Makoto HIROTA and Youichi MIE Chapter 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance 155 Igor HOVEIJN and Oleg N. KIRILLOV Chapter 9. Index Theorems for Polynomial Pencils 177 Richard KOLLÁR and Radomír BOSÁK Chapter 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches 203 Paolo LUZZATTO-FEGIZ and Charles H.K. WILLIAMSON Chapter 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows 223 Sherwin A. MASLOWE Chapter 12. Continuum Hamiltonian Hopf Bifurcation I 247 Philip J. MORRISON and George I. HAGSTROM Chapter 13. Continuum Hamiltonian Hopf Bifurcation II 283 George I. HAGSTROM and Philip J. MORRISON Chapter 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model 311 Philip J. MORRISON, Emanuele TASSI and Cesare TRONCI Chapter 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators 331 Francis NIER Chapter 16. Stability Optimization for Polynomials and Matrices 351 Michael L. OVERTON Chapter 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations 377 Dmitry E. PELINOVSKY Chapter 18. Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities 401 List of Authors 421 Index 425

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Book SynopsisThis volume encompasses prototypical, innovative and emerging examples and benchmarks of Differential-Algebraic Equations (DAEs) and their applications, such as electrical networks, chemical reactors, multibody systems, and multiphysics models, to name but a few. Each article begins with an exposition of modelling, explaining whether the model is prototypical and for which applications it is used. This is followed by a mathematical analysis, and if appropriate, a discussion of the numerical aspects including simulation. Additionally, benchmark examples are included throughout the text.Mathematicians, engineers, and other scientists, working in both academia and industry either on differential-algebraic equations and systems or on problems where the tools and insight provided by differential-algebraic equations could be useful, would find this book resourceful. Trade Review“The book can be of special interest to mathematicians, STEM students, and engineers working in multidisciplinary industry settings where the insight provided by differential-algebraic equations can be determinant in decision making.” (Andrzej Sokolowski, MAA Reviews, August 11, 2019)Table of ContentsGeneral Nonlinear Differential Algebraic Equations and Tracking Problems: A Robotics Example.- DAE Aspects in Vehicle Dynamics and Mobile Robotics.- Open-loop Control of Underactuated Mechanical Systems Using Servo-constraints: Analysis and Some Examples.- Systems of Differential Algebraic Equations in Computational Electromagnetics.- Gas Network Benchmark Models.- Topological Index Analysis Applied to Coupled Flow Networks.- Nonsmooth DAEs with Applications in Modeling Phase Changes.- Continuous, Semi-Discrete, and Fully Discretized Navier-Stokes Equations.

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Book SynopsisThis book in the BCAM SpringerBriefs series is a treatise on the p-Laplace equation. It is based on lectures by the author that were originally delivered at the Summer School in Jyväskylä, Finland, in August 2005 and have since been updated and extended to cover various new topics, including viscosity solutions and asymptotic mean values. The p-Laplace equation is a far-reaching generalization of the ordinary Laplace equation, but it is non-linear and degenerate (p>2) or singular (p<2). Thus it requires advanced methods. Many fascinating properties of the Laplace equation are, in some modified version, extended to the p-Laplace equation. Nowadays the theory is almost complete, although some challenging problems remain open.Trade Review“The book will provide the reader with an up-to-date overview on the p-Laplace equation and will uncover ideas behind some concepts and involved results in the field.” (Vladimir Bobkov, Mathematical Reviews, December, 2019)“The book is a very useful contribution to the growing literature on this circle of ideas. I wholeheartedly recommend this book both as a textbook, as well as for independent study.” (Vicenţiu D. Rădulescu, zbMATH 1421.35002, 2019)Table of Contents1 Introduction.- 2 The Dirichlet problem and weak solutions.- 3 Regularity theory.- 4 Differentiability.- 5 On p-superharmonic functions.- 6 Perron's method.- 7 Some remarks in the complex plane.- 8 The infinity Laplacian.- 9 Viscosity solutions.- 10 Asymptotic mean values.- 11 Some open problems.- 12 Inequalities for vectors.

