Mathematical modelling Books

Book SynopsisMathematical Models for Structural Reliability Analysis offers mathematical models for describing load and material properties in solving structural engineering problems. Examples are provided, demonstrating how the models are implemented, and the limitations of the models are clearly stated. Analytical solutions are also discussed, and methods are clearly distinguished from models. The authors explain both theoretical models and practical applications in a clear, concise, and readable fashion.Table of ContentsStochastic Process Models (F. Casciati and M. Di Paola)IntroductionThe Orthogonal-Increment ModelThe Correlation-Stationary Model Time-Invariant Linear Systems Models of Common UseThe Evolutionary Model Time-Invariant Linear SystemsMarkov Processes A Model of Common Use Itô Stochastic Differential Equation Some Examples Approximation of Mechanical Processes: Physical versus Itô EquationsThe Random Pulse Train Model The Delta-Correlated Model Fokker Planck and Moment Equations for Parametric Delta Correlated Input Quasi-Linear Systems Simulation of Delta Correlated Processes and Response Simulation of Normal White Noise Input and Response Orthogonal-Increment Model for Delta Correlated ProcessesMultidegree-of-Freedom Systems Under Parametric Delta Correlated Input Moment Equation Approach for MDOF Systems Simulation of Multivariate Delta Correlated Processes and ResponseConclusions and ReferencesAppendix Characterization of Random Variables Joint Characterization of Random Variables Operation on Stochastic Processes Kronecker Algebra: Some FundamentalsDimension Reduction and Discretization in Stochastic Problems by Regression Method (O. Ditlevsen)IntroductionLinear RegressionNormal DistributionNon-Gaussian Distributions and Linear RegressionMarginally Transformed Gaussian Processes and FieldsDiscretized Fields Defined by Linear Regression on a Finite Set of Field ValuesDiscretization Defined by Linear Regression on a Finite Set of Linear FunctionalsPoisson Load Field ExampleStochastic Finite Element Methods and Reliability CalculationsClassical versus Statistical-Stochastic Interpolation Formulated on the Basis of the Principle of Maximum LikelihoodComputational Practicability of the Statistical-Stochastic Interpolation MethodField Modeling on the Basis of Measured Noisy DataDiscretization Defined by L

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Book SynopsisUnique in its clarity, examples, and range, Physical Mathematics explains simply and succinctly the mathematics that graduate students and professional physicists need to succeed in their courses and research. The book illustrates the mathematics with numerous physical examples drawn from contemporary research. This second edition has new chapters on vector calculus, special relativity and artificial intelligence and many new sections and examples. In addition to basic subjects such as linear algebra, Fourier analysis, complex variables, differential equations, Bessel functions, and spherical harmonics, the book explains topics such as the singular value decomposition, Lie algebras and group theory, tensors and general relativity, the central limit theorem and Kolmogorov''s theorems, Monte Carlo methods of experimental and theoretical physics, Feynman''s path integrals, and the standard model of cosmology.Trade Review'Cahill has given us a concise and mathematically clear text, one that adds many contemporary topics to a classic selection.' Roy J. Glauber, Harvard University, Massachusetts'Kevin Cahill's book collects and presents the mathematical methods of physics succinctly, and will be much welcomed by students and researchers alike.' A. Zee, University of California, Santa Barbara'This book provides a comprehensive coverage of the mathematics necessary for advanced study in physics. The key notions are presented concisely and accurately with a wealth of beautiful examples borrowed from physics, biophysics, and probability theory. The inclusion of advanced topics such as differential forms and group theory makes this book a most useful resource for graduate students and researchers in theoretical physics.' Alain Comtet, Université Pierre et Marie Curie'I am impressed by the extremely clear and direct style with which this book has been written … The author has distilled his experience and obvious enjoyment of the subject, and presented mathematics in a remarkably human and accessible form.' David Waxman, Fudan University, Shanghai'The book places itself as a reference for the undergraduate students, the graduate students and the researchers who need to refresh some topics. As the author indicates, the book covers all the mathematics needed for a physicist. Also, the instructor who gives a course on mathematical physics can find useful material in the book. The topics are explained with little stress on the physical meaning which is meant to come spontaneously progressing in the book. The completeness of topics is preferred to proofs and deep explanations … this book is indicated for the student who wants a solid and complete reference for mathematical tool in physics. The instructor can use the book as a reference for basic or advanced courses of mathematical physics, as well as for more theoretically based courses.' Stefano Scali, Contemporary PhysicsTable of ContentsPreface; 1. Linear algebra; 2. Vector calculus; 3. Fourier series; 4. Fourier and Laplace transforms; 5. Infinite series; 6. Complex-variable theory; 7. Differential equations; 8. Integral equations; 9. Legendre polynomials and spherical harmonics; 10. Bessel functions; 11. Group theory; 12. Special relativity; 13. Tensors and general relativity; 14. Forms; 15. Probability and statistics; 16. Monte Carlo methods; 17. Artificial intelligence; 18. Order, chaos, and fractals; 19. Functional derivatives; 20. Path integrals; 21. Renormalization group; 22. Strings; References; Index.

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Book SynopsisThe goal of this book is to encourage the teaching and learning of mathematical model building relatively early in the undergraduate program. The text introduces the student to a number of important mathematical topics and to a variety of models in the social sciences, life sciences, and humanities.Table of ContentsPreface viii Acknowledgements xiii 1 Mathematical Models 1 I. Mathematical Systems and Models 1 II. An Example: Modeling Free Fall 4 III. Discrete Examples: Credit Cards and Populations 10 IV. Classification of Mathematical Models 16 V. Uses and Limitations of Mathematical Models 18 Exercises 19 Suggested Projects 21 2 Stable and Unstable Arms Races 23 I. The Real-World Setting 23 II. Constructing a Deterministic Model 25 III. A Simple Model for an Arms Race 25 IV. The Richardson Model 28 V. Interpreting and Testing the Richardson Model 45 VI. Obtaining an Exact Solution 53 Exercises 59 Suggested Projects 63 3 Ecological Models: Single Species 65 I. Introduction 65 II. The Pure Birth Process 65 III. Exponential Decay 71 IV. Logistic Population Growth 72 V. The Discrete Model of Logistic Growth and Chaos 80 VI. The Allee Effect 87 VII. Historical