Applied mathematics Books

Book SynopsisAiming to foster future collaborations between researchers in algorithms, bioinformatics, and molecular biology, this book serves as an up-to-date survey of the most important recent developments and computational challenges in various application areas of next-generation sequencing technologies.Table of ContentsCONTRIBUTORS xix PREFACE xxiii ABOUT THE COMPANION WEBSITE xxv PART I COMPUTING AND EXPERIMENTAL INFRASTRUCTURE FOR NGS 1 1 Cloud Computing for Next-Generation Sequencing Data Analysis 3Xuan Guo, Ning Yu, Bing Li, and Yi Pan 2 Introduction to the Analysis of Environmental Sequence Information Using Metapathways 25Niels W. Hanson, Kishori M. Konwar, Shang-Ju Wu, and Steven J. Hallam 3 Pooling Strategy for Massive Viral Sequencing 57Pavel Skums, Alexander Artyomenko, Olga Glebova, Sumathi Ramachandran, David S. Campo, Zoya Dimitrova, Ion I. Mândoiu, Alexander Zelikovsky, and Yury Khudyakov 4 Applications of High-Fidelity Sequencing Protocol to RNA Viruses 85Serghei Mangul, Nicholas C. Wu, Ekaterina Nenastyeva, Nicholas Mancuso, Alexander Zelikovsky, Ren Sun, and Eleazar Eskin PART II GENOMICS AND EPIGENOMICS 105 5 Scaffolding Algorithms 107Igor Mandric, James Lindsay, Ion I.Mândoiu, and Alexander Zelikovsky 6 Genomic Variants Detection and Genotyping 133Jorge Duitama 7 Discovering and Genotyping Twilight Zone Deletions 149Tobias Marschall and Alexander Schönhuth 8 Computational Approaches for Finding Long Insertions and Deletions with NGS Data 175Jin Zhang, Chong Chu, and Yufeng Wu 9 Computational Approaches in Next-Generation Sequencing Data Analysis for Genome-Wide DNA Methylation Studies 197Jeong-Hyeon Choi and Huidong Shi 10 Bisulfite-Conversion-Based Methods for DNA Methylation Sequencing Data Analysis 227Elena Harris and Stefano Lonardi PART III TRANSCRIPTOMICS 245 11 Computational Methods for Transcript Assembly from RNA-SEQ Reads 247Stefan Canzar and Liliana Florea 12 An Overview And Comparison of Tools for RNA-Seq Assembly 269Rasiah Loganantharaj and Thomas A. Randall 13 Computational Approaches for Studying Alternative Splicing in Nonmodel Organisms From RNA-SEQ Data 287Sing-Hoi Sze 14 Transcriptome Quantification and Differential Expression From NGS Data 301Olga Glebova, Yvette Temate-Tiagueu, Adrian Caciula, Sahar Al Seesi, Alexander Artyomenko, Serghei Mangul, James Lindsay, Ion I. M¢andoiu, and Alexander Zelikovsky PART IV MICROBIOMICS 329 15 Error Correction of NGS Reads from Viral Populations 331Pavel Skums, Alexander Artyomenko, Olga Glebova, David S. Campo, Zoya Dimitrova, Alexander Zelikovsky, and Yury Khudyakov 16 Probabilistic Viral Quasispecies Assembly 355Armin Töpfer and Niko Beerenwinkel 17 Reconstruction of Infectious Bronchitis Virus Quasispecies from NGS Data 383Bassam Tork, Ekaterina Nenastyeva, Alexander Artyomenko, Nicholas Mancuso, Mazhar I. Khan, Rachel O’Neill, Ion I. Mândoiu, and Alexander Zelikovsky 18 Microbiome Analysis: State of the Art and Future Trends 401Mitch Fernandez, Vanessa Aguiar-Pulido, Juan Riveros, Wenrui Huang, Jonathan Segal, Erliang Zeng, Michael Campos, Kalai Mathee, and Giri Narasimhan INDEX 425

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Book SynopsisFeatures new results and up-to-date advances in modeling and solving differential equations Introducing the various classes of functional differential equations, Functional Differential Equations: Advances and Applications presents the needed tools and topics to study the various classes of functional differential equations and is primarily concerned with the existence, uniqueness, and estimates of solutions to specific problems. The book focuses on the general theory of functional differential equations, provides the requisite mathematical background, and details the qualitative behavior of solutions to functional differential equations. The book addresses problems of stability, particularly for ordinary differential equations in which the theory can provide models for other classes of functional differential equations, and the stability of solutions is useful for the application of results within various fields of science, engineering, and economics. FunctionTrade Review"This monograph deals with several aspects of the functional differential equations theory, viz., the problem of existence (local and global) and uniqueness of solutions, stability, and oscillatory motions (periodic and almost periodic)...This book will be useful to people working on functional differential equations and their applications to science, engineering and economics." (Mathematical Reviews/MathSciNet June 2017)Table of ContentsPREFACE xi ACKNOWLEDGMENTS xv 1 Introduction, Classification, Short History, Auxiliary Results, and Methods 1 1.1 Classical and New Types of FEs 2 1.2 Main Directions in the Study of FDE 4 1.3 Metric Spaces and Related Concepts 11 1.4 Functions Spaces 15 1.5 Some Nonlinear Auxiliary Tools 21 1.6 Further Types of FEs 25 2 Existence Theory for Functional Equations 37 2.1 Local Existence for Continuous or Measurable Solutions 38 2.2 Global Existence for Some Classes of Functional Differential Equations 43 2.3 Existence for a Second-Order Functional Differential Equation 50 2.4 The Comparison Method in Obtaining Global Existence Results 55 2.5 A Functional Differential Equation with Bounded Solutions on the Positive Semiaxis 59 2.6 An Existence Result for Functional Differential Equations with Retarded Argument 64 2.7 A Second Order Functional Differential Equation with Bounded Solutions on the Positive Semiaxis 68 2.8 A Global Existence Result for a Class of First-Order Functional Differential Equations 72 2.9 A Global Existence Result in a Special Function Space and a Positivity Result 76 2.10 Solution Sets for Causal Functional Differential Equations 81 2.11 An Application to Optimal Control Theory 87 2.12 Flow Invariance 92 2.13 Further Examples/Applications/Comments 95 2.14 Bibliographical Notes 98 3 Stability Theory of Functional Differential Equations 105 3.1 Some Preliminary Considerations and Definitions 106 3.2 Comparison Method in Stability Theory of Ordinary Differential Equations 111 3.3 Stability under Permanent Perturbations 115 3.4 Stability for Some Functional Differential Equations 126 3.5 Partial Stability 133 3.6 Stability and Partial Stability of Finite Delay Systems 139 3.7 Stability of Invariant Sets 147 3.8 Another Type of Stability 155 3.9 Vector and Matrix Liapunov Functions 160 3.10 A Functional Differential Equation 163 3.11 Brief Comments on the Start and Evolution of the Comparison Method in Stability 168 3.12 Bibliographical Notes 169 4 Oscillatory Motion, with Special Regard to the Almost Periodic Case 175 4.1 Trigonometric Polynomials and APr-Spaces 176 4.2 Some Properties of the Spaces APr(R,C) 183 4.3 APr-Solutions to Ordinary Differential Equations 190 4.4 APr-Solutions to Convolution Equations 196 4.5 Oscillatory Solutions Involving the Space B 202 4.6 Oscillatory Motions Described by Classical Almost Periodic Functions 207 4.7 Dynamical Systems and Almost Periodicity 217 4.8 Brief Comments on the Definition of APr(R,C) Spaces and Related Topics 221 4.9 Bibliographical Notes 224 5 Neutral Functional Differential Equations 231 5.1 Some Generalities and Examples Related to Neutral Functional Equations 232 5.2 Further Existence Results Concerning Neutral First-Order Equations 240 5.3 Some Auxiliary Results 243 5.4 A Case Study, I 248 5.5 Another Case Study, II 256 5.6 Second-Order Causal Neutral Functional Differential Equations, I 261 5.7 Second-Order Causal Neutral Functional Differential Equations, II 268 5.8 A Neutral Functional Equation with Convolution 276 5.9 Bibliographical Notes 278 Appendix A On the Third Stage of Fourier Analysis 281 A.1 Introduction 281 A.2 Reconstruction of Some Classical Spaces 282 A.3 Construction of Another Classical Space 288 A.4 Constructing Spaces of Oscillatory Functions: Examples and Methods 290 A.5 Construction of Another Space of Oscillatory Functions 295 A.6 Searching Functional Exponents for Generalized Fourier Series 297 A.7 Some Compactness Problems 304 BIBLIOGRAPHY 307 INDEX 341