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Book SynopsisThis book presents a history of differential equations, both ordinary and partial, as well as the calculus of variations, from the origins of the subjects to around 1900. Topics treated include the wave equation in the hands of d’Alembert and Euler; Fourier’s solutions to the heat equation and the contribution of Kovalevskaya; the work of Euler, Gauss, Kummer, Riemann, and Poincaré on the hypergeometric equation; Green’s functions, the Dirichlet principle, and Schwarz’s solution of the Dirichlet problem; minimal surfaces; the telegraphists’ equation and Thomson’s successful design of the trans-Atlantic cable; Riemann’s paper on shock waves; the geometrical interpretation of mechanics; and aspects of the study of the calculus of variations from the problems of the catenary and the brachistochrone to attempts at a rigorous theory by Weierstrass, Kneser, and Hilbert. Three final chapters look at how the theory of partial differential equations stood around 1900, as they were treated by Picard and Hadamard. There are also extensive, new translations of original papers by Cauchy, Riemann, Schwarz, Darboux, and Picard. The first book to cover the history of differential equations and the calculus of variations in such breadth and detail, it will appeal to anyone with an interest in the field. Beyond secondary school mathematics and physics, a course in mathematical analysis is the only prerequisite to fully appreciate its contents. Based on a course for third-year university students, the book contains numerous historical and mathematical exercises, offers extensive advice to the student on how to write essays, and can easily be used in whole or in part as a course in the history of mathematics. Several appendices help make the book self-contained and suitable for self-study.Trade Review“This book is a very good example of a text for a course in the history of mathematics. … the author provides for students and readers a historical overview of how mathematics, physics, celestial mechanics and difficult problems to tackle from differential equations as well as applications were intertwined, and the resulting dialogues between mathematicians, physicists and astronomers. This book is a successful attempt to fill in some of the gaps on the history of differential equations.” (‪Clara Silvia Roero, Mathematical Reviews, September, 2022)Table of Contents1 The First Ordinary Differential Equations.- 2 Variational Problems and the Calculus.- 3 The Partial Differential Calculus.- 4 Rational Mechanics.- 5 Partial Differential Equations.- 6 Lagrange's General Theory.- 7 The Calculus of Variations.- 8 Monge and Solutions to Partial Differential Equations.- 9 Revision.- 10 The Heat Equation.- 11 Gauss and the Hypergeometric Equation.- 12 Existence Theorem.- 13 Riemann and Complex Function Theory.- 14 Riemann and the Hypergeometric Equation.- 15 Schwarz and the Complex Hypergeometric Equation.- 16 Complex Ordinary Differential Equations: Poincaré.- 17 More General Partial Differential Equations.- 18 Green's Functions and Dirichlet's Principle.- 19 Attempts on Laplace's Equation.- 20 Applied Wave Equations.- 21 Revision.- 22 Riemann's Shock Wave Paper.- 23 The Example of Minimal Surfaces.- 24 Partial Differential Equations and Mechanics.- 25 Geometrical Interpretations of Mechanics.- 26 The Calculus of Variations in the 19th Century.- 27 Poincaré and Mathematical Physics.- 28 Elliptic Equations and Regular Variational Problems.- 29 Hyperbolic Equations.- 30 Revision.- 32 Translations.- A Newton's Principia Mathematica.- B Characteristics.- C First-order Non-linear Equations.- D Green's Theorem and Heat Conduction.- E Complex Analysis.- F Möbius Transformations.- G Lipschitz and Picard.- H The Assessment.- Bibliography.- Index.

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Book SynopsisThis book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.The research on wave-type problems is a fascinating and emerging field in mathematical research with many challenging applications in sciences and engineering. Profound investigations on waves require a strong interaction of several mathematical disciplines including functional analysis, partial differential equations, mathematical modeling, mathematical physics, numerical analysis, and scientific computing.The goal of this book is to present a comprehensive introduction to the research on wave phenomena. Starting with basic models for acoustic, elastic, and electro-magnetic waves, topics such as the existence of solutions for linear and some nonlinear material laws, efficient discretizations and solution methods in space and time, and the application to inverse parameter identification problems are covered. The aim of this book is to intertwine analysis and numerical mathematics for wave-type problems promoting thus cooperative research projects in this field.Table of ContentsSpace-time approximations for linear acoustic, elastic, and electro-magnetic wave equations.- Local wellposedness and long-time behavior of quasilinear Maxwell equations.- Error analysis of second-order time integration methods for discontinuous Galerkin discretizations of Friedrichs’ systems.- An abstract framework for inverse wave problems with applications.