and Biographical Notes 89 Exercises 100 Suggested Projects 104 Biographical References 105 4 Ecological Models: Interacting Species 106 I. Introduction 106 II. Two Real-World Situations 106 III. Autonomous Systems 108 IV. The Competitive Hunters Model 116 V. The Predator-Prey Model 123 VI. Concluding Remarks on Simple Models in Population Dynamics 131 VII. Biographical Sketches 133 Exercises 137 Suggested Projects 139 5 Tumor Growth Models 141 I. Introduction 141 II. A General Tumor Growth Model 142 III. The Gompertz Model 145 IV. Modeling Colorectal Cancer 155 V. Historical and Biographical Notes 167 Exercises 176 Suggested Projects 177 6 Social Choice and Voting Procedures 179 I. Three Voting Situations 179 II. Two Voting Mechanisms 180 III. An Axiomatic Approach 185 IV. Arrow’s Impossibility Theorem 187 V. The Liberal Paradox and the Theorem of the Gloomy Alternatives 191 VI. Instant Runoff Voting 197 VII. Approval Voting 203 VIII. Topological Social Choice 207 IX. Historical and Biographical Notes 212 Exercises 224 Suggested Projects 229 7 Foundations of Measurement Theory 232 I. The Registrar’s Problem 232 II. What Is Measurement? 233 III. Simple Measures on Finite Sets 238 IV. Perception of Differences 240 V. An Alternative Approach 242 VI. Some Historical Notes 245 Exercises 245 Suggested Projects 247 8 Introduction to Utility Theory 249 I. Introduction 249 II. Gambles 250 III. Axioms of Utility Theory 251 IV. Existence and Uniqueness of Utility 254 V. Classification of Scales 257 VI. Interpersonal Comparison of Utility 259 VII. Historical and Biographical Notes 261 Exercises 265 Suggested Projects 266 9 Equilibrium in an Exchange Economy 268 I. Introduction 268 II. A Two-Person Economy with Two Commodities 268 III. An m-Person Economy 276 IV. Existence of Economic Equilibrium 283 V. Some Remaining Questions 293 VI. Historical and Biographical Notes 294 Exercises 298 Suggested Projects 301 VII. Additional Historical and Biographical Notes 302 10 Elementary Probability 303 I. The Need for Probability Models 303 II. What Is Probability? 304 III. A Probabilistic Model 322 IV. Stochastic Processes 325 Exercises 331 Suggested Projects 335 11 Markov Processes 336 I. Markov Chains 336 II. Matrix Operations and Markov Chains 341 III. Regular Markov Chains 347 IV. Absorbing Markov Chains 357 V. Historical and Biographical Notes 369 Exercises 371 Suggested Projects 374 12 Two Models of Cultural Stability 375 I. Introduction 375 II. The Gadaa System 375 III. A Deterministic Model 378 IV. A Probabilistic Model 381 V. Criticisms of the Models 383 VI. Hans Hoffmann 384 Exercises 386 Suggested Projects 387 13 Paired-Associate Learning 388 I. The Learning Problem 388 II. The Model 389 III. Testing the Model 397 IV. Historical and Biographical Notes 401 Exercises 404 Suggested Projects 406 14 Epidemics 407 I. Introduction 407 II. Deterministic Models 411 III. A Probabilistic Approach 449 IV. Historical and Biographical Notes 455 Exercises 459 Suggested Projects 463 15 Roulette Wheels and Hospital Beds: A Computer Simulation of Operating and Recovery Room Usage 464 I. Introduction 464 II. The Problems of Interest 468 III. Projecting the Number of Surgical Procedures 468 IV. Estimating Operating Room Demands 469 V. The Simulation Model 474 VI. Other Examples of Simulation 480 VII. Historical and Biographical Notes 484 Exercises 487 Suggested Projects 488 16 Game Theory 490 I. Two Difficult Decisions 490 II. Game Theory Basics 492 III. The Binding of Isaac 502 IV. Tosca and the Prisoners’ Dilemma 507 V. Nash Equilibrium 511 VI. Dynamic Solutions 515 VII. Historical and Biographical Notes 519 Exercises 522 Suggested Projects 526 Appendices Appendix I: Sets 613 Appendix II: Matrices 617 Appendix III: Solving Systems of Equations 631 Appendix IV: Functions of Two Variables 645 Appendix V: Differential Equations 648 Index 657

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Book SynopsisThis set includes: Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel and Solutions Manual to Accompany Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel. With a focus on mathematical models based on real and current data, Models for Life: An Introduction to Discrete Mathematical Modeling with Microsoft Office Excel guides readers in the solution of relevant, practical problems by introducing both mathematical and Excel techniques. The book begins with a step-by-step introduction to discrete dynamical systems, which are mathematical models that describe how a quantity changes from one point in time to the next. Readers are taken through the process, language, and notation required for the construction of such models as well as their implementation in Excel. The book examines single-compartment models in contexts such as population growth, personal financ

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Book SynopsisThe engineering of materials with advanced features is driving the research towards the design of innovative materials with high performances. New materials often deliver the best solution for structural applications, precisely contributing towards the finest combination of mechanical properties and low weight.Table of ContentsPreface xiii Part 1 Engineering of Materials, Characterizations, and Applications 1 Mechanical Behavior and Resistance of Structural Glass Beams in Lateral–Torsional Buckling (LTB) with Adhesive Joints 3 Chiara Bedon and Jan Belis 1.1 Introduction 4 1.2 Overview on Structural Glass Applications in Buildings 5 1.3 Glass Beams in LTB 5 1.3.1 Susceptibility of Glass Structural Elements to Buckling Phenomena 5 1.3.2 Mechanical and Geometrical Influencing Parameters in Structural Glass Beams 8 1.3.3 Mechanical Joints 9 1.3.4 Adhesive Joints 10 1.4 Theoretical Background for Structural Members in LTB 14 1.4.1 General LTB Method for Laterally Unrestrained (LU) Members 14 1.4.2 LTB Method for Laterally Unrestrained (LU) Glass Beams 17 1.4.2.1 Equivalent Thickness Methods for Laminated Glass Beams 18 1.4.3 Laterally Restrained (LR) Beams in LTB 23 1.4.3.1 Extended Literature Review on LR Beams 23 1.4.3.2 Closed-form Formulation for LR Beams in LTB 24 1.4.3.3 LR Glass Beams Under Positive Bending Moment My 28 1.5 Finite-element Numerical Modeling 31 1.5.1 FE Solving Approach and Parametric Study 32 1.5.1.1 Linear Eigenvalue Buckling Analyses (lba) 32 1.5.1.2 Incremental Nonlinear Analyses (inl) 35 1.6 LTB Design Recommendations 38 1.6.1 LR Beams Under Positive Bending Moment My 38 1.6.2 Further Extension and Developments of the Current Outcomes 39 1.7 Conclusions 42 References 44 2 Room Temperature Mechanosynthesis of Nanocrystalline Metal Carbides and Their Microstructure Characterization 49 S.K. Pradhan and H. Dutta 2.1 Introduction 50 2.1.1 Application 50 2.1.2 Different Methods for Preparation of Metal Carbide 50 2.1.3 Mechanical Alloying 51 2.1.4 Planetary Ball Mill 51 2.1.5 The Merits and Demerits of Planetary Ball Mill 52 2.1.6 Review of Works on Metal Carbides by Other Authors 53 2.1.7 Significance of the Study 54 2.1.8 Objectives of the Study 55 2.2 Experimental 56 2.3 Theoretical Consideration 58 2.3.1 Microstructure Evaluation by X-ray Diffraction 58 2.3.2 General Features of Structure 60 2.4 Results and Discussions 60 2.4.1 XRD Pattern Analysis 60 2.4.2 Variation of Mol Fraction 65 2.4.3 Phase Formation Mechanism 69 2.4.4 Is Ball-milled Prepared Metal Carbide Contains Contamination? 