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Book SynopsisThis book illustrates how analytical sociology is progressively refining its theoretical framework and how powerful this framework is in explaining a large array of social phenomena.Table of ContentsPreface and Acknowledgments xiii About the Editor xv List of Contributors xvii Introduction 1 Editor's Introduction to Chapter 1 2 1 Data, Generative Models, and Mechanisms: More on the Principles of Analytical Sociology 4 Gianluca Manzo 1.1 Introduction 4 1.2 The Principles of Analytical Sociology 7 1.3 Clarity (P1) 10 1.4 Description (P2) 12 1.5 Generative Models (P3) 14 1.6 Structural Methodological Individualism (P4a) 17 1.7 Logics of Action (P4b) 21 1.8 Structural Interdependency (P4c) 27 1.9 Agent-Based Modeling (P5) 29 1.10 Back to Data (P6 and P7) 35 1.11 Concluding Remarks 37 1.12 How to Read this Book 40 Part I ACTIONS 53 Foundational Issues 54 Editor's Introduction to Chapter 2 55 2 Analytical Sociology and Rational-Choice Theory 57 Peter Hedström and Petri Ylikoski 2.1 Rational-Choice Theory 58 2.2 Sociological Rational-Choice Theory 59 2.3 Analytical Sociology as a Meta-Theory 60 2.4 The Key Ideas of Analytical Sociology 61 2.4.1 Mechanism-Based Explanation 61 2.4.2 Realism 62 2.4.3 Theories of Middle Range 63 2.4.4 Theory of Action 64 2.5 The Puzzle 64 2.6 The Assumed Special Role of RCT 65 2.7 Conclusion 67 3 Why Crime Happens: A Situational Action Theory 74 Per-Olof H. Wikström 3.1 Situational Action Theory 75 3.2 Explaining Crime 76 3.3 The Situational Model 77 3.4 The Situational Process 78 3.4.1 Motivation 79 3.4.2 Perception of Action Alternatives: The Moral Filter 80 3.4.3 The Process of Choice: Habits and Deliberation 80 3.4.4 Controls: Self-Control and Deterrence 82 3.5 The Social Model 82 3.6 Integrating the Social and Situational Models 84 3.7 Testing SAT 85 3.7.1 The Peterborough Adolescent and Young Adult Development Study 85 3.7.2 Measuring Crime, Crime Propensity and Criminogenic Exposure 86 3.7.3 Crime Involvement by Crime Propensity and Criminogenic Exposure 87 3.7.4 The Impact of Criminogenic Exposure on Crime for Groups with Different Levels of Crime Propensity 88 3.8 Explaining Crime Concentrations (Hot Spots) 90 3.9 Coda 92 4 Frames, Scripts, and Variable Rationality: An Integrative Theory of Action 97 Clemens Kroneberg 4.1 Introduction 97 4.2 The Model of Frame Selection (MFS) 99 4.2.1 Frames, Scripts, and Actions 99 4.2.2 Dual-processes: Spontaneous vs. Reflected Modes of Selection 100 4.2.3 The Determinants of Variable Rationality 104 4.3 Hypotheses and Previous Applications 106 4.4 An Exemplary Application Using Survey Data: Explaining Voter Participation 108 4.4.1 Theory 108 4.4.2 Data and Measures 112 4.4.3 Results 113 4.5 Applying the MFS to Study Social Dynamics 115 4.5.1 The MFS and the Study of Social Movements and Collective Action 116 4.5.2 Strategic Interaction with Variable Rationality and Framing 117 4.6 Conclusion 118 5 Analytical Sociology and Quantitative Narrative Analysis: Explaining Lynchings in Georgia (1875–1930) 127 Roberto Franzosi 5.1 Strange Fruits on Southern Trees 127 5.2 Analytical Sociology 128 5.3 Quantitative Narrative Analysis (QNA) 129 5.3.1 Step 1: Story Grammars 130 5.3.2 Step 2: PC-ACE (Program for Computer-Assisted Coding of Events) 132 5.3.3 Step 3: Data Analysis: Actor-Centered vs. Variable-Centered Tools of Analysis 134 5.4 Of Sequences 139 5.5 Of Time and Space 142 5.6 Conclusions 144 6 Identity and Opportunity in Early Modern Politics: How Job Vacancies Induced Witch Persecutions in Scotland, 1563–1736 151 Anna Mitschele 6.1 Introduction 151 6.2 Theories about Witches and Research on State Making 153 6.3 Towards a Theory of Persecution 155 6.3.1 Communities 156 6.3.2 Elite Social Structure and Government 157 6.4 Witch-Hunting in Scotland 157 6.5 Findings 159 6.5.1 Prosecution as Career Device I: Waves of Witch-Hunting and their Historical Correlates 159 6.5.2 Prosecution as Career Device II: Witch-Hunters Become Justices of the Peace 161 6.5.3 Competing Explanations I: The Godly State Ideology 162 6.5.4 Competing Explanations II: Witches as Scapegoats for Disaster 163 6.6 Discussion 164 7 Mechanisms of Cooperation 172 Davide Barrera 7.1 Introduction 172 7.2 Cooperation Problems in Dyadic Settings 174 7.2.1 Models of Trust Problem 175 7.2.2 Cooperation Mechanisms in Embedded Settings 178 7.2.3 Empirical Research on Trust in Embedded Settings 179 7.2.4 Dyadic Embeddedness 180 7.2.5 Network Embeddedness 180 7.3 Cooperation Problems Involving More than Two Actors 181 7.3.1 Reciprocity and Non-Standard Utility Models 183 7.3.2 Empirical Evidence on Heterogeneous Preferences 184 7.4 Discussion and Concluding Remarks 187 8 The Impact of Elections on Cooperation: Evidence from a Lab-in-the-Field Experiment in Uganda 201 Guy Grossman and Delia Baldassarri 8.1 Theoretical Framework and Hypotheses 203 8.2 Research Site, Sampling, and Experimental Design 206 8.3 Research Site 207 8.4 Sampling and Data Collection 208 8.5 Experimental Design 208 8.6 Experimental Findings 210 8.7 Monitors’ Sanctioning Behavior 214 8.8 Discussion of the Experimental Part 216 8.9 Observational Data 217 8.10 Comparing Behavior in the Experiment and Real Life 219 8.11 Conclusion 221 Part II NETWORKS 233 Collective Action 234 Editor's Introduction to Chapter 9 235 9 Social Networks and Agent-Based Modelling 237 Meredith Rolfe 9.1 Social Network Properties 238 9.1.1 Surveys of Personal Networks 239 9.2 Network Construction Techniques 243 9.2.1 Global Reference or Full Information 243 9.2.2 Random Graph Local Networks 243 9.2.3 Two-Dimensional Lattices or Grid-Based Networks 244 9.2.4 One-Dimensional Lattice or Small-World Method 245 9.2.5 Biased or Structured Random Networks 245 9.3 Networks as Pipes: A Basic Demonstration 246 9.3.1 Global Networks and Group Size 248 9.3.2 Results with Network Construction Methods 251 9.4 Discussion 256 10 Online Networks and the Diffusion of Protest 263 Sandra Gonzalez-Bailón, Javier Borge-Holthoefer, and Yamir Moreno 10.1 Diffusion Dynamics 264 10.1.1 Models of Diffusion 264 10.1.2 Case Study 266 10.2 Thresholds and Critical Mass 268 10.3 Networks and Social Influence 271 10.4 Conclusion: Digital Data and Analytical Sociology 275 11 Liability to Rupture: Multiple Mechanisms and Subgroup Formation. An Exploratory Theoretical Study 282 Peter Abell 11.1 Introduction 282 11.2 A Formal Framework 283 11.3 Balance Theory 284 11.4 Homophily (H-theory) 287 11.5 Baseline Structures 288 11.6 Developing a Dynamic Mechanism for Balance Theory 289 11.7 Developing a Dynamic Mechanism for H-theory 291 11.8 The Dynamic Interaction of Balance and H-theories 293 11.9 Conclusions 294 12 Network Size and Network Homophily: Same-Sex Friendships in 595 Scandinavian Schools 299 Thomas Grund 12.1 Introduction 299 12.2 Theoretical Considerations 301 12.2.1 Biased Urn Model Without Replacement for Network Formation 301 12.2.2 Role of Group Size for Homophily 305 12.3 Empirical Application: Same-Sex Ties in School Classes 308 12.3.1 Hypotheses 308 12.3.2 Data and Method 309 12.4 Results 310 12.5 Conclusion 312 13 Status and Participation in Online Task Groups: An Agent-Based Model 317 Simone Gabbriellini 13.1 Introduction 317 13.2 Previous Models 319 13.3 E-state Structuralism: A Very Brief Review with an Add-On 321 13.4 Case Study: Strategies and Discussions in Massively Multi-Player Online Games 324 13.5 Analysis of the Model 326 13.6 Empirical Test/Validation of the Model 331 13.7 Conclusions 336 14 Turbulent Careers: Social Networks, Employer Hiring Preferences, and Job Instability 342 Christine Fountain and Katherine Stovel 14.1 Introduction 342 14.2 Background 343 14.2.1 The Rise of Turbulence in Individual Employment Trajectories 343 14.2.2 Inequality in Insecurity 344 14.3 Networks 346 14.3.1 Network Structure and Inequality in Information 346 14.3.2 Our Approach 348 14.4 Methods 349 14.4.1 The Simulation Environment 349 14.4.2 Implementation 350 14.4.3 Experimental Structure 353 14.5 Results 355 14.6 Summary and Conclusions 362 15 Employer Networks, Priming, and Discrimination in Hiring: An Experiment 373 Karoly Takacs, Flaminio Squazzoni, Giangiacomo Bravo, and Marco Castellani 15.1 Introduction 373 15.2 Method 376 15.2.1 Experimental Design 376 15.2.2 Manipulations 378 15.2.3 Subjects 378 15.3 Results 379 15.3.1 Index Values 379 15.3.2 Hierarchical Models 382 15.3.3 From Traditional Testing Toward Finding Indicators for Mechanisms 385 15.4 Discussion 391 16 The Duality of Organizations and Audiences 400 Balazs Kovacs 16.1 Introduction 400 16.2 Similarity and the Duality of Organizations and their Audiences 401 16.3 Organizational Similarity, Audiences, and Arguments for Extending Structural Equivalence 403 16.4 A Representation for Dual Similarity of Organizations and their Audiences 406 16.5 Empirical Illustration: The Duality of Restaurants and their Reviewers 407 16.6 Similarity as a Basis for Prediction: Validating the Model 408 16.7 Discussion, Implications, and Limitations 412 16.8 Connections to Analytical Sociology 415 References 415 Further Reading 418 Coda 419 Problem Shift in Sociology: Mechanisms, Generic Instruments, and Fractals 420 Gianluca Manzo Index 427