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Book SynopsisThis book addresses the state of the art of reduced order methods for modelling and computational reduction of complex parametrised systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in various fields.Consisting of four contributions presented at the CIME summer school, the book presents several points of view and techniques to solve demanding problems of increasing complexity. The focus is on theoretical investigation and applicative algorithm development for reduction in the complexity – the dimension, the degrees of freedom, the data – arising in these models.The book is addressed to graduate students, young researchers and people interested in the field. It is a good companion for graduate/doctoral classes.Table of Contents- 1. The Reduced Basis Method in Space and Time: Challenges, Limits and Perspectives. - 2. Inverse Problems: A Deterministic Approach Using Physics-Based Reduced Models. - 3. Model Order Reduction for Optimal Control Problems. - 4. Machine Learning Methods for Reduced Order Modeling.

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Book SynopsisThe first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. In includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(Peter Lax, SIAM review, June 1998)Table of ContentsContents of Volumes II and III.- Preface.- 1 Basic Theory of ODE and Vector Fields.- 2 The Laplace Equation and Wave Equation.- 3 Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE.- 4 Sobolev Spaces.- 5 Linear Elliptic Equation.- 6 Linear Evolution Equations.- A Outline of Functional Analysis.- B Manifolds, Vector Bundles, and Lie Groups.- Index.

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Book SynopsisThe third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L^p Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis. The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.”(Peter Lax, SIAM review, June 1998)Table of ContentsContents of Volumes I and II.- Preface.- 13 Function Space and Operator Theory for Nonlinear Analysis.- 14 Nonlinear Elliptic Equations.- 15 Nonlinear Parabolic Equations.- 16 Nonlinear Hyperbolic Equations.- 17 Euler and Navier–Stokes Equations for Incompressible Fluids.- 18 Einstein’s Equations.- Index.

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Book SynopsisThis book offers a modern introduction to Nevanlinna theory and its intricate relation to the theory of normal families, algebraic functions, asymptotic series, and algebraic differential equations.Following a comprehensive treatment of Nevanlinna’s theory of value distribution, the author presents advances made since Hayman’s work on the value distribution of differential polynomials and illustrates how value- and pair-sharing problems are linked to algebraic curves and Briot–Bouquet differential equations. In addition to discussing classical applications of Nevanlinna theory, the book outlines state-of-the-art research, such as the effect of the Yosida and Zalcman–Pang method of re-scaling to algebraic differential equations, and presents the Painlevé–Yosida theorem, which relates Painlevé transcendents and solutions to selected 2D Hamiltonian systems to certain Yosida classes of meromorphic functions.Aimed at graduate students interested in recent developments in the field and researchers working on related problems, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations will also be of interest to complex analysts looking for an introduction to various topics in the subject area. With examples, exercises and proofs seamlessly intertwined with the body of the text, this book is particularly suitable for the more advanced reader.Trade Review“The book by Steinmetz is clearly written, including a substantial number of exercises related to and complementing the actual text.” (Ilpo Laine, Mathematical Reviews, June, 2018)“The list of references contains more than 200 items including very recent results of the author and other. … I recommend this book to any person who is interested in complex analysis, in particular, in value distribution theory and complex differential equations.” (Igor Chyzhykov, zbMATH, 2018)Table of ContentsIntroduction and preface.- Selected Topics in Complex Analysis.- Nevanlinna Theory.- Selected Applications of Nevanlinna Theory.- Normal Families.- Algebraic Differential Equations.- Higher-Order Algebraic Differential Equations.- Index.

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Book SynopsisThis book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.Table of Contents1. Notes on Weyl Algebras and D-modules.- 2. Inverse Systems of Local Rings.- 3. Lectures on the Representation Type of a Projective Variety.- 4. Simplicial Toric Varieties which are set-theoretic Complete Intersections.

