Differential calculus and equations Books

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 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 SynopsisSuitable for students in the fields of mathematics, science, and engineering, this title provides a theoretical approach to dynamical systems and chaos. It helps them to analyze the types of differential equations that arise in their area of study.Table of Contents1. First-Order Equations 2. Planar Linear Systems 3. Phase Portraits 4. Classification of Planar Systems 5. Higher Dimension Linear Algebra 6. Higher Dimension Linear Systems 7. Nonlinear Systems 8. Equilibria in Nonlinear Systems 9. Global Nonlinear Techniques 10. Closed Orbits and Limit Sets 11. Applications in Biology 12. Applications in Circuit Theory 13. Applications in Mechanics 14. The Lorenz System 15. Discrete Dynamical Systems 16. Homoclinic Phenomena 17. Existence and Uniqueness Revisited

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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 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 SynopsisStraightforward and easy to read, A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATIONS, 11E, INTERNATIONAL METRIC EDITION, gives you a thorough overview of the topics typically taught in a first course in differential equations. Your study of differential equations and its applications will be supported by a bounty of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, definitions, and more.Table of Contents1. INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminology. Initial-Value Problems. Differential Equations as Mathematical Models. Chapter 1 in Review. 2. FIRST-ORDER DIFFERENTIAL EQUATIONS. Solution Curves Without a Solution. Separable Variables. Linear Equations. Exact Equations and Integrating Factors. Solutions by Substitutions. A Numerical Method. Chapter 2 in Review. 3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS. Linear Models. Nonlinear Models. Modeling with Systems of First-Order Differential Equations. Chapter 3 in Review. 4. HIGHER-ORDER DIFFERENTIAL EQUATIONS. Preliminary Theory-Linear Equations. Reduction of Order. Homogeneous Linear Equations with Constant Coefficients. Undetermined Coefficients-Superposition Approach. Undetermined Coefficients-Annihilator Approach. Variation of Parameters. Cauchy-Euler Equation. Solving Systems of Linear Differential Equations by Elimination. Nonlinear Differential Equations. Chapter 4 in Review. 5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS. Linear Models: Initial-Value Problems. Linear Models: Boundary-Value Problems. Nonlinear Models. Chapter 5 in Review. 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Review of Power Series Solutions About Ordinary Points. Solutions About Singular Points. Special Functions. Chapter 6 in Review. 7. LAPLACE TRANSFORM. Definition of the Laplace Transform. Inverse Transform and Transforms of Derivatives. Operational Properties I. Operational Properties II. Dirac Delta Function. Systems of Linear Differential Equations. Chapter 7 in Review. 8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. Preliminary Theory. Homogeneous Linear Systems. Nonhomogeneous Linear Systems. Matrix Exponential. Chapter 8 in Review. 9. NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS. Euler Methods. Runge-Kutta Methods. Multistep Methods. Higher-Order Equations and Systems. Second-Order Boundary-Value Problems. Chapter 9 in Review. Appendix I. Gamma Function. Appendix II. Matrices. Appendix III. Laplace Transforms. Answers for Selected Odd-Numbered Problems.

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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 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 textbook proposes an informal access to the most important issues of multidimensional differential and integral calculus. To situate the contents within a historical perspective, the book is accompanied by a number of links to the biographies of all scientists mentioned as leading actors in the development of the theory.

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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 SynopsisThe finite element, an approximation method for solving differential equations of mathematical physics, is a highly effective technique in the analysis and design, or synthesis, of structural dynamic systems. Starting from the system differential equations and its boundary conditions, what is referred to as a weak form of the problem (elaborated in the text) is developed in a variational sense. This variational statement is used to define elemental properties that may be written as matrices and vectors as well as to identify primary and secondary boundaries and all possible boundary conditions. Specific equilibrium problems are also solved. This book clearly reveals the effectiveness and great significance of the finite element method available and the essential role it will play in the future as further development occurs.Table of Contents1. Finite Element Techniques for Nonlinear Postbuckling and Collapse of Elastic Structures 2. Boundary Element Methods in Structural Dynamic System Problems hhhNewton-Raphson Techniques in Finite Ele ment Methods for Nonlinear Structural Problems 3. The Nonlinear Stress Analysis of Metallic and Reinforced Concrete Structures 4. Co-Rational Formulation for Nonlinear Dynamic Analysis of Beam Structures

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Book SynopsisThis book introduces the reader to the study of Hamiltonian systems, focusing on the stability of autonomous and periodic systems and expanding to topics that are usually not covered by the canonical literature in the field. It emerged from lectures and seminars given at the Federal University of Pernambuco, Brazil, known as one of the leading research centers in the theory of Hamiltonian dynamics. This book starts with a brief review of some results of linear algebra and advanced calculus, followed by the basic theory of Hamiltonian systems. The study of normal forms of Hamiltonian systems is covered by Ch.3, while Chapters 4 and 5 treat the normalization of Hamiltonian matrices. Stability in non-linear and linear systems are topics in Chapters 6 and 7. This work finishes with a study of parametric resonance in Ch. 8. All the background needed is presented, from the Hamiltonian formulation of the laws of motion to the application of the Krein-Gelfand-Lidskii theory of strongly stable systems. With a clear, self-contained exposition, this work is a valuable help to advanced undergraduate and graduate students, and to mathematicians and physicists doing research on this topic.Table of ContentsForeword.- Preliminaries on Advanced Calculus.- Hamiltonian Systems Theory.- Normal Forms of Hamiltonian Systems.- Spectral Decomposition of Hamiltonian Matrices.- The General Linear Normalization.- Stability of Equilibria.- Stability of Linear Hamiltonian Systems.- Parametric Resonance.- References.- 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 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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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 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 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 N-body problem has been investigated since Isaac Newton, however vast tracts of the problem remain open. Showcasing the vibrancy of the problem, this book describes four open questions and explores progress made over the last 20 years. After a comprehensive introduction, each chapter focuses on a different open question, highlighting how the stance taken and tools used vary greatly depending on the question. Progress on question one, ''Are the central configurations finite?'', uses tools from algebraic geometry. Two, ''Are there any stable periodic orbits?'', is dynamical and requires some understanding of the KAM theorem. The third, ''Is every braid realised?'', requires topology and variational methods. The final question, ''Does a scattered beam have a dense image?'', is quite new and formulating it precisely takes some effort. An excellent resource for students and researchers of mathematics, astronomy, and physics interested in exploring state-of-the-art techniques and perspectives on this classical problem.