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Book SynopsisTable of Contents1. Functions, Graphs, and Limits. 2. Differentiation. 3. Applications of the Derivative. 4. Exponential and Logarithmic Functions. 5. Integration and Its Applications. 6. Techniques of Integration. 7. Functions of Several Variables. 8. Trigonometric Functions. 9. Probability and Calculus. 11. Differential Equations. Appendix A. Precalculus Review. The Real Number Line and Order. Absolute Value of a Real Number. Exponents and Radicals. Factoring Polynomials. Fractions and Rationalization. Appendix B. Alternative Introduction to the Fundamental Theorem of Calculus. Appendix C. Formulas.

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Book SynopsisA physics-driven exploration of sports played on ice and snow that is truly fun and informative, Gliding for Gold is the perfect primer for understanding the science behind cold weather athletics.Trade ReviewOne of the best [books on snow sports] I've ever read. -- Eric H. Book Bargains and Previews From snowboarding to curling and skating, this places physics explanations in the world of sports and offers fans and competitors alike a solid understanding of the physics of sports. Midwest Book Review Denny, a theoretical physicist and popular science writer provides an enthusiastic, almost breezy tour of the rules, art, and science of skating, hockey, curling, skiing, and snowboarding... For the scientifically inclined reader it provides an interesting window on the science of winter sports. Choice Both travel and science collections will find it packed with lively history, cultural observations, and fun insights. Midwest Book ReviewTable of ContentsAcknowledgmentsThe Start Lines1. Solid Water—Sports and SciencePart I: Ice Sports2. Skating on Thin Ice3. Down the Slippery Slope4. Pucks and RocksPart II: Snow Sports5. Skiing—On the Slopes and on the Level6. Ski Jumping and Snowboarding—On Snow and AirThe Finish LinesPonderablesTechnical NotesBibliographyIndex

