Mathematics Books

Book SynopsisMany industries use control systems to insure that parameters such as temperature or altitude behave in a desirable way over time. This book offers an alternative failure detection approach that addresses two of the fundamental problems in the safe and efficient operation of modern control systems: failure detection and model identification.Trade Review"Aimed at a broad audience that includes graduate students in engineering and applied mathematics, the book is notable for its emphasis and focus on mathematical intuition and numerical issues. It is very well written, with an attention to detail and rigor... [T]he material is accessible to a wide audience with interests in areas such as control theory of differential equations."--Bogdan I. Epureanu, IEEE Control SystemsTable of ContentsPreface vii Chapter 1. Introduction 1 1.1 The Basic Question 1 1.2 Failure etection 3 1.3 Failure Identification 9 1.4 Active Approach versus Passive Approach 10 1.5 Outline of the Book 13 Chapter 2. Failure Detection 14 2.1 Introduction 14 2.2 Static Case 15 2.3 Continuous-Time Systems 25 2.4 iscrete-Time Systems 36 2.5 Real-Time Implementation Issues 42 2.6 Useful Results 44 Chapter 3. Multimodel Formulation 59 3.1 Introduction 59 3.2 Static Case 60 3.3 Continuous-Time Case 76 3.4 Case of On-line Measured Input 90 3.5 More GeneralCost Functions 92 3.6 iscrete-Time Case 99 3.7 Suspension Example 102 3.8 Asymptotic Behavior 111 3.9 Useful Results 112 Chapter 4. Direct Optimization Formulations 122 4.1 Introduction 122 4.2 Optimization Formulation for Two Models 123 4.3 General-ModelCase 138 4.4 Early etection 142 4.5 Other Extensions 150 4.6 Systems with Delays 155 4.7 Setting Error Bounds 172 4.8 Model Uncertainty 173 Chapter 5. Remaining Problems and Extensions 176 5.1 Direct Extensions 177 5.2 Hybrid and Sampled Data Systems 179 5.3 Relation to Stochastic Modeling 179 Chapter 6. Scilab Programs 181 6.1 Introduction 181 6.2 Riccati-based Solution 181 6.3 The Block iagonalization Approach 185 6.4 Getting Scilab and the Programs 188 Appendix A. List of Symbols 189 Bibliography 193 Index 201

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Book SynopsisPresents an introduction to optimization through the use of illustrations and applications. This book focuses on analytically solving optimization problems with a finite number of continuous variables. It also provides introductions to classical and modern numerical methods of optimization and to dynamic optimization.Trade Review"The authors provide a very nice and interesting textbook on the theory and the application of mathematical optimization... The book is written as well as for beginners and for experts... Both types of readers can profit from the given shortcuts and royal roads which jump over some theoretical explanations and lead directly to the applications."--Jorg Thierfelder, Zentralblatt MATHTable of ContentsPreface xi 0.1 Optimization: insights and applications xiii 0.2 Lunch, dinner, and dessert xiv 0.3 For whom is this book meant? xvi 0.4 What is in this book? xviii 0.5 Special features xix Necessary Conditions: What Is the Point? 1 Chapter 1. Fermat: One Variable without Constraints 3 1.0 Summary 3 1.1 Introduction 5 1.2 The derivative for one variable 6 1.3 Main result: Fermat theorem for one variable 14 1.4 Applications to concrete problems 30 1.5 Discussion and comments 43 1.6 Exercises 59 Chapter 2. Fermat: Two or More Variables without Constraints 85 2.0 Summary 85 2.1 Introduction 87 2.2 The derivative for two or more variables 87 2.3 Main result: Fermat theorem for two or more variables 96 2.4 Applications to concrete problems 101 2.5 Discussion and comments 127 2.6 Exercises 128 Chapter 3. Lagrange: Equality Constraints 135 3.0 Summary 135 3.1 Introduction 138 3.2 Main result: Lagrange multiplier rule 140 3.3 Applications to concrete problems 152 3.4 Proof of the Lagrange multiplier rule 167 3.5 Discussion and comments 181 3.6 Exercises 190 Chapter 4. Inequality Constraints and Convexity 199 4.0 Summary 199 4.1 Introduction 202 4.2 Main result: Karush-Kuhn-Tucker theorem 204 4.3 Applications to concrete problems 217 4.4 Proof of the Karush-Kuhn-Tucker theorem 229 4.5 Discussion and comments 235 4.6 Exercises 250 Chapter 5. Second Order Conditions 261 5.0 Summary 261 5.1 Introduction 262 5.2 Main result: second order conditions 262 5.3 Applications to concrete problems 267 5.4 Discussion and comments 271 5.5 Exercises 272 Chapter 6. Basic Algorithms 273 6.0 Summary 273 6.1 Introduction 275 6.2 Nonlinear optimization is difficult 278 6.3 Main methods of linear optimization 283 6.4 Line search 286 6.5 Direction of descent 299 6.6 Quality of approximation 301 6.7 Center of gravity method 304 6.8 Ellipsoid method 307 6.9 Interior point methods 316 Chapter 7. Advanced Algorithms 325 7.1 Introduction 325 7.2 Conjugate gradient method 325 7.3 Self-concordant barrier methods 335 Chapter 8. Economic Applications 363 8.1 Why you should not sell your house to the highest bidder 363 8.2 Optimal speed of ships and the cube law 366 8.3 Optimal discounts on airline tickets with a Saturday stayover 368 8.4 Prediction of ows of cargo 370 8.5 Nash bargaining 373 8.6 Arbitrage-free bounds for prices 378 8.7 Fair price for options: formula of Black and Scholes 380 8.8 Absence of arbitrage and existence of a martingale 381 8.9 How to take a penalty kick, and the minimax theorem 382 8.10 The best lunch and the second welfare theorem 386 Chapter 9. Mathematical Applications 391 9.1 Fun and the quest for the essence 391 9.2 Optimization approach to matrices 392 9.3 How to prove results on linear inequalities 395 9.4 The problem of Apollonius 397 9.5 Minimization of a quadratic function: Sylvester's criterion and Gram's formula 409 9.6 Polynomials of least deviation 411 9.7 Bernstein inequality 414 Chapter 10. Mixed Smooth-Convex Problems 417 10.1 Introduction 417 10.2 Constraints given by inclusion in a cone 419 10.3 Main result: necessary conditions for mixed smooth-convex problems 422 10.4 Proof of the necessary conditions 430 10.5 Discussion and comments 432 Chapter 11. Dynamic Programming in Discrete Time 441 11.0 Summary 441 11.1 Introduction 443 11.2 Main result: Hamilton-Jacobi-Bellman equation 444 11.3 Applications to concrete problems 446 11.4 Exercises 471 Chapter 12. Dynamic Optimization in Continuous Time 475 12.1 Introduction 475 12.2 Main results: necessary conditions of Euler, Lagrange, Pontrya-gin, and Bellman 478 12.3 Applications to concrete problems 492 12.4 Discussion and comments 498 Appendix A. On Linear Algebra: Vector and Matrix Calculus 503 A.1 Introduction 503 A.2 Zero-sweeping or Gaussian elimination, and a formula for the dimension of the solution set 503 A.3 Cramer's rule 507 A.4 Solution using the inverse matrix 508 A.5 Symmetric matrices 510 A.6 Matrices of maximal rank 512 A.7 Vector notation 512 A.8 Coordinate free approach to vectors and matrices 513 Appendix B. On Real Analysis 519 B.1 Completeness of the real numbers 519 B.2 Calculus of differentiation 523 B.3 Convexity 528 B.4 Differentiation and integration 535 Appendix C. The Weierstrass Theorem on Existence of Global Solutions 537 C.1 On the use of the Weierstrass theorem 537 C.2 Derivation of the Weierstrass theorem 544 Appendix D. Crash Course on Problem Solving 547 D.1 One variable without constraints 547 D.2 Several variables without constraints 548 D.3 Several variables under equality constraints 549 D.4 Inequality constraints and convexity 550 Appendix E. Crash Course on Optimization Theory: Geometrical Style 553 E.1 The main points 553 E.2 Unconstrained problems 554 E.3 Convex problems 554 E.4 Equality constraints 555 E.5 Inequality constraints 556 E.6 Transition to infinitely many variables 557 Appendix F. Crash Course on Optimization Theory: Analytical Style 561 F.1 Problem types 561 F.2 Definitions of differentiability 563 F.3 Main theorems of differential and convex calculus 565 F.4 Conditions that are necessary and/or sufficient 567 F.5 Proofs 571 Appendix G. Conditions of Extremum from Fermat to Pontryagin 583 G.1 Necessary first order conditions from Fermat to Pontryagin 583 G.2 Conditions of extremum of the second order 593 Appendix H. Solutions of Exercises of Chapters 1-4 601 Bibliography 645 Index 651

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Book SynopsisIntroduces quaternions for scientists and engineers, and shows how they can be used in a variety of practical situations. This book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. It also presents the conventional and familiar 3 x 3 (9-element) matrix rotation operator.Trade Review"This book will appeal to anyone with an interest in three-dimensional geometry. It is a competent and comprehensive survey... This book is unique in that it is probably the only modern book to treat quaternions seriously... A valuable asset."--Aeronautical Journal "[A] splendid book ... everything one could wish for in a primer. It is also beautifully set out with an attractive layout, clear diagrams, and wide margins with explanatory notes where appropriate. It must be strongly recommended to all students of physics, engineering or computer science."--Peter Rowlands, Contemporary PhysicsTable of ContentsList of FiguresAbout This BookAcknowledgements1Historical Matters32Algebraic Preliminaries133Rotations in 3-space454Rotation Sequences in R[superscript 3]775Quaternion Algebra1036Quaternion Geometry1417Algorithm Summary1558Quaternion Factors1779More Quaternion Applications20510Spherical Trigonometry23511Quaternion Calculus for Kinematics and Dynamics25712Rotations in Phase Space27713A Quaternion Process30314Computer Graphics333Further Reading and References365Index367

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Book SynopsisApplies model theoretic methods to the study of certain finite permutation groups, the automorphism groups of structures for a fixed finite language with a bounded number of orbits on 4-tuples. This book treats the general case by developing analogs of the model theoretic methods of geometric stability theory.

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Book SynopsisProvides surveys of the uses and concepts of entropy in diverse areas of mathematics and the physical sciences. This book starts by providing basic concepts and terminology, illustrated by examples from both the macroscopic and microscopic lines of thought. In-depth surveys covering the macroscopic, microscopic and probabilistic approaches follow.Trade Review"This book is that rare thing: an edited volume that will be a lasting contribution to the literature."—Ray Streater, King's CollegeTable of ContentsPreface xi List of Contributors xiii Chapter 1. Introduction A.Greven, G.Keller, G.Warnecke 1 1.1 Outline of the Book 4 1.2 Notations 14 PART 1. FUNDAMENTAL CONCEPTS 17 Chapter 2. Entropy: a Subtle Concept in Thermodynamics I. Muller 19 2.1 Origin of Entropy in Thermodynamics 19 2.2 Mechanical Interpretation of Entropy in the Kinetic Theory of Gases 23 2.2.1 Configurational Entropy 25 2.3 Entropy and Potential Energy of Gravitation 28 2.3.1 Planetary Atmospheres 28 2.3.2 Pfeffer Tube 29 2.4 Entropy and Intermolecular Energies 30 2.5 Entropy and Chemical Energies 32 2.6 Omissions 34 References 35 Chapter 3. Probabilistic Aspects of Entropy H. -O.Georgii 37 3.1 Entropy as a Measure of Uncertainty 37 3.2 Entropy as a Measure of Information 39 3.3 Relative Entropy as a Measure of Discrimination 40 3.4 Entropy Maximization under Constraints 43 3.5 Asymptotics Governed by Entropy 45 3.6 Entropy Density of Stationary Processes and Fields 48 References 52 PART 2.ENTROPY IN THERMODYNAMICS 55 Chapter 4. Phenomenological Thermodynamics and Entropy Principles K.Hutter and Y.Wang 57 4.1 Introduction 57 4.2 A Simple Classification of Theories of Continuum Thermodynamics 58 4.3 Comparison of Two Entropy Principles 63 4.3.1 Basic Equations 63 4.3.2 Generalized Coleman-Noll Evaluation of the Clausius-Duhem Inequality 66 4.3.3 Muller-Liu's Entropy Principle 71 4.4 Concluding Remarks 74 References 75 Chapter 5. Entropy in Nonequilibrium I. Muller 79 5.1 Thermodynamics of Irreversible Processes and Rational Thermodynamics for Viscous, Heat-Conducting Fluids 79 5.2 Kinetic Theory of Gases, the Motivation for Extended Thermodynamics 82 5.2.1 A Remark on Temperature 82 5.2.2 Entropy Density and Entropy Flux 83 5.2.3 13-Moment Distribution. Maximization of Nonequilibrium Entropy 83 5.2.4 Balance Equations for Moments 84 5.2.5 Moment Equations for 13 Moments. Stationary Heat Conduction 85 5.2.6 Kinetic and Thermodynamic Temperatures 87 5.2.7 Moment Equations for 14 Moments. Minimum Entropy Production 89 5.3 Extended Thermodynamics 93 5.3.1 Paradoxes 93 5.3.2 Formal Structure 95 5.3.3 Pulse Speeds 98 5.3.4 Light Scattering 101 5.4 A Remark on Alternatives 103 References 104 Chapter 6. Entropy for Hyperbolic Conservation Laws C.M.Dafermos 107 6.1 Introduction 107 6.2 Isothermal Thermoelasticity 108 6.3 Hyperbolic Systems of Conservation Laws 110 6.4 Entropy 113 6.5 Quenching of Oscillations 117 References 119 Chapter 7. Irreversibility and the Second Law of Thermodynamics J.Uffink 121 7.1 Three Concepts of (Ir)reversibility 121 7.2 Early Formulations of the Second Law 124 7.3 Planck 129 7.4 Gibbs 132 7.5 Caratheodory 133 7.6 Lieb and Yngvason 140 7.7 Discussion 143 References 145 Chapter 8. The Entropy of Classical Thermodynamics E. H. Lieb, J. Yngvason 147 8.1 A Guide to Entropy and the Second Law of Thermodynamics 148 8.2 Some Speculations and Open Problems 190 8.3 Some Remarks about Statistical Mechanics 192 References 193 PART 3.ENTROPY IN STOCHASTIC PROCESSES 197 Chapter 9. Large Deviations and Entropy S. R. S. Varadhan 199 9.1 Where Does Entropy Come From? 199 9.2 Sanov's Theorem 201 9.3 What about Markov Chains? 202 9.4 Gibbs Measures and Large Deviations 203 9.5 Ventcel-Freidlin Theory 205 9.6 Entropy and Large Deviations 206 9.7 Entropy and Analysis 209 9.8 Hydrodynamic Scaling: an Example 211 References 214 Chapter 10. Relative Entropy for Random Motion in a Random Medium F. den Hollander 215 10.1 Introduction 215 10.1.1 Motivation 215 10.1.2 A Branching Random Walk in a Random Environment 217 10.1.3 Particle Densities and Growth Rates 217 10.1.4 Interpretation of the Main Theorems 219 10.1.5 Solution of the Variational Problems 220 10.1.6 Phase Transitions 223 10.1.7 Outline 224 10.2 Two Extensions 224 10.3 Conclusion 225 10.4 Appendix: Sketch of the Derivation of the Main Theorems 226 10.4.1 Local Times of Random Walk 226 10.4.2 Large Deviations and Growth Rates 228 10.4.3 Relation between the Global and the Local Growth Rate 230 References 231 Chapter 11. Metastability and Entropy E. Olivieri 233 11.1 Introduction 233 11.2 van der Waals Theory 235 11.3 Curie-Weiss Theory 237 11.4 Comparison between Mean-Field and Short-Range Models 237 11.5 The 'Restricted Ensemble' 239 11.6 The Pathwise Approach 241 11.7 Stochastic Ising Model. Metastability and Nucleation 241 11.8 First-Exit Problem for General Markov Chains 244 11.9 The First Descent Tube of Trajectories 246 11.10 Concluding Remarks 248 References 249 Chapter 12. Entropy Production in Driven Spatially Extended Systems C. Maes 251 12.1 Introduction 251 12.2 Approach to Equilibrium 252 12.2.1 Boltzmann Entropy 253 12.2.2 Initial Conditions 254 12.3 Phenomenology of Steady-State Entropy Production 254 12.4 Multiplicity under Constraints 255 12.5 Gibbs Measures with an Involution 258 12.6 The Gibbs Hypothesis 261 12.6.1 Pathspace Measure Construction 262 12.6.2 Space-Time Equilibrium 262 12.7 Asymmetric Exclusion Processes 263 12.7.1 MEP for ASEP 263 12.7.2 LFT for ASEP 264 References 266 Chapter 13. Entropy: a Dialogue J. L. Lebowitz, C. Maes 269 References 275 PART 4.ENTROPY AND INFORMATION 277 Chapter 14. Classical and Quantum Entropies:Dynamics and Information F. Benatti 279 14.1 Introduction 279 14.2 Shannon and von Neumann Entropy 280 14.2.1 Coding for Classical Memoryless Sources 281 14.2.2 Coding for Quantum Memoryless Sources 282 14.3 Kolmogorov-Sinai Entropy 283 14.3.1 KS Entropy and Classical Chaos 285 14.3.2 KS Entropy and Classical Coding 285 14.3.3 KS Entropy and Algorithmic Complexity 286 14.4 Quantum Dynamical Entropies 287 14.4.1 Partitions of Unit and Decompositions of States 290 14.4.2 CNT Entropy: Decompositions of States 290 14.4.3 AF Entropy: Partitions of Unit 292 14.5 Quantum Dynamical Entropies: Perspectives 293 14.5.1 Quantum Dynamical Entropies and Quantum Chaos 295 14.5.2 Dynamical Entropies and Quantum Information 296 14.5.3 Dynamical Entropies and Quantum Randomness 296 References 296 Chapter 15. Complexity and Information in Data J. Rissanen 299 15.1 Introduction 299 15.2 Basics of Coding 301 15.3 Kolmogorov Sufficient Statistics 303 15.4 Complexity 306 15.5 Information 308 15.6 Denoising with Wavelets 311 References 312 Chapter 16. Entropy in Dynamical Systems L. -S. Young 313 16.1 Background 313 16.1.1 Dynamical Systems 313 16.1.2 Topological and Metric Entropies 314 16.2 Summary 316 16.3 Entropy, Lyapunov Exponents, and Dimension 317 16.3.1 Random Dynamical Systems 321 16.4 Other Interpretations of Entropy 322 16.4.1 Entropy and Volume Growth 322 16.4.2 Growth of Periodic Points and Horseshoes 323 16.4.3 Large Deviations and Rates of Escape 325 References 327 Chapter 17. Entropy in Ergodic Theory M. Keane 329 References 335 Combined References 337 Index 351