71 2.4.5 Variation of Particle Size 72 2.4.6 Variation of Strain 74 2.4.7 High-Resolution Transmission Electron Microscopy Study 76 2.4.8 Comparison Study between Binary and Ternary Ti-based Metal Carbides 76 2.5 Conclusion 80 Acknowledgment 80 References 80 3 Toward a Novel SMA-reinforced Laminated Glass Panel 87 Chiara Bedon and Filipe Amarante dos Santos 3.1 Introduction 87 3.2 Glass in Buildings 89 3.2.1 Actual Reinforcement Techniques for Structural Glass Applications 92 3.3 Structural Engineering Applications of Shape-Memory Alloys (SMAs) 93 3.4 The Novel SMA-Reinforced Laminated Glass Panel Concept 94 3.4.1 Design Concept 94 3.4.2 Exploratory Finite-Element (FE) Numerical Study 96 3.4.2.1 General FE Model Assembly Approach and Solving Method 96 3.4.2.2 Mechanical Characterization of Materials 98 3.5 Discussion of Parametric FE Results 101 3.5.1 Roof Glass Panel (M1) 101 3.5.1.1 Short-term Loads and Temperature Variations 102 3.5.1.2 First-cracking Configuration 106 3.5.2 Point-supported Façade Panel (M2) 109 3.5.2.1 Short-term Loads and Temperature Variations 111 3.6 Conclusions 114 References 117 4 Sustainable Sugarcane Bagasse Cellulose for Papermaking 121 Noé Aguilar-Rivera 4.1 Pulp and Paper Industry 122 4.2 Sugar Industry 123 4.3 Sugarcane Bagasse 124 4.4 Advantageous Utilizations of SCB 129 4.5 Applications of SCB Wastes 130 4.6 Problematic of Nonwood Fibers in Papermaking 131 4.7 SCB as Raw Material for Pulp and Paper 134 4.8 Digestion 135 4.9 Bleaching 135 4.10 Properties of Bagasse Pulps 136 4.10.1 Pulp Strength 137 4.10.2 Pulp Properties 137 4.10.3 Washing Technology 138 4.10.4 Paper Machine Operation 138 4.11 Objectives 138 4.12 Old Corrugated Container Pulps 139 4.13 Synergistic Delignification SCB–OCC 141 4.14 Elemental Chlorine-Free Bleaching of SCB Pulps 150 4.15 Conclusions 156 References 158 5 Bio-inspired Composites: Using Nature to Tackle Composite Limitations 165 F. Libonati 5.1 Introduction 166 5.2 Bio-inspiration: Bone as Biomimetic Model 169 5.3 Case Studies Using Biomimetic Approach 172 5.3.1 Fiber-reinforced Bone-inspired Composites 172 5.3.2 Fiber-reinforced Bone-inspired Composites with CNTs 176 5.3.3 Bone-inspired Composites via 3D Printing 177 5.4 Methods 179 5.4.1 Composite Lamination 180 5.4.2 Additive Manufacturing 181 5.4.3 Computational Modeling 182 5.5 Conclusions 183 References 185 Part 2 Computational Modeling of Materials 6 On the Electronic Structure and Band Gap of ZnSxSe1–x 193 Ghassan H. E. Al-Shabeeb and A. K. Arof 6.1 Introduction 193 6.2 Computational Method 194 6.3 The k·p Perturbation Theory with the Effect of Spin–Orbit Interaction 197 6.4 Results and Discussion 202 Acknowledgment 205 References 205 7 Application of First Principles Theory to the Design of Advanced Titanium Alloys 207 Y. Song, J. H. Dai, and R. Yang 7.1 Introduction 207 7.2 Basic Concepts of First Principles 208 7.3 Theoretical Models of Alloy Design 211 7.3.1 The Hume-Rothery Theory 211 7.3.2 Discrete Variational Method and d-Orbital Method 216 7.3.2.1 Discrete Variational Method 216 7.3.2.2 d-Electrons Alloy Theory 218 7.4 Applications 219 7.4.1 Phase Stability 219 7.4.1.1 Binary Alloy 219 7.4.1.2 Multicomponent Alloys 222 7.4.2 Elastic Properties 223 7.4.3 Examples 226 7.4.3.1 Gum Metal 226 7.4.3.2 Ti2448 (Ti–24Nb–4Zr–8Sn) 227 7.5 Conclusions 230 Acknowledgment 230 References 230 8 Digital Orchid: Creating Realistic Materials 233 Iftikhar B. Abbasov 8.1 Introduction 234 8.2 Conclusion 243 References 243 9 Transformation Optics-based Computational Materials for Stochastic Electromagnetics 245 Ozlem Ozgun and Mustafa Kuzuoglu 9.1 Introduction 246 9.2 Theory of Transformation Optics 249 9.3 Scattering from Rough Sea Surfaces 252 9.3.1 Numerical Validation and Monte Carlo Simulations 256 9.4 Scattering from Obstacles with Rough Surfaces or Shape Deformations 258 9.4.1 Numerical Validation and Monte Carlo Simulations 263 9.4.2 Combining Perturbation Theory and Transformation Optics for Weakly Perturbed Surfaces 264 9.5 Scattering from Randomly Positioned Array of Obstacles 268 9.5.1 Separate Transformation Media 269 9.5.1.1 Numerical Validation & Monte Carlo Simulations 271 9.5.2 A Single Transformation Medium 273 9.5.2.1 Numerical Validation & Monte Carlo Simulations 275 9.5.3 Recurring Scaling and Translation Transformations 276 9.5.3.1 Numerical Validation & Monte Carlo Simulations 278 9.6 Propagation in a Waveguide with Rough or Randomly Varying Surface 278 9.3.1 Numerical Validation and Monte Carlo Simulations 283 9.7 Conclusion 287 References 288 10 Superluminal Photons Tunneling through Brain Microtubules Modeled as Metamaterials and Quantum Computation 291 Luigi Maxmilian Caligiuri and Takaaki Musha 10.1 Introduction 292 10.2 QED Coherence in Water: A Brief Overview 295 10.3 “Electronic” QED Coherence in Brain Microtubules 301 10.4 Evanescent Field of Coherent Photons and Their Superluminal Tunneling through MTs 305 10.5 Coupling between Nearby MTs and their Superluminal Interaction through the Exchange of Virtual Superradiant Photons 312 10.6 Discussion 316 10.7 Brain Microtubules as “Natural” Metamaterials and the Amplification of Evanescent Tunneling Wave Amplitude 319 10.8 Quantum Computation by Means of Superluminal Photons 325 10.9 Conclusions 329 References 330 11 Advanced Fundamental-solution-based Computational Methods for Thermal Analysis of Heterogeneous Materials 335 Hui Wang and Qing-Hua Qin 11.1 Introduction 336 11.2 Basic Formulation of MFS 338 11.2.1 Standard MFS 338 11.2.2 Modified MFS 340 11.2.2.1 RBF Interpolation for the Particular Solution 341 11.2.2.2 MFS for the Homogeneous Solution 342 11.2.2.3 Complete Solution 343 11.3 Basic Formulation of HFS-FEM 344 11.3.1 Problem Statement 344 11.3.2 Implementation of the HFS-FEM 346 11.3.4 Recovery of Rigid-body Motion 349 11.4 Applications in Functionally Graded Materials 349 11.4.1 Basic Equations in Functionally Graded Materials 349 11.4.2 MFS for Functionally Graded Materials 350 11.4.3 HFS-FEM for Functionally Graded Materials 353 11.5 Applications in Composite Materials 357 11.5.1 Basic Equations of Composite Materials 357 11.5.2 MFS for Composite Materials 360 11.5.2.1 MFS for the Matrix Domain 360 11.5.2.2 MFS for the Fiber Domain 360 11.5.2.3 Complete Linear Equation System 361 11.5.3 HFS-FEM for Composite Materials 362 11.5.3.1 Special Fundamental Solutions 362 11.5.3.2 Special n-Sided Fiber/Matrix Elements 363 11.6 Conclusions 365 Acknowledgments 366 Conflict of Interest 366 References 366 12 Understanding the SET/RESET Characteristics of Forming Free TiOx/TiO2–x Resistive-Switching Bilayer Structures through Experiments and Modeling 373 P. Bousoulas and D. Tsoukalas 12.1 Introduction 374 12.2 Experimental Methodology 376 12.3 Bipolar Switching Model 378 12.3.1 Resistive-Switching Performance 378 12.3.2 Resistive-Switching Model 383 12.4 RESET Simulations 389 12.4.1 I–V Response 389 12.4.2 Influence of TE on the CFs Broken Region 393 12.5 