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Book SynopsisA physics-driven exploration of sports played on ice and snow that is truly fun and informative, Gliding for Gold is the perfect primer for understanding the science behind cold weather athletics.Trade ReviewOne of the best [books on snow sports] I've ever read. -- Eric H. Book Bargains and Previews From snowboarding to curling and skating, this places physics explanations in the world of sports and offers fans and competitors alike a solid understanding of the physics of sports. Midwest Book Review Denny, a theoretical physicist and popular science writer provides an enthusiastic, almost breezy tour of the rules, art, and science of skating, hockey, curling, skiing, and snowboarding... For the scientifically inclined reader it provides an interesting window on the science of winter sports. Choice Both travel and science collections will find it packed with lively history, cultural observations, and fun insights. Midwest Book ReviewTable of ContentsAcknowledgmentsThe Start Lines1. Solid Water—Sports and SciencePart I: Ice Sports2. Skating on Thin Ice3. Down the Slippery Slope4. Pucks and RocksPart II: Snow Sports5. Skiing—On the Slopes and on the Level6. Ski Jumping and Snowboarding—On Snow and AirThe Finish LinesPonderablesTechnical NotesBibliographyIndex

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Book SynopsisAlong the way, he tells us what various cultures knew about math and how they came to learn it, providing instructors with a wonderful way to incorporate multicultural mathematics into the middle school, high school, and college classroom.Trade ReviewSwetz has collected word problems, or story problems, used to teach mathematics around the world and throughout history, so mathematics teachers in middle and secondary schools can use them today. University students of mathematics and its history might also find them useful as well as entertaining. Reference and Research Book News Mathematical Expeditions is a wonderful resource for any teacher who would like to use old problems in a course to help students understand the context of mathematical ideas. -- Victor J. Katz Mathematical Reviews The book is well thought-out and is recommended to readers interested in the history of mathematics. -- E. Keith Lloyd London Mathematical Society Newsletter One of my graduate students, who is majoring in mathematics, was excited when I showed her a sample of problems in the book. A month later, she asked whether I had finished my review-she wanted to borrow the book! -- Winifred A. Mallam Mathematics TeacherTable of ContentsPreface1. Word Problems: Footprints from the History of Mathematics2. Problems, Problems: A Resource for Teaching3. Ancient Babylonia (2002–1000 BCE)4. Ancient Egypt5. Ancient Greece6. Ancient China7. India8. Islam9. Medieval Europe10. Renaissance Europe11. Japanese Temple Problems12. The Ladies Diary (1704–1841)13. Nineteenth-Century Victorian Problems14. Eighteenth- and Nineteenth-Century American Problems15. Problems from the Farmer's Almanac16. Nineteenth-Century Calculus Problems17. Some Sample Problem Solution Methods18. Where to from Here? Where Do You Want to Go?AcknowledgmentsAnswers to Numbered ProblemsGlossary of Strange and Exotic Terms: Measurements, Monetary Units, and Culturally Relevant WordsBibliographyIndex

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Book SynopsisWhile the Jesuits claimed Xu as a convert, he presented the Jesuits as men from afar who had traveled from the West to China to serve the emperor.Trade ReviewOverall, this book is interesting for the analytical framework it suggests for approaching area-based global historical questions and it is very original in some of its historiographic claims... The Math IntelligencerTable of Contents1. Introduction2. Science as the Measure of Civilizations3. From Copula to Incommensurable Worlds4. Mathematical Texts in Historical Context5. Tracing Practices Purloined by the Three Pillars6. Xu Guangqi, Grand Guardian7. ConclusionsAcknowledgmentsAppendix A: Zhu Zaiyu's New Theory of CalculationAppendix B: Xu Guangqi's Right Triangles, MeaningsAppendix C: Xu Guangqi's WritingsBibliographyIndex

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Book SynopsisIt's a creative and forward-thinking approach to math instruction.Topics include:; First-Order Differential Equations; Incorporation of Newtonian Mechanics; Second-Order Differential Equations; The Annihilator Method; Using Linear Algebra with Differential Equations; Nonlinear Systems; Partial Differential Equations; Romeo and JulietTrade ReviewThis book, with its many practice examples, would be ideal for those in the first year of a mathematics degree course and for those studying for working in physics or any area requiring a good knowledge of differential equations. -- Adrian Hamilton Institute of Mathematics and its Applications ... [ Introduction to Differential Equations Using Sage] provides a nice mix of theory and symbolic computation. The pedagogy is excellent and the exercises are models of their kind. Gazette of the Australian Mathematical SocietyTable of ContentsPrefaceAcknowledgments1. First-order differential equations1.1. Introduction to DEs1.2. Initial value problems1.3. Existence of solutions to ODEs1.3.1. First-order ODEs1.3.2. Second-order homogeneous ODEs1.4. First-order ODEs: Separable and linear cases1.4.1. Separable DEs1.4.2. Autonomous ODEs1.4.3. Substitution methods1.4.4. Linear first-order ODEs1.5. Isoclines and direction fields1.6. Numerical solutions: Euler's and improved Euler's method1.6.1. Euler's method1.6.2. Improved Euler's method1.6.3. Euler's method for systems and higher-order DEs1.7. Numerical solutions II: Runge-Kutta and other methods1.7.1. Fourth-order Runge-Kutta method1.7.2. Multistep methods: Adams-Bashforth1.7.3. Adaptive step size1.8. Newtonian mechanics1.9. Application to mixing problems1.10. Application to cooling problems2. Second-order differential equations2.1. Linear differential equations2.1.1. Solving homogeneous constant-coefficient ODEs2.2. Linear differential equations, revisited2.3. Linear differential equations, continued2.4. Undetermined coefficients method2.4.1. Simple case2.4.2. Nonsimple case2.5. Annihilator method2.6. Variation of parameters2.6.1. The Leibniz rule2.6.2. The method2.7. Applications of DEs: Spring problems2.7.1. Introduction: Simple harmonic case2.7.2. Simple harmonic case2.7.3. Free damped motion2.7.4. Spring-mass systems with an external force2.8. Applications to simple LRC circuits2.9. The power of series method2.9.1. Part 12.9.2. Part 22.10. The Laplace transform method2.10.1. Part 12.10.2. Part 22.10.3. Part 33. Matrix theory and systems of DEs3.1. Quick survey of linear algebra3.1.1. Matrix arithmetic3.2. Row reduction and solving systems of equations3.2.1. The Gauss elimination game3.2.2. Solving systems using inverses3.2.3. Computing inverses using row reduction3.2.4. Solving higher-dimensional linear systems3.2.5. Determinants3.2.6. Elementary matrices and computation of determinants3.2.7. Vector spaces3.2.8. Bases, dimension, linear independence, and span3.3. Application: Solving systems of DEs3.3.1. Modeling battles using Lanchester's equations3.3.2. Romeo and Juliet3.3.3. Electrical networks using Laplace transforms3.4. Eigenvalue method for systems of DEs3.4.1. Motivation3.4.2. Computing eigenvalues3.4.3. The eigenvalue method3.4.4. Examples of the eigenvalue method3.5. Introduction to variation of parameters for systems3.5.1. Motivation3.5.2. The method3.6. Nonlinear systems3.6.1. Linearizing near equilibria3.6.2. The nonlinear pendulum3.6.3. The Lorenz equations3.6.4. Zombies attack4. Introduction to partial differential equations4.1. Introduction to separation of variables4.1.1. The transport or advection equation4.1.2. The heat equation4.2. The method of superposition4.3. Fourier, sine, and cosine series4.3.1. Brief history4.3.2. Motivation4.3.3. Definitions4.4. The heat equation4.4.1. Method for zero ends4.4.2. Method for insulated ends4.4.3. Explanation via separation of variables4.5. The wave equation in one dimension4.5.1. Methods4.6. The Schrodinger equation4.6.1. MethodBibliographyIndex