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Book SynopsisProvides a collection of English translations of mathematical texts from five important ancient and medieval non-Western mathematical cultures, and puts them into historical and mathematical context. This book is intended for mathematics teachers who want to use non-Western mathematical ideas in the classroom.Trade ReviewJoseph Warren Dauben, Winner of the 2012 Albert Leon Whiteman Memorial Prize, American Mathematical Society "This pioneering work provides English translations of mathematical texts from each of these regions and cultures, and a better understanding of their contributions to mathematics. There are nuggets of information difficult to find elsewhere. The use of non-mathematical sources, particularly letters and administrative documents from Egypt and Mesopotamia, reveals the practical applications of mathematics and the scribes who composed and used the documents...An essential resource for anyone wishing to know more about how the mathematics of the different regions influenced and shaped the development of world mathematics."--George Gheverghese Joseph, Nature "We're aware that the ancient cultures were mathematically advanced. Now translations of early texts from five key regions are available together for the first time, and put into context by experts."--Nature Physics "The corrections to the Eurocentrism that understandably characterized Western assays of the intellectual history of the planet early on must inevitably be applied to the history of mathematics. Editor Katz and his scholarly coauthors have greatly advanced the process with this one-volume sourcebook...The introductory essays that precede each section are lucidly written, well within the reach of an undergraduate math major. Katz asks more or less rhetorically 'how much effect the mathematics of these civilizations had on what is now world mathematics of the twenty-first century.' This invaluable book will help significantly in formulating an answer."--M. Schiff, Choice "This book is an essential resource for anyone with at least an undergraduate degree in mathematics who wants to learn about non-Western mathematical developments and how they helped shape and enrich world mathematics. The book is also an indispensable guide for mathematics teachers who want to use non-Western mathematical ideas in the classroom."--L'Enseignement Mathematique "The Mathematics of Egypt, Mesopotamia, China, India, and Islam is a wonderful collection, for which Victor Katz is to be commended. This book is a one-stop source for numerous original mathematical texts in translation. I cannot overemphasize how wonderful it is to have this large, exquisite selection of ... mathematical texts together in one volume... Every history of mathematics teacher will want a copy of this book in their personal library as well as in the library of their college or university."--James V. Rauff, Mathematics and Computer Education "What we have here is a useful selection, one that should be of interest to specialists in world history or in the history of the sciences in any of these culture areas and, in particular, to scholars who are engaged with the history of mathematics as specialists or because of its role as a tool."--Tom Archibald, Isis "[This] is the biggest sourcebook containing the newest fruit of historical research and that is why the book can replace older sources for the history of mathematics."--EMS NewsletterTable of ContentsPreface ix Permissions xi Introduction 1 Chapter 1: Egyptian Mathematics Annette Imhausen Preliminary Remarks 7 I. Introduction 9 a. Invention of writing and number systems 13 b. Arithmetic 14 c.Metrology 17 II. Hieratic Mathematical Texts 17 a. Table texts 18 b. Problem texts 24 III. Mathematics in Administrative Texts 40 a. Middle Kingdom texts: The Reisner papyri 40 b. New Kingdom texts: Ostraca from Deir el Medina 44 IV. Mathematics in the Graeco-Roman Period 46 a. Context 46 b. Table texts 47 c. Problem texts 48 V. Appendices 52 a. Glossary of Egyptian terms 52 b. Sources 52 c. References 54 Chapter 2: Mesopotamian Mathematics Eleanor Robson I. Introduction 58 a. Mesopotamian mathematics through Western eyes 58 b.Mathematics and scribal culture in ancient Iraq 62 c. From tablet to translation 65 d. Explananda 68 II. The Long Third Millennium, c. 3200-2000 BCE 73 a. Uruk in the late fourth millennium 73 b. Shuruppag in the mid-third millennium 74 c. Nippur and Girsu in the twenty-fourth century BCE 76 d. Umma and Girsu in the twenty-first century BCE 78 III. The Old Babylonian Period, c. 2000-1600 BCE 82 a. Arithmetical and metrological tables 82 b. Mathematical problems 92 c. Rough work and reference lists 142 IV. Later Mesopotamia, c. 1400-150 BCE 154 V. Appendices 180 a. Sources 180 b. References 181 Chapter 3: Chinese Mathematics Joseph W. Dauben Preliminary Remarks 187 I. China: The Historical and Social Context 187 II. Methods and Procedures: Counting Rods, The "Out-In" Principle 194 III. Recent Archaeological Discoveries: The Earliest Yet-Known Bamboo Text 201 IV. Mathematics and Astronomy: The Zhou bi suan jing and Right Triangles (The Gou-gu or "Pythagorean" Theorem) 213 V. The Chinese "Euclid", Liu Hui 226 a. The Nine Chapters 227 b. The Sea Island Mathematical Classic 288 VI. The "Ten Classics" of Ancient Chinese Mathematics 293 a. Numbers and arithmetic: The Mathematical Classic of Master Sun 295 b. The Mathematical Classic of Zhang Qiujian 302 VII. Outstanding Achievements of the Song and Yuan Dynasties (960-1368 CE) 308 a. Qin Jiushao 309 b. Li Zhi (Li Ye) 323 c. Yang Hui 329 d. Zhu Shijie 343 VIII. Matteo Ricci and Xu Guangxi, "Prefaces" to the First Chinese Edition of Euclid's Elements (1607) 366 IX. Conclusion 375 X. Appendices 379 a. Sources 379 b. Bibliographic guides 379 c. References 380 Chapter 4: Mathematics in India Kim Plofker I. Introduction: Origins of Indian Mathematics 385 II. Mathematical Texts in Ancient India 386 a. The Vedas 386 b. The S'ulbasutras 387 c. Mathematics in other ancient texts 393 d. Number systems and numerals 395 III. Evolution of Mathematics in Medieval India 398 a.Mathematics chapters in Siddhanta texts 398 b. Transmission of mathematical ideas to the Islamic world 434 c. Textbooks on mathematics as a separate subject 435 d. The audience for mathematics education 477 e. Specialized mathematics: Astronomical and cosmological problems 478 IV. The Kerala School 480 a. Madhava, his work, and his school 480 b. Infinite series and the role of demonstrations 481 c. Other mathematical interests in the Kerala school 493 V. Continuity and Transition in the Second Millennium 498 a. The ongoing development of Sanskrit mathematics 498 b. Scientific exchanges at the courts of Delhi and Jaipur 504 c. Assimilation of ideas from Islam; mathematical table texts 506 VI. Encounters with Modern Western Mathematics 507 a. Early exchanges with European mathematics 507 b. European versus "native" mathematics education in British India 508 c. Assimilation into modern global mathematics 510 VII. Appendices 511 a. Sources 511 b. References 512 Chapter 5: Mathematics in Medieval Islam J. Lennart Berggren I. Introduction 515 II. Appropriation of the Ancient Heritage 520 III. Arithmetic 525 IV. Algebra 542 V. Number Theory 560 VI. Geometry 564 a. Theoretical geometry 564 b. Practical geometry 610 VII. Trigonometry 621 VIII. Combinatorics 658 IX. On mathematics 666 X. Appendices 671 a. Sources 671 b. References 674 Contributors 677 Index 681

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Book SynopsisCovers local search and its variants from both a theoretical and practical point of view. This book is suitable for students and researchers in discrete mathematics, computer science, operations research, industrial engineering, and management science.Trade Review"A truly remarkable and unique collection of work... Invaluable."--Informs "The world of local search has changed dramatically in the last decade and Aarts and Lenstra's book is a tribute to this development... A very useful source."--Optima

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Book SynopsisOffers an account of Kiyosi Ito's program. This book offers an account of integral curves on the space of probability measures. It provides a systematic development of Ito's theory of stochastic integration: first for Brownian motion and then for continuous martingales.Table of Contents*Frontmatter, pg. i*Contents, pg. vii*Preface, pg. xi*Chapter 1. Finite State Space, a Trial Run, pg. 1*Chapter 2. Moving to Euclidean Space, the Real Thing, pg. 35*Chapter 3. Ito's Approach in the Euclidean Setting, pg. 73*Chapter 4. Further Considerations, pg. 111*Chapter 5. Ito's Theory of Stochastic Integration, pg. 125*Chapter 6. Applications of Stochastic Integration to Brownian Motion, pg. 151*Chapter 7. The Kunita-Watanabe Extension, pg. 189*Chapter 8. Stratonovich's Theory, pg. 221*Notation, pg. 260*References, pg. 263*Index, pg. 265

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Book SynopsisUses the phenomenon called 'six degrees of separation' as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network? This book is intended for a variety of fields, including physics and mathematics, as well as sociology, economics, and biology.Trade Review"An engaging and informative introduction."--Science "Playfully and clearly written... [Watts] uses examples adroitly, and mixes abstract theory with real-world anecdotes with superb skill... I have not enjoyed reading a book this much in a long time."--Peter Kareiva, Quarterly Review of Biology "[Small Worlds] will be seized on by those seeking a first rough map of this fascinating new mathematical land. Those entering can expect to find some amazing connections between areas of research with apparently nothing in common, such as neurology to business studies. But then, it's a small world."--Robert Matthews, New Scientist "Informally written and aimed at a wide audience, this book shows how mathematics yields new vistas on ubiquitous and seemingly familiar aspects of our world."--ChoiceTable of ContentsPREFACE xiii 1 Kevin Bacon, the Small World, and Why It All Matters 3 PART I STRUCTURE 9 2 An Overview of the Small-World Phenomenon 11 2.1 Social Networks and the Small World 11 2.1.1 A Brief History of the Small World 12 2.1.2 Difficulties with the Real World 20 2.1.3 Reframing the Question to Consider All Worlds 24 2.2 Background on the Theory of Graphs 25 2.2.1 Basic Definitions 25 2.2.2 Length and Length Scaling 27 2.2.3 Neighbourhoods and Distribution Sequences 31 2.2.4 Clustering 32 2.2.5 "Lattice Graphs" and Random Graphs 33 2.2.6 Dimension and Embedding of Graphs 39 3 Big Worlds and Small Worlds: Models of Graphs 41 3.1 Relational Graphs 42 3.1.1 a-Graphs 42 3.1.2 A Stripped-Down Model: B-Graphs 66 3.1.3 Shortcuts and Contractions: Model Invariance 70 3.1.4 Lies, Damned Lies, and (More) Statistics 87 3.2 Spatial Graphs 91 3.2.1 Uniform Spatial Graphs 93 3.2.2 Gaussian Spatial Graphs 98 3.3 Main Points in Review 100 4 Explanations and Ruminations 101 4.1 Going to Extremes 101 4.1.1 The Connected-Caveman World 102 4.1.2 Moore Graphs as Approximate Random Graphs 109 4.2 Transitions in Relational Graphs 114 4.2.1 Local and Global Length Scales 114 4.2.2 Length and Length Scaling 116 4.2.3 Clustering Coefficient 117 4.2.4 Contractions 118 4.2.5 Results and Comparisons with B-Model 120 4.3 Transitions in Spatial Graphs 127 4.3.1 Spatial Length versus Graph Length 127 4.3.2 Length and Length Scaling 128 4.3.3 Clustering 130 4.3.4 Results and Comparisons 132 4.4 Variations on Spatial and Relational Graphs 133 4.5 Main Points in Review 136 5 "It's a Small World after All": Three Real Graphs 138 5.1 Making Bacon 140 5.1.1 Examining the Graph 141 5.1.2 Comparisons 143 5.2 The Power of Networks 147 5.2.1 Examining the System 147 5.2.2 Comparisons 150 5.3 A Worm's Eye View 153 5.3.1 Examining the System 154 5.3.2 Comparisons 156 5.4 Other Systems 159 5.5 Main Points in Review 161 PART II DYNAMICS 163 6 The Spread of Infectious Disease in Structured Populations 165 6.1 A Brief Review of Disease Spreading 166 6.2 Analysis and Results 168 6.2.1 Introduction of the Problem 168 6.2.2 Permanent-Removal Dynamics 169 6.2.3 Temporary-Removal Dynamics 176 6.3 Main Points in Review 180 7 Global Computation in Cellular Automata 181 7.1 Background 181 7.1.1 Global Computation 184 7.2 Cellular Automata on Graphs 187 7.2.1 Density Classification 187 7.2.2 Synchronisation 195 7.3 Main Points in Review 198 8 Cooperation in a Small World: Games on Graphs 199 8.1 Background 199 8.1.1 The Prisoner's Dilemma 200 8.1.2 Spatial Prisoner's Dilemma 204 8.1.3 N-Player Prisoner's Dilemma 206 8.1.4 Evolution of Strategies 207 8.2 Emergence of Cooperation in a Homogeneous Population 208 8.2.1 Generalised Tit-for-Tat 209 8.2.2 Win-Stay, Lose-Shift 214 8.3 Evolution of Cooperation in a Heterogeneous Population 219 8.4 Main Points in Review 221 9 Global Synchrony in Populations of Coupled Phase Oscillators 223 9.1 Background 223 9.2 Kuramoto Oscillators on Graphs 228 9.3 Main Points in Review 238 10 Conclusions 240 NOTES 243 BIBLIOGRAPHY 249 INDEX 257

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Book SynopsisAddressing the topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. It includes fixed point theorems and applications to functional equations and optimization theory.Trade Review"The book is intended as a textbook on real analysis for graduate students in economics. It is largely graduate level mathematics, and the students should have a solid undergraduate real analysis background... The author's writing style is ... in general quite attractive. The book should be quite successful for its intended purpose."--Gerald A. Heuer, Zentralblatt MATH "Important and commendable, this indispensable resource should be highly prized by all concerned with courses on mathematics for economists and by graduate students working on economic theory. Rarely do books meet such high aspirations and carry out their aims, yet this one certainly does. Well written in an engaging style and impressively researched in the requirements of graduate students of economics and finance, Real Analysis with Economic Applications is sure to become the definitive work for its intended audience. Real Analysis with Economic Applications with its large number of economics applications and variety of exercises represents the single most important mathematical source for students of economics applications and it will be the book, for a long time to come, to which they will turn with confidence, as well as pleasure, in all questions of economic applications."--Current Engineering PracticeTable of ContentsPreface xvii Prerequisites xxvii Basic Conventions xxix Part I: SET THEORY 1 Chapter A: Preliminaries of Real Analysis 3 A.1 Elements of Set Theory 4 A.1.1 Sets 4 A.1.2 Relations 9 A.1.3 Equivalence Relations 11 A.1.4 Order Relations 14 A.1.5 Functions 20 A.1.6 Sequences, Vectors, and Matrices 27 A.1.7* A Glimpse of Advanced Set Theory: The Axiom of Choice 29 A.2 Real Numbers 33 A.2.1 Ordered Fields 33 A.2.2 Natural Numbers, Integers, and Rationals 37 A.2.3 Real Numbers 39 A.2.4 Intervals and R 44 A.3 Real Sequences 46 A.3.1 Convergent Sequences 46 A.3.2 Monotonic Sequences 50 A.3.3 Subsequential Limits 53 A.3.4 Infinite Series 56 A.3.5 Rearrangement of Infinite Series 59 A.3.6 Infinite Products 61 A.4 Real Functions 62 A.4.1 Basic Definitions 62 A.4.2 Limits, Continuity, and Differentiation 64 A.4.3 Riemann Integration 69 A.4.4 Exponential, Logarithmic, and Trigonometric Functions 74 A.4.5 Concave and Convex Functions 77 A.4.6 Quasiconcave and Quasiconvex Functions 80 Chapter B: Countability 82 B.1 Countable and Uncountable Sets 82 B.2 Losets and Q 90 B.3 Some More Advanced Set Theory 93 B.3.1 The Cardinality Ordering 93 B.3.2* The Well-Ordering Principle 98 B.4 Application: Ordinal Utility Theory 99 B.4.1 Preference Relations 100 B.4.2 Utility Representation of Complete Preference Relations 102 B.4.3* Utility Representation of Incomplete Preference Relations 107 Part II: ANALYSIS ON METRIC SPACES 115 Chapter C: Metric Spaces 117 C.1 Basic Notions 118 C.1.1 Metric Spaces: Definition and Examples 119 C.1.2 Open and Closed Sets 127 C.1.3 Convergent Sequences 132 C.1.4 Sequential Characterization of Closed Sets 134 C.1.5 Equivalence of Metrics 136 C.2 Connectedness and Separability 138 C.2.1 Connected Metric Spaces 138 C.2.2 Separable Metric Spaces 140 C.2.3 Applications to Utility Theory 145 C.3 Compactness 147 C.3.1 Basic Definitions and the Heine-Borel Theorem 148 C.3.2 Compactness as a Finite Structure 151 C.3.3 Closed and Bounded Sets 154 C.4 Sequential Compactness 157 C.5 Completeness 161 C.5.1 Cauchy Sequences 161 C.5.2 Complete Metric Spaces: Definition and Examples 163 C.5.3 Completeness versus Closedness 167 C.5.4 Completeness versus Compactness 171 C.6 Fixed Point Theory I 172 C.6.1 Contractions 172 C.6.2 The Banach Fixed Point Theorem 175 C.6.3* Generalizations of the Banach Fixed Point Theorem 179 C.7 Applications to Functional Equations 183 C.7.1 Solutions of Functional Equations 183 C.7.2 Picard's Existence Theorems 187 C.8 Products of Metric Spaces 192 C.8.1 Finite Products 192 C.8.2 Countably Infinite Products 193 Chapter D: Continuity I 200 D.1 Continuity of Functions 201 D.1.1 Definitions and Examples 201 D.1.2 Uniform Continuity 208 D.1.3 Other Continuity Concepts 210 D.1.4* Remarks on the Differentiability of Real Functions 212 D.1.5 A Fundamental Characterization of Continuity 213 D.1.6 Homeomorphisms 216 D.2 Continuity and Connectedness 218 D.3 Continuity and Compactness 222 D.3.1 Continuous Image of a Compact Set 222 D.3.2 The Local-to-Global Method 223 D.3.3 Weierstrass' Theorem 225 D.4 Semicontinuity 229 D.5 Applications 237 D.5.1* Caristi's Fixed Point Theorem 238 D.5.2 Continuous Representation of a Preference Relation 239 D.5.3* Cauchy's Functional Equations: Additivity on Rn 242 D.5.4* Representation of Additive Preferences 247 D.6 CB(T) and Uniform Convergence 249 D.6.1 The Basic Metric Structure of CB(T) 249 D.6.2 Uniform Convergence 250 D.6.3* The Stone-Weierstrass Theorem and Separability of C(T) 257 D.6.4* The Arzela-Ascoli Theorem 262 D.7* Extension of Continuous Functions 266 D.8 Fixed Point Theory II 272 D.8.1 The Fixed Point Property 273 D.8.2 Retracts 274 D.8.3 The Brouwer Fixed Point Theorem 277 D.8.4 Applications 280 Chapter E: Continuity II 283 E.1 Correspondences 284 E.2 Continuity of Correspondences 287 E.2.1 Upper Hemicontinuity 287 E.2.2 The Closed Graph Property 294 E.2.3 Lower Hemicontinuity 297 E.2.4 Continuous Correspondences 300 E.2.5* The Hausdorff Metric and Continuity 302 E.3 The Maximum Theorem 306 E.4 Application: Stationary Dynamic Programming 311 E.4.1 The Standard Dynamic Programming Problem 312 E.4.2 The Principle of Optimality 315 E.4.3 Existence and Uniqueness of an Optimal Solution 320 E.4.4 Application: The Optimal Growth Model 324 E.5 Fixed Point Theory III 330 E.5.1 Kakutani's Fixed Point Theorem 331 E.5.2* Michael's Selection Theorem 333 E.5.3* Proof of Kakutani's Fixed Point Theorem 339 E.5.4* Contractive Correspondences 341 E.6 Application: The Nash Equilibrium 343 E.6.1 Strategic Games 343 E.6.2 The Nash Equilibrium 346 E.6.3* Remarks on the Equilibria of Discontinuous Games 351 Part III: ANALYSIS ON LINEAR SPACES 355 Chapter F: Linear Spaces 357 F.1 Linear Spaces 358 F.1.1 Abelian Groups 358 F.1.2 Linear Spaces: Definition and Examples 360 F.1.3 Linear Subspaces, Affine Manifolds, and Hyperplanes 364 F.1.4 Span and Affine Hull of a Set 368 F.1.5 Linear and Affine Independence 370 F.1.6 Bases and Dimension 375 F.2 Linear Operators and Functionals 382 F.2.1 Definitions and Examples 382 F.2.2 Linear and Affine Functions 386 F.2.3 Linear Isomorphisms 389 F.2.4 Hyperplanes, Revisited 392 F.3 Application: Expected Utility Theory 395 F.3.1 The Expected Utility Theorem 395 F.3.2 Utility Theory under Uncertainty 403 F.4* Application: Capacities and the Shapley Value 409 F.4.1 Capacities and Coalitional Games 410 F.4.2 The Linear Space of Capacities 412 F.4.3 The Shapley Value 415 Chapter G: Convexity 422 G.1 Convex Sets 423 G.1.1 Basic Definitions and Examples 423 G.1.2 Convex Cones 428 G.1.3 Ordered Linear Spaces 432 G.1.4 Algebraic and Relative Interior of a Set 436 G.1.5 Algebraic Closure of a Set 447 G.1.6 Finitely Generated Cones 450 G.2 Separation and Extension in Linear Spaces 454 G.2.1 Extension of Linear Functionals 455 G.2.2 Extension of Positive Linear Functionals 460 G.2.3 Separation of Convex Sets by Hyperplanes 462 G.2.4 The External Characterization of Algebraically Closed and Convex Sets 471 G.2.5 Supporting Hyperplanes 473 G.2.6* Superlinear Maps 476 G.3 Reflections on Rn 480 G.3.1 Separation in Rn 480 G.3.2 Support in Rn 486 G.3.3 The Cauchy-Schwarz Inequality 488 G.3.4 Best Approximation from a Convex Set in Rn 489 G.3.5 Orthogonal Complements 492 G.3.6 Extension of Positive Linear Functionals, Revisited 496 Chapter H: Economic Applications 498 H.1 Applications to Expected Utility Theory 499 H.1.1 The Expected Multi-Utility Theorem 499 H.1.2* Knightian Uncertainty 505 H.1.3* The Gilboa-Schmeidler Multi-Prior Model 509 H.2 Applications to Welfare Economics 521 H.2.1 The Second Fundamental Theorem of Welfare Economics 521 H.2.2 Characterization of Pareto Optima 525 H.2.3* Harsanyi's Utilitarianism Theorem 526 H.3 An Application to Information Theory 528 H.4 Applications to Financial Economics 535 H.4.1 Viability and Arbitrage-Free Price Functionals 535 H.4.2 The No-Arbitrage Theorem 539 H.5 Applications to Cooperative Games 542 H.5.1 The Nash Bargaining Solution 542 H.5.2* Coalitional Games without Side Payments 546 Part IV: ANALYSIS ON METRIC/NORMED LINEAR SPACES 551 Chapter I: Metric Linear Spaces 553 I.1 Metric Linear Spaces 554 I.2 Continuous Linear Operators and Functionals 561 I.2.1 Examples of (Dis-)Continuous Linear Operators 561 I.2.2 Continuity of Positive Linear Functionals 567 I.2.3 Closed versus Dense Hyperplanes 569 I.2.4 Digression: On the Continuity of Concave Functions 573 I.3 Finite-Dimensional Metric Linear Spaces 577 I.4* Compact Sets in Metric Linear Spaces 582 I.5 Convex Analysis in Metric Linear Spaces 587 I.5.1 Closure and Interior of a Convex Set 587 I.5.2 Interior versus Algebraic Interior of a Convex Set 590 I.5.3 Extension of Positive Linear Functionals, Revisited 594 I.5.4 Separation by Closed Hyperplanes 594 I.5.5* Interior versus Algebraic Interior of a Closed and Convex Set 597 Chapter J: Normed Linear Spaces 601 J.1 Normed Linear Spaces 602 J.1.1 A Geometric Motivation 602 J.1.2 Normed Linear Spaces 605 J.1.3 Examples of Normed Linear Spaces 607 J.1.4 Metric versus Normed Linear Spaces 611 J.1.5 Digression: The Lipschitz Continuity of Concave Maps 614 J.2 Banach Spaces 616 J.2.1 Definition and Examples 616 J.2.2 Infinite Series in Banach Spaces 618 J.2.3* On the "Size" of Banach Spaces 620 J.3 Fixed Point Theory IV 623 J.3.1 The Glicksberg-Fan Fixed Point Theorem 623 J.3.2 Application: Existence of the Nash Equilibrium, Revisited 625 J.3.3* The Schauder Fixed Point Theorems 626 J.3.4* Some Consequences of Schauder's Theorems 630 J.3.5* Applications to Functional Equations 634 J.4 Bounded Linear Operators and Functionals 638 J.4.1 Definitions and Examples 638 J.4.2 Linear Homeomorphisms, Revisited 642 J.4.3 The Operator Norm 644 J.4.4 Dual Spaces 648 J.4.5* Discontinuous Linear Functionals, Revisited 649 J.5 Convex Analysis in Normed Linear Spaces 650 J.5.1 Separation by Closed Hyperplanes, Revisited 650 J.5.2* Best Approximation from a Convex Set 652 J.5.3 Extreme Points 654 J.6 Extension in Normed Linear Spaces 661 J.6.1 Extension of Continuous Linear Functionals 661 J.6.2* Infinite-Dimensional Normed Linear Spaces 663 J.7* The Uniform Boundedness Principle 665 Chapter K: Differential Calculus 670 K.1 Frechet Differentiation 671 K.1.1 Limits of Functions and Tangency 671 K.1.2 What Is a Derivative? 672 K.1.3 The Frechet Derivative 675 K.1.4 Examples 679 K.1.5 Rules of Differentiation 686 K.1.6 The Second Frechet Derivative of a Real Function 690 K.1.7 Differentiation on Relatively Open Sets 694 K.2 Generalizations of the Mean Value Theorem 698 K.2.1 The Generalized Mean Value Theorem 698 K.2.2* The Mean Value Inequality 701 K.3 Frechet Differentiation and Concave Maps 704 K.3.1 Remarks on the Differentiability of Concave Maps 704 K.3.2 Frechet Differentiable Concave Maps 706 K.4 Optimization 712 K.4.1 Local Extrema of Real Maps 712 K.4.2 Optimization of Concave Maps 716 K.5 Calculus of Variations 718 K.5.1 Finite-Horizon Variational Problems 718 K.5.2 The Euler-Lagrange Equation 721 K.5.3* More on the Sufficiency of the Euler-Lagrange Equation 733 K.5.4 Infinite-Horizon Variational Problems 736 K.5.5 Application: The Optimal Investment Problem 738 K.5.6 Application: The Optimal Growth Problem 740 K.5.7* Application: The Poincare-Wirtinger Inequality 743 Hints for Selected Exercises 747 References 777 Glossary of Selected Symbols 789 Index 793