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Book SynopsisThe second of two volumes, this edited proceedings book features research presented at the XVI International Conference on Hyperbolic Problems held in Aachen, Germany in summer 2016. It focuses on the theoretical, applied, and computational aspects of hyperbolic partial differential equations (systems of hyperbolic conservation laws, wave equations, etc.) and of related mathematical models (PDEs of mixed type, kinetic equations, nonlocal or/and discrete models) found in the field of applied sciences.Table of ContentsChapter 54: Jingwei Hu, Shi Jin and Ruiwen Shu: A stochastic Galerkin method for the Fokker-Planck-Landau Equation with random uncertainties Chapter 55: Guanghui Hu, Xucheng Meng and Tao Tang: On robust and adaptive finite volume methods for steady Euler equations Chapter 56: John K. Hunter: The Burgers-Hilbert equation Chapter 57: Alexander Jaust and Jochen Schutz: General linear methods for time-dependent PDEs Chapter 58: Yi Jiang and Hailiang Liu; An Invariant-Region-Preserving (IRP) limiter to DG methods for compressible Euler equations Chapter 59: Nan Jiang: β -Schemes with Source Terms and the Convergence Analysis Chapter 60: Bugra Kabil: Existence of undercompressive shock wave solutions to the Euler equations Chapter 61: Touria Karite, Ali Boutoulout, and Fatima Zahrae El Alaoui: Some numerical results of regional boundary controllability with output constraints Chapter 62: Rukhsana Kausar and Stephan Trenn: Water hammer modeling for water networks via hyperbolic PDEs and switched DAEs Chapter 63: Yuya Kiri and Yoshihiro Ueda: Stability criteria for some system of delay differential equations Chapter 64: Matej Klima, Milan Kucharik, Mikhail Shashkov and Jan Velechovsky: Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics Chapter 65: Christian Klingenberg and Andrea Thomann: On computing compressible Euler equations with gravity Chapter 66: Christian Klingenberg, Jens Klotzky and Nicolas Seguin: On well-posedness for a multi-particle-fluid model Chapter 67: Christian Klingenberg, Qin Li and Marlies Pirner: On quantifying uncertainties for the linearized BGK kinetic equation Chapter 68: Christian Klingenberg, Marlies Pirner and Gabriella Puppo: Kinetic ES-BGK models for a multi-component gas mixture Chapter 69: Christian Klingenberg, Gero Schnücke, Yinhua Xia: An arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Entropy stability Chapter 70: Julian Koellermeier and Manuel Torrilhon: Simplified Hyperbolic Moment Equations Chapter 71: Andrea Korsch: Weakly coupled systems of conservation laws on moving surfaces Chapter 72: Mirko Krankel, Dietmar Kröner: A phasefield model for flows with phasetransition Chapter 73: W. J. Lambert, A. C. Alvarez, D. Marchesin and J. Bruining: Mathematical theory of two phase geochemical flow with chemical species Chapter 74: Min-Gi Lee, Theodoros Katsaounis and Athanasios E. Tzavaras: Localization of Adiabatic Deformations in Thermoviscoplastic Materials Chapter 75: Philippe G. LeFloch: The global nonlinear stability of Minkowski spacetime for self-gravitating massive fields Chapter 76: Jim Magiera and Christian Rohde: A Particle-based Multiscale Solver for Compressible Liquid-Vapor Flow Chapter 77: Corrado Mascia and Thinh Tien Nguyen: Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D Chapter 78: Clement Mifsud and Bruno Despres: A numerical approach of Friedrichs’ systems under constraints in bounded domains Chapter 79: Stefano Modena: Lagrangian representation for systems of conservation laws: an overview Chapter 80: R. Murti, S. Baskar, and P. Prasad: Kinematical conservation laws in inhomogeneous media Chapter 81: Philipp Offner, Jan Glaubitz, Hendrik Ranocha, and Thomas Sonar: Artificial Viscosity for Correction Procedure via Reconstruction Using Summation-by-Parts Operators Chapter 82: Masashi Ohnawa: On a relation between shock profiles and stabilization mechanisms in a radiating gas model Chapter 83: Evgeny Yu. Panov: On the long time behavior of almost periodic entropy solutions to scalar conservations laws Chapter 84: Lorenzo Pareschi and Mattia Zanella: Structure preserving schemes for mean-field equations of collective behavior Chapter 85: Marica Pelanti, Keh-Ming Shyue and Tore Flatten: A Numerical Model for Three-Phase Liquid-Vapor-Gas Flows with Relaxation Processes Chapter 86: Gilbert Peralta: Feedback Stabilization of a Linear Fluid-Membrane System with Time-Delay Chapter 87: Ilya Peshkov, Evgeniy Romenski and Michael Dumbser: A unified hyperbolic formulation for viscous fluids and elastoplastic solids Chapter 88: T. Pichard, B. Dubroca, S. Brull and M. Frank: On the transverse diffusion of beams of photons in radiation therapy Chapter 89: Marin Prebeg: Numerical Viscosity in Large Time Step HLL-type Schemes Chapter 90: Hendrik Ranocha, Philipp Offner, and Thomas Sonar: Correction procedure via reconstruction Using summation-by-parts operators Chapter 91: Deep Ray: A third-order entropy stable scheme for the compressible Euler equations Chapter 92: Philip Roe: Did Numerical Methods for Hyperbolic Problems Take a Wrong Turning? Chapter 93: Friedrich K. Röpke: Astrophysical fluid dynamics and applications to stellar modeling Chapter 94: Olga S. Rozanova and Marko K. Turzynsky: Nonlinear stability of localized and non-localized vortices in rotating compressible media Chapter 95: Smita Sahu: Coupled scheme for Hamilton-Jacobi equations Chapter 96: Nicolas Seguin: Compressible heterogeneous two-phase flows Chapter 97: Chi-Wang Shu: Bound-preserving high order schemes for hyperbolic equations: survey and recent developments Chapter 98: Aleksey Sikstel, Anne Kusters, Markus Frings, Sebastian Noelle and Stefanie Elgeti: Comparison of shallow water models for rapid channel flows Chapter 99: Veronika Straub, Sigrun Ortleb, Philipp Birken and Andreas Meister: On stability and conservation properties of (s)EPIRK integrators in the context of discretized PDEs Chapter 100: Tian-Yi Wang: Compactness on Multidimensional Steady Euler Equations Chapter 101: Franziska Weber: A constraint preserving finite difference method for the damped wave map equation to the sphere Chapter 102: Karen Yagdjian: Integral transform approach to solving Klein-Gordon equation with variable coefficients Chapter 103: Hamed Zakerzadeh: Asymptotic consistency of the RS-IMEX scheme for the low-Froude shallow water equations: Analysis and numerics Chapter 104: Mohammad Zakerzadeh and Georg May: Class of Space-Time Entropy Stable DG Schemes for Systems of Convection-Diffusion Chapter 105: Kevin Zumbrun: Invariant manifolds for a class of degenerate evolution equations and structure of kinetic shock layers