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Book SynopsisNonlinearity, Complexity and Randomness in Economics presents a variety of papers by leading economists, scientists, and philosophers who focus on different aspects of nonlinearity, complexity and randomness, and their implications for economics. A theme of the book is that economics should be based on algorithmic, computable mathematical foundations. Features an interdisciplinary collection of papers by economists, scientists, and philosophers Presents new approaches to macroeconomic modelling, agent-based modelling, financial markets, and emergent complexity Reveals how economics today must be based on algorithmic, computable mathematical foundations Table of ContentsNotes on Contributors vii 1. Introduction 1 Stefano Zambelli 2. Towards an Algorithmic Revolution in Economic Theory 7 K. Vela Velupillai 3. An Algorithmic Information-Theoretic Approach to the Behaviour of Financial Markets 37 Hector Zenil and Jean-Paul Delahaye 4. Complexity and Randomness in Mathematics: Philosophical Reflections on the Relevance for Economic Modelling 69 Sundar Sarukkai 5. Behavioural Complexity 85 Sami Al-Suwailem 6. Bounded Rationality and the Emergence of Simplicity Amidst Complexity 111 Cassey Lee 7. Emergent Complexity in Agent-Based Computational Economics 131 Shu-Heng Chen and Shu G.Wang 8. Non-Linear Dynamics, Complexity and Randomness: Algorithmic Foundations 151 K. Vela Velupillai 9. Stock-Flow Interactions, Disequilibrium Macroeconomics and the Role of Economic Policy 173 Toichiro Asada, Carl Chiarella, Peter Flaschel, Tarik Mouakil, Christian Proa˜no andWilli Semmler 10. Equilibrium Versus Market Efficiency: Randomness versus Complexity in Finance Markets 203 Joseph L. McCauley 11. Flexible Accelerator Economic Systems as Coupled Oscillators 211 Stefano Zambelli 12. Shifting Sands: Non-Linearity, Complexity and Randomness in Economics 237 Donald A.R. George Index 241

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Book Synopsis See the world in a completely new way as an esteemed mathematician shows how math powers the world—from technology to health care and beyond. Almost all of us have sat in a math class, wondering when we'd ever need to know how to find the roots of a polynomial or graph imaginary numbers. And in one sense, we were right: if we needed to, we'd use a computer. But as Ian Stewart argues in What's the Use?, math isn't just about boring computations. Rather, it offers us new and profound insights into our world, allowing us to accomplish feats as significant as space exploration and organ donation. From the trigonometry that keeps a satellite in orbit to the prime numbers used by the world's most advanced security systems to the imaginary numbers that enable augmented reality, math isn't just relevant to our lives. It is the very fabric of our existence.

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Book SynopsisThis textbook develops the fundamental skills of numerical analysis: designing numerical methods, implementing them in computer code, and analyzing their accuracy and efficiency. A number of mathematical problems—interpolation, integration, linear systems, zero finding, and differential equations—are considered, and some of the most important methods for their solution are demonstrated and analyzed. Notable features of this book include the development of Chebyshev methods alongside more classical ones; a dual emphasis on theory and experimentation; the use of linear algebra to solve problems from analysis, which enables students to gain a greater appreciation for both subjects; and many examples and exercises.Numerical Analysis: Theory and Experiments is designed to be the primary text for a junior- or senior-level undergraduate course in numerical analysis for mathematics majors. Scientists and engineers interested in numerical methods, particularly those seeking an accessible introduction to Chebyshev methods, will also be interested in this book.

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Book SynopsisThe field of computer sciences is broad, deep, and continually growing, and impacts nearly every quarter of our modern lives. Computer scientists, therefore, need to be lifelong learners.Problem Solving with Python takes an explicit problem-solving approach to the introduction of computer programming and computational thinking, emphasizing the programmer's role in problem definition, expressing, evaluating, and implementing solutions, and evaluating results.Designed to lay the groundwork for further study in computer science, Problem Solving with Python: Encourages curiosity and guides exploration, experimentation, and the meaningful evaluation of results. Uses turtle graphics and interesting computation problems in the Python 3 programming language to introduce fundamental imperative programming constructs and patterns. Includes chapters on input-processing-output programs, conditional and iterative structures, functions, recursion, file processing, lists and strings, and contains an introduction to secure programming. Presents programming constructs as tools for the solution of classes of problems, using the problems as motivation for mastering the tools. Try This and Challenge Accepted! problem sections are suitable for laboratory exercises, starting points for classroom discussion, or to help the reader develop the habits of mind they will need for success as a computer programmer. Table of Contents 1. Getting started with Python 1.1 Problem solving 1.2 Let's get started! 1.3 Meet the turtle 1.4 Designing solutions 1.5 When things go wrong 1.6 Challenge accepted! 2. Input, processing, output programs 2.1 Your programming tool kit 2.2 Literals and data types 2.3 Variables 2.4 Variable names and keywords 2.5 Constants 2.6 Evaluating expressions 2.7 Operators and operands 2.8 Order of operations 2.9 Math functions and mathematical expressions in Python 2.10 Input and type conversion functions 2.11 Formatting output 2.12 Comments 2.13 Using your tool kit 2.14 Challenge accepted! 3. Logic and conditional statements 3.1 Changing the flow 3.2 Relational operators and Boolean expressions 3.3 Boolean operators and short-circuiting 3.4 Checking ranges 3.5 Boolean variables 3.6 What else? 3.7 Complementing logical expressions 3.8 Testing assumptions with assert statements 3.9 Series if statements 3.10 Nested if statements 3.11 Desk checking and debugging conditional statements 3.12 Data representation 3.13 The random library 3.14 Challenge accepted! 4. Loops and files 4.1 Finding repetition in a problem 4.2 Desk-checking loops 4.3 The loop control variable 4.4 Loop design errors 4.5 Augmented assignment operators 4.6 Loop patterns 4.7 Input validation loops 4.7 Sentinel-controlled loops 4.8 File processing 4.9 Counting loops and ranges 4.10 Nested loops 4.11 Finding repetition in a problem (again) 4.12 Challenge accepted! 5. Functions 5.1 Defining functions 5.2 Functional decomposition 5.3 Flow of control 5.4 Parameters 5.5 Returning values 5.6 Input-processing-output (IPO) charts and structure charts 5.7 Code blocks, namespaces, and scope 5.8 The math module, revisited 5.9 Input validation functions 5.10 Problem-solving with functions 5.11 Challenge accepted! 6. Lists 6.1 Collections and indexing 6.2 Loops and lists 6.3 Slicing, concatenating, and copying 6.4 Operations on lists 6.5 Lists and functions 6.6 Lists and files 6.7 Two-dimensional lists 6.8 Parallel lists and records 6.9 Strings, revisited 6.10 Problem-solving with lists 6.11 Challenge accepted! 7. Supplementary chapter: Number Systems 7.1 What is a number base? 7.2 Binary numbers 7.3 Hexadecimal numbers 7.4 Conversions 7.5 Seven-bit ASCII, 8-bit ASCII, Unicode: How many digits do we need? 8. Supplementary chapter: Introduction to security-conscious coding 8.1 What you already know 8.2 What do malicious actors do? 8.3 Know your data 8.4 Validating type with exception handling 8.5 Restricting input size with files 8.6 Validating input with string functions and regular expressions 9. Supplementary chapter: Recursion 9.1 Recursive problems 9.2 The call stack 9.3 Designing recursive solutions References Index