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Book SynopsisPresents an area of mathematical research that combines topology, geometry, and logic. This book seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity.Trade Review"This is a terrific book. It does no less than introduce an entire new field of mathematics - a truly astounding development. It will be widely read, I think, as much because of the masterful exposition as for the beautiful mathematics. Weinberger gives very clear and accessible descriptions of all the relevant tools from computability, topology, and geometry, in a friendly and engaging style. He has done the mathematical community a great service indeed." - Robin Forman, Rice University; "This book represents a very exciting new area of research at the interface of topology and logic. Written in a quite readable style, and presenting the more accessible cases in detail while giving references for the more involved results, it is a book whose methods and ideas will surely have many more significant applications over the next several years." - Kevin M. Whtye, University of Illinois at Chicago"

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Book SynopsisPresents the theory and the methods of nonlinear optimization, with proofs illustrated by examples and figures. This book covers convex analysis, the theory of optimality conditions, duality theory, and numerical methods for solving unconstrained and constrained optimization problems. It is aimed at graduate students and researchers.Trade Review"This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods. With no doubt the major strength of this book is the clear and intuitive structure and systematic style of presentation. This book can be recommended as a material for both self study and teaching purposes, but because of its rigorous style it works also as a valuable reference for research purposes."--Mathematical Modeling and Operational Research "This is one of the best textbooks on nonlinear optimization I know. Focus is on both theory and algorithmic solution of convex as well as of differentiable programming problems."--Stephan Dempe, Zentralblatt MATH Database "In summary, this book competes with the topmost league of books on optimization. The wide range of topics covered and the thorough theoretical treatment of algorithms make it not only a good prospective textbook, but even more a reference text (which I am happy to have on my shelf.)"--Franz Rendl, Operations Research Letters "Throughout the book the writing style is very clear, compact and easy to follow, but at the same time mathematically rigorous. The proofs are easy to follow because the author usually carefully explains every move. In addition the meaning of the most central results is usually demonstrated with examples and in many cases explanations are also supported by visualizations...This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods...Recommended as a material for both self study and teaching purposes"--Petri Eskelinen, Mathematical Methods of Operation ResearchTable of ContentsPreface xi Chapter 1. Introduction 1 PART 1. THEORY 15 Chapter 2. Elements of Convex Analysis 17 2.1 Convex Sets 17 2.2 Cones 25 2.3 Extreme Points 39 2.4 Convex Functions 44 2.5 Subdifferential Calculus 57 2.6 Conjugate Duality 75 Chapter 3. Optimality Conditions 88 3.1 Unconstrained Minima of Differentiable Functions 88 3.2 Unconstrained Minima of Convex Functions 92 3.3 Tangent Cones 98 3.4 Optimality Conditions for Smooth Problems 113 3.5 Optimality Conditions for Convex Problems 125 3.6 Optimality Conditions for Smooth-Convex Problems 133 3.7 Second Order Optimality Conditions 139 3.8 Sensitivity 150 Chapter 4. Lagrangian Duality 160 4.1 The Dual Problem 160 4.2 Duality Relations 166 4.3 Conic Programming 175 4.4 Decomposition 180 4.5 Convex Relaxation of Nonconvex Problems 186 4.6 The Optimal Value Function 191 4.7 The Augmented Lagrangian 196 PART 2. METHODS 209 Chapter 5. Unconstrained Optimization of Differentiable Functions 211 5.1 Introduction to Iterative Algorithms 211 5.2 Line Search 213 5.3 The Method of Steepest Descent 218 5.4 Newton's Method 233 5.5 The Conjugate Gradient Method 240 5.6 Quasi-Newton Methods 257 5.7 Trust Region Methods 266 5.8 Nongradient Methods 275 Chapter 6. Constrained Optimization of Differentiable Functions 286 6.1 Feasible Point Methods 286 6.2 Penalty Methods 297 6.3 The Basic Dual Method 308 6.4 The Augmented Lagrangian Method 311 6.5 Newton's Method 324 6.6 Barrier Methods 331 Chapter 7. Nondifferentiable Optimization 343 7.1 The Subgradient Method 343 7.2 The Cutting Plane Method 357 7.3 The Proximal Point Method 366 7.4 The Bundle Method 372 7.5 The Trust Region Method 384 7.6 Constrained Problems 389 7.7 Composite Optimization 397 7.8 Nonconvex Constraints 406 Appendix A. Stability of Set-Constrained Systems 411 A.1 Linear-Conic Systems 411 A.2 Set-Constrained Linear Systems 415 A.3 Set-Constrained Nonlinear Systems 418 Further Reading 427 Bibliography 431 Index 445

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Book SynopsisThis is the first full-scale biography of Leonhard Euler (1707-83), one of the greatest mathematicians and theoretical physicists of all time. In this comprehensive and authoritative account, Ronald Calinger connects the story of Euler's eventful life to the astonishing achievements that place him in the company of Archimedes, Newton, and Gauss. DrTrade ReviewOne of Choice's Outstanding Academic Titles for 2016 "Ronald Calinger's impressively detailed biography memorably portrays an extraordinarily able scientist rather than a hero of the Enlightenment as it is conventionally conceived."--Ulinka Rublack, Times Literary Supplement "Leonhard Euler, written by historian of mathematics Ronald Calinger, is perhaps the first biography that attempts to offer a panoramic view of this immense body of work... This impressively researched tome will be of great value to anyone with a serious interest in the history of mathematics and the Enlightenment."--David Castelvecchi, Nature "[A]n impressive work of scientific biography... A fascinating portrait of Euler, his work and the world around him."--The Economist "The book is so rich in information that it makes it the best reference work on Euler that is currently available... This book will be a standard for many years to come."--Adhemar Bultheel, European Mathematical Society "An excellent new biography."--Mark Ronan, Standpoint "This work befits Leonhard Euler, one of the greatest mathematicians ever, and fills in details that raise his stature even further. Calinger is respected for his texts that explore the history of mathematics. This focused effort is his best! Calinger successfully embeds Euler's mathematical and physics results in a rich context--cultural, political, religious, and intellectual--while providing great insight into Euler as a person... Read this book about Euler and be enlightened!"--Choice "This biography gives a complete picture of the person Euler and his scientific work. For the interested reader a must."--Eos "Written in a masterly manner."--Eberhard Knobloch, Zentralblatt MATH "The present monograph is an important contribution on Euler's life and on his achievements in various areas of human knowledge, being of interest to all people interested in the development of science in historical perspective."--S. Cobza, Studia Universitatis Babes-Bolyai MethematicaTable of ContentsPreface ix Acknowledgments xv Author's Notes xvii Introduction 1 1. The Swiss Years: 1707 to April 1727 4 "Das alte ehrwurdige Basel" (Worthy Old Basel) 4 Lineage and Early Childhood 8 Formal Education in Basel 14 Initial Publications and the Search for a Position 27 2. "Into the Paradise of Scholars": April 1727 to 1730 38 Founding Saint Petersburg and the Imperial Academy of Sciences 40 A Fledgling Camp Divided 53 The Entrance of Euler 65 3. Departures, and Euler in Love: 1730 to 1734 82 Courtship and Marriage 87 Groundwork Research and Massive Computations 90 4. Reaching the "Inmost Heart of Mathematics": 1734 to 1740 113 The Basel Problem and the Mechanica 118 The Konigsberg Bridges and More Foundational Work in Mathematics 130 Scientia navalis, Polemics, and the Prix de Paris 140 Pedagogy and Music Theory 150 Daniel Bernoulli and Family 160 5. Life Becomes Rather Dangerous: 1740 to August 1741 165 Another Paris Prize, a Textbook, and Book Sales 165 Health, Interregnum Dangers, and Prussian Negotiations 169 6. A Call to Berlin: August 1741 to 1744 176 "Ex Oriente Lux": Toward a Frederician Era for the Sciences 176 The Arrival of the Grand Algebraist 185 The New Royal Prussian Academy of Sciences 189 Europe's Mathematician, Whom Others Wished to Emulate 200 Relations with the Petersburg Academy of Sciences 211 7. "The Happiest Man in the World": 1744 to 1746 215 Renovation, Prizes, and Leadership 215 Investigating the Fabric of the Universe 224 Contacts with the Petersburg Academy of Sciences 234 Home, Chess, and the King 237 8. The Apogee Years, I: 1746 to 1748 239 The Start of the New Royal Academy 241 The Monadic Dispute, Court Relations, and Accolades 247 Exceeding the Pillars of Hercules in the MathematicalSciences 255 Academic Clashes in Berlin, and Euler's Correspondence with the Petersburg Academy 279 The Euler Family 282 9. The Apogee Years, II: 1748 to 1750 285 The Introductio and Another Paris Prize 287 Competitions and Disputes 292 Decrial, Tasks, and Printing Scientia navalis 298 A Sensational Retraction and Discord 303 State Projects and the "Vanity of Mathematics" 308 The Konig Visit and Daily Correspondence 313 Family Affairs 316 10. The Apogee Years, III: 1750 to 1753 318 Competitions in Saint Petersburg, Paris, and Berlin 320 Maupertuis's Cosmologie and Selected Research 325 Academic Administration 329 Family Life and Philidor 333 Rivalries: Euler, d'Alembert, and Clairaut 335 The Maupertuis-Konig Affair: The Early Second Phase 337 Two Camps, Problems, and Inventions 344 Botany and Maps 348 The Maupertuis-Konig Affair: The Late Second and Early Third Phases 350 Planetary Perturbations and Mechanics 359 Music, Rameau, and Basel 360 Strife with Voltaire and the Academy Presidency 363 11. Increasing Precision and Generalization in the Mathematical Sciences: 1753 to 1756 368 The Dispute over the Principle of Least Action: The Third Phase 369 Administration and Research at the Berlin Academy 374 The Charlottenburg Estate 384 Wolff, Segner, and Mayer 385 A New Correspondent and Lessons for Students 391 Institutiones calculi differentialis and Fluid Mechanics 395 A New Telescope, the Longitude Prize, Haller, and Lagrange 399 Anleitung zur Nauturlehre and Electricity and Optimism Prizes 401 12. War and Estrangement, 1756 to July 1766 404 The Antebellum Period 404 Into the Great War and Beyond 409 Losses, Lessons, and Leadership 415 Rigid-Body Disks, Lambert, and Better Optical Instruments 427 The Presidency of the Berlin Academy 430 What Soon Happened, and Denouement 432 13. Return to Saint Petersburg: Academy Reform and Great Productivity, July 1766 to 1773 451 Restoring the Academy: First Efforts 452 The Grand Geometer: A More Splendid Oeuvre 456 A Further Research Corpus: Relentless Ingenuity 471 The Kulibin Bridge, the Great Fire, and One Fewer Distraction 485 Persistent Objectives: To Perfect, to Create, and to Order 488 14. Vigorous Autumnal Years: 1773 to 1782 495 The Euler Circle 496 Elements of Number Theory and Second Ship Theory 497 The Diderot Story and Katharina's Death 499 The Imperial Academy: Projects and Library 502 The Russian Navy, Turgot's Request, and a Successor 504 At the Academy: Technical Matters and a New Director 506 A Second Marriage and Rapprochement with Frederick II 509 End of Correspondence and Exit from the Academy 515 Mapmaking and Prime Numbers 517 A Notable Visit and Portrait 518 Magic Squares and Another Honor 520 15. Toward "a More Perfect State of Dreaming": 1782 to October 1783 526 The Inauguration of Princess Dashkova 526 1783 Articles 529 Final Days 530 Major Eulogies and an Epilogue 532 Notes 537 General Bibliography of Works Consulted 571 Register of Principal Names 625 General Index 657

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Book SynopsisFocuses on heat equations associated with non self-adjoint uniformly elliptic operators. This book provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. It then treats Lp properties of solutions to a wide class of heat equations.Trade Review"This book is both an excellent introduction for those learning about heat operators for the first time, and a reference work for the mathematician searching for information. The author has presented an especially lucid exposition of the subject." - Alan McIntosh, Australian National University; "This book contains very interesting material, starting with the basics and progressing to lively trends of current research." - Thierry Coulhon, Cergy-Pontoise University"Table of ContentsPreface ix Notation xiii Chapter 1. SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS 1 1.1 Bounded sesquilinear forms 1 1.2 Unbounded sesquilinear forms and their associated operators 3 1.3 Semigroups and unbounded operators 18 1.4 Semigroups associated with sesquilinear forms 29 1.5 Correspondence between forms, operators, and semigroups 38 Chapter 2. CONTRACTIVITY PROPERTIES 43 2.1 Invariance of closed convex sets 44 2.2 Positive and Lp-contractive semigroups 49 2.3 Domination of semigroups 58 2.4 Operations on the form-domain 64 2.5 Semigroups acting on vector-valued functions 68 2.6 Sesquilinear forms with nondense domains 74 Chapter 3. INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS 79 3.1 Sub-Markovian semigroups and Kato type inequalities 79 3.2 Further inequalities and the corresponding domain in Lp 88 3.3 Lp-holomorphy of sub-Markovian semigroups 95 Chapter 4. UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS 99 4.1 Examples of boundary conditions 99 4.2 Positivity and irreducibility 103 4.3 L1-contractivity 107 4.4 The conservation property 120 4.5 Domination 125 4.6 Lp-contractivity for 1 134 4.7 Operators with unbounded coefficients 137 Chapter 5. DEGENERATE-ELLIPTIC OPERATORS 143 5.1 Symmetric degenerate-elliptic operators 144 5.2 Operators with terms of order 1 145 Chapter 6. GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS 155 6.1 Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities 155 6.2 Holder-continuity estimates of the heat kernel 160 6.3 Gaussian upper bounds 163 6.4 Sharper Gaussian upper bounds 174 6.5 Gaussian bounds for complex time and Lp-analyticity 180 6.6 Weighted gradient estimates 185 Chapter 7. GAUSSIAN UPPER BOUNDS AND Lp-SPECTRAL THEORY 193 7.1 Lp-bounds and holomorphy 196 7.2 Lp-spectral independence 204 7.3 Riesz means and regularization of the Schrodinger group 208 7.4 Lp-estimates for wave equations 214 7.5 Singular integral operators on irregular domains 228 7.6 Spectral multipliers 235 7.7 Riesz transforms associated with uniformly elliptic operators 240 7.8 Gaussian lower bounds 245 Chapter 8. A REVIEW OF THE KATO SQUARE ROOT PROBLEM 253 8.1 The problem in the abstract setting 253 8.2 The Kato square root problem for elliptic operators 257 8.3 Some consequences 261 Bibliography 265 Index 283

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Book SynopsisPresenting mathematical ideas of people from a variety of small-scale and traditional cultures, this book humanizes our view of mathematics and expands our conception of what is mathematical. It demonstrates that traditional cultures have mathematical ideas that are far more substantial and sophisticated than is generally acknowledged.Trade ReviewHonorable Mention for the 2003 Award for Best Professional/Scholarly Book in Mathematics and Statistics, Association of American Publishers "A useful reminder of how universal mathematical and logical structures are in any culture. Mathematicians will enjoy seeing the subject they love cropping up in apparently unexpected contexts. Non-mathematicians should be encouraged to realize that some of the processes that seem to appear naturally in everyday life do in fact have a mathematical content."--John O'Connor, Nature "For a mathematician, Mathematics Elsewhere will expand the universe; for a non-mathematician, the expansion will just take a little more time. The book succeeds well in presenting and explaining very different ways of doing math both within particular cultural contexts and in terms of modern mathematics... The author is clearly an excellent teacher and a wonderful explainer. Every time I felt a bit lost, the next sequence would present the same concept in different words or with another example. She is adept at moving from the general to the specific, from narrative to figurative."--Helaine Selin, Science "This interesting book is a fundamental work in the area of ethnomathematics... [T]he author opens numerous doors and directions in which one finds interesting, nontrivial mathematics. Persons interested in investigating the mathematics of non-Western cultures can use this book as a motivation to look beyond the obvious."--Thomas E. Golsdorf, Mathematical Reviews "Ascher illustrates that non-Western cultures have developed sophisticated mathematical ideas often without having any formal concept of mathematics. This stimulating book deserves a wide audience, especially among those involved in teaching the subject."--Andrew Bowler, New Scientist "In a follow-up to Ascher's highly recommended Ethnomathematics, this scholarly work describes the anthropology of mathematical ideas in traditional societies and shows how the same ideas might be expressed by standard mathematical expressions... It is particularly interesting to see how people with no separate mathematical language made practical use of sophisticated mathematical ideas."--Library Journal "All throughout the book, I was struck by how many uses human cultures have found for modular arithmetic... [I]t appears that mathematics may be an essential survival skill for the human species rather than an extraneous one. The descriptions in this book describe so many different applications, that it becomes hard to deny that something more fundamental is responsible for the many ways we find to person mathematical operations."--Charles Ashbacher, MAA Online "Ascher's spendid book is rich in possibilities for raising readers' horizons: anthropological, educational, mathematical, and philosophical."--Philip J. Davis, SIAM News "Ascher's book is at once a scholarly progress report and an introduction for the curious general reader to a relatively new area of study known as ethnomathematics... Ascher offers a new way of understanding the customs and traditions of non-Western people, adding the lens of mathematics to those of literature, anthropology, and sociology... [She] proves adept at illuminating the connections between local and global mathematics... Part of what makes the volume accessible to the general reader ... is Ascher's evident love for her subject. The mathematics she includes clearly serves a larger purpose: to enhance and illuminate the anecdotes that are the foundation of genuine cultural understanding."--James V. Rauff, Natural HistoryTable of ContentsPreface ix Introduction 1 CHAPTER 1: The Logic of Divination 5 CHAPTER 2: Marking Time 39 CHAPTER 3: Cycles of Time 59 CHAPTER 4: Models and Maps 89 CHAPTER 5: Systems of Relationships 127 CHAPTER 6: Figures on the Threshold 161 CHAPTER 7: Epilogue 191 Index 205