SET Simulations 398 12.6 Simulation of Time-dependent SET/RESET Processes 401 12.7 Conclusions 403 Acknowledgments 404 References 404 13 Advanced Materials and Three-dimensional Computer-aided Surgical Workflow in Cranio-maxillofacial Reconstruction 411 Luis Miguel Gonzalez-Perez, Borja Gonzalez-Perez-Somarriba Gabriel Centeno, Carpóforo Vallellano, and Juan Jose Egea-Guerrero 13.1 Introduction 412 13.2 Methodology 413 13.3 Findings 418 13.4 Discussion 427 References 436 14 Displaced Multiwavelets and Splitting Algorithms 439 Boris M. Shumilov 14.1 An Algorithm with Splitting of Wavelet Transformation of Splines of the First Degree 443 14.1.1 “Lazy” Wavelets 444 14.1.2 Examples of Wavelet Decomposition of a Signal of Length 8 447 14.1.3 “Orthonormal” Wavelets 450 14.1.4 An Example of Function of Harten 454 14.2 An Algorithm for Constructing Orthogonal to Polynomials Multiwavelet Bases 456 14.2.1 Creation of System of Basic Multiwavelets of Any Odd Degree on a Closed Interval 456 14.2.2 Creation of the Block of Filters 459 14.2.3 Example of Orthogonal to Polynomials Multiwavelet Bases 461 14.2.4 The Discussion of Approximation on a Closed Interval 463 14.3 The Tridiagonal Block Matrix Algorithm 464 14.3.1 Inverse of the Block of Filters 464 14.3.2 Example of the Hermite Quintic Spline Function Supported on [−1, 1] 465 14.3.3 Example of the Hermite Septimus Spline Function Supported on [−1, 1] 467 14.3.4 Numerical Example of Approximation of Polynomial Function 470 14.3.5 Numerical Example with Two Ruptures of the First Kind and a Corner 471 14.4 Problem of Optimization of Wavelet Transformation of Hermite Splines of Any Odd Degree 475 14.4.1 An Algorithm with Splitting for Wavelet Transformation of Hermite Splines of Fifth Degree 478 14.4.2 Examples 485 14.5 Application to Data Processing of Laser Scanning of Roads490 14.5.1 Calculation of Derivatives on Samples 490 14.5.2 Example of Wavelet Compression of One Track of Data of Laser Scanning 490 14.5.3 Modeling of Surfaces 490 14.5.4 Functions of a Package of Applied Programs for Modeling of Routes and Surfaces of Highways 492 14.6 Conclusions 494 References 494

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Book SynopsisA comprehensive introduction to the multidisciplinary applications of mathematical methods, revised and updated The second edition of Essentials of Mathematical Methods in Science and Engineering offers an introduction to the key mathematical concepts of advanced calculus, differential equations, complex analysis, and introductory mathematical physics for students in engineering and physics research. The book's approachable style is designed in a modular format with each chapter covering a subject thoroughly and thus can be read independently. This updated second edition includes two new and extensive chapters that cover practical linear algebra and applications of linear algebra as well as a computer file that includes Matlab codes. To enhance understanding of the material presented, the text contains a collection of exercises at the end of each chapter. The author offers a coherent treatment of the topics with a style that makes the essential mathematicTable of ContentsPreface xxiii Acknowledgments xxix 1 Functional Analysis 1 1.1 Concept of Function 1 1.2 Continuity and Limits 3 1.3 Partial Differentiation 6 1.4 Total Differential 8 1.5 Taylor Series 9 1.6 Maxima and Minima of Functions 13 1.7 Extrema of Functions with Conditions 17 1.8 Derivatives and Differentials of Composite Functions 21 1.9 Implicit Function Theorem 23 1.10 Inverse Functions 28 1.11 Integral Calculus and the Definite Integral 30 1.12 Riemann Integral 32 1.13 Improper Integrals 35 1.14 Cauchy Principal Value Integrals 38 1.15 Integrals Involving a Parameter 40 1.16 Limits of Integration Depending on a Parameter 44 1.17 Double Integrals 45 1.18 Properties of Double Integrals 47 1.19 Triple and Multiple Integrals 48 References 49 Problems 49 2 Vector Analysis 55 2.1 Vector Algebra: Geometric Method 55 2.1.1 Multiplication of Vectors 57 2.2 Vector Algebra: Coordinate Representation 60 2.3 Lines and Planes 65 2.4 Vector Differential Calculus 67 2.4.1 Scalar Fields and Vector Fields 67 2.4.2 Vector Differentiation 69 2.5 Gradient Operator 70 2.5.1 Meaning of the Gradient 71 2.5.2 Directional Derivative 72 2.6 Divergence and Curl Operators 73 2.6.1 Meaning of Divergence and the Divergence Theorem 75 2.7 Vector Integral Calculus in Two Dimensions 79 2.7.1 Arc Length and Line Integrals 79 2.7.2 Surface Area and Surface Integrals 83 2.7.3 An Alternate Way to Write Line Integrals 84 2.7.4 Green’s Theorem 86 2.7.5 Interpretations of Green’s Theorem 88 2.7.6 Extension to Multiply Connected Domains 89 2.8 Curl Operator and Stokes’s Theorem 92 2.8.1 On the Plane 92 2.8.2 In Space 96 2.8.3 Geometric Interpretation of Curl 99 2.9 Mixed Operations with the Del Operator 99 2.10 Potential Theory 102 2.10.1 Gravitational Field of a Star 105 2.10.2 Work Done by Gravitational Force 106 2.10.3 Path Independence and Exact Differentials 108 2.10.4 Gravity and Conservative Forces 109 2.10.5 Gravitational Potential 111 2.10.6 Gravitational Potential Energy of a System 113 2.10.7 Helmholtz Theorem 115 2.10.8 Applications of the Helmholtz Theorem 116 2.10.9 Examples from Physics 120 References 123 Problems 123 3 Generalized Coordinates and Tensors 133 3.1 Transformations between Cartesian Coordinates 134 3.1.1 Basis Vectors and Direction Cosines 134 3.1.2 Transformation Matrix and Orthogonality 136 3.1.3 Inverse Transformation Matrix 137 3.2 Cartesian Tensors 139 3.2.1 Algebraic Properties of Tensors 141 3.2.2 Kronecker Delta and the Permutation Symbol 145 3.3 Generalized Coordinates 148 3.3.1 Coordinate Curves and Surfaces 148 3.3.2 Why Upper and Lower Indices 152 3.4 General Tensors 153 3.4.1 Einstein Summation Convention 156 3.4.2 Line Element 157 3.4.3 Metric Tensor 157 3.4.4 How to Raise and Lower Indices 158 3.4.5 Metric Tensor and the Basis Vectors 160 3.4.6 Displacement Vector 161 3.4.7 Line Integrals 162 3.4.8 Area Element in Generalized Coordinates 164 3.4.9 Area of a Surface 165 3.4.10 Volume Element in Generalized Coordinates 169 3.4.11 Invariance and Covariance 171 3.5 Differential Operators in Generalized Coordinates 171 3.5.1 Gradient 171 3.5.2 Divergence 172 3.5.3 Curl 174 3.5.4 Laplacian 178 3.6 Orthogonal Generalized Coordinates 178 3.6.1 Cylindrical Coordinates 179 3.6.2 Spherical Coordinates 184 References 189 Problems 189 4 Determinants and Matrices 197 4.1 Basic Definitions 197 4.2 Operations with Matrices 198 4.3 Submatrix and Partitioned Matrices 204 4.4 Systems of Linear Equations 207 4.5 Gauss’s Method of Elimination 208 4.6 Determinants 211 4.7 Properties of Determinants 214 4.8 Cramer’s Rule 216 4.9 Inverse of a Matrix 221 4.10 Homogeneous Linear Equations 224 References 225 Problems 225 5 Linear Algebra 233 5.1 Fields and Vector Spaces 233 5.2 Linear Combinations, Generators, and Bases 236 5.3 Components 238 5.4 Linear Transformations 241 5.5 Matrix Representation of Transformations 242 5.6 Algebra of Transformations 244 5.7 Change of Basis 246 5.8 Invariants under Similarity Transformations 247 5.9 Eigenvalues and Eigenvectors 