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Book SynopsisThis book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.Table of ContentsFoundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement.- Adaptive Multiscale Methods for the Numerical Treatment of Systems of PDEs.- Generalized Locally Toeplitz Sequences: A Spectral Analysis Tool for Discretized Differential Equations.- Isogeometric Analysis: Mathematical and Implementational Aspects, with Applications.

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Book SynopsisDifferenzialgleichungen sind Ihnen ein Buch mit sieben Siegeln? Kein Problem! Im ersten Teil liefert Ihnen dieses Buch wirklich alles, was Sie an Handwerkszeug zum Lösen von Differenzialgleichungen benötigen. Anschließend erfahren Sie, was Differenzialgleichungen überhaupt sind und mit welchen Methoden man sie lösen kann. Im dritten Teil wird es ernst: Sie werden einfache Differenzialgleichungen rechnerisch lösen. Aber keine Sorge: Vielfältige Beispiele geben Ihnen die Gelegenheit, die Verfahren gründlich zu üben. Und damit Sie wissen, warum Sie sich all diesen Mühen unterziehen, werden Sie zuletzt auf berühmte Differenzialgleichungen aus Biologie, Chemie, Physik und Ökonomie treffen.

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Book SynopsisFrom the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student." --New Zealand Mathematical Society, 1985Trade Review“This book is a bibliographical monument to the theory of both theoretical and applied PDEs that has not acquired any flaws due to its age. On the contrary, it remains a crucial and essential tool for the active research in the field.” (Francesco Petitta, SIAM Review, Vol. 61 (4), December, 2019)From the reviews:"The aim of the book is to present "the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process". The book is divided into two parts. The first (Chapters 2-8) is devoted to the linear theory, the second (Chapters 9-15) to the theory of quasilinear partial differential equations. These 14 chapters are preceded by an Introduction (Chapter 1) which expounds the main ideas and can serve as a guide to the book. ...The authors have succeeded admirably in their aims; the book is a real pleasure to read".Mathematical Reviews,1986 "Advanced students and professionals are snapping up this paperback text on linear and quasilinear partial differential equations. Whether you use their book as textbook or reference, the authors give you plenty to think about and work on, including an epilogue summarizing the latest research."Amazon.com delivers Mathematics and Statistics e-bulletin, July 2001Table of ContentsChapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem; the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Hölder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Hölder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Hölder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions; the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions; the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Hölder Continuity 8.10 Local Estimates at the Boundary 8.11 Hölder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Hölder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 hölder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Hölder Estimates for

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