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Book SynopsisIn a manner accessible to beginning undergraduates, this book introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, and more. Showing how experiments are used to test conjectures and prove theorems, it allows students to do original work on such problems.Trade Review"This is a great book... [I]t is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory."--Henry Ricardo, MAA Reviews "This is the first text to present Random Matrix Theory and the Circle Method for German primes. This well-written book supplements classic texts by showing connections between seemingly diverse topics, by making the subject accessible to beginning students and by whetting their appetite for continuing in mathematics"--Mathematical Reviews "I would highly recommend this book to anybody interested in number theory, from an undergraduate student to an established expert, since everybody will be able to find in this book lots of new interesting material, tempting problems, and interesting computational challenges. It could also be used as a textbook for a graduate course in number theory. To promote and stimulate independent research, it contains many very interesting exercises and even suggestions for research projects."--Igor Shparlinski, SIAM ReviewTable of ContentsForeword xi Preface xiii Notation xix PART 1. BASIC NUMBER THEORY 1 Chapter 1. Mod p Arithmetic, Group Theory and Cryptography 3 Chapter 2. Arithmetic Functions 29 Chapter 3. Zeta and L-Functions 47 Chapter 4. Solutions to Diophantine Equations 81 PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS 107 Chapter 5. Algebraic and Transcendental Numbers 109 Chapter 6. The Proof of Roth's Theorem 137 Chapter 7. Introduction to Continued Fractions 158 PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION 189 Chapter 8. Introduction to Probability 191 Chapter 9. Applications of Probability: Benford's Law and Hypothesis Testing 216 Chapter 10. Distribution of Digits of Continued Fractions 231 Chapter 11. Introduction to Fourier Analysis 255 Chapter 12. f n k g and Poissonian Behavior 278 PART 4. THE CIRCLE METHOD 301 Chapter 13. Introduction to the Circle Method 303 Chapter 14. Circle Method: Heuristics for Germain Primes 326 PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357 Chapter 15. From Nuclear Physics to L-Functions 359 Chapter 16. Random Matrix Theory: Eigenvalue Densities 391 Chapter 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues 405 Chapter 18. The Explicit Formula and Density Conjectures 421 Appendix A. Analysis Review 439 Appendix B. Linear Algebra Review 455 Appendix C. Hints and Remarks on the Exercises 463 Appendix D. Concluding Remarks 475 Bibliography 476 Index 497

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Book SynopsisPresents an overview of the developments in the area of localization for quasi-periodic lattice Schrodinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. This book emphasises on so-called 'non-perturbative' methods and the role of subharmonic function theory and semi-algebraic set methods.Trade Review"This text is an up to date introduction to localization problems for lattice Schrodinger operations with deterministic ergodic potentials by one of the leading experts... I can recommend it to any graduate student or researcher in the field."--G. Teschl, Monatschefte fur MathematikTable of ContentsAcknowledgment v CHAPTER 1: Introduction 1 CHAPTER 2: Transfer Matrix and Lyapounov Exponent 11 CHAPTER 3: Herman's Subharmonicity Method 15 CHAPTER 4: Estimates on Subharmonic Functions 19 CHAPTER 5: LDT for Shift Model 25 CHAPTER 6: Avalanche Principle in SL2( R ) 29 CHAPTER 7: Consequences for Lyapounov Exponent, IDS, and Green's Function 31 CHAPTER 8: Refinements 39 CHAPTER 9: Some Facts about Semialgebraic Sets 49 CHAPTER 10: Localization 55 CHAPTER 11: Generalization to Certain Long-Range Models 65 CHAPTER 12: Lyapounov Exponent and Spectrum 75 CHAPTER 13: Point Spectrum in Multifrequency Models at Small Disorder 87 CHAPTER 14: A Matrix-Valued Cartan-Type Theorem 97 CHAPTER 15: Application to Jacobi Matrices Associated with Skew Shifts 105 CHAPTER 16: Application to the Kicked Rotor Problem 117 CHAPTER 17: Quasi-Periodic Localization on the Z d -lattice ( d > 1) 123 CHAPTER 18: An Approach to Melnikov's Theorem on Persistency of Non-resonant Lower Dimension Tori 133 CHAPTER 19: Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrodinger Equations 143 CHAPTER 20: Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations 159 Appendix 169

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Book SynopsisOffers an introduction to mathematical models in epidemiology and shows how they can be used to predict and control the geographic spread of major infectious diseases. This work explains the key concepts in infectious disease modeling, guides readers from simple mathematical models to more complex ones, and explores their strengths and weaknesses.Trade Review"Sattenspiel and Lloyd do a first-rate job of making a lot of material accessible to a broad audience. They focus on a handful of examples and provide comprehensive insights. I found this book to be tightly and cogently written, supplying a level of detail that will be really useful for advanced undergraduates, graduate students, and researchers. It is one I would certainly recommend."—Andrew P. Dobson, Princeton UniversityTable of ContentsPreface ix Chapter 1. Introduction 1 1.1 Mathematical Models and the Geographic Spread of Epidemics 5 1.2 Structure of this Book 11 Chapter 2. The Art of Epidemic Modeling: Concepts and Basic Structures 12 2.1 Essential Biological and Epidemiological Concepts 12 2.2 The Cornerstone of Many Epidemic Models | the SIR Model 16 2.3 Demography and Epidemic Models 23 2.4 More Complex Models 25 2.5 The Basic Reproductive Number Revisited 53 Chapter 3. Modeling the Geographic Spread of Inuenza Epidemics 58 3.1 A Brief Overview of the Biology of Inuenza 58 3.2 Population-based Inuenza Models 61 3.3 Individual-based Inuenza Models 77 3.4 So What Kind of Model Should One Use to Study Inuenza Transmission? 84 Chapter 4. Modeling Geographic Spread I: Population-based Approaches 86 4.1 Spatial Structure and Disease Transmission: Basic Themes 86 4.2 Spatial Modeling Frameworks 89 4.3 Metapopulation Models 90 4.4 Spatially Continuous Models 102 Chapter 5. Spatial Heterogeneity and Endemicity: The Case of Measles 117 5.1 The Persistence and Long-term Cycling of Measles 122 5.2 Spatial Heterogeneity, Synchrony, and the Spatial Spread of Measles 125 Chapter 6. Modeling Geographic Spread II: Individual-based Approaches 134 6.1 Historical Underpinnings of the Use of Networks in Epidemiology 137 6.2 The Nature of Networks 140 6.3 The Language of Network Analysis 142 6.4 Major Classes of Networks 150 6.5 The Inuence of Networks on the Dynamics of Epidemic Spread 159 6.6 Theoretical Analysis of Network Models 162 6.7 The Basic Reproductive Number in Network Models 168 6.8 Infection Control on Networks 171 6.9 Why Aren't There More Applications of Network Models for Spatial Spread? 173 Chapter 7. Spatial Models and the Control of Foot-and-Mouth Disease 176 7.1 Modeling the Geographic Spread of FMD 180 7.2 The Official Response to the Epidemic and Its Aftermath 185 Chapter 8. Maps, Projections, and GIS: Geographers' Approaches 191 8.1 Mapping Methods 191 8.2 Identifying Patterns of Disease Di_usion 195 8.3 Epidemic Projections 204 8.4 Detection of Disease Clustering 208 8.5 New and Potential Directions 211 Chapter 9. Revisiting SARS and Looking to the Future 215 9.1 Did Mathematical Modeling Help to Stop the 2003 SARS Epidemic? 215 9.2 Modeling the Geographic Spread of Past, Present, and Future Infectious Disease Epidemics: Lessons and Advice 223 Bibliography 237 Index 279

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Book SynopsisOffers a different approach to introductory scientific computing. This title aims to make students comfortable using computers to do science, to provide them with the computational tools and knowledge they need throughout their college careers and into their professional careers, and to show how all the pieces can work together.Trade ReviewOne of Choice's Outstanding Academic Titles for 2005 "Essential... Rubin Landau offers a practical introduction to the world of scientific computing or numerical analysis. He introduces not only the concepts of numerical analysis, but also more importantly the tools that can be used to perform scientific computing... The presentation is particularly useful because real-life examples with real code and results are included."--Choice "Not only is [this book] an invaluable learning text and an essential reference for students of mathematics, engineering, physics, and other sciences, but it is also a consummate model for future textbooks in computational science and engineering courses."--Mathematical Reviews "The contents can be taught in lab-based courses at the undergraduate level. Much of the material covered is usually addressed in separate books. Therefore, the book is also suitable for independent study by graduate students and professional who wish to learn one or more of the languages in a comprehensive way with the emphasis kept on problem-solving."--Frits Agterberg, Computers and Geosciences "The colloquial and tutorial approach might help alleviate the many practical problems associated with incorporating computational applications into a more traditional lecture environment. The text provides many concrete and programming examples in action and illustrates how much you can accomplish with a few well-chosen tools... [S]tudents impressed with the text's workbook style and reference book quality will add it to their bookshelves and return to it often."--Michael Jay Schillaci, IEEE Computing in Science and EngineeringTable of ContentsList of Figures xv List of Tables xix Preface xxi Chapter 1. Introduction 1 1.1 Nature of Scientific Computing 1 1.2 Talking to Computers 2 1.3 Instructional Guide 4 1.4 Exercises to Come Back To 6 PART 1. MAPLE (OR MATHEMATICA) BY DOING 7 Chapter 2. Getting Started with Maple 9 2.1 Setting Up Your Work Space 9 2.2 Maple's Problem-Solving Environment 10 2.3 Maple's Command Structure 14 2.4 Sums and sums 16 2.5 Execution Groups 21 2.6 Key Words and Concepts 22 2.7 Supplementary Exercises 23 Chapter 3. Numbers, Expressions, Functions; Rocket Golf 25 3.1 Problem: Viewing Rocket Golf 25 3.2 Theory: Einstein's Special Relativity 26 3.3 Math: Integer, Rational and Irrational Numbers 27 3.4 CS: Floating-Point Numbers 29 3.5 Complex Numbers 31 3.6 Expressions 32 3.7 Assignment Statements 34 3.8 Equality (rhs, lhs) 36 3.9 Functions 36 3.10 User-Defined Functions 39 3.11 Reexpressing Answers 39 3.12 CS: Overflow, Underflow, and Round-Off Error 44 3.13 Solution: Viewing Rocket Golf 45 3.14 Extension: Tachyons* 50 3.15 Key Words and Concepts 51 3.16 Supplementary Exercises 51 Chapter 4. Visualizing Data, Abstract Types; Electric Fields 55 4.1 Why Visualization? 55 4.2 Problem: Stable Points in Electric Fields 56 4.3 Theory: Stability Criteria and Potential Energy 56 4.4 Basic 2-D Plots: plot 58 4.5 Compound (Abstract) Data Types: [Lists] and {Sets } 63 4.6 3-D (Surface) Plots of Analytic Functions 69 4.7 Solution: Dipole and Quadrupole Fields 73 4.8 Exploration: The Tripole 76 4.9 Extension: Yet More Plot Types* 76 4.10 Visualizing Numerical Data 91 4.11 Plotting a Matrix: matrixplot* 97 4.12 Animations of Data* 102 4.13 Key Words and Concepts 104 4.14 Supplementary Exercises 105 Chapter 5. Solving Equations, Differentiation; Towers 107 5.1 Problem: Maximum Height of a Tower 107 5.2 Model: Block Stacking 107 5.3 Math: Equations as Challenges 109 5.4 Solving a Single Equation: solve, fsolve 110 5.5 Solving Simultaneous Equations (Sets) 113 5.6 Solution to Tower Problem 115 5.7 Differentiation: limit, diff, D 117 5.8 Numerical Derivatives* 126 5.9 Alternate Solution: Maximum Tower Height 127 5.10 Assessment and Exploration 128 5.11 Auxiliary Problem: Nonlinear Oscillations 129 5.12 Key Words and Concepts 131 5.13 Supplementary Exercises 131 Chapter 6. Integration; Power and Energy Usage (Also 14) 134 6.1 Problem: Relating Power and Energy Usage 134 6.2 Empirical Models 134 6.3 Theory: Power and Energy Definitions 136 6.4 Maple: Tools for Integration 136 6.5 Problem Solution: Energy from Power 139 6.6 Key Words and Concepts 143 6.7 Supplementary Exercises 144 Chapter 7. Matrices and Vectors; Rotation 145 7.1 Problem: Rigid-Body Rotation 145 7.2 Math: Vectors and Matrices 147 7.3 Theory: Angular Momentum Dynamics 149 7.4 Maple: Linear Algebra Tools 151 7.5 Matrix Arithmetic and Operations 157 7.6 Solution: Rotating Rigid Bodies 171 7.7 Exploration: Principal Axes of Rotation* 176 7.8 Key Words and Concepts 181 7.9 Supplementary Exercises 182 Chapter 8. Searching, Programming; Dipsticks 184 8.1 Problem: Volume of Liquid in Spherical Tanks 184 8.2 Math: Volume Integration 184 8.3 Algorithm: Bisection Searches 185 8.4 Programming in Maple 187 8.5 Solution: Volume from Dipstick Height 194 8.6 Key Words and Concepts 195 8.7 Supplementary Exercises 196 PART 2. JAVA (OR FORTRAN90) BY DOING 197 Chapter 9. Getting Started with Java 199 9.1 Compiled Languages 199 9.2 Java Program Pieces 201 9.3 Entering and Running Your First Program 202 9.4 Looking Inside Area.java 205 9.5 Key Words 207 9.6 Supplementary Exercises 207 Chapter 10. Data Types, Limits, Methods; Rocket Golf 208 10.1 Problem and Theory (Same as Chapter 3) 208 10.2 Java's Primitive Data Types 208 10.3 Methods (Functions) and Modular Programming 215 10.4 Solution: Viewing Rocket Golf 219 10.5 Your Problem: Modify Golf.java 223 10.6 Coercion and Overloading* 224 10.7 Key Words 229 Chapter 11. Visualization with Java, Classes, Packages 232 11.1 2-D Graphs within Java: PtPlot 232 11.2 Installing PtPlot: See Appendix C* 238 11.3 Classes and Packages* 238 11.4 Gnuplot Basics 240 11.5 Java Archives: jar* 244 Chapter 12. Flow Control via Logic; Projectiles 247 12.1 Problem: Frictionless Projectile Motion 247 12.2 Theory: Kinematics 248 12.3 Computer Science: Designing Structured Programs 249 12.4 Flow Control via Logic 251 12.5 Implementation: Projectile.java 258 12.6 Solution: Projectile Trajectories 259 12.7 Key Words 259 12.8 Supplementary Exercises 260 Chapter 13. Java Input and Output* 262 13.1 Basic Input with Scanner 263 13.2 Streams: Standard Output, Input, and Error 263 13.3 I/O Exceptions: FileCatchThrow.java 272 13.4 Automatic Code Documentation: javadoc 274 13.5 Nonstandard Formatted Output: printf 275 Chapter 14. Numerical Integration; Power and Energy Usage 281 14.1 Problem (Same as Chapter 6): Power and Energy 281 14.2 Algorithms: Trapezoid and Simpson's Rules 282 14.3 Assessment: Which Rule Is Better? 288 14.4 Key Words and Concepts 289 14.5 Supplementary Exercises 289 Chapter 15. Differential Equations with Java and Maple* 290 15.1 Problem: Projectile Motion with Drag 290 15.2 Model: Velocity-Dependent Drag 291 15.3 Algorithm: Numerical Differentiation 292 15.4 Math: Solving Differential Equations 292 15.5 Assessment: Balls Falling Out of the Sky? 295 15.6 Maple: Differential-Equation Tools 297 15.7 Maple Solution: Drag Velocity 302 15.8 Extract Operands 303 15.9 Drag v2 (Exercise) 306 15.10 Drag v3/2 306 15.11 Exploration: Planetary Motion* 310 15.12 Key Words 311 15.13 Supplementary Exercises 311 Chapter 16. Object-Oriented Programming; Complex Currents 313 16.1 Problem: Resonance in RLC Circuit 313 16.2 Math: Complex Numbers 313 16.3 Theory: Resistance Becomes Impedance 317 16.4 CS: Abstract Data Types, Objects 319 16.5 Java Solution: Complex Currents 329 16.6 Maple Solution: Complex Currents 330 16.7 Explorations: OOP Worked Examples* 334 16.8 Key Words 340 16.9 Java and Maple Exercises 340 Chapter 17. Arrays: Vectors, Matrices; Rigid-Body Rotations 341 17.1 Problem: Rigid-Body Rotations 341 17.2 Theory: Angular-Momentum Dynamics 343 17.3 CS, Math: Arrays, Vectors, and Matrices 344 17.4 Implementation: Inertia.java, Inertia3D.java 347 17.5 Jama: Java Matrix Library* 349 17.6 Key Words 353 17.7 Supplementary Exercises 353 Chapter 18. Advanced Objects; Baton Projectiles* 355 18.1 Problem: Trajectory of Thrown Baton 355 18.2 Theory: Combined Translation and Rotation 356 18.3 CS: OOP Design Concepts 359 18.4 Key Words 377 18.5 Supplementary Exercises 377 Chapter 19. Discrete Math, Arrays as Bins; Bug Dynamics* 378 19.1 Problem: Variability of Bug Populations 378 19.2 Theory: Self-Limiting Growth, Discrete Maps 378 19.3 Assessment: Properties of Nonlinear Maps 380 19.4 Exploration: Bifurcation Diagram, BugSort.java* 381 19.5 Exploration: Other Discrete Maps* 384 Chapter 20. 2-D Arrays: File I/O, PDEs; Realistic Capacitor 385 20.1 Problem: Field of Realistic Capacitor 385 20.2 Theory and Model: Electrostatics and PDEs 385 20.3 Algorithm: Finite Differences 387 20.4 Implementation: Laplace.java 389 20.5 Exploration: 2-D Capacitor 391 20.6 Exploration: 3-D Capacitor* 393 20.7 Key Words 393 Chapter 21. Web Computing, Applets, Primitive Graphics 394 21.1 What Is Web Computing? 394 21.2 Implementation: Get This to Work First 396 21.3 Exploration: Modify Applet1.java 401 21.4 Extension: PtPlot as Applet* 402 21.5 Extension: Applet with Button Input* 403 21.6 Extension: AWT, JFC, and Swing* 405 21.7 Example: Baton Applet, Jparabola.java* 407 21.8 Key Words 410 21.9 Supplementary Exercises 410 PART 3. LATEX SURVIVAL GUIDE 411 Chapter 22. LATEX for Text 413 22.1 Why LATEX? 413 22.2 Structure of a LATEXDocument 414 22.3 Sample Input File (Sample.tex) 414 22.4 Sample LATEXOutput 416 22.5 Fonts for Text 420 22.6 Environments 422 22.7 Lists 422 22.8 Sections 425 Chapter 23. LATEX for Mathematics 427 23.1 Entering Mathematics: Math Mode 427 23.2 Mathematical Symbols and Greek 428 23.3 Math Accents 431 23.4 Superscripts and Subscripts 431 23.5 Calculus and Sums 431 23.6 Changing Math Fonts 432 23.7 Math Functions 432 23.8 Fractions 432 23.9 Roots 433 23.10 Brackets (Delimiters) 433 23.11 Multiline Equations 434 23.12 Matrices and Math Arrays 435 23.13 Including Graphics 436 23.14 Exercise: Putting It All Together 438 Appendix A. Glossary 441 Appendix B. Maple Quick Reference, Debugging Help 450 Appendix C. Java Quick Reference and Installing Software 461 C.1 Java Elements 461 C.2 Transferring Files from the CD 465 C.3 Using our Maple Worksheets 466 C.4 Using our Java Programs 466 C.5 Installing PtPlot (or Other) Packages 467 C.6 Installing Java Developer's Kit 469 Bibliography 471 Index 477

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Book SynopsisSurveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. This work considers every proposed fix, each with its distinctive philosophical advantages and drawbacks.Trade ReviewCo-Winner of the 2007 Shoenfield Prize, Association for Symbolic Logic "Fixing Frege fills a serious gap in the Frege's literature (always increasing but perhaps with an excessive attention paid to semantics and the philosophy of language) and should remain for a long time a necessary reference for scholars in the field."--Ignacio Angelelli, Review of Modern LogicTable of ContentsAcknowledgments ix CHAPTER 1: Frege, Russell, and After 1 CHAPTER 2: Predicative Theories 86 CHAPTER 3: Impredicative Theories 146 Tables 215 Notes 227 References 241 Index 249