248 5.10 Moment of Inertia Tensor 257 5.11 Inner Product Spaces 262 5.12 The Inner Product 262 5.13 Orthogonality and Completeness 265 5.14 Gram–Schmidt Orthogonalization 267 5.15 Eigenvalue Problem for Real Symmetric Matrices 268 5.16 Presence of Degenerate Eigenvalues 270 5.17 Quadratic Forms 276 5.18 Hermitian Matrices 279 5.19 Matrix Representation of Hermitian Operators 283 5.20 Functions of Matrices 284 5.21 Function Space and Hilbert Space 286 5.22 Dirac’s Bra and Ket Vectors 287 References 288 Problems 289 6 Practical Linear Algebra 293 6.1 Systems of Linear Equations 294 6.1.1 Matrices and Elementary Row Operations 295 6.1.2 Gauss-Jordan Method 295 6.1.3 Information From the Row-Echelon Form 300 6.1.4 Elementary Matrices 301 6.1.5 Inverse by Gauss-Jordan Row-Reduction 302 6.1.6 Row Space, Column Space, and Null Space 303 6.1.7 Bases for Row, Column, and Null Spaces 307 6.1.8 Vector Spaces Spanned by a Set of Vectors 310 6.1.9 Rank and Nullity 312 6.1.10 Linear Transformations 315 6.2 Numerical Methods of Linear Algebra 317 6.2.1 Gauss-Jordan Row-Reduction and Partial Pivoting 317 6.2.2 LU-Factorization 321 6.2.3 Solutions of Linear Systems by Iteration 325 6.2.4 Interpolation 328 6.2.5 Power Method for Eigenvalues 331 6.2.6 Solution of Equations 333 6.2.7 Numerical Integration 343 References 349 Problems 350 7 Applications of Linear Algebra 355 7.1 Chemistry and Chemical Engineering 355 7.1.1 Independent Reactions and Stoichiometric Matrix 356 7.1.2 Independent Reactions from a Set of Species 359 7.2 Linear Programming 362 7.2.1 The Geometric Method 363 7.2.2 The Simplex Method 367 7.3 Leontief Input–Output Model of Economy 375 7.3.1 Leontief Closed Model 375 7.3.2 Leontief Open Model 378 7.4 Applications to Geometry 381 7.4.1 Orbit Calculations 382 7.5 Elimination Theory 383 7.5.1 Quadratic Equations and the Resultant 384 7.6 Coding Theory 388 7.6.1 Fields and Vector Spaces 388 7.6.2 Hamming (7,4) Code 390 7.6.3 Hamming Algorithm for Error Correction 393 7.7 Cryptography 396 7.7.1 Single-Key Cryptography 396 7.8 Graph Theory 399 7.8.1 Basic Definition 399 7.8.2 Terminology 400 7.8.3 Walks, Trails, Paths and Circuits 402 7.8.4 Trees and Fundamental Circuits 404 7.8.5 Graph Operations 404 7.8.6 Cut Sets and Fundamental Cut Sets 405 7.8.7 Vector Space Associated with a Graph 407 7.8.8 Rank and Nullity 409 7.8.9 Subspaces in WG 410 7.8.10 Dot Product and Orthogonal vectors 411 7.8.11 Matrix Representation of Graphs 413 7.8.12 Dominance Directed Graphs 417 7.8.13 Gray Codes in Coding Theory 419 References 419 Problems 420 8 Sequences and Series 425 8.1 Sequences 426 8.2 Infinite Series 430 8.3 Absolute and Conditional Convergence 431 8.3.1 Comparison Test 431 8.3.2 Limit Comparison Test 431 8.3.3 Integral Test 431 8.3.4 Ratio Test 432 8.3.5 Root Test 432 8.4 Operations with Series 436 8.5 Sequences and Series of Functions 438 8.6 M-Test for Uniform Convergence 441 8.7 Properties of Uniformly Convergent Series 441 8.8 Power Series 443 8.9 Taylor Series and Maclaurin Series 446 8.10 Indeterminate Forms and Series 447 References 448 Problems 448 9 Complex Numbers and Functions 453 9.1 The Algebra of Complex Numbers 454 9.2 Roots of a Complex Number 458 9.3 Infinity and the Extended Complex Plane 460 9.4 Complex Functions 463 9.5 Limits and Continuity 465 9.6 Differentiation in the Complex Plane 467 9.7 Analytic Functions 470 9.8 Harmonic Functions 471 9.9 Basic Differentiation Formulas 474 9.10 Elementary Functions 475 9.10.1 Polynomials 475 9.10.2 Exponential Function 476 9.10.3 Trigonometric Functions 477 9.10.4 Hyperbolic Functions 478 9.10.5 Logarithmic Function 479 9.10.6 Powers of Complex Numbers 481 9.10.7 Inverse Trigonometric Functions 483 References 483 Problems 484 10 Complex Analysis 491 10.1 Contour Integrals 492 10.2 Types of Contours 494 10.3 The Cauchy–Goursat Theorem 497 10.4 Indefinite Integrals 500 10.5 Simply and Multiply Connected Domains 502 10.6 The Cauchy Integral Formula 503 10.7 Derivatives of Analytic Functions 505 10.8 Complex Power Series 506 10.8.1 Taylor Series with the Remainder 506 10.8.2 Laurent Series with the Remainder 510 10.9 Convergence of Power Series 514 10.10 Classification of Singular Points 514 10.11 Residue Theorem 517 References 522 Problems 522 11 Ordinary Differential Equations 527 11.1 Basic Definitions for Ordinary Differential Equations 528 11.2 First-Order Differential Equations 530 11.2.1 Uniqueness of Solution 530 11.2.2 Methods of Solution 532 11.2.3 Dependent Variable is Missing 532 11.2.4 Independent Variable is Missing 532 11.2.5 The Case of Separable f(x, y) 532 11.2.6 Homogeneous f(x, y) of Zeroth Degree 533 11.2.7 Solution When f(x, y) is a Rational Function 533 11.2.8 Linear Equations of First-order 535 11.2.9 Exact Equations 537 11.2.10 Integrating Factors 539 11.2.11 Bernoulli Equation 542 11.2.12 Riccati Equation 543 11.2.13 Equations that Cannot Be Solved for y' 546 11.3 Second-Order Differential Equations 548 11.3.1 The General Case 549 11.3.2 Linear Homogeneous Equations with Constant Coefficients 551 11.3.3 Operator Approach 556 11.3.4 Linear Homogeneous Equations with Variable Coefficients 557 11.3.5 Cauchy–Euler Equation 560 11.3.6 Exact Equations and Integrating Factors 561 11.3.7 Linear Nonhomogeneous Equations 564 11.3.8 Variation of Parameters 564 11.3.9 Method of Undetermined Coefficients 566 11.4 Linear Differential Equations of Higher Order 569 11.4.1 With Constant Coefficients 569 11.4.2 With Variable Coefficients 570 11.4.3 Nonhomogeneous Equations 570 11.5 Initial Value Problem and Uniqueness of the Solution 571 11.6 Series Solutions: Frobenius Method 571 11.6.1 Frobenius Method and First-order Equations 581 References 582 Problems 582 12 Second-Order Differential Equations and Special Functions 589 12.1 Legendre Equation 590 12.1.1 Series Solution 590 12.1.2 Effect of Boundary Conditions 593 12.1.3 Legendre Polynomials 594 12.1.4 Rodriguez Formula 596 12.1.5 Generating Function 597 12.1.6 Special Values 599 12.1.7 Recursion Relations 600 12.1.8 Orthogonality 601 12.1.9 Legendre Series 603 12.2 Hermite Equation 606 12.2.1 Series Solution 606 12.2.2 Hermite Polynomials 610 12.2.3 Contour Integral Representation 611 12.2.4 Rodriguez Formula 612 12.2.5 Generating Function 613 12.2.6 Special Values 614 12.2.7 Recursion Relations 614 12.2.8 Orthogonality 616 12.2.9 Series Expansions in Hermite Polynomials 618 12.3 Laguerre Equation 619 12.3.1 Series Solution 620 12.3.2 Laguerre Polynomials 621 12.3.3 Contour Integral Representation 622 12.3.4 Rodriguez Formula 623 12.3.5 Generating Function 623 12.3.6 Special Values and Recursion Relations 624 12.3.7 Orthogonality 624 12.3.8 Series Expansions in Laguerre Polynomials 625 References 626 Problems 626 13 Bessel’s Equation and Bessel Functions 629 13.1 Bessel’s Equation and Its Series Solution 630 13.1.1 Bessel Functions J±m(x), Nm(x), and H(1,2)m (x) 634 13.1.2 Recursion Relations 639 13.1.3 Generating Function 639 13.1.4 Integral Definitions 641 13.1.5 Linear Independence of Bessel Functions 642 13.1.6 Modified Bessel Functions Im(x) and Km(x) 644 13.1.7 Spherical Bessel Functions jl(x), nl(x), and h(1,2)l (x) 645 13.2 Orthogonality and the Roots of