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Book Synopsis'A wise, witty and insightful guide to clear thinking amid a deluge of percentages and probabilities.' Ian StewartLike it or not, our lives are dominated by mathematics. Our daily diet of news regales us with statistical forecasts, opinion polls, risk assessments, inflation figures, weather and climate predictions and all sorts of political decisions and advice backed up by supposedly accurate numbers. Most of us do not even pause and question such figures even to ask what they really mean and whether they raise more questions than they answer. In this simple guide for anyone numbed by numbers, William Hartston reveals with clarity and humour why the figures being flung at us may not tell the whole story. Along the way he explains commonly misused mathematical terms, solves everyday mathematical problems and shows how to steer a safe path through the minefield of mathematics that surrounds us.Trade ReviewIf you're not one of the lucky few who can say, "Oh, mathematics was my best subject at school," then you might find yourself stumped by the news, or personal finance, or chaos and catastrophe (yes, William Hartston shows us there's maths involved there, too). Luckily, Numb and Number is able to explain these things and more, in a way that's easy to understand and even enjoyable to read. * BBC Science Focus *A wise, witty, and insightful guide to clear thinking amid a deluge of percentages and probabilities. Learn to spot the fake formulas and the spurious statistics. Up to 100% of readers will find this book utterly fascinating. Recommended by 92.53% of mathematicians. -- Ian Stewart, author of Do Dice Play God?Table of Contents1: The Number of Our Days 2: Surveying the Scene 3: Risk and Behaviour 4: The Mathematics of Sport 5: Saved You! 6: Numbers Large and Small 7: The Insignificance of Significance 8: Cause and Effect 9: Percentages and More Misleading Mathematics 10: Chaotic Butterflies 11: Torpedoes, Toilets and True Love 12: Formula Milking 13: Monkey Maths 14: Pandemic Pandemonium

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Book SynopsisThis book provides a detailed study of Financial Mathematics. In addition to the extraordinary depth the book provides, it offers a study of the axiomatic approach that is ideally suited for analyzing financial problems. This book is addressed to MBA's, Financial Engineers, Applied Mathematicians, Banks, Insurance Companies, and Students of Business School, of Economics, of Applied Mathematics, of Financial Engineering, Banks, and more.Table of ContentsPreface xvii Part I. Deterministic Models 1 Chapter 1. Introductory Elements to Financial Mathematics 3 Chapter 2. Theory of Financial Laws 13 Chapter 3. Uniform Regimes in Financial Practice 41 Chapter 4. Financial Operations and their Evaluation: Decisional Criteria 91 Chapter 5. Annuities-Certain and their Value at Fixed Rate 147 Chapter 6. Loan Amortization and Funding Methods 211 Chapter 7. Exchanges and Prices on the Financial Market 289 Chapter 8. Annuities, Amortizations and Funding in the Case of Term Structures 331 Chapter 9. Time and Variability Indicators, Classical Immunization 363 Part II. Stochastic Models 409 Chapter 10. Basic Probabilistic Tools for Finance 411 Chapter 11. Markov Chains 457 Chapter 12. Semi-Markov Processes 481 Chapter 13. Stochastic or Itô Calculus 517 Chapter 14. Option Theory 553 Chapter 15. Markov and Semi-Markov Option Models 607 Chapter 16. Interest Rate Stochastic Models – Application to the Bond Pricing Problem 641 Chapter 17. Portfolio Theory 687 Chapter 18. Value at Risk (VaR) Methods and Simulation 703 Chapter 19. Credit Risk or Default Risk 743 Chapter 20. Markov and Semi-Markov Reward Processes and Stochastic Annuities 791 References 831 Index 839