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Book SynopsisIn this book, the author introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. It begins with brief introductions to informal set theory and general topology, and avoids advanced algebra; thus it is self-contained and suitable for readers with little background in mathematics.Trade Review"We warmly welcome this book as an example of how the mathematical way of thinking can be made available and pleasant to a large group of students."--Solomon Marcus, Zentralblatt MATHTable of ContentsA Preliminaries 1 Chapter 1: Introduction for teachers 3 Purpose and intended audience, 3 Topics in the book, 6 Why pluralism?, 13 Feedback, 18 Acknowledgments, 19 Chapter 2: Introduction for students 20 Who should study logic?, 20 Formalism and certication, 25 Language and levels, 34 Semantics and syntactics, 39 Historical perspective, 49 Pluralism, 57 Jarden's example (optional), 63 Chapter 3: Informal set theory 65 Sets and their members, 68 Russell's paradox, 77 Subsets, 79 Functions, 84 The Axiom of Choice (optional), 92 Operations on sets, 94 Venn diagrams, 102 Syllogisms (optional), 111 Infinite sets (postponable), 116 Chapter 4: Topologies and interiors (postponable) 126 Topologies, 127 Interiors, 133 Generated topologies and finite topologies (optional), 139 Chapter 5: English and informal classical logic 146 Language and bias, 146 Parts of speech, 150 Semantic values, 151 Disjunction (or), 152 Conjunction (and), 155 Negation (not), 156 Material implication, 161 Cotenability, fusion, and constants (postponable), 170 Methods of proof, 174 Working backwards, 177 Quantifiers, 183 Induction, 195 Induction examples (optional), 199 Chapter 6: Definition of a formal language 206 The alphabet, 206 The grammar, 210 Removing parentheses, 215 Defined symbols, 219 Prefix notation (optional), 220 Variable sharing, 221 Formula schemes, 222 Order preserving or reversing subformulas (postponable), 228 B Semantics 233 Chapter 7: Definitions for semantics 235 Interpretations, 235 Functional interpretations, 237 Tautology and truth preservation, 240 Chapter 8: Numerically valued interpretations 245 The two-valued interpretation, 245 Fuzzy interpretations, 251 Two integer-valued interpretations, 258 More about comparative logic, 262 More about Sugihara's interpretation, 263 Chapter 9: Set-valued interpretations 269 Powerset interpretations, 269 Hexagon interpretation (optional), 272 The crystal interpretation, 273 Church's diamond (optional), 277 Chapter 10: Topological semantics (postponable) 281 Topological interpretations, 281 Examples, 282 Common tautologies, 285 Nonredundancy of symbols, 286 Variable sharing, 289 Adequacy of finite topologies (optional), 290 Disjunction property (optional), 293 Chapter 11: More advanced topics in semantics 295 Common tautologies, 295 Images of interpretations, 301 Dugundji formulas, 307 C Basic syntactics 311 Chapter 12: Inference systems 313 Chapter 13: Basic implication 318 Assumptions of basic implication, 319 A few easy derivations, 320 Lemmaless expansions, 326 Detachmental corollaries, 330 Iterated implication (postponable), 332 Chapter 14: Basic logic 336 Further assumptions, 336 Basic positive logic, 339 Basic negation, 341 Substitution principles, 343 D One-formula extensions 349 Chapter 15: Contraction 351 Weak contraction, 351 Contraction, 355 Chapter 16: Expansion and positive paradox 357 Expansion and mingle, 357 Positive paradox (strong expansion), 359 Further consequences of positive paradox, 362 Chapter 17: Explosion 365 Chapter 18: Fusion 369 Chapter 19: Not-elimination 372 Not-elimination and contrapositives, 372 Interchangeability results, 373 Miscellaneous consequences of not-elimination, 375 Chapter 20: Relativity 377 E Soundness and major logics 381 Chapter 21: Soundness 383 Chapter 22: Constructive axioms: avoiding not-elimination 385 Constructive implication, 386 Herbrand-Tarski Deduction Principle, 387 Basic logic revisited, 393 Soundness, 397 Nonconstructive axioms and classical logic, 399 Glivenko's Principle, 402 Chapter 23: Relevant axioms: avoiding expansion 405 Some syntactic results, 405 Relevant deduction principle (optional), 407 Soundness, 408 Mingle: slightly irrelevant, 411 Positive paradox and classical logic, 415 Chapter 24: Fuzzy axioms: avoiding contraction 417 Axioms, 417 Meredith's chain proof, 419 Additional notations, 421 Wajsberg logic, 422 Deduction principle for Wajsberg logic, 426 Chapter 25: Classical logic 430 Axioms, 430 Soundness results, 431 Independence of axioms, 431 Chapter 26: Abelian logic 437 F Advanced results 441 Chapter 27: Harrop's principle for constructive logic 443 Meyer's valuation, 443 Harrop's principle, 448 The disjunction property, 451 Admissibility, 451 Results in other logics, 452 Chapter 28: Multiple worlds for implications 454 Multiple worlds, 454 Implication models, 458 Soundness, 460 Canonical models, 461 Completeness, 464 Chapter 29: Completeness via maximality 466 Maximal unproving sets, 466 Classical logic, 470 Wajsberg logic, 477 Constructive logic, 479 Non-finitely-axiomatizable logics, 485 References 487 Symbol list 493 Index 495

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Book SynopsisRelates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. This book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings.Trade Review"[Richard Epstein] never gives only the technical side of the matter, but...always offers intuitive motivations and explains basic decisions which constitute the whole approach, and which build a bridge to the students' experiences, with natural language as well as with standard 'elementary' mathematics. The book is a self-contained textbook, requiring as background only some facility in mathematics...This makes the book particularly suitable as a textbook for self-study."--Siegfried J. Gottwald, Zentralblatt MATHTable of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. xvii*Acknowledgments, pg. xix*Introduction, pg. xxi*I. Classical Propositional Logic, pg. 1*II. Abstracting and Axiomatizing Classical Propositional Logic, pg. 27*III. The Language of Predicate Logic, pg. 53*IV. The Semantics of Classical Predicate Logic, pg. 69*V. Substitutions and Equivalences, pg. 99*VI. Equality, pg. 113*VII. Examples of Formalization, pg. 121*VIII. Functions, pg. 139*IX. The Abstraction of Models, pg. 153*X. Axiomatizing Classical Predicate Logic, pg. 167*XI. The Number of Objects in the Universe of a Model, pg. 183*XII. Formalizing Group Theory, pg. 191*XIII. Linear Orderings, pg. 207*XIV. Second-Order Classical Predicate Logic, pg. 225*XV. The Natural Numbers, pg. 263*XVI. The Integers and Rationals, pg. 291*XVII. The Real Numbers, pg. 303*XVIII. One-Dimensional Geometry, pg. 331*XIX. Two-Dimensional Euclidean Geometry, pg. 363*XX. Translations within Classical Predicate Logic, pg. 403*XXI. Classical Predicate Logic with Non-Referring Names, pg. 413*XXII. The Liar Paradox, pg. 437*XXIII. On Mathematical Logic and Mathematics, pg. 461*Appendix: The Completeness of Classical Predicate Logic Proved by Godel's Method, pg. 465*Summary of Formal Systems, pg. 475*Bibliography, pg. 487*Index of Notation, pg. 495*Index, pg. 499

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Book SynopsisServes as a how-to guide for developing mathematical models in biology. Starting at an elementary level of mathematical modeling, this title gradually builds from classic models in ecology and evolution to more intricate class-structured and probabilistic models. It provides primers with instructive exercises.Trade ReviewHonorable Mention for the 2007 Best Professional/Scholarly Book in Biological Sciences, Association of American Publishers "A gentle but thorough introduction to the mathematical techniques employed in ecological and evolutionary theory. Readers who ... finish this well-written book will be prepared to read and understand a sizeable fraction of the current literature."--Donald L. DeAngelis, Quarterly Review of Biology "At long last, Sally Otto and Troy Day have provided relief for biologists and epidemiologists in search of an easily read, practical, and thorough starting point from which to learn mathematical modeling... We would recommend this book over shorter texts that are labeled as 'introductory'... The depth and detail that Otto and Day have included in this text arc appealing rather than intimidating, and the structure of the text is empowering rather than didactic or formulaic."--Sanjay Basu and Alison P. Galvani, Siam Review "[T]he great value of the Otto/Day book is that it attempts pedagogical soundness, and so is useful for teaching. Besides being perfectly readable, it engages and impresses the reader quickly not only with the subject matter, but also with the quality of printing and layout which have to be seen to be believed. These praises may sound lavish by many a reader of these columns but first see the book or better still buy the volume and you will see our passion and rage for going all out in praise of this volume."--Current Engineering Practice "I highly recommend this book for every university biology department because it provides both a unique, and often uplifting, introduction and a comprehensive reference of techniques for building and analysing mathematical models."--Volker Grimm, Basic and Applied Ecology "I cannot help but think that future textbook authors will want to have Otto and Day front and center on the work desk, for this is a valuable source of material... This book stands out, and its contribution is quite apparent. In sum, this book is a valuable contribution to the literature, and one to which I expect to refer regularly in connection with my teaching and writing duties."--Steven G. Krantz, UMAP Journal "[A] great textbook... [M]asterful use of figures and illustrations and exercises ... provide the reader with valuable practice in constructing models and implementing related mathematical techniques. I certainly recommend this text and can attest to its usefulness for budding researchers in the biological sciences."--Jason M. Graham, MAA ReviewsTable of ContentsPreface ix Chapter 1: Mathematical Modeling in Biology 1 1.1 Introduction 1 1.2 HIV 2 1.3 Models of HIV/AIDS 5 1.4 Concluding Message 14 Chapter 2: How to Construct a Model 17 2.1 Introduction 17 2.2 Formulate the Question 19 2.3 Determine the Basic Ingredients 19 2.4 Qualitatively Describe the Biological System 26 2.5 Quantitatively Describe the Biological System 33 2.6 Analyze the Equations 39 2.7 Checks and Balances 47 2.8 Relate the Results Back to the Question 50 2.9 Concluding Message 51 Chapter 3: Deriving Classic Models in Ecology and Evolutionary Biology 54 3.1 Introduction 54 3.2 Exponential and Logistic Models of Population Growth 54 3.3 Haploid and Diploid Models of Natural Selection 62 3.4 Models of Interactions among Species 72 3.5 Epidemiological Models of Disease Spread 77 3.6 Working Backward--Interpreting Equations in Terms of the Biology 79 3.7 Concluding Message 82 Primer 1: Functions and Approximations 89 P1.1 Functions and Their Forms 89 P1.2 Linear Approximations 96 P1.3 The Taylor Series 100 Chapter 4: Numerical and Graphical Techniques--Developing a Feeling for Your Model 110 4.1 Introduction 110 4.2 Plots of Variables Over Time 111 4.3 Plots of Variables as a Function of the Variables Themselves 124 4.4 Multiple Variables and Phase-Plane Diagrams 133 4.5 Concluding Message 145 Chapter 5: Equilibria and Stability Analyses--One-Variable Models 151 5.1 Introduction 151 5.2 Finding an Equilibrium 152 5.3 Determining Stability 163 5.4 Approximations 176 5.5 Concluding Message 184 Chapter 6: General Solutions and Transformations--One-Variable Models 191 6.1 Introduction 191 6.2 Transformations 192 6.3 Linear Models in Discrete Time 193 6.4 Nonlinear Models in Discrete Time 195 6.5 Linear Models in Continuous Time 198 6.6 Nonlinear Models in Continuous Time 202 6.7 Concluding Message 207 Primer 2: Linear Algebra 214 P2.1 An Introduction to Vectors and Matrices 214 P2.2 Vector and Matrix Addition 219 P2.3 Multiplication by a Scalar 222 P2.4 Multiplication of Vectors and Matrices 224 P2.5 The Trace and Determinant of a Square Matrix 228 P2.6 The Inverse 233 P2.7 Solving Systems of Equations 235 P2.8 The Eigenvalues of a Matrix 237 P2.9 The Eigenvectors of a Matrix 243 Chapter 7: Equilibria and Stability Analyses--Linear Models with Multiple Variables 254 7.1 Introduction 254 7.2 Models with More than One Dynamic Variable 255 7.3 Linear Multivariable Models 260 7.4 Equilibria and Stability for Linear Discrete-Time Models 279 7.5 Concluding Message 289 Chapter 8: Equilibria and Stability Analyses--Nonlinear Models with Multiple Variables 294 8.1 Introduction 294 8.2 Nonlinear Multiple-Variable Models 294 8.3 Equilibria and Stability for Nonlinear Discrete-Time Models 316 8.4 Perturbation Techniques for Approximating Eigenvalues 330 8.5 Concluding Message 337 Chapter 9: General Solutions and Tranformations--Models with Multiple Variables 347 9.1 Introduction 347 9.2 Linear Models Involving Multiple Variables 347 9.3 Nonlinear Models Involving Multiple Variables 365 9.4 Concluding Message 381 Chapter 10: Dynamics of Class-Structured Populations 386 10.1 Introduction 386 10.2 Constructing Class-Structured Models 388 10.3 Analyzing Class-Structured Models 393 10.4 Reproductive Value and Left Eigenvectors 398 10.5 The Effect of Parameters on the Long-Term Growth Rate 400 10.6 Age-Structured Models--The Leslie Matrix 403 10.7 Concluding Message 418 Chapter 11: Techniques for Analyzing Models with Periodic Behavior 423 11.1 Introduction 423 11.2 What Are Periodic Dynamics? 423 11.3 Composite Mappings 425 11.4 Hopf Bifurcations 428 11.5 Constants of Motion 436 11.6 Concluding Message 449 Chapter 12: Evolutionary Invasion Analysis 454 12.1 Introduction 454 12.2 Two Introductory Examples 455 12.3 The General Technique of Evolutionary Invasion Analysis 465 12.4 Determining How the ESS Changes as a Function of Parameters 478 12.5 Evolutionary Invasion Analyses in Class-Structured Populations 485 12.6 Concluding Message 502 Primer 3: Probability Theory 513 P3.1 An Introduction to Probability 513 P3.2 Conditional Probabilities and Bayes' Theorem 518 P3.3 Discrete Probability Distributions 521 P3.4 Continuous Probability Distributions 536 P3.5 The (Insert Your Name Here) Distribution 553 Chapter 13: Probabilistic Models 567 13.1 Introduction 567 13.2 Models of Population Growth 568 13.3 Birth-Death Models 573 13.4 Wright-Fisher Model of Allele Frequency Change 576 13.5 Moran Model of Allele Frequency Change 581 13.6 Cancer Development 584 13.7 Cellular Automata--A Model of Extinction and Recolonization 591 13.8 Looking Backward in Time--Coalescent Theory 594 13.9 Concluding Message 602 Chapter 14: Analyzing Discrete Stochastic Models 608 14.1 Introduction 608 14.2 Two-State Markov Models 608 14.3 Multistate Markov Models 614 14.4 Birth-Death Models 631 14.5 Branching Processes 639 14.6 Concluding Message 644 Chapter 15: Analyzing Continuous Stochastic Models--Diffusion in Time and Space 649 15.1 Introduction 649 15.2 Constructing Diffusion Models 649 15.3 Analyzing the Diffusion Equation with Drift 664 15.4 Modeling Populations in Space Using the Diffusion Equation 684 15.5 Concluding Message 687 Epilogue: The Art of Mathematical Modeling in Biology 692 Appendix 1: Commonly Used Mathematical Rules 695 A1.1 Rules for Algebraic Functions 695 A1.2 Rules for Logarithmic and Exponential Functions 695 A1.3 Some Important Sums 696 A1.4 Some Important Products 696 A1.5 Inequalities 697 Appendix 2: Some Important Rules from Calculus 699 A2.1 Concepts 699 A2.2 Derivatives 701 A2.3 Integrals 703 A2.4 Limits 704 Appendix 3: The Perron-Frobenius Theorem 709 A3.1: Definitions 709 A3.2: The Perron-Frobenius Theorem 710 Appendix 4: Finding Maxima and Minima of Functions 713 A4.1 Functions with One Variable 713 A4.2 Functions with Multiple Variables 714 Appendix 5: Moment-Generating Functions 717 Index of Definitions, Recipes, and Rules 725 General Index 727

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Book SynopsisQuantal Response Equilibrium presents a stochastic theory of games that unites probabilistic choice models developed in psychology and statistics with the Nash equilibrium approach of classical game theory. Nash equilibrium assumes precise and perfect decision making in games, but human behavior is inherently stochastic and people realize that theTrade Review"This book brings together two decades of scholarship on an important model of boundedly rational behavior in strategic decision-making settings. Including numerous important applications in economics, political science, and pure game theory, this unified treatment will be valuable to a wide range of scholars."—Timothy Cason, Purdue University"Quantal response equilibrium is a standard tool for game theorists and has numerous connections to other tools and applications. This book collects and extends existing material on QRE and is a significant contribution to pure, and especially applied, game theory. No other books explicate QRE systematically beyond the introductory level and these authors are the right team for pulling the core material together."—Daniel Friedman, University of California, Santa Cruz"Well-written and easy to follow, this book covers the topic of quantal response equilibrium. The notion of stochastic equilibrium has changed the way game theorists think about long-run and short-run equilibrium. Written by three leading experts, this book is of great importance to researchers in economic theory and political science, and to graduate students."—David K. Levine, European University InstituteTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. ix*1. Introduction and Background, pg. 1*2. Quantal Response Equilibrium in Normal-Form Games, pg. 10*3. Quantal Response Equilibrium in Extensive-Form Games, pg. 63*4. Heterogeneity, pg. 88*5. Dynamics and Learning, pg. 112*6. QRE as a Structural Model for Estimation, pg. 141*7. Applications to Game Theory, pg. 161*8. Applications to Political Science, pg. 206*9. Applications to Economics, pg. 248*10. Epilogue: Some Thoughts about Future Research, pg. 281*References, pg. 291*Index, pg. 301

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Book SynopsisSeeks to describe the development in recent decades of sieve methods able to detect prime numbers. This work covers topics such as; primes in short intervals, the greatest prime factor of the sequence of shifted primes, Goldbach numbers in short intervals, the distribution of Gaussian primes, and the recent work of John Friedlander, and Iwaniec.Trade Review"This book provides a very nice introduction to a very active and important area of research. Several chapters include discussion of the limitations of the given methods; this is an unusual feature but a very useful one to readers. There is also helpful discussion of historical developments of the given methods. This a valuable book, both for researchers and for advanced graduate students in analytic number theory."--S. W Graham, Mathematical Reviews "[T]his book contains a valuable compendium of methods and results, and it will be of interest to aficionados of prime number theory."--Harold G. Diamond, SIAM Review "The book is written in a very accessible style for a wide spectrum of readers... Besides mathematical ideas, the presentation also contains many important historical comments, which make the book useful for a general mathematical audience trying to orient themselves in the evolution of the main techniques applied in sieve methods."--EMS NewsletterTable of ContentsPreface xi Notation xiii Chapter 1. Introduction 1 Chapter 2. The Vaughan Identity 25 Chapter 3. The Alternative Sieve 47 Chapter 4. The Rosser-Iwaniec Sieve 65 Chapter 5. Developing the Alternative Sieve 83 Chapter 6. An Upper-Bound Sieve 103 Chapter 7. Primes in Short Intervals 119 Chapter 8. The Brun-Titchmarsh Theorem on Average 157 Chapter 9. Primes in Almost All Intervals 189 Chapter 10. Combination with the Vector Sieve 201 Chapter 11. Generalizing to Algebraic Number Fields 231 Chapter 12. Variations on Gaussian Primes 265 Chapter 13. Primes of the Form x3 + 2y3 303 Chapter 14. Epilogue 335 Appendix 337 Bibliography 349 Index 361

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Book SynopsisFeaturing a challenging exposition of calculus in the European style, this text is suitable for a first-year university honors course or for a third-year analysis course. It uses applications of derivatives and integrals to show how calculus is applied in these disciplines. It also offers both routine and demanding exercises.Trade Review"MacCluer's book ... is calculus 'done right.' And in a mere 162 pages and 1.1 lb! ... The approach is axiomatic, but the writing style is breezy and inviting. The student will see right away that this is not simply a rehash of material seen in high school--there are some genuinely new ideas here. These include topology, compactness, quantum mechanics, differential equations, and a host of other fascinating topics... This could be an exciting book from which to teach. This is a book that will allow the instructor to build a fascinating course in a variety of different ways. Teaching from this book should be a joy for all."--Steven G. Krantz, UMAP JournalTable of ContentsPreface xi Acknowledgments xiii Chapter 1: Functions on Sets 1.1 Sets 1 1.2 Functions 2 1.3 Cardinality 5 Exercises 6 Chapter 2: The Real Numbers 2.1 The Axioms 12 2.2 Implications 14 2.3 Latter-Day Axioms 16 Exercises 16 Chapter 3: Metric Properties 3.1 The Real Line 19 3.2 Distance 20 3.3 Topology 21 3.4 Connectedness 22 3.5 Compactness 23 Exercises 27 Chapter 4: Continuity 4.1 The Definition 30 4.2 Consequences 31 4.3 Combinations of Continuous Functions 33 4.4 Bisection 36 4.5 Subspace Topology 37 Exercises 38 Chapter 5: Limits and Derivatives 5.1 Limits 41 5.2 The Derivative 43 5.3 Mean Value Theorem 46 5.4 Derivatives of Inverse Functions 48 5.5 Derivatives of Trigonometric Functions 50 Exercises 53 Chapter 6: Applications of the Derivative 6.1 Tangents 60 6.2 Newton's Method 63 6.3 Linear Approximation and Sensitivity 65 6.4 Optimization 66 6.5 Rate of Change 67 6.6 Related Rates 68 6.7 Ordinary Differential Equations 69 6.8 Kepler's Laws 71 6.9 Universal Gravitation 73 6.10 Concavity 76 6.11 Differentials 79 Exercises 80 Chapter 7: The Riemann Integral 7.1 Darboux Sums 89 7.2 The Fundamental Theorem of Calculus 91 7.3 Continuous Integrands 92 7.4 Properties of Integrals 94 7.5 Variable Limits of Integration 95 7.6 Integrability 96 Exercises 97 Chapter 8: Applications of the Integral 8.1 Work 100 8.2 Area 102 8.3 Average Value 104 8.4 Volumes 105 8.5 Moments 106 8.6 Arclength 109 8.7 Accumulating Processes 110 8.8 Logarithms 110 8.9 Methods of Integration 112 8.10 Improper Integrals 113 8.11 Statistics 115 8.12 Quantum Mechanics 117 8.13 Numerical Integration 118 Exercises 121 Chapter 9: Infinite Series 9.1 Zeno's Paradoxes 134 9.2 Convergence of Sequences 134 9.3 Convergence of Series 136 9.4 Convergence Tests for Positive Series 138 9.5 Convergence Tests for Signed Series 140 9.6 Manipulating Series 142 9.7 Power Series 145 9.8 Convergence Tests for Power Series 147 9.9 Manipulation of Power Series 149 9.10 Taylor Series 151 Exercises 154 References 163 Index 165