Bessel Functions 648 13.2.1 Expansion Theorem 652 13.2.2 Boundary Conditions for the Bessel Functions 652 References 656 Problems 656 14 Partial Differential Equations and Separation of Variables 661 14.1 Separation of Variables in Cartesian Coordinates 662 14.1.1 Wave Equation 665 14.1.2 Laplace Equation 666 14.1.3 Diffusion and Heat Flow Equations 671 14.2 Separation of Variables in Spherical Coordinates 673 14.2.1 Laplace Equation 677 14.2.2 Boundary Conditions for a Spherical Boundary 678 14.2.3 Helmholtz Equation 682 14.2.4 Wave Equation 683 14.2.5 Diffusion and Heat Flow Equations 684 14.2.6 Time-Independent Schrödinger Equation 685 14.2.7 Time-Dependent Schrödinger Equation 685 14.3 Separation of Variables in Cylindrical Coordinates 686 14.3.1 Laplace Equation 688 14.3.2 Helmholtz Equation 689 14.3.3 Wave Equation 690 14.3.4 Diffusion and Heat Flow Equations 691 References 701 Problems 701 15 Fourier Series 705 15.1 Orthogonal Systems of Functions 705 15.2 Fourier Series 711 15.3 Exponential Form of the Fourier Series 712 15.4 Convergence of Fourier Series 713 15.5 Sufficient Conditions for Convergence 715 15.6 The Fundamental Theorem 716 15.7 Uniqueness of Fourier Series 717 15.8 Examples of Fourier Series 717 15.8.1 Square Wave 717 15.8.2 Triangular Wave 719 15.8.3 Periodic Extension 720 15.9 Fourier Sine and Cosine Series 721 15.10 Change of Interval 722 15.11 Integration and Differentiation of Fourier Series 723 References 724 Problems 724 16 Fourier and Laplace Transforms 727 16.1 Types of Signals 727 16.2 Spectral Analysis and Fourier Transforms 730 16.3 Correlation with Cosines and Sines 731 16.4 Correlation Functions and Fourier Transforms 735 16.5 Inverse Fourier Transform 736 16.6 Frequency Spectrums 736 16.7 Dirac-Delta Function 738 16.8 A Case with Two Cosines 739 16.9 General Fourier Transforms and Their Properties 740 16.10 Basic Definition of Laplace Transform 743 16.11 Differential Equations and Laplace Transforms 746 16.12 Transfer Functions and Signal Processors 748 16.13 Connection of Signal Processors 750 References 753 Problems 753 17 Calculus of Variations 757 17.1 A Simple Case 758 17.2 Variational Analysis 759 17.2.1 Case I: The Desired Function is Prescribed at the End Points 761 17.2.2 Case II: Natural Boundary Conditions 762 17.3 Alternate Form of Euler Equation 763 17.4 Variational Notation 765 17.5 A More General Case 767 17.6 Hamilton’s Principle 772 17.7 Lagrange’s Equations of Motion 773 17.8 Definition of Lagrangian 777 17.9 Presence of Constraints in Dynamical Systems 779 17.10 Conservation Laws 783 References 784 Problems 784 18 Probability Theory and Distributions 789 18.1 Introduction to Probability Theory 790 18.1.1 Fundamental Concepts 790 18.1.2 Basic Axioms of Probability 791 18.1.3 Basic Theorems of Probability 791 18.1.4 Statistical Definition of Probability 794 18.1.5 Conditional Probability and Multiplication Theorem 795 18.1.6 Bayes’ Theorem 796 18.1.7 Geometric Probability and Buffon’s Needle Problem 798 18.2 Permutations and Combinations 800 18.2.1 The Case of Distinguishable Balls with Replacement 800 18.2.2 The Case of Distinguishable Balls without Replacement 801 18.2.3 The Case of Indistinguishable Balls 802 18.2.4 Binomial and Multinomial Coefficients 803 18.3 Applications to Statistical Mechanics 804 18.3.1 Boltzmann Distribution for Solids 805 18.3.2 Boltzmann Distribution for Gases 807 18.3.3 Bose–Einstein Distribution for Perfect Gases 808 18.3.4 Fermi–Dirac Distribution 810 18.4 Statistical Mechanics and Thermodynamics 811 18.4.1 Probability and Entropy 811 18.4.2 Derivation of β 812 18.5 Random Variables and Distributions 814 18.6 Distribution Functions and Probability 817 18.7 Examples of Continuous Distributions 819 18.7.1 Uniform Distribution 819 18.7.2 Gaussian or Normal Distribution 820 18.7.3 Gamma Distribution 821 18.8 Discrete Probability Distributions 821 18.8.1 Uniform Distribution 822 18.8.2 Binomial Distribution 822 18.8.3 Poisson Distribution 824 18.9 Fundamental Theorem of Averages 825 18.10 Moments of Distribution Functions 826 18.10.1 Moments of the Gaussian Distribution 827 18.10.2 Moments of the Binomial Distribution 827 18.10.3 Moments of the Poisson Distribution 829 18.11 Chebyshev’s Theorem 831 18.12 Law of Large Numbers 832 References 833 Problems 834 19 Information Theory 841 19.1 Elements of Information Processing Mechanisms 844 19.2 Classical Information Theory 846 19.2.1 Prior Uncertainty and Entropy of Information 848 19.2.2 Joint and Conditional Entropies of Information 851 19.2.3 Decision Theory 854 19.2.4 Decision Theory and Game Theory 856 19.2.5 Traveler’s Dilemma and Nash Equilibrium 862 19.2.6 Classical Bit or Cbit 866 19.2.7 Operations on Cbits 869 19.3 Quantum Information Theory 871 19.3.1 Basic Quantum Theory 872 19.3.2 Single-Particle Systems and Quantum Information 878 19.3.3 Mach–Zehnder Interferometer 880 19.3.4 Mathematics of the Mach–Zehnder Interferometer 882 19.3.5 Quantum Bit or Qbit 886 19.3.6 The No-Cloning Theorem 889 19.3.7 Entanglement and Bell States 890 19.3.8 Quantum Dense Coding 895 19.3.9 Quantum Teleportation 896 References 900 Problems 901 Further Reading 907 Index 915

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Book SynopsisIntroduction to Modeling and Simulation An essential introduction to engineering system modeling and simulation from a well-trusted source in engineering and education This new introductory-level textbook provides thirteen self-contained chapters, each covering an important topic in engineering systems modeling and simulation. The importance of such a topic cannot be overstated; modeling and simulation will only increase in importance in the future as computational resources improve and become more powerful and accessible, and as systems become more complex. This resource is a wonderful mix of practical examples, theoretical concepts, and experimental sessions that ensure a well-rounded education on the topic. The topics covered in Introduction to Modeling and Simulation are timeless fundamentals that provide the necessary background for further and more advanced study of one or more of the topics. The text includes topics such as linear and nonlinear dynamical systems, continuous-time and discrete-time systems, stability theory, numerical methods for solution of ODEs, PDE models, feedback systems, optimization, regression and more. Each chapter provides an introduction to the topic to familiarize students with the core ideas before delving deeper. The numerous tools and examples help ensure students engage in active learning, acquiring a range of tools for analyzing systems and gaining experience in numerical computation and simulation systems, from an author prized for both his writing and his teaching over the course of his over-40-year career. Introduction to Modeling and Simulation readers will also find: Numerous examples, tools, and programming tips to help clarify points made throughout the textbook, with end-of-chapter problems to further emphasize the material As systems become more complex, a chapter devoted to complex networks including small-world and scale-free networks a unique advancement for textbooks