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Book SynopsisThe concept of an inverse problem is a familiar one to most scientists and engineers, particularly in the field of signal and image processing, imaging systems (medical, geophysical, industrial non-destructive testing, etc.), and computer vision. In imaging systems, the aim is not just to estimate unobserved images but also their geometric characteristics from observed quantities that are linked to these unobserved quantities by a known physical or mathematical relationship. In this manner techniques such as image enhancement or addition of hidden detail can be delivered. This book focuses on imaging and vision problems that can be clearly described in terms of an inverse problem where an estimate for the image and its geometrical attributes (contours and regions) is sought. The book uses a consistent methodology to examine inverse problems such as: noise removal; restoration by deconvolution; 2D or 3D reconstruction in X-ray, tomography or microwave imaging; reconstruction of the surface of a 3D object using X-ray tomography or making use of its shading; reconstruction of the surface of a 3D landscape based on several satellite photos; super-resolution; motion estimation in a sequence of images; separation of several images mixed using instruments with different sensitivities or transfer functions; and much more.Trade Review"Apart from the high price I can recommend this book if you are interested in imaging or artificial vision." (I Programmer, 3 February 2011)Table of ContentsPreface 13 Chapter 1. Introduction to Inverse Problems in Imaging and Vision 15 Ali MOHAMMAD-DJAFARI 1.1. Inverse problems 16 1.2. Specific vision problems 21 1.3. Models for time-dependent quantities 26 1.4. Inverse problems with multiple inputs and multiple outputs (MIMO) 27 1.5. Non-linear inverse problems 30 1.6. 3D reconstructions 33 1.7. Inverse problems with multimodal observations 33 1.8. Classification of inversion methods: analytical or algebraic 34 1.9. Standard deterministic methods 40 1.10. Probabilistic methods 44 1.11. Problems specific to vision 50 1.12. Introduction to the various chapters of the book 52 1.13. Bibliography 55 Chapter 2. Noise Removal and Contour Detection 59 Pierre CHARBONNIER and Christophe COLLET 2.1. Introduction 61 2.2. Statistical segmentation of noisy images 72 2.3. Multi-band multi-scale Markovian regularization 79 2.4. Bibliography 88 Chapter 3. Blind Image Deconvolution 97 Laure BLANC-FÉRAUD, Laurent MUGNIER and André JALOBEANU 3.1. Introduction 97 3.2. The blind deconvolution problem 98 3.3. Joint estimation of the PSF and the object 103 3.4. Marginalized estimation of the impulse response 107 3.5. Various other approaches 112 3.6. Multi-image methods and phase diversity 114 3.7. Conclusion 115 3.8. Bibliography 116 Chapter 4. Triplet Markov Chains and Image Segmentation 123 Wojciech PIECZYNSKI 4.1. Introduction 124 4.2. Pairwise Markov chains (PMCs) 127 4.3. Copulas in PMCs 130 4.4. Parameter estimation 132 4.5. Triplet Markov chains (TMCs) 136 4.6. TMCs and non-stationarity 139 4.7. Hidden Semi-Markov chains (HSMCs) and TMCs 140 4.8. Auxiliary multivariate chains 144 4.9. Conclusions and outlook 148 4.10. Bibliography 149 Chapter 5. Detection and Recognition of a Collection of Objects in a Scene 155 Xavier DESCOMBES, Ian JERMYN and Josiane ZERUBIA 5.1. Introduction 155 5.2. Stochastic approaches 156 5.3. Variational approaches 167 5.4. Bibliography 184 Chapter 6. Apparent Motion Estimation and Visual Tracking 191 Etienne MÉMIN and Patrick PÉREZ 6.1. Introduction: from motion estimation to visual tracking 191 6.2. Instantaneous estimation of apparent motion 193 6.3. Visual tracking 219 6.4. Conclusions 240 6.5. Bibliography 241 Chapter 7. Super-resolution 251 Ali MOHAMMAD-DJAFARI and Fabrice HUMBLOT 7.1. Introduction 251 7.2. Modeling the direct problem 252 7.3. Classical SR methods 257 7.4. SR inversion methods 261 7.5. Methods based on a Bayesian approach 265 7.6. Simulation results 271 7.7. Conclusion 272 7.8. Bibliography 274 Chapter 8. Surface Reconstruction from Tomography Data 277 Charles SOUSSEN and Ali MOHAMMAD-DJAFARI 8.1. Introduction 277 8.2. Reconstruction of localized objects 280 8.3. Use of deformable contours for 3D reconstruction 284 8.4. Appropriate surface models and algorithmic considerations 293 8.5. Reconstruction of a polyhedric active contour 298 8.6. Conclusion 303 8.7. Bibliography 305 Chapter 9. Gauss-Markov-Potts Prior for Bayesian Inversion in Microwave Imaging 309 Olivier FÉRON, Bernard DUCHÊNE and Ali MOHAMMAD-DJAFARI 9.1. Introduction 310 9.2. Experimental configuration and modeling of the direct problem 311 9.3. Inversion in the linear case 315 9.4. Inversion in the non-linear case 325 9.5. Conclusion 335 9.6. Bibliography 336 Chapter 10. Shape from Shading 339 Jean-Denis DUROU 10.1. Introduction 339 10.2. Modeling of shape from shading 340 10.3. Resolution of shape from shading 353 10.4. Conclusion 371 10.5. Bibliography 372 Chapter 11. Image Separation 377 Hichem SNOUSSI and Ali MOHAMMAD-DJAFARI 11.1. General introduction 377 11.2. Blind image separation 378 11.3. Bayesian formulation 384 11.4. Stochastic algorithms 390 11.5. Simulation results 398 11.6. Conclusion 401 11.7. Appendix 1: a posteriori distributions 407 11.8. Bibliography 409 Chapter 12. Stereo Reconstruction in Satellite and Aerial Imaging 411 Julie DELON and Andrés ALMANSA 12.1. Introduction 411 12.2. Principles of satellite stereovision 412 12.3. Matching 415 12.4. Regularization 421 12.5. Numerical considerations 425 12.6. Conclusion 432 12.7. Bibliography 434 Chapter 13. Fusion and Multi-modality 437 Christophe COLLET, Farid FLITTI, Stéphanie BRICQ and André JALOBEANU 13.1. Fusion of optical multi-detector images without loss of information 437 13.2. Fusion of multi-spectral images using hidden Markov trees 438 13.3. Segmentation of multimodal cerebral MRI using an a priori probabilistic map 448 13.4. Bibliography 458 List of Authors 461 Index 463