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Book SynopsisExplains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. This book shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis.Table of Contents*FrontMatter, pg. i*Contents, pg. vii*Preface, pg. ix*Chapter One. Introduction: Diffusive Motion and Where It Leads, pg. 1*Chapter Two. Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys, pg. 45*Chapter Three. Ed Nelson's Work in Quantum Theory, pg. 75*Chapter Four Symanzik, Nelson, and Self-Avoiding Walk, pg. 95*Chapter Five. Stochastic Mechanics: A Look Back and a Look Ahead, pg. 117*Chapter Six. Current Trends in Optimal Transportation: A Tribute to Ed Nelson, pg. 141*Chapter Seven. Internal Set Theory and Infinitesimal Random Walks, pg. 157*Chapter Eight. Nelson's Work on Logic and Foundations and Other Reflections on the Foundations of Mathematics, pg. 183*Chapter Nine. Some Musical Groups: Selected Applications of Group Theory in Music, pg. 209*Chapter Ten. Afterword, pg. 229*Appendix A. Publications by Edward Nelson, pg. 233*Index, pg. 241

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Book SynopsisA study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers.Trade Review"This book represents a major milestone for research at the intersection of arithmetic geometry and automorphic forms. The results will shape the research in this area for some time to come."--Jens Funke, Mathematical ReviewsTable of ContentsAcknowledgments ix Chapter 1. Introduction 1 Bibliography 21 Chapter 2. Arithmetic intersection theory on stacks 27 Chapter 3. Cycles on Shimura curves 45 Chapter 4. An arithmetic theta function 71 Chapter 5. The central derivative of a genus two Eisenstein series 105 Chapter 6. The generating function for 0-cycles 167 Chapter 6 Appendix. The case p = 2, p | D (B) 181 Chapter 7. An inner product formula 205 Chapter 8. On the doubling integral 265 Chapter 9. Central derivatives of L-functions 351 Index 371

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Book SynopsisSuitable for students of all mathematics and art levels, this title focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. It encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery.Trade Review"The book goes a long way trying to convey to its audience--through both theory and practice--professional techniques that could not fail but empower students to make accurate, sophisticated drawings. The book presents an elegant fusion of mathematical ideas and practical aspects of fine art."--Cut the Knot "[T]his is an excellent text that I will certainly consider using for a future class. The material on perspective is accessible, thorough and well-written, and the text is designed for a hands-on pedagogy that is well-suited to the intended audience. And as an elementary, but thorough, discussion of both the mathematics and practice of perspective drawing, it deserves a place in any collection of books on mathematics and the arts."--Blake Mellor, Journal of Mathematics and the Arts "The writing is extremely clear, the material is fresh, and the many excellent diagrams clarify the ideas under discussion. The authors use relevant artwork to illustrate the mathematical principles... The exercises are original and promote active learning... This is an excellent work for academic curricula and an outstanding resource for self-study in mathematical perspective, fractals, and the relationship between art and mathematics."--Choice "This is not a book to read passively and, indeed, you will want to read this book with a pencil in hand. The text is designed to be experienced first hand, sketching out examples whilst following the text, as well as doing the exercises at the end of each chapter that develop the material well... Prerequisites for this book are a minimum, effectively being an understanding of basic coordinate geometry. I would recommend this book to anyone who is interested in the interplay between mathematics and art."--George Matthews, Mathematics TodayTable of ContentsPreface vii Acknowledgments ix Chapter 1: Introduction to Perspective and Space Coordinates 1 Artist Vignette: Sherry Stone 9 Chapter 2: Perspective by the Numbers 13 Artist Vignette: Peter Galante 25 Chapter 3: Vanishing Points and Viewpoints 29 Artist Vignette: Jim Rose 39 Chapter 4: Rectangles in One-Point Perspective 43 What's My Line?: A Perspective Game 55 Chapter 5: Two-Point Perspective 59 Artist Vignette: Robert Bosch 77 Chapter 6: Three-Point Perspective and Beyond 85 Artist Vignette: Dick Termes 113 Chapter 7: Anamorphic Art 117 Viewpoints at the Movies: The Hitchcock Zoom 135 Plates follow page 138 Chapter 8: Introduction to Fractal Geometry 139 Artist Vignette: Teri Wagner 157 Chapter 9: Fractal Dimension 161 Artist Vignette: Kerry Mitchell 193 Answers to Selected Exercises 197 Appendix: Information for Instructors 215 Annotated References 223 Index 229

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Book SynopsisFrom rainbows, river meanders, and shadows to spider webs, honeycombs, the visible world is full of patterns that can be described mathematically. Examining such readily observable phenomena, this book introduces readers to the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature.Trade ReviewWinner of the 2003 for Professional/Scholarly Award in Mathematics and Statistics, Association of American Publishers One of Choice's Outstanding Academic Titles for 2004 "Mathematics in Nature is an excellent resource for bringing a greater variety of patterns into the mathematical study of nature, as well as for teaching students to think about describing natural phenomena mathematically... [T]he breadth of patterns studied is phenomenal."--Will Wilson, American Scientist "John Adam has combined his interest in the great outdoors and applied mathematics to compile one surprising example after another of how mathematics can be used to explain natural phenomena. And what examples! ... [He] has done a great deal of reading and exposition, indulging his passions to create this compilation of mathematical models of natural phenomena, and the sheer number of examples he manages to cram into this book is testament to his efforts. There are other texts on the market which explore the connection between mathematics and nature ... but none this wide-ranging."--Steven Morics, MAA Online "Adam has laced his mathematical models with popular descriptions of the phenomena selected... Mathematics in Nature can accordingly be read for pleasure and instruction by the select laity who are not afraid of reading between the lines of equations."--Philip J. Davis, SIAM News "John Adam's quest is a very simple one: that is, to invite one to look around and observe the wonders of nature, both natural and biological; to ponder them; and to try to explain them at various levels with, for the most part, quite elementary mathematical concepts and techniques."--Brian D. Sleeman, Notices of the American Mathematical Association "Reading this book progressively creates a course in mathematical modeling built around familiar, tangible, human-scale examples, with a trajectory that takes readers from dimensional estimates through geometrical modeling, linear and nonlinear dynamics, to pattern formation."--Choice "John Adam's Mathematics in Nature illustrates how, in a friendly and lucid manner, mathematicians think about nature. Adam lets us see how mathematics is not only an ally, but is perhaps the very language that nature uses to express the beautiful... This is a book that will challenge while it intrigues and excites."--Stanley David Gedzelman, Weatherwise "Although Mathematics in Nature has not been written as a textbook, availability of such a manual shall help instructors who choose this delightful book for teaching a course in applied mathematics or mathematical modeling."--Yuri V. Rogovchenko, Zentralblatt Math "Spanning a range of mathematical levels, this book can be used as an undergraduate textbook, a source of high school math enrichment, or can be read for pleasure by folks with an appreciation of nature but without advanced mathematical background."--Southeastern NaturalistTable of ContentsPreface: The motivation for the book; Acknowledgments; Credits xiii Prologue: Why I Might Never Have Written This Book xxi CHAPTER ONE: The Confluence of Nature and Mathematical Modeling 1 CHAPTER TWO: Estimation: The Power of Arithmetic in Solving Fermi Problems 17 CHAPTER THREE: Shape, Size, and Similarity: The Problem of Scale 31 CHAPTER FOUR: Meteorological Optics I: Shadows, Crepuscular Rays, and Related Optical Phenomena 57 CHAPTER FIVE: Meteorological Optics II: A "Calculus I" Approach to Rainbows, Halos, and Glories 80 CHAPTER SIX: Clouds, Sand Dunes, and Hurricanes 118 CHAPTER SEVEN: (Linear) Waves of All Kinds 139 CHAPTER EIGHT: Stability 173 CHAPTER NINE: Bores and Nonlinear Waves 194 CHAPTER TEN: The Fibonacci Sequence and the Golden Ratio 213 CHAPTER ELEVEN: Bees, Honeycombs, Bubbles, and Mud Cracks 231 CHAPTER TWELVE: River Meanders, Branching Patterns, and Trees 254 CHAPTER THIRTEEN: Bird Flight 295 CHAPTER FOURTEEN: HowDid the Leopard Get Its Spots? 309 APPENDIX: Fractals: An Appetite Whetter... 336 BIBLIOGRAPHY 341 INDEX 357

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Book SynopsisHow does mathematics enable us to send pictures from space back to Earth? Where does the bell-shaped curve come from? Drawing on areas of mathematics from probability theory, number theory, and geometry, this work highlights how ideas, mostly from pure math, can answer these questions. It includes puzzles and problems of varying difficulty.Trade ReviewOne of Choice's Outstanding Academic Titles for 2004 "Keith Ball demonstrated that though math may not be laugh-out-loud hilarious, it is deeply and gloriously satisfying... Ball's style is pacy and informal, and he does far more than just show off polished results. This is math with the hood up and the engine running."--Ben Longstaff, New Scientist "A recreational math book with enough heft to give its intended audience a series of mental workouts, ranging from the rough equivalent of a stroll to the corner mailbox to a hard mile run. The writing style is open and engaging."--Choice "A gem... Each topic is taken up in a setting that immediately generates interest ... Ball's achievement is to have come up with a selection of topics which are fresh and unusual... It is a pleasure to report that the book is written in limpid, graceful, elegant English prose--nowadays a nearly vanished species."--Stacy G. Langton, MAA Online "The author's writing style is informal, inviting, and clear... This book gives a lively and carefully written treatment of a number of interesting topics... The range of topics is wide, so even the experienced mathematician may learn something new."--Harold R. Parks, Notices of the American Mathematical Society "[I]f you salivate at the thought of working those calculations, then run don't walk to the bookshop--for once they've produced a book just for you."--Peter Spitz, Popular ScienceTable of ContentsPreface xi Acknowledgements xiii Chapter One Shannon's Free Lunch 1 1.1 The ISBN Code 1 1.2 Binary Channels 5 1.3 The Hunt for Good Codes 7 1.4 Parity-Check Construction 11 1.5 Decoding a Hamming Code 13 1.6 The Free Lunch Made Precise 19 1.7 Further Reading 21 1.8 Solutions 22 Chapter Two Counting Dots 25 2.1 Introduction 25 2.2 Why Is Pick's Theorem True?27 2.3 An Interpretation 31 2.4 Pick's Theorem and Arithmetic 32 2.5 Further Reading 34 2.6 Solutions 35 Chapter Three Fermat's Little Theorem and Infinite Decimals 41 3.1 Introduction 41 3.2 The Prime Numbers 43 3.3 Decimal Expansions of Reciprocals of Primes 46 3.4 An Algebraic Description of the Period 48 3.5 The Period Is a Factor of p 150 3.6 Fermat's Little Theorem 55 3.7 Further Reading 56 3.8 Solutions 58 Chapter Four Strange Curves 63 4.1 Introduction 63 4.2 A Curve Constructed Using Tiles 65 4.3 Is the Curve Continuous? 70 4.4 Does the Curve Cover the Square? 71 4.5 Hilbert's Construction and Peano's Original 73 4.6 A Computer Program 75 4.7 A Gothic Frieze 76 4.8 Further Reading 79 4.9 Solutions 80 Chapter Five Shared Birthdays, Normal Bells 83 5.1 Introduction 83 5.2 What Chance of a Match? 84 5.3 How Many Matches? 89 5.4 How Many People Share? 91 5.5 The Bell-Shaped Curve 93 5.6 The Area under a Normal Curve 100 5.7 Further Reading 105 5.8 Solutions 106 Chapter Six Stirling Works 109 6.1 Introduction 109 6.2 A First Estimate for n 110 6.3 A Second Estimate for n 114 6.4 A Limiting Ratio 117 6.5 Stirling's Formula 122 6.6 Further Reading 124 6.7 Solutions 125 Chapter Seven Spare Change, Pools of Blood 127 7.1 Introduction 127 7.2 The Coin-Weighing Problem 128 7.3 Back to Blood 131 7.4 The Binary Protocol for a Rare Abnormality 134 7.5 A Refined Binary Protocol 139 7.6 An Eficiency Estimate Using Telephones 141 7.7 An Eficiency Estimate for Blood Pooling 144 7.8 A Precise Formula for the Binary Protocol 147 7.9 Further Reading 149 7.10 Solutions 151 Chapter Eight Fibonacci's Rabbits Revisited 153 8.1 Introduction 153 8.2 Fibonacci and the Golden Ratio 154 8.3 The Continued Fraction for the Golden Ratio 158 8.4 Best Approximations and the Fibonacci Hyperbola 161 8.5 Continued Fractions and Matrices 165 8.6 Skipping down the Fibonacci Numbers 169 8.7 The Prime Lucas Numbers 174 8.8 The Trace Problem 178 8.9 Further Reading 181 8.10 Solutions 182 Chapter Nine Chasing the Curve 189 9.1 Introduction 189 9.2 Approximation by Rational Functions 193 9.3 The Tangent 202 9.4 An Integral Formula 207 9.5 The Exponential 210 9.6 The Inverse Tangent 213 9.7 Further Reading 214 9.8 Solutions 215 Chapter Ten Rational and Irrational 219 10.1 Introduction 219 10.2 Fibonacci Revisited 220 10.3 The Square Root of d 223 10.4 The Box Principle 225 10.5 The Numbers e and p 230 10.6 The Irrationality of e 233 10.7 Euler's Argument 236 10.8 The Irrationality of p 238 10.9 Further Reading 242 10.10 Solutions 243 Index 247

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Book SynopsisConcentrates on thirty highlights of pure and applied mathematics. This book opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four important open mathematical problems of the twenty-first century.Trade Review"Odifreddi's overview is of course a personal one, but it is hard to argue with either his choices or his organization. This is a perfect handle on an otherwise bewildering proliferation of ideas."--Ben Longstaff, New Scientist "Odifreddi clearly and concisely describes important 20th-century developments in pure and applied mathematics... Unlike similar volumes, this book keeps descriptions general and contains a short section on the philosophical foundations of mathematics to help non-mathematicians easily navigate the material."--Library Journal "This is an astonishingly readable, succinct, and wonderful account of twentieth-century mathematics! It is a great book for mathematics majors, students in liberal-arts courses in mathematics, and the general public. I am amazed at how easily the author has set out the achievements in a broad array of mathematical fields. The writing appears effortless."--Paul Campbell, Mathematics Magazine "Piergiogio Odifreddi's book successfully portrays the major developments in 20th century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years... [T]he literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification."--P.N. Ruane, MathDL "Odifreddi ... has an engaging and effective style and a knack for compact but comprehensible summaries, making his presentation seem effortless. The Mathematical Century can be dabbled in, read through, or perhaps even used as a quick reference."--Danny Yee, Danny ReviewsTable of ContentsForeword xi Acknowledgments xvii Introduction 1 CHAPTER 1: THE FOUNDATIONS 8 1.1. The 1920s: Sets 10 1.2. The 1940s: Structures 14 1.3. The 1960s: Categories 17 1.4. The 1980s: Functions 21 CHAPTER TWO: PURE MATHEMATICS 25 2.1. Mathematical Analysis: Lebesgue Measure (1902) 29 2.2. Algebra: Steinitz Classification of Fields (1910) 33 2.3. Topology: Brouwer's Fixed-Point Theorem (1910) 37 2.4. Number Theory: Gelfand Transcendental Numbers (1929) 39 2.5. Logic: Godel's Incompleteness Theorem (1931) 43 2.6. The Calculus of Variations: Douglas's Minimal Surfaces (1931) 47 2.7. Mathematical Analysis: Schwartz's Theory of Distributions (1945) 52 2.8. Differential Topology: Milnor's Exotic Structures (1956) 56 2.9. Model Theory: Robinson's Hyperreal Numbers (1961) 59 2.10. Set Theory: Cohen's Independence Theorem (1963) 63 2.11. Singularity Theory: Thom's Classification of Catastrophes (1964) 66 2.12. Algebra: Gorenstein's Classification of Finite Groups (1972) 71 2.13. Topology: Thurston's Classification of 3-Dimensional Surfaces (1982) 78 2.14. Number Theory: Wiles's Proof of Fermat's Last Theorem (1995) 82 2.15. Discrete Geometry: Hales's Solution of Kepler's Problem (1998) 87 CHAPTER THREE: APPLIED MATHEMATICS 92 3.1. Crystallography: Bieberbach's Symmetry Groups (1910) 98 3.2. Tensor Calculus: Einstein's General Theory of Relativity (1915) 104 3.3. Game Theory: Von Neumann's Minimax Theorem (1928) 108 3.4. Functional Analysis: Von Neumann's Axiomatization of Quantum Mechanics (1932) 112 3.5. Probability Theory: Kolmogorov's Axiomatization (1933) 116 3.6. Optimization Theory: Dantzig's Simplex Method (1947) 120 3.7. General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954) 122 3.8. The Theory of Formal Languages: Chomsky's Classification (1957) 125 3.9. Dynamical Systems Theory: The KAM Theorem (1962) 128 3.10. Knot Theory: Jones Invariants (1984) 132 CHAPTER FOUR: MATHEMATICS AND THE COMPUTER 139 4.1. The Theory of Algorithms: Turing's Characterization (1936) 145 4.2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950) 148 4.3. Chaos Theory: Lorenz's Strange Attractor (1963) 151 4.4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976) 154 4.5. Fractals: The Mandelbrot Set (1980) 159 CHAPTER FIVE: OPEN PROBLEMS 165 5.1. Arithmetic: The Perfect Numbers Problem (300 BC) 166 5.2. Complex Analysis: The Riemann Hypothesis (1859) 168 5.3. Algebraic Topology: The Poincare Conjecture (1904) 172 5.4. Complexity Theory: The P=NP Problem (1972) 176 Conclusion 181 References and Further Reading 187 Index 189

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Book SynopsisResolution of singularities is a powerful and frequently used tool in algebraic geometry. This book provides a comprehensive treatment of the characteristic 0 case. It describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether.Trade Review"Throughout his lectures, Kollar uses plenty of motivations and examples, and the text is very readable. Any graduate student or mathematicians who wishes to learn about the subject would be well-served to use this book as a starting point."--Darren Glass, MAA Review "People are already using this book. I am using this book now. I expect it will be used well into the future."--Dan Abramovich, Mathematical Reviews "The book will be an invaluable tool not only for graduate student, but also for algebraic geometers. Mathematicians working in different fields will also enjoy the clarity of the exposition and the wealth of ideas included. This will become, I'm sure, as it happened to most books in this series, one of the classics of modern mathematics."--Paul Blaga, MathematicaTable of ContentsIntroduction 1 Chapter 1. Resolution for Curves 5 1.1. Newton's method of rotating rulers 5 1.2. The Riemann surface of an algebraic function 9 1.3. The Albanese method using projections 12 1.4. Normalization using commutative algebra 20 1.5. Infinitely near singularities 26 1.6. Embedded resolution, I: Global methods 32 1.7. Birational transforms of plane curves 35 1.8. Embedded resolution, II: Local methods 44 1.9. Principalization of ideal sheaves 48 1.10. Embedded resolution, III: Maximal contact 51 1.11. Hensel's lemma and the Weierstrass preparation theorem 52 1.12. Extensions of K((t)) and algebroid curves 58 1.13. Blowing up 1-dimensional rings 61 Chapter 2. Resolution for Surfaces 67 2.1. Examples of resolutions 68 2.2. The minimal resolution 73 2.3. The Jungian method 80 2.4. Cyclic quotient singularities 83 2.5. The Albanese method using projections 89 2.6. Resolving double points, char 6= 2 97 2.7. Embedded resolution using Weierstrass' theorem 101 2.8. Review of multiplicities 110 Chapter 3. Strong Resolution in Characteristic Zero 117 3.1. What is a good resolution algorithm? 119 3.2. Examples of resolutions 126 3.3. Statement of the main theorems 134 3.4. Plan of the proof 151 3.5. Birational transforms and marked ideals 159 3.6. The inductive setup of the proof 162 3.7. Birational transform of derivatives 167 3.8. Maximal contact and going down 170 3.9. Restriction of derivatives and going up 172 3.10. Uniqueness of maximal contact 178 3.11. Tuning of ideals 183 3.12. Order reduction for ideals 186 3.13. Order reduction for marked ideals 192 Bibliography 197 Index 203