within modeling and simulation A complementary website that hosts a complete set of lecture slides, a solution manual for end-of-chapter problems, MATLAB files, and case-study exercises Introduction to Modeling and Simulation is aimed at undergraduate and first-year graduate engineering students studying systems, in diverse avenues within the field: electrical, mechanical, mathematics, aerospace, bioengineering, physics, and civil and environmental engineering. It may also be of interest to those in mathematical modeling courses, as it provides in-depth material on MATLAB simulation and contains appendices with brief reviews of linear algebra, real analysis, and probability theory.Table of ContentsPreface xiii About the Companion Website xvii 1 Introduction 1 1.1 Introduction 1 1.1.1 Systems Engineering 1 1.1.2 The Input/Output Viewpoint 2 1.1.3 Some Examples 2 1.2 Model Classification 5 1.2.1 Static and Dynamic Systems 5 1.2.2 Linear and Nonlinear Systems 5 1.2.3 Distributed-Parameter Systems 6 1.2.4 Hybrid and Discrete-Event Systems 6 1.2.5 Deterministic and Stochastic Systems 7 1.2.6 Large-Scale Systems 7 1.3 Simulation Languages 9 1.4 Outline of the Text 10 Problems 11 2 Second-Order Systems 15 2.1 Introduction 15 2.2 State-Space Representation 19 2.3 Trajectories and Phase Portraits 22 2.4 The Direction Field 27 2.5 Equilibria 30 2.6 Linear Systems 33 2.7 Linearization of Nonlinear Systems 41 2.8 Periodic Trajectories and Limit Cycles 45 2.8.1 Relaxation Oscillators 45 2.8.2 Bendixson’s Theorem 49 2.8.3 Poincaré–Bendixson Theorem 51 2.9 Coupled Second-Order Systems 53 Problems 55 3 System Fundamentals 61 3.1 Introduction 61 3.2 Existence and Uniqueness of Solution 61 3.3 The Matrix Exponential 64 3.4 The Jordan Canonical Form 67 3.5 Linearization 71 3.6 The Hartman–Grobman Theorem 72 3.7 Singular Perturbations 73 Problems 79 4 Compartmental Models 83 4.1 Introduction 83 4.2 Exponential Growth and Decay 84 4.3 The Logistic Equation 87 4.4 Models of Epidemics 88 4.5 Predator–Prey System 95 Problems 97 5 Stability 101 5.1 Introduction 101 5.2 Lyapunov Stability 102 5.3 Basin of Attraction 109 5.4 The Invariance Principle 110 5.5 Linear Systems and Linearization 113 Problems 116 6 Discrete-Time Systems 119 6.1 Introduction 119 6.2 Stability of Discrete-Time Systems 123 6.3 Stability of Discrete-Time Linear Systems 124 6.4 Moving-Average Filter 126 6.5 Cobweb Diagrams 128 6.5.1 Cobweb Diagrams in Economics 130 6.5.2 The Discrete Logistic Equation 131 Problems 134 7 Numerical Methods 137 7.1 Introduction 137 7.2 Numerical Differentiation 138 7.3 Numerical Integration 141 7.4 Numerical Solution of ODEs 147 7.4.1 Euler Predictor–Corrector Method 150 7.4.2 Runge–Kutta Methods 152 7.5 Stiff Systems 155 7.6 Event Detection 160 7.7 Simulink 163 7.8 Summary 168 Problems 169 8 Optimization 173 8.1 Introduction 173 8.2 Unconstrained Optimization 177 8.2.1 Iterative Search 179 8.2.2 Gradient Descent 180 8.2.3 Newton’s Method 184 8.3 Case Study: Numerical Inverse Kinematics 187 8.4 Constrained Optimization 191 8.4.1 Equality Constraints 191 8.4.2 Inequality Constraints 196 8.5 Convex Optimization 200 Problems 204 9 System Identification 209 9.1 Introduction 209 9.2 Least Squares 209 9.3 Regression 212 9.4 Recursive Least Squares 217 9.5 Logistic Regression 220 9.6 Neural Networks 224 Problems 230 10 Stochastic Systems 233 10.1 Markov Chains 233 10.1.1 Regular and Ergodic Markov Chains 240 10.1.2 Absorbing Markov Chains 244 10.2 Monte Carlo Methods 249 10.2.1 Random Number Generation 250 10.2.2 Monte Carlo Integration 253 10.2.3 Monte Carlo Optimization 255 10.2.4 Monte Carlo Simulation 255 Problems 258 11 Feedback Systems 261 11.1 Introduction 261 11.2 Transfer Functions 263 11.3 Feedback Control 269 11.4 State-Space Models 273 11.4.1 Minimal Realizations 274 11.4.2 Pole Placement 280 11.4.3 State Estimation 283 11.4.4 The Separation Principle 285 11.5 Optimal Control 288 11.6 Control of Nonlinear Systems 289 Problems 292 12 Partial Differential Equation Models 297 12.1 Introduction 297 12.1.1 Existence and Uniqueness of Solutions 297 12.1.2 Classification of Linear Second-Order PDEs 298 12.2 The Wave Equation 299 12.2.1 The D’Alembert Solution 300 12.2.2 Initial-Value Problem 300 12.2.3 Separation of Variables 302 12.3 The Heat Equation 310 12.4 Laplace’s Equation 313 12.5 Numerical Solution of PDEs 315 Problems 319 13 Complex Networks 321 13.1 Introduction 321 13.1.1 Examples of Complex Networks 322 13.2 Graph Theory: Basic Concepts 324 13.2.1 Graph Isomorphism 327 13.2.2 Connectivity 327 13.2.3 Trees 331 13.2.4 Bipartite Graphs 332 13.2.5 Planar Graphs 333 13.2.6 Graphs and Matrices 335 13.3 Matlab Graph Functions 341 13.4 Network Metrics 343 13.4.1 Degree Distribution 343 13.4.2 Centrality 347 13.4.3 Clustering 350 13.5 Random Graphs 354 13.5.1 Erdős–Rényi Networks 354 13.5.2 Small-World Networks 358 13.5.3 Scale-Free Networks 360 13.6 Synchronization in Networks 362 Problems 366 Appendix A Linear Algebra 371 A. 1 Vectors 371 A. 2 Matrices 373 A. 3 Eigenvalues and Eigenvectors 375 Appendix B Real Analysis 379 B. 1 Set Theory 379 B. 2 Vector Fields 380 B. 3 Jacobian 381 B. 4 Scalar Functions 381 B. 5 Taylor’s Theorem 382 B. 6 Extreme-Value Theorem 383 Appendix C Probability 385 C.1 Discrete Probability 385 C.2 Conditional Probability 386 C.3 Random Variables 389 C.4 Continuous Probability 391 Appendix D Proofs of Selected Results 395 D. 1 Proof of Theorem 2.2 395 D. 2 Proof of Theorem 5.1 395 D. 3 Proof of Theorem 5.5 396 D. 4 Proof of Theorem 13.3 397 D. 5 Proof of Corollary 13.2 397 D. 6 Proof of Proposition 13.2 398 D. 7 Proof of Proposition 13.3 398 Appendix E Matlab Command Reference 399 References 403 Index 407

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Book SynopsisThis book is dedicated to modern approaches to mathematical modeling of reflexive processes in control. The authors consider reflexive games that describe the gametheoretical interaction of agents making decisions based on a hierarchy of beliefs regarding (1) essential parameters (informational reflexion), (2) decision principles used by opponents (strategic reflexion), (3) beliefs about beliefs, and so on. Informational and reflexive equilibria in reflexive games generalize a series of well-known equilibrium concepts in noncooperative games and models of collective behavior. These models allow posing and solving the problems of informational and reflexive control in organizational, economic, social and other systems, in military applications, etc. (the interested reader will find in the book over 30 examples of possible applications in these fields) and describing uniformly many psychological/sociological phenomena connected with reflexion, viz., implicit control, informational control via the mass media, reflexion in chess, art works, etc. The present book is intended for experts in decision making and control of systems of an interdisciplinary nature, as well as for undergraduates and postgraduates.Table of ContentsIntroduction. 1. Reflexion in decision-making 2. Informational reflexion and control 3. Strategic reflexion and control 4. Applied models of informational and reflexive control. Conclusion.