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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 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 brief explores the Krasnosel'skiĭ-Man (KM) iterative method, which has been extensively employed to find fixed points of nonlinear methods. Table of Contents1. Introduction.- 2. Notation and Mathematical Foundations.-3. The Krasnoselskii-Mann Iteration.- 4. Relations of the Krasnosel'skii-Mann Iteration and the Operator Splitting Methods.- 5. The Inertial Krasnoselskii-Mann Iteration.- 6. The Multi-step Inertial Krasnoselskii-Mann Iteration.- 7. Relaxation Parameters of the Krasnoselskii-Mann Iteration.- 8. Two Applications.

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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 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 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 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 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 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 SynopsisThe Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.Table of ContentsPreface; 1. Basic notions; 2. Capacity; 3. Boundary behavior; 4. Zero sets; 5. Multipliers; 6. Conformal invariance; 7. Harmonically weighted Dirichlet spaces; 8. Invariant subspaces; 9. Cyclicity; Appendix A. Hardy spaces; Appendix B. The Hardy–Littlewood maximal function; Appendix C. Positive definite matrices; Appendix D. Regularization and the rising-sun lemma; References; Index of notation; Index.

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Book SynopsisThis monograph is the first to present the theory of global attractors of Hamiltonian partial differential equations. A particular focus is placed on the results obtained in the last three decades, with chapters on the global attraction to stationary states, to solitons, and to stationary orbits. The text includes many physically relevant examples and will be of interest to graduate students and researchers in both mathematics and physics. The proofs involve novel applications of methods of harmonic analysis, including Tauberian theorems, Titchmarsh''s convolution theorem, and the theory of quasimeasures. As well as the underlying theory, the authors discuss the results of numerical simulations and formulate open problems to prompt further research.Table of ContentsIntroduction; 1. Global attraction to stationary states; 2. Global attraction to solitons; 3. Global attraction to stationary orbits; 4. Asymptotic stability of stationary orbits and solitons; 5. Adiabatic effective dynamics of solitons; 6. Numerical simulation of solitons; 7. Dispersive decay; 8. Attractors and quantum mechanics; References; Index.

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Book SynopsisIn its Second Edition, this in-depth study of the vibrations of fractal strings interlinks number theory, spectral geometry and fractal geometry. Includes a geometric reformulation of the Riemann hypothesis and a new final chapter on recent topics and results.Trade Review“This interesting volume gives a thorough introduction to an active field of research and will be very valuable to graduate students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)“In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a self-contained manner the results … are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)“The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a self-contained manner, the results (including some fundamental ones) are completely proved. … the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory.” (Nicolae-Adrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH"This highly original self-contained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." -- Mathematical Reviews (Review of previous book by authors)"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is self-contained, intelligent and well paced." -- Bulletin of the London Mathematical Society (Review of previous book by authors)"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics." -- Simulation News Europe (Review of previous book by authors)Table of ContentsPreface.- Overview.- Introduction.- 1. Complex Dimensions of Ordinary Fractal Strings.- 2. Complex Dimensions of Self-Similar Fractal Strings.- 3. Complex Dimensions of Nonlattice Self-Similar Strings.- 4. Generalized Fractal Strings Viewed as Measures.- 5. Explicit Formulas for Generalized Fractal Strings.- 6. The Geometry and the Spectrum of Fractal Strings.- 7. Periodic Orbits of Self-Similar Flows.- 8. Fractal Tube Formulas.- 9. Riemann Hypothesis and Inverse Spectral Problems.- 10. Generalized Cantor Strings and their Oscillations.- 11. Critical Zero of Zeta Functions.- 12 Fractality and Complex Dimensions.- 13. Recent Results and Perspectives.- Appendix A. Zeta Functions in Number Theory.- Appendix B. Zeta Functions of Laplacians and Spectral Asymptotics.- Appendix C. An Application of Nevanlinna Theory.- Bibliography.- Author Index.- Subject Index.- Index of Symbols.- Conventions.- Acknowledgements.

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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 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 presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions.The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics.The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis. Trade Review“The book is carefully prepared and well presented, and I recommend the book … for students who have just mastered vector calculus and Maxwellian electromagnetism.” (Hirokazu Nishimura, zbMATH 1433.58001, 2020)Table of ContentsPrelude: Linear algebra.- Exterior algebra.- Clifford algebra.- Mappings of inner product spaces.- Spinors in inner product spaces.- Interlude: Analysis.- Exterior calculus.- Hodge decompositions.- Hypercomplex analysis.- Dirac equations.- Multivector calculus on manifolds.- Two index theorems.

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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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