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Book SynopsisRevealing the mathematical side of Benjamin Franklin, this book explains the mathematics behind Franklin's popular "Poor Richard's Almanac", which featured such things as population estimates and a host of mathematical digressions. It includes optional math problems that challenge readers to match wits with the Founding Father himself.Trade Review"Pasles...speculates gleefully on the oft-denied mathematical genius of Benjamin Franklin...Drawing on Franklin's letters and journals as well as modern-day reconstructions of his library, Pasles touches on Franklin's fondness for magazines of mathematical diversions; publication of arithmetic problems in Poor Richard's Almanac; startlingly accurate projections of population growth and cost-benefit arguments against slavery."--Publisher's Weekly "In Franklin's Numbers, a book mixing intellectual history and mathematical puzzles (with solutions appended), Paul Pasles brings out a less-celebrated sphere of Franklin's intellect. He makes the case for the founding father as a mathematician."--Jared Wunsch, Nature "Pasles delivers surprising news to Sudoku lovers: Benjamin Franklin once shared their passion...Pasles illuminates Franklin's innovative use of mathematical logic in settling moral questions and in assessing population trends. Franklin's mathematical pursuits thus emerge as a complement to his much-lauded work in politics and science. An unexpected but welcome perspective on the genial genius of Philadelphia."--Bryce Christensen, Booklist "There is hardly a discipline on which Franklin did not stamp his mark during the 18th century. But the role that mathematics played in his life has been overlooked, argues Paul Pasles. Franklin, for instance, was fascinated with magic squares, and this book provides plenty of background to help the reader admire his interest."--New Scientist "[This is] a book that is an easy read for the innumerate but which also provides nourishment for those more skilled in the niceties of math...Also included are some contemporary puzzles that offer the reader the chance to contest skills with Franklin himself."--James Srodes, The Washington Times "Making frequent use of Franklin's writings as well as mathematical brainteasers of the type that Franklin enjoyed, Benjamin Franklin's Numbers is an engaging and thoroughly unique biography of a singular figure in American history."--Ray Bert, Civil Engineering "I thoroughly enjoyed reading this book. It is written in a pleasant, conversational style and the author's enthusiasm for his subject is infectious. The text is richly embroidered with colorful details, both mathematical and historical."--Eugene Boman, Convergence: A Magazine of the Mathematical Association of America "Pasles has succeeded in writing a book dealing with mathematics that is accessible to readers at all levels, yet thoroughly referenced and scholarly enough to satisfy researchers. His endeavor was eased by the fact that the bulk of the material concerns Franklin's magic squares and circles, which only require that the reader have the ability to add. Unexpectedly, Pasles contributes much that is new; he corrects the errors of previous authors and presents new ideas through literary sleuthing and mathematical analysis."--C. Bauer, Choice "Pasles makes a convincing case for Franklin as the last true Renaissance man in what is an entertaining and informative book that will even appeal to readers with only limited knowledge of mathematics."--Physics World "With seven years of diligent study, by going through a vast amount of archive material, references including primary sources and books and research papers, the author has produced a carefully documented and fascinating account to substantiate the theme he makes, namely, that Franklin 'possessed a mathematical mind.'"--Man Keung Siu, Mathematical Reviews "[Paul C. Pasles] and the publisher should ... be commended for producing a highly aesthetically pleasing book, with a color centerpiece showing many of Franklin's beloved magic squares in their full glory."--Eli Maor, SIAM Review "This book will appeal to readers with an interdisciplinary interest in both history and mathematics. Teachers who enjoy showing students the many ways in which they can draw on mathematics to construct logical, real-world arguments will find useful examples for the classroom. The book also includes a variety of number puzzles that can be used to challenge students."--Michelle Cirillo, Mathematics Teacher "I found Benjamin Franklin's Numbers a delightful book. I enjoyed studying and playing with the magic squares and patterns, and I was fascinated by the biographical tidbits about Franklin. This book is very well written, and I highly recommend it to anyone with an interest in mathematics or in Benjamin Franklin."--James V. Rauff, Mathematics and Computer EducationTable of ContentsPreface ix Chapter 1: The Book Franklin Never Wrote 1 Chapter 2: A Brief History of Magic 20 Chapter 3: Almanacs and Assembly 61 Interlude: Philomath Math 83 Chapter 4: Publisher, Theorist, Inventor, Innovator 87 Chapter 5: A Visit to the Country 117 Chapter 6: The Mutation Spreads (Adventures Among the English) 141 Chapter 7: Circling the Square 158 Chapter 8: Newly Unearthed Discoveries 191 Chapter 9: Legacy 226 Acknowledgements 243 Appendix 245 Index 253

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Book SynopsisPresents the findings on one of the most intensely investigated subjects in computational mathematics - the travelling salesman problem. This book describes the method and computer code used to solve a range of large-scale problems, and demonstrates the interplay of applied mathematics with increasingly powerful computing platforms.Trade ReviewWinner of the 2007 Lanchester Prize, Informs "The authors have done a wonderful job of explaining how they developed new techniques in response to the challenges posed by ever larger instances of the Traveling Salesman Problem."--MAA Online "By bringing together the best work from a wide array of researchers, advancing the field where needed, describing their findings in a book, and implementing everything in an extremely well-written computer program, the authors show how research in computational combinatorial optimization should be done."--Michael Trick, Operations Research Letters "The book is certainly a must for every researcher in practical TSP-computation."--Ulrich Faigle, Mathematical Reviews "It is very well written and clearly structured. Many examples are provided, which help the reader to better understand the presented results. The authors succeed in describing the TSP problem, beginning with its history, and the first approaches, and ending with the state of the art."--Stefan Nickel, Zentralblatt MATH "[T]the text read[s] more like a best-seller than a tome of mathematics... The resulting book provides not only a map for understanding TSP computation, but should be the starting point for anyone interested in launching a computational assault on any combinatorial optimization problem."--Jan Karel Lenstra, SIAM Review "By bringing together the best work from a wide array of researchers, advancing the field where needed, describing their findings in a book, and implementing everything in an extremely well-written computer program, the authors show how research in computational combinatorial optimization should be done."--Michael Trick, ScienceDirect "[T]he book provides a comprehensive treatment of the traveling salesman problem and I highly recommend it not only to specialists in the area but to anyone interested in combinatorial optimization."--EMS NewsletterTable of ContentsPreface xi Chapter 1: The Problem 1 1.1 Traveling Salesman 1 1.2 Other Travelers 5 1.3 Geometry 15 1.4 Human Solution of the TSP 31 1.5 Engine of Discovery 40 1.6 Is the TSP Hard? 44 1.7 Milestones in TSP Computation 50 1.8 Outline of the Book 56 Chapter 2: Applications 59 2.1 Logistics 59 2.2 Genome Sequencing 63 2.3 Scan Chains 67 2.4 Drilling Problems 69 2.5 Aiming Telescopes and X-Rays 75 2.6 Data Clustering 77 2.7 Various Applications 78 Chapter 3: Dantzig, Fulkerson, and Johnson 81 3.1 The 49-City Problem 81 3.2 The Cutting-Plane Method 89 3.3 Primal Approach 91 Chapter 4: History of TSP Computation 93 4.1 Branch-and-Bound Method 94 4.2 Dynamic Programming 101 4.3 Gomory Cuts 102 4.4 The Lin-Kernighan Heuristic 103 4.5 TSP Cuts 106 4.6 Branch-and-Cut Method 117 4.7 Notes 125 Chapter 5: LP Bounds and Cutting Planes 129 5.1 Graphs and Vectors 129 5.2 Linear Programming 131 5.3 Outline of the Cutting-Plane Method 137 5.4 Valid LP Bounds 139 5.5 Facet-Inducing Inequalities 142 5.6 The Template Paradigm for Finding Cuts 145 5.7 Branch-and-Cut Method 148 5.8 Hypergraph Inequalities 151 5.9 Safe Shrinking 153 5.10 Alternative Calls to Separation Routines 156 Chapter 6: Subtour Cuts and PQ-Trees 159 6.1 Parametric Connectivity 159 6.2 Shrinking Heuristic 164 6.3 Subtour Cuts from Tour Intervals 164 6.4 Padberg-Rinaldi Exact Separation Procedure 170 6.5 Storing Tight Sets in PQ-trees 173 Chapter 7: Cuts from Blossoms and Blocks 185 7.1 Fast Blossoms 185 7.2 Blocks of G1/2 187 7.3 Exact Separation of Blossoms 191 7.4 Shrinking 194 Chapter 8: Combs from Consecutive Ones 199 8.1 Implementation of Phase 2 202 8.2 Proof of the Consecutive Ones Theorem 210 Chapter 9: Combs from Dominoes 221 9.1 Pulling Teeth from PQ-trees 223 9.2 Nonrepresentable Solutions also Yield Cuts 229 9.3 Domino-Parity Inequalities 231 Chapter 10: Cut Metamorphoses 241 10.1 Tighten 243 10.2 Teething 248 10.3 Naddef-Thienel Separation Algorithms 256 10.4 Gluing 261 Chapter 11: Local Cuts 271 11.1 An Overview 271 11.2 Making Choices of V and sigma 272 11.3 Revisionist Policies 274 11.4 Does phi(chi*) Lie Outside the Convex Hull of T ? 275 11.5 Separating phi(chi*) from T : The Three Phases 289 11.6 PHASE 1: From T* to T" 291 11.7 PHASE 2: From T" to T' 315 11.8 Implementing ORACLE 326 11.9 PHASE 3: From T' to T 329 11.10 Generalizations 339 Chapter 12: Managing the Linear Programming Problems 345 12.1 The Core LP 345 12.2 Cut Storage 354 12.3 Edge Pricing 362 12.4 The Mechanics 367 Chapter 13: The Linear Programming Solver 373 13.1 History 373 13.2 The Primal Simplex Algorithm 378 13.3 The Dual Simplex Algorithm 384 13.4 Computational Results: The LP Test Sets 390 13.5 Pricing 404 Chapter 14: Branching 411 14.1 Previous Work 411 14.2 Implementing Branch and Cut 413 14.3 Strong Branching 415 14.4 Tentative Branching 417 Chapter 15: Tour Finding 425 15.1 Lin-Kernighan 425 15.2 Flipper Routines 436 15.3 Engineering Lin-Kernighan 449 15.4 Chained Lin-Kernighan on TSPLIB Instances 458 15.5 Helsgaun's LKH Algorithm 466 15.6 Tour Merging 469 Chapter 16: Computation 489 16.1 The Concorde Code 489 16.2 Random Euclidean Instances 493 16.3 The TSPLIB 500 16.4 Very Large Instances 506 16.5 The World TSP 524 Chapter 17: The Road Goes On 531 17.1 Cutting Planes 531 17.2 Tour Heuristics 534 17.3 Decomposition Methods 539 Bibliography 541 Index 583

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Book SynopsisBy gathering together the information a physicist needs to know about mathematics, this guide takes on the question: What can a physicist gain by studying mathematics? It shows graduate students and researchers the vital benefits of integrating mathematics into their study and experience of the physical world.Trade Review"Throughout the book Appel maintains a nice balance between rigorous mathematics and physical applications."--William J. Satzer, MAA Review "The majority of applied mathematical fields presently require so much specialization that mathematics often takes a back seat to the particular field of study. This book not only contains a great deal of the mathematics necessary to seriously study physics but also encourages physicists and potential physicists to embrace mathematics."--J.T. Zerger, Choice "Walter Appel, a theoretical physicist and mathematics educator who currently teaches mathematics at the Henri Poincare School in France, seeks in his book--appearing here in translation--to cover all the mathematics that a physicist requires above undergraduate level, including recent topics such as convergence, Green functions and Fourier series, as well as offering biographical sketches of mathematicians and problem sets."--Times Higher Education "There is a law, if not a physical law, that ensures that whenever one is using a standard mathematical technique for a physical problem it is always the special case or a first principles argument that is required. Nothing is straightforward! For such a situation this book is ideal. It presents clear definitions and the rationale for such definitions. The style of the book is very readable and an interesting biographical asides of the mathematicians associated with the topics provide light relief from the depth of the the analysis. The book is both a valuable reference book and is a good pedagogic treatment of mathematical physics. It is a book that should be on many bookshelves."--Brian L. Burrows, Zentralblatt Math "Mathematics for Physics and Physicists is a well-organized resource today's physicists must have to strengthen their math skills and to gain otherwise unattainable insights into their fields of study. Mathematics has always been and is still a precious... One is delighted to see Appel's book maintains a nice balance between rigorous mathematics and physical applications... It will lead potential physicists to embrace mathematics and they will benefit substantially."--Current Engineering PracticeTable of ContentsA book's apology xviii Index of notation xxii Chapter 1: Reminders: convergence of sequences and series 1 Chapter 2: Measure theory and the Lebesgue integral 51 Chapter 3: Integral calculus 73 Chapter 4: Complex Analysis I 87 Chapter 5: Complex Analysis II 135 Chapter 6: Conformal maps 155 Chapter 7: Distributions I 179 Chapter 8: Distributions II 223 Chapter 9: Hilbert spaces; Fourier series 249 Chapter 10: Fourier transform of functions 277 Chapter 11: Fourier transform of distributions 299 Chapter 12: The Laplace transform 331 Chapter 13: Physical applications of the Fourier transform 355 Chapter 14: Bras, kets, and all that sort of thing 377 Chapter 15: Green functions 407 Chapter 16: Tensors 433 Chapter 17: Differential forms 463 Chapter 18: Groups and group representations 489 Chapter 19: Introduction to probability theory 509 Chapter 20: Random variables 521 Chapter 21: Convergence of random variables: central limit theorem 553 Appendices A: Reminders concerning topology and normed vector spaces 573 B: Elementary reminders of differential calculus 585 C: Matrices 593 D: A few proofs 597 Tables Fourier transforms 609 Laplace transforms 613 Probability laws 616 Further reading 617 References 621 Portraits 627 Sidebars 629 Index 631

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Book SynopsisMany problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It is of interest to applied mathematicians, and computer scientists.Trade Review"This book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first."--Anders Linner, Mathematical Reviews "[T]his book is succinct but essentially self-contained; it includes an appendix with background material as well as an extensive bibliography. The algorithmic techniques developed may be useful anytime a model leads to a mathematical optimization problem where the domain naturally is a manifold, particularly if the manifold is a matrix manifold. The book follows the usual definition-theorem-proof style but it is not intended for traditional course work so there are no exercises. A reader with limited exposure to manifold theory and differential geometry most likely will benefit from consulting standard texts on those subjects first."--Anders Linner, American Mathematical Society "The book is very well and carefully written. Every chapter starts with a page-long introduction clearly outlining its goals and how they are achieved together with possible relations to other chapters. I find the material very well explained and supported with appropriate examples. It is a pleasure to work with such a book."--Nickolay T. Trendafilov, Foundations of Computational MathematicsTable of ContentsList of Algorithms xi Foreword, by Paul Van Dooren xiii Notation Conventions xv Chapter 1. Introduction 1 Chapter 2. Motivation and Applications 5 2.1 A case study: the eigenvalue problem 5 2.1.1 The eigenvalue problem as an optimization problem 7 2.1.2 Some benefits of an optimization framework 9 2.2 Research problems 10 2.2.1 Singular value problem 10 2.2.2 Matrix approximations 12 2.2.3 Independent component analysis 13 2.2.4 Pose estimation and motion recovery 14 2.3 Notes and references 16 Chapter 3. Matrix Manifolds: First-Order Geometry 17 3.1 Manifolds 18 3.1.1 Definitions: charts, atlases, manifolds 18 3.1.2 The topology of a manifold* 20 3.1.3 How to recognize a manifold 21 3.1.4 Vector spaces as manifolds 22 3.1.5 The manifolds Rn x p and Rn x p 22 3.1.6 Product manifolds 23 3.2 Differentiable functions 24 3.2.1 Immersions and submersions 24 3.3 Embedded submanifolds 25 3.3.1 General theory 25 3.3.2 The Stiefel manifold 26 3.4 Quotient manifolds 27 3.4.1 Theory of quotient manifolds 27 3.4.2 Functions on quotient manifolds 29 3.4.3 The real projective space RPn x 1 30 3.4.4 The Grassmann manifold Grass(p, n) 30 3.5 Tangent vectors and differential maps 32 3.5.1 Tangent vectors 33 3.5.2 Tangent vectors to a vector space 35 3.5.3 Tangent bundle 36 3.5.4 Vector fields 36 3.5.5 Tangent vectors as derivations? 37 3.5.6 Differential of a mapping 38 3.5.7 Tangent vectors to embedded submanifolds 39 3.5.8 Tangent vectors to quotient manifolds 42 3.6 Riemannian metric, distance, and gradients 45 3.6.1 Riemannian submanifolds 47 3.6.2 Riemannian quotient manifolds 48 3.7 Notes and references 51 Chapter 4. Line-Search Algorithms on Manifolds 54 4.1 Retractions 54 4.1.1 Retractions on embedded submanifolds 56 4.1.2 Retractions on quotient manifolds 59 4.1.3 Retractions and local coordinates* 61 4.2 Line-search methods 62 4.3 Convergence analysis 63 4.3.1 Convergence on manifolds 63 4.3.2 A topological curiosity* 64 4.3.3 Convergence of line-search methods 65 4.4 Stability of fixed points 66 4.5 Speed of convergence 68 4.5.1 Order of convergence 68 4.5.2 Rate of convergence of line-search methods* 70 4.6 Rayleigh quotient minimization on the sphere 73 4.6.1 Cost function and gradient calculation 74 4.6.2 Critical points of the Rayleigh quotient 74 4.6.3 Armijo line search 76 4.6.4 Exact line search 78 4.6.5 Accelerated line search: locally optimal conjugate gradient 78 4.6.6 Links with the power method and inverse iteration 78 4.7 Refining eigenvector estimates 80 4.8 Brockett cost function on the Stiefel manifold 80 4.8.1 Cost function and search direction 80 4.8.2 Critical points 81 4.9 Rayleigh quotient minimization on the Grassmann manifold 83 4.9.1 Cost function and gradient calculation 83 4.9.2 Line-search algorithm 85 4.10 Notes and references 86 Chapter 5. Matrix Manifolds: Second-Order Geometry 91 5.1 Newton's method in Rn 91 5.2 Affine connections 93 5.3 Riemannian connection 96 5.3.1 Symmetric connections 96 5.3.2 Definition of the Riemannian connection 97 5.3.3 Riemannian connection on Riemannian submanifolds 98 5.3.4 Riemannian connection on quotient manifolds 100 5.4 Geodesics, exponential mapping, and parallel translation 101 5.5 Riemannian Hessian operator 104 5.6 Second covariant derivative* 108 5.7 Notes and references 110 Chapter 6. Newton's Method 111 6.1 Newton's method on manifolds 111 6.2 Riemannian Newton method for real-valued functions 113 6.3 Local convergence 114 6.3.1 Calculus approach to local convergence analysis 117 6.4 Rayleigh quotient algorithms 118 6.4.1 Rayleigh quotient on the sphere 118 6.4.2 Rayleigh quotient on the Grassmann manifold 120 6.4.3 Generalized eigenvalue problem 121 6.4.4 The nonsymmetric eigenvalue problem 125 6.4.5 Newton with subspace acceleration: Jacobi-Davidson 126 6.5 Analysis of Rayleigh quotient algorithms 128 6.5.1 Convergence analysis 128 6.5.2 Numerical implementation 129 6.6 Notes and references 131 Chapter 7. Trust-Region Methods 136 7.1 Models 137 7.1.1 Models in Rn 137 7.1.2 Models in general Euclidean spaces 137 7.1.3 Models on Riemannian manifolds 138 7.2 Trust-region methods 140 7.2.1 Trust-region methods in Rn 140 7.2.2 Trust-region methods on Riemannian manifolds 140 7.3 Computing a trust-region step 141 7.3.1 Computing a nearly exact solution 142 7.3.2 Improving on the Cauchy point 143 7.4 Convergence analysis 145 7.4.1 Global convergence 145 7.4.2 Local convergence 152 7.4.3 Discussion 158 7.5 Applications 159 7.5.1 Checklist 159 7.5.2 Symmetric eigenvalue decomposition 160 7.5.3 Computing an extreme eigenspace 161 7.6 Notes and references 165 Chapter 8. A Constellation of Superlinear Algorithms 168 8.1 Vector transport 168 8.1.1 Vector transport and affine connections 170 8.1.2 Vector transport by differentiated retraction 172 8.1.3 Vector transport on Riemannian submanifolds 174 8.1.4 Vector transport on quotient manifolds 174 8.2 Approximate Newton methods 175 8.2.1 Finite difference approximations 176 8.2.2 Secant methods 178 8.3 Conjugate gradients 180 8.3.1 Application: Rayleigh quotient minimization 183 8.4 Least-square methods 184 8.4.1 Gauss-Newton methods 186 8.4.2 Levenberg-Marquardt methods 187 8.5 Notes and references 188 A. Elements of Linear Algebra, Topology, and Calculus 189 A.1 Linear algebra 189 A.2 Topology 191 A.3 Functions 193 A.4 Asymptotic notation 194 A.5 Derivatives 195 A.6 Taylor's formula 198 Bibliography 201 Index 221