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Book SynopsisThis book contains theoretical and application-oriented methods to treat models of dynamical systems involving non-smooth nonlinearities. The theoretical approach that has been retained and underlined in this work is associated with differential inclusions of mainly finite dimensional dynamical systems and the introduction of maximal monotone operators (graphs) in order to describe models of impact or friction. The authors of this book master the mathematical, numerical and modeling tools in a particular way so that they can propose all aspects of the approach, in both a deterministic and stochastic context, in order to describe real stresses exerted on physical systems. Such tools are very powerful for providing reference numerical approximations of the models. Such an approach is still not very popular nevertheless, even though it could be very useful for many models of numerous fields (e.g. mechanics, vibrations, etc.). This book is especially suited for people both in research and industry interested in the modeling and numerical simulation of discrete mechanical systems with friction or impact phenomena occurring in the presence of classical (linear elastic) or non-classical constitutive laws (delay, memory effects, etc.). It aims to close the gap between highly specialized mathematical literature and engineering applications, as well as to also give tools in the framework of non-smooth stochastic differential systems: thus, applications involving stochastic excitations (earthquakes, road surfaces, wind models etc.) are considered. Contents 1. Some Simple Examples. 2. Theoretical Deterministic Context. 3. Stochastic Theoretical Context. 4. Riemannian Theoretical Context. 5. Systems with Friction. 6. Impact Systems. 7. Applications–Extensions. About the Authors Jérôme Bastien is Assistant Professor at the University Lyon 1 (Centre de recherche et d'Innovation sur le sport) in France. Frédéric Bernardin is a Research Engineer at Département Laboratoire de Clermont-Ferrand (DLCF), Centre d'Etudes Techniques de l'Equipement (CETE), Lyon, France. Claude-Henri Lamarque is Head of Laboratoire Géomatériaux et Génie Civil (LGCB) and Professor at Ecole des Travaux Publics de l'Etat (ENTPE), Vaulx-en-Velin, France.Table of ContentsIntroduction xi Chapter 1. Some Simple Examples 1 1.1. Introduction 1 1.2. Frictions 1 1.2.1. Coulomb’s law 1 1.2.2. Differential equation with univalued operator and usual sign 3 1.2.3. Differential equation with multivalued term: differential inclusion 11 1.2.4. Other friction laws 12 1.3. Impact 16 1.3.1. Difficulties with writing the differential equation 16 1.3.2. Ill-posed problems 19 1.4. Probabilistic context 22 Chapter 2. Theoretical Deterministic Context 27 2.1. Introduction 27 2.2. Maximal monotone operators and first result on differential inclusions (in R) 27 2.2.1. Graphs (operators) definitions 28 2.2.2. Maximal monotone operators 29 2.2.3. Convex function, subdifferentials and operators 33 2.2.4. Resolvent and regularization 38 2.2.5. Taking the limit 40 2.2.6. First result of existence and uniqueness for a differential inclusion 40 2.3. Extension to any Hilbert space 45 2.4. Existence and uniqueness results in Hilbert space 57 2.5. Numerical scheme in a Hilbert space 59 2.5.1. The numerical scheme 59 2.5.2. State of the art summary and results shown in this publication 60 2.5.3. Convergence (general results and order 1/2) 61 2.5.4. Convergence (order one) 67 2.5.5. Change of scalar product 72 2.5.6. Resolvent calculation 74 2.5.7. More regular schemes 76 Chapter 3. Stochastic Theoretical Context 79 3.1. Introduction 79 3.2. Stochastic integral 79 3.2.1. The stochastic processes background 80 3.2.2. Stochastic integral 84 3.3. Stochastic differential equations 90 3.3.1. Existence and uniqueness of strong solution 91 3.3.2. Existence and uniqueness of weak solution 92 3.3.3. Kolmogorov and Fokker–Planck equations 95 3.4. Multivalued stochastic differential equations 101 3.4.1. Problem statement 101 3.4.2. Uniqueness and existence results 103 3.5. Numerical scheme 104 3.5.1. Which convergence: weak or strong? 106 3.5.2. Strong convergence results 108 3.5.3. Weak convergence results 122 Chapter 4. Riemannian Theoretical Context 129 4.1. Introduction 129 4.2. First or second order 129 4.3. Differential geometry 131 4.3.1. Sphere case 131 4.3.2. General case 132 4.4. Dynamics of the mechanical systems 139 4.4.1. Definition of mechanical system 139 4.4.2. Equation of the dynamics 141 4.5. Connection, covariant derivative, geodesics and parallel transport 144 4.6. Maximal monotone term 148 4.7. Stochastic term 149 4.8. Results on the existence and uniqueness of a solution 151 Chapter 5. Systems with Friction 155 5.1. Introduction 155 5.2. Examples of frictional systems with a finite number of degrees of freedom 155 5.2.1. General framework 155 5.2.2. Two elementary models 156 5.2.3. Assembly and results in finite dimensions 165 5.2.4. Conclusion 193 5.2.5. Examples of numerical simulation 194 5.2.6. Identification of the generalized Prandtl model (principles and simulation) 205 5.3. Another example: the case of a pendulum with friction 215 5.3.1. Formulation of the problem, existence and uniqueness 215 5.3.2. Numerical scheme 218 5.3.3. Numerical estimation of the order 219 5.3.4. Example of numerical simulations 221 5.3.5. Free oscillations 221 5.3.6. Forced oscillations 221 5.3.7. Transition matrix and calculation of the Lyapunov exponents 222 5.3.8. Melnikov’s method, transitory chaos and Lyapunov exponents 230 5.4. Elastoplastic oscillator under a stochastic forcing 231 5.4.1. Introduction 231 5.4.2. Modeling 232 5.4.3. Numerical scheme 236 5.4.4. Numerical results 238 5.5. Spherical pendulum under a stochastic external force 243 5.5.1. Establishment of the model 243 5.5.2. Numerical aspects 248 5.6. Gephyroidal model 255 5.6.1. Introduction 255 5.6.2. Description and transformation of the model 256 5.6.3. Quasi-static problems 263 5.6.4. Numerical simulations 265 5.6.5. Conclusion 267 5.7. Chain 268 5.7.1. Introduction 268 5.7.2. Description of the model 270 5.7.3. Transformation of the equations 271 5.7.4. Conclusion 283 5.8. An infinity of internal variables: continuous generalized Prandtl model 283 5.8.1. Introduction 283 5.8.2. Description of the continuous model 284 5.8.3. Existence, uniqueness and regularity results 287 5.8.4. Application to the discrete case, and convergence of the discrete model to the continuous model 289 5.8.5. Numerical scheme 291 5.8.6. Study of hysteresis loops 293 5.8.7. Numerical simulations 301 5.9. Locally Lipschitz continuous spring 301 5.9.1. Introduction 301 5.9.2. The studied model 301 5.9.3. Results for the existence and uniqueness of the solutions 303 5.9.4. Convergence results for the numerical schemes 311 5.9.5. The locally Lipschitz continuous case 313 5.9.6. Identification of the parameters from the hysteresis loops 314 5.9.7. Numerical simulations 320 Chapter 6. Impact Systems 325 6.1. Existence and uniqueness for simple problems (one degree of freedom) 326 6.1.1. The work of Schatzman–Paoli 326 6.1.2. Simple case with one degree of freedom, forcing and impact: piecewise analytical solutions 327 6.1.3. Adaptation of some classical methods 329 6.1.4. Movement with the accumulation of impacts and a sticking phase 333 6.1.5. Behavior of the numerical methods 337 6.1.6. Convergence and order of one-step numerical methods applied to non-smooth differential systems 338 6.1.7. Results of numerical experiments 343 6.2. A particular behavior: grazing bifurcation 348 6.2.1. Approximation of the map in the general case 349 6.2.2. Particular case 350 6.2.3. Stability of the non-differentiable fixed point 351 6.2.4. Numerical example 353 Chapter 7. Applications–Extensions 355 7.1. Oscillators with piecewise linear coupling and passive control 355 7.1.1. Description of the model 356 7.1.2. Free oscillations of the system 356 7.1.3. Order 1 362 7.1.4. Case of periodic forcing 366 7.1.5. Conclusion 377 7.2. Friction and passive control 378 7.2.1. Introduction 378 7.2.2. Introduction to the models: smooth and non-smooth systems 379 7.3. The billiard ball 386 7.3.1. Maximal monotone framework 386 7.3.2. More realistic but non-maximal monotone framework 389 7.4. An industrial application: the case of a belt tensioner 390 7.4.1. The theory 390 7.4.2. The tensioner used 392 7.4.3. Identification of the parameters 392 7.4.4. Validation 393 7.5. Problems with delay and memory 396 7.5.1. Theory 396 7.5.2. Applications 399 7.6. Other friction forces 400 7.6.1. More general forms (variable dynamical coefficient) 401 7.6.2. With a variable static coefficient 419 7.6.3. With variable static and dynamical coefficients 421 7.7. With the viscous dissipation term 423 7.8. Ill-posed problems 424 7.8.1. First model: limit of a well-posed friction law 426 7.8.2. Second model: a differential inclusion without uniqueness 427 7.8.3. Conclusion 429 Appendix 1. Mathematical Reminders 431 A1.1. Two Gronwall’s lemmas 431 A1.2. Norms, scalar products, normed vector space, Banach and Hilbert space 432 A1.2.1. Scalar products, norms 432 A1.2.2. Banach and Hilbert space, separable space 433 A1.3. Symmetric positive definite matrices 435 A1.4. Differentiable function 435 A1.5. Weak limit 436 A1.6. Continuous function spaces 436 A1.7. Lp space of integrable functions 437 A1.7.1. Lp(Ω) space 437 A1.7.2. Lp(Ω, Rq ) space 438 A1.7.3. Lp(Ω; H) spaces 438 A1.8. Distributions 439 A1.8.1. Real values distributions 439 A1.8.2. Distributions with values in Rq 440 A1.8.3. Distributions with values in Hilbert space 440 A1.9. Sobolev space definition 441 A1.9.1. Functions with real values 441 A1.9.2. Functions with values in Hilbert space 441 Appendix 2. Convex Functions 443 A2.1. Functions defined on R 443 A2.2. Functions defined on Hilbert space 446 A2.2.1. Any Hilbert space 446 A2.2.2. Particular case of the finite dimension 446 Appendix 3. Proof of Theorem 2.20 447 Appendix 4. Proof of Theorem 3.18 455 Appendix 5. Research of Convex Potential 467 A5.1. Method used 467 A5.2. Lemma 5.1 468 A5.3. Lemma 5.4 473 A5.4. Lemma 7.1 476 Bibliography 477 Index 495

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