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Book SynopsisExplains how to execute computationally intensive analysis on very large data sets. This book shows readers how to determine some of the best methods for solving a variety of different problems, how to create and debug statistical models, and how to run an analysis and evaluate the results.Trade Review"This book presents an original, cheap and powerful solution to the problem of analysis of large data sets... The book is devoted mainly to the practitioner of Statistics, but is also useful to mathematicians, computer scientists, researchers and students in the biology, economics and social sciences."--Radu Trimbitas, StudiaUBBTable of ContentsPreface xi Chapter 1. Statistics in the modern day 1 PART I COMPUTING 15 Chapter 2. C 17 2.1 Lines 18 2.2 Variables and their declarations 28 2.3 Functions 34 2.4 The debugger 43 2.5 Compiling and running 48 2.6 Pointers 53 2.7 Arrays and other pointer tricks 59 2.8 Strings 65 2.9 *Errors 69 Chapter 3. Databases 74 3.1 Basic queries 76 3.2 *Doing more with queries 80 3.3 Joins and subqueries 87 3.4 On database design 94 3.5 Folding queries into C code 98 3.6 Maddening details 103 3.7 Some examples 108 Chapter 4. Matrices and models 113 4.1 The GSL's matrices and vectors 114 4.2 apo_da t120 4.3 Shunting data 123 4.4 Linear algebra 129 4.5 Numbers 135 4.6 *gsl_matrixand gsl_ve torinternals 140 4.7 Models 143 Chapter 5. Graphics 157 5.1 plot 160 5.2 *Some common settings 163 5.3 From arrays to plots 166 5.4 A sampling of special plots 171 5.5 Animation 177 5.6 On producing good plots 180 5.7 *Graphs--nodes and flowcharts 182 5.8 Printing and LATEX 185 Chapter 6. *More coding tools 189 6.1 Function pointers 190 6.2 Data structures 193 6.3 Parameters 203 6.4 *Syntactic sugar 210 6.5 More tools 214 PART II STATISTICS 217 Chapter 7. Distributions for description 219 7.1 Moments 219 7.2 Sample distributions 235 7.3 Using the sample distributions 252 7.4 Non-parametric description 261 Chapter 8. Linear projections 264 8.1 *Principal component analysis 265 8.2 OLS and friends 270 8.3 Discrete variables 280 8.4 Multilevel modeling 288 Chapter 9. Hypothesis testing with the CLT 295 9.1 The Central Limit Theorem 297 9.2 Meet the Gaussian family 301 9.3 Testing a hypothesis 307 9.4 ANOVA 312 9.5 Regression 315 9.6 Goodness of fit 319 Chapter 10. Maximum likelihood estimation 325 10.1 Log likelihood and friends 326 10.2 Description: Maximum likelihood estimators 337 10.3 Missing data 345 10.4 Testing with likelihoods 348 Chapter 11. Monte Carlo 356 11.1 Random number generation 357 11.2 Description: Finding statistics for a distribution 364 11.3 Inference: Finding statistics for a parameter 367 11.4 Drawing a distribution 371 11.5 Non-parametric testing 375 Appendix A: Environments and makefiles 381 A.1 Environment variables 381 A.2 Paths 385 A.3 Make 387 Appendix B: Text processing 392 B.1 Shell scripts 393 B.2 Some tools for scripting 398 B.3 Regular expressions 403 B.4 Adding and deleting 413 B.5 More examples 415 Appendix C: Glossary 419 Bibliography 435 Index 443

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Book SynopsisExplores important aspects of Markov and hidden Markov processes and the applications of these ideas to various problems in computational biology. This book provides a range of exercises, including drills to familiarize the reader with concepts and more advanced problems that require deep thinking about the theory.Trade Review"This book will serve as a solid and invaluable reference."--Byung-Jun Yoon, Quarterly Review of BiologyTable of ContentsPreface xi PART 1. PRELIMINARIES 1 Chapter 1. Introduction to Probability and Random Variables 3 1.1 Introduction to Random Variables 3 1.1.1 Motivation 3 1.1.2 Definition of a Random Variable and Probability 4 1.1.3 Function of a Random Variable, Expected Value 8 1.1.4 Total Variation Distance 12 1.2 Multiple Random Variables 17 1.2.1 Joint and Marginal Distributions 17 1.2.2 Independence and Conditional Distributions 18 1.2.3 Bayes' Rule 27 1.2.4 MAP and Maximum Likelihood Estimates 29 1.3 Random Variables Assuming Infinitely Many Values 32 1.3.1 Some Preliminaries 32 1.3.2 Markov and Chebycheff Inequalities 35 1.3.3 Hoeffding's Inequality 38 1.3.4 Monte Carlo Simulation 41 1.3.5 Introduction to Cramer's Theorem 43 Chapter 2. Introduction to Information Theory 45 2.1 Convex and Concave Functions 45 2.2 Entropy 52 2.2.1 Definition of Entropy 52 2.2.2 Properties of the Entropy Function 53 2.2.3 Conditional Entropy 54 2.2.4 Uniqueness of the Entropy Function 58 2.3 Relative Entropy and the Kullback-Leibler Divergence 61 Chapter 3. Nonnegative Matrices 71 3.1 Canonical Form for Nonnegative Matrices 71 3.1.1 Basic Version of the Canonical Form 71 3.1.2 Irreducible Matrices 76 3.1.3 Final Version of Canonical Form 78 3.1.4 Irreducibility, Aperiodicity, and Primitivity 80 3.1.5 Canonical Form for Periodic Irreducible Matrices 86 3.2 Perron-Frobenius Theory 89 3.2.1 Perron-Frobenius Theorem for Primitive Matrices 90 3.2.2 Perron-Frobenius Theorem for Irreducible Matrices 95 PART 2. HIDDEN MARKOV PROCESSES 99 Chapter 4. Markov Processes 101 4.1 Basic Definitions 101 4.1.1 The Markov Property and the State Transition Matrix 101 4.1.2 Estimating the State Transition Matrix 107 4.2 Dynamics of Stationary Markov Chains 111 4.2.1 Recurrent and Transient States 111 4.2.2 Hitting Probabilities and Mean Hitting Times 114 4.3 Ergodicity of Markov Chains 122 Chapter 5. Introduction to Large Deviation Theory 129 5.1 Problem Formulation 129 5.2 Large Deviation Property for I.I.D. Samples: Sanov's Theorem 134 5.3 Large Deviation Property for Markov Chains 140 5.3.1 Stationary Distributions 141 5.3.2 Entropy and Relative Entropy Rates 143 5.3.3 The Rate Function for Doubleton Frequencies 148 5.3.4 The Rate Function for Singleton Frequencies 158 Chapter 6. Hidden Markov Processes: Basic Properties 164 6.1 Equivalence of Various Hidden Markov Models 164 6.1.1 Three Different-Looking Models 164 6.1.2 Equivalence between the Three Models 166 6.2 Computation of Likelihoods 169 6.2.1 Computation of Likelihoods of Output Sequences 170 6.2.2 The Viterbi Algorithm 172 6.2.3 The Baum-Welch Algorithm 174 Chapter 7. Hidden Markov Processes: The Complete Realization Problem 177 7.1 Finite Hankel Rank: A Universal Necessary Condition 178 7.2 Nonsuffciency of the Finite Hankel Rank Condition 180 7.3 An Abstract Necessary and Suffcient Condition 190 7.4 Existence of Regular Quasi-Realizations 195 7.5 Spectral Properties of Alpha-Mixing Processes 205 7.6 Ultra-Mixing Processes 207 7.7 A Sufficient Condition for the Existence of HMMs 211 PART 3. APPLICATIONS TO BIOLOGY 223 Chapter 8. Some Applications to Computational Biology 225 8.1 Some Basic Biology 226 8.1.1 The Genome 226 8.1.2 The Genetic Code 232 8.2 Optimal Gapped Sequence Alignment 235 8.2.1 Problem Formulation 236 8.2.2 Solution via Dynamic Programming 237 8.3 Gene Finding 240 8.3.1 Genes and the Gene-Finding Problem 240 8.3.2 The GLIMMER Family of Algorithms 243 8.3.3 The GENSCAN Algorithm 246 8.4 Protein Classification 247 8.4.1 Proteins and the Protein Classification Problem 247 8.4.2 Protein Classification Using Profile Hidden Markov Models 249 Chapter 9. BLAST Theory 255 9.1 BLAST Theory: Statements of Main Results 255 9.1.1 Problem Formulations 255 9.1.2 The Moment Generating Function 257 9.1.3 Statement of Main Results 259 9.1.4 Application of Main Results 263 9.2 BLAST Theory: Proofs of Main Results 264 Bibliography 273 Index 285

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Book SynopsisPresents and develops an extensive treatment of stability analysis and control design of nonlinear dynamical systems, with an emphasis on Lyapunov-based methods. This graduate-level textbook is suitable for applied mathematicians, dynamical systems theorists, control theorists, and engineers.Trade ReviewWassim Haddad, Winner of the 2014 Pendray Aerospace Literature Award, American Institute of Aeronautics and Astronautics "The book is lucid and well written and contains numerous worked examples for specific applications to important classes of systems as well as numerous problems and suggestions for further study at the end of the main chapters. This book will be an excellent source of reference materials for graduate students of applied mathematics, control theorists and engineers studying the stability theory of dynamical systems and controls. It will also be a rich source of materials for self study by researchers and practitioners interested in systems theory of engineering, controls, computer science, chemistry, life sciences and economics."--Olusola Akinyele, Mathematical ReviewsTable of ContentsConventions and Notation xv Preface xxi Chapter 1. Introduction 1 Chapter 2. Dynamical Systems and Differential Equations 9 Chapter 3. Stability Theory for Nonlinear Dynamical Systems 135 Chapter 4. Advanced Stability Theory 207 Chapter 5. Dissipativity Theory for Nonlinear Dynamical Systems 325 Chapter 6. Stability and Optimality of Feedback Dynamical Systems 411 Chapter 7. Input-Output Stability and Dissipativity 471 Chapter 8. Optimal Nonlinear Feedback Control 511 Chapter 9. Inverse Optimal Control and Integrator Backstepping 557 Chapter 10. Disturbance Rejection Control for Nonlinear Dynamical Systems 603 Chapter 11. Robust Control for Nonlinear Uncertain Systems 649 Chapter 12. Structured Parametric Uncertainty and Parameter-Dependent Lyapunov Functions 719 Chapter 13. Stability and Dissipativity Theory for Discrete-Time Nonlinear Dynamical Systems 763 Chapter 14. Discrete-Time Optimal Nonlinear Feedback Control 845 Bibliography 901 Index 939

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Book SynopsisA student in class asks the math teacher: "Shouldn't minus times minus make minus?" Teachers soon convince most students that it does not. Yet the innocent question brings with it a germ of mathematical creativity. What happens if we encourage that thought, odd and ungrounded though it may seem? Few books in the field of mathematics encourage suchTrade Review"Alberto A. Martinez ... shows that the concept of negative numbers has perplexed not just young students but also quite a few notable mathematicians... The rule that minus times minus makes plus is not in fact grounded in some deep and immutable law of nature. Martinez shows that it's possible to construct a fully consistent system of arithmetic in which minus times minus makes minus. It's a wonderful vindication for the obstinate smart-aleck kid in the back of the class."--Greg Ross, American Scientist "Alberto Martinez ... has written an entire book about the fact that the product of two negative numbers is considered positive. He begins by reminding his readers that it need not be so... The book is written in a relaxed, conversational manner... It can be recommended to anyone with an interest in the way algebra was developed behind the scenes, at a time when calculus and analytic geometry were the main focus of mathematical interest."--James Case, SIAM News "[Negative Math] is very readable and the style is entertaining. Much is done through examples rather than formal proofs. The writer avoids formal mathematical logic and the more esoteric abstract algebras such as group theory."--Mathematics MagazineTable of ContentsFigures ix Chapter 1: Introduction 1 Chapter 2: The Problem 10 Chapter 3: History: Much Ado About Less than Nothing 18 The Search for Evident Meaning 36 Chapter 4: History: Meaningful and Meaningless Expressions 43 Impossible Numbers? 66 Chapter 5: History: Making Radically New Mathematics 80 From Hindsight to Creativity 104 Chapter 6: Math Is Rather Flexible 110 Sometimes -1 Is Greater than Zero 112 Traditional Complications 115 Can Minus Times Minus Be Minus? 131 Unity in Mathematics 166 Chapter 7: Making a Meaningful Math 174 Finding Meaning 175 Designing Numbers and Operations 186 Physical Mathematics? 220 Notes 235 Further Reading 249 Acknowledgments 259 Index 261

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Book SynopsisPlotting humankind's efforts to visualize data, this book discusses atheoretical plotting of data to reveal suggestive patterns. It includes chapters illustrating the uses and abuses of this invention (plotting), from a murder trial in Connecticut to the Vietnam War's effect on college admissions.Trade ReviewOne of Choice's Outstanding Academic Titles for 2005 "Well written and innovative... The book is fascinating with its wide view, including introductions to historical personalities, analyses of statistical paradoxes, and well-documented discussions of actual uses of visual data to mislead viewers."--Choice "During a dairyman's strike in 19th century New England, when there was suspicion of milk being watered down, Henry David Thoreau wrote, 'Sometimes circumstantial evidence can be quite convincing; like when you find a trout in the milk.' Howard Wainer uses this as a metaphor in his entertaining, informative, and persuasive book on graphs, or the visual communication of information. Sometimes a well-designed graph tells a very convincing story."--Raymond N. Greenwell, MAA Online "Wainer's wit and broad intellect make this a very entertaining book."--Linda Pickle, ,American Statistician "[A] personalized and readable jaunt through the history of charting."--The Economist "This book may be seen as a chronology of graphic date presentation beginning with Playfair to the present and pointing toward the future... It is a remarkable value that every practitioner of statistics can afford."--Malcolm James Ree, Personnel Psychology "Graphic Discovery is a welcome addition to the literature on investigation and effective communication through graphic display. It contains a wealth of information and opinions, which are motivated and illustrated through a plethora of real life examples which can be easily incorporated into any educational setting: classroom, seminar, self-enhancement... This book will be useful to and it can be mastered by a diverse readership."--Thomas E. Bradstreet, Computational StatisticsTable of ContentsPreface xiii Introduction 1 In the sixteenth century, the bubonic plague provided the motivation for the English to begin gathering data on births, marriages, and deaths. These data, the Bills of Mortality, were the grist that Dr. John Arbuthnot used to prove the existence of God. Unwittingly, he also provided strong evidence that data graphs were not yet part of a scientist's tools. Part I: William Playfair and the Origins of Graphical Display Chapter 1: Why Playfair? 9 All of the pieces were in place for the invention of statistical graphics long before Playfair was born. Why didn't anyone else invent them? Why did Playfair? Chapter 2: Who Was Playfair? 20 by Ian Spence and Howard Wainer William Playfair (1759-1823) was an inventor and ardent advocate of statistical graphics. Here we tell a bit about his life. Chapter 3: William Playfair: A Daring Worthless Fellow 24 by Ian Spence and Howard Wainer Audacity was an important personality trait for the invention of graphics because the inventor had to move counter to the Cartesian approach to science. We illustrate this quality in Playfair by describing his failed attempt to blackmail one of the richest lords of Great Britain. Chapter 4: Scaling the Heights (and Widths) 28 The message conveyed by a statistical graphic can be distorted by manipulating the aspect ratio, the ratio of a graph's width to its height. Playfair deployed this ability in a masterly way, providing a guide to future display technology. Chapter 5: A Priestley View of International Currency Exchanges 39 A recent plot of the operating hours of international currency exchanges confuses matters terribly. Why? We find that when we use a different graphical form, developed by Joseph Priestley in 1765, the structure becomes clear. We also learn how Priestley discovered the latent graphicacy in his (and our) audiences. Chapter 6: Tom's Veggies and the American Way 44 European intellectuals were not the only ones graphing data. During a visit to Paris (and prompted by letters from Benjamin Franklin), Thomas Jefferson learned of this invention and he later put it to a more practical use than the depiction of the life spans of heroes from classical antiquity. Chapter 7: The Graphical Inventions of Dubourg and Ferguson: Two Precursors to William Playfair 47 Although he developed the line chart independently, Priestley was not the first to do so. The earliest seems to be the Parisian physician Jacques Barbeau-Dubourg (1709-1779), who created a wonderful graphical scroll in 1753. Graphical representation must have been in the air, for the Scottish philosopher Adam Ferguson (1723-1816) added his version of time lines to the mix in 1780. Chapter 8: Winds across Europe: Francis Galton and the Graphic Discovery of Weather Patterns 52 In 1861, Francis Galton organized weather observatories throughout Western Europe to gather data in a standardized way. He organized these data and presented them as a series of ninety-three maps and charts, from which he confirmed the existence of the anticyclonic movement of winds around a low-pressure zone. Part II: Using Graphical Displays to Understand the Modern World Chapter 9: A Graphical Investigation of the Scourge of Vietnam 59 During the Vietnam War, average SAT scores went down for those students who were not in the military. In addition, the average ASVAB scores (the test used by the military to classify all members of the military) also declined. This Lake Wobegon-like puzzle is solved graphically. Chapter 10: Two Mind-Bending Statistical Paradoxes 63 The odd phenomenon observed with test scores during the Vietnam War is not unusual. We illustrate this seeming paradox with other instances, show how to avoid them, and then discuss an even subtler statistical pitfall that has entrapped many illustrious would-be data analysts. Chapter 11: Order in the Court 72 How one orders the elements of a graph is critical to its comprehensibility. We look at a New York Times graphic depicting the voting records of U.S. Supreme Court justices and show that reordering the graphic provides remarkable insight into the operation of the court. Chapter 12: No Order in the Court 78 We examine one piece of the evidence presented in the 1998 murder trial of State v. Gibbs and show how the defense attorneys, by misordering the data in the graph shown to the judge, miscommunicated a critical issue in their argument. Chapter 13: Like a Trout in the Milk 81 Thoreau pointed out that sometimes circumstantial evidence can be quite convincing, as when you find a trout in the milk. We examine a fascinating graph that provides compelling evidence of industrial malfeasance. Chapter 14: Scaling the Market 86 We examine the stock market and show that different kinds of scalings provide the answers to different levels of questions. One long view suggests a fascinating conjecture about the trade-offs between investing in stocks and investing in real estate. Chapter 15: Sex, Smoking, and Life Insurance: A Graphical View 90 We examine two risk factors for life insurance--sex and smoking--and uncover the implicit structure that underlies insurance premiums. Chapter 16: There They Go Again! 97 The New York Times is better than most media sources for statistical graphics, but even the Times has occasional relapses to an earlier time in which confusing displays ran rampant over its pages. We discuss some recent slips and compare them with prior practice. Chapter 17: Sex and Sports: How Quickly Are Women Gaining? 103 A simple graph of winning times in the Boston Marathon augmented by a fitted line provides compelling, but incorrect, evidence for the relative gains that women athletes have made over the past few decades. A more careful analysis provides a better notion of the changing size of the sex differences in athletic performances. Chapter 18: Clear Thinking Made Visible: Redesigning Score Reports for Students 109 Too often communications focus on what the transmitter thinks is important rather than on what the receiver is most critically interested in. The standard SAT score report that is sent to more than one million high school students annually is one such example. Here we revise this report using principles abstracted from another missive sent to selected high school students. Part III: Graphical Displays in the Twenty-first Century The three chapters of this section grew out of a continuing conversation with John W. Tukey, the renowned Princeton polymath, on the graphical tools that were likely to be helpful when data were displayed on a computer screen rather than a piece of paper. These conversations began shortly after Tukey's eighty-fourth birthday and continued for more than a year, ending the night before he died. Chapter 19: John Wilder Tukey: The Father of Twenty-first-Century Graphical Display 117 Chapter 20: Graphical Tools for the Twenty-first Century: I. Spinning and Slicing 125 Chapter 21: Graphical Tools for the Twenty-first Century: II. Nearness and Smoothing Engines 134 Chapter 22: Epilogue: A Selection of Selection Anomalies 142 Graphical displays are only as good as the data from which they are composed. In this final chapter we examine an all too frequent data flaw. The effects of nonsampling errors deserve greater attention, especially when randomization is absent. Formal statistical analysis treats only some of the uncertainties. In this chapter we describe three examples of how flawed inferences can be made from nonrandomly obtained samples and suggest a strategy to guard against flawed inferences. Conclusion 150 Dramatis Personae 151 This graphical epic has more than one hundred characters. Some play major roles, but most are cameos. To help keep straight who is who, this section contains thumbnail biographies of all the players. Notes 173 References 177 Index 185

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