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Book SynopsisStochastic Communities presents a theory of biodiversity by analyzing the distribution of abundances among species in the context of a community. The basis of this theory is a distribution called the J distribution. This distribution is a pure hyperbola and mathematically implied by the stochastic species hypothesis assigning equal probabilities of birth and death within the population of each species over varying periods of time. The J distribution in natural communities has strong empirical support resulting from a meta-study and strong theoretical support from a theorem that is mathematically implied by the stochastic species hypothesis.Trade Review"The science of ecology suffers from a disconnect between theory and direct observation. Mathematicians have thought that simple equations could explain ecology. Field ecologists have assumed the mathematicians are right. Thus, empirical ecological understanding and prediction have suffered. Dewdney is an exceptions; he does field work and he is a mathematician. In his wonderful book, he takes advantage of both parts of ecology. And if Dewdney has done his math right, this book opens a whole new door to understanding biodiversity and its myriad causes."- Daniel Botkin, Professor Emeritus in the Department of Ecology, Evolution, and Marine Biology at the University of California, Santa Barbara, and President of the Center for the Study of the Environment"… development of a theory and guide to sampling, … it will be of great interest to both empirical and theoretical ecologists." - Trends in Ecology and EvolutionTable of ContentsThe J-curve and the J distribution. The J-distribution and its variations. Sampling in practice and in theory. Compiling and analysing field data. Predictions from data. Extending the sample. Stochastic systems and the stochastic community. The metastudy: A review. Fossil J-curves. Summary of theory and open problems. Appendix A: Mathematical Notes and Computer Tools. Appendix B: Results of the metastudy for the J distribution. Appendix C: Results of the test for the J distribution in taxonomic data.

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Book SynopsisStochastic Modeling for Medical Image Analysis provides a brief introduction to medical imaging, stochastic modeling, and model-guided image analysis.Today, image-guided computer-assisted diagnostics (CAD) faces two basic challenging problems. The first is the computationally feasible and accurate modeling of images from different modalities to obtain clinically useful information. The second is the accurate and fast inferring of meaningful and clinically valid CAD decisions and/or predictions on the basis of model-guided image analysis.To help address this, this book details original stochastic appearance and shape models with computationally feasible and efficient learning techniques for improving the performance of object detection, segmentation, alignment, and analysis in a number of important CAD applications.The book demonstrates accurate descriptions of visual appearances and shapes of the goal objects and their background to help solve aTable of ContentsMedical Imaging Modalities. From Images to Graphical Models. IRF Models: Estimating Marginals. Markov-Gibbs Random Field Models: Estimating Signal Interactions. Applications: Image Alignment. Segmenting Multimodal Images. Segmenting with Deformable Models. Segmenting with Shape and Appearance Priors. Cine Cardiac MRI Analysis. Sizing Cardiac Pathologies.

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Book SynopsisIntroduction to Mathematical Modeling helps students master the processes used by scientists and engineers to model real-world problems, including the challenges posed by space exploration, climate change, energy sustainability, chaotic dynamical systems and random processes.Primarily intended for students with a working knowledge of calculus but minimal training in computer programming in a first course on modeling, the more advanced topics in the book are also useful for advanced undergraduate and graduate students seeking to get to grips with the analytical, numerical, and visual aspects of mathematical modeling, as well as the approximations and abstractions needed for the creation of a viable model.Table of ContentsThe Process of Mathematical Modeling. Modeling with ODEs. Systems of ODES. Stability. Bifurcations. Modeling with PDES. Modeling of Fluid flow. Geophysical Modeling. Nonlinear PDEs. Variational Principles.

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Book SynopsisThe book presents qualitative results for different classes of fractional equations, including fractional functional differential equations, fractional impulsive differential equations, and fractional impulsive functional differential equations, which have not been covered by other books. It manifests different constructive methods by demonstrating how these techniques can be applied to investigate qualitative properties of the solutions of fractional systems. Since many applications have been included, the demonstrated techniques and models can be used in training students in mathematical modeling and in the study and development of fractional-order models.Table of ContentsIntroduction. Preliminary Notes. Qualitative Properties Definitions. Lyapunov Functions and their Fractional Derivatives. Fractional Comparison Lemmas. Stability and Boundedness. Lyapunov Stability. Theorems on Boundedness. Global Stability. Mittag-Leffler Stability. Practical Stability. Lipschitz Stability. Stability of Sets. Stability of Integral Manifolds. Almost Periodicity. Almost Periodic Solutions. Lyapunov Method for Almost Periodic Solutions. Uncertain Fractional Differential Systems. Applications. Fractional Impulsive Neural Networks. Stability and Synchronization. Almost Periodic Solutions. The Uncertain Case. Fractional Impulsive Biological Models. Lasota-Wazewska Models. Lotka-Volterra Models. Kolmogorov-type Models. Fractional Impulsive Models in Economics.

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Book SynopsisElastic plates form a class of very important mechanical structures that appear in a wide range of practical applications, from building bodies to microchip production. As the sophistication of industrial designs has increased, so has the demand for greater accuracy in analysis. This in turn has led modelers away from Kirchoff's classical theory for thin plates and toward increasingly refined models that yield not only the deflection of the middle section, but also account for transverse shear deformation. The improved performance of these models is achieved, however, at the expense of a much more complicated system of governing equations and boundary conditions.In this Monograph, the authors conduct a rigorous mathematical study of a number of boundary value problems for the system of partial differential equations that describe the equilibrium bending of an elastic plate with transverse shear deformation. Specifically, the authors explore the existence, uniqueness, and continuous dependence of the solution on the data. In each case, they give the variational formulation of the problems and discuss their solvability in Sobolev spaces. They then seek the solution in the form of plate potentials and reduce the problems to integral equations on the contour of the domain.This treatment covers an extensive range of problems and presents the variational method and the boundary integral equation method applied side-by-side. Readers will find that this feature of the book, along with its clear exposition, will lead to a firm and useful understanding of both the model and the methods.Trade Review"It is amazing that the authors have managed to cover so many fundamental boundary-value problems and present the variational method and the boundary integral equation method applied side-by-side in a single volume…This feature of the book will certainly strengthen understanding of both the model and the methods. The writing style is very clear, the book is self-contained and easy to read, and it should be extremely valuable to researchers interested in applied analysis and mathematical models in elasticity."-Proceedings of the Edinburgh Mathematical Society (2002, vol. 45) "This book will be useful for mathematicians, theoretical engineers, and all interested in mathematical modeling in elasticity."-European Mathematical Society Newsletter, No. 40 (June 2001)Table of ContentsIntroduction. Formulation of the Problems. Variational Formulation of the Dirichlet and Neumann Problems. Boundary Integral Equations for the Dirichlet and Neumann Problems. Transmission Boundary Value Problems. Plate Weakened by a Crack. Boundary Value Problems with Other Types of Boundary Conditions. Plate on a Generalized Elastic Foundation. Appendix.

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Book Synopsis"If Augustin Cournot had still been alive, he could have won the Nobel Memorial Prize in Economics on at least three different occasions", exclaimed Nobel Laureate Robert Aumann during the 2005 Cournot Centre conference.From his earliest publications, Cournot broke from tradition with his predecessors in applying mathematical modelling to the social sphere. Consequently, he was the first to affirm the mathematization of social phenomena as an essential principle. The fecundity of Cournot's works stems not only from this departure, but also from a richness that irrigated the social sciences of the twentieth century. In this collection, the contributors - including two Nobel laureates in economics - highlight Cournot's profound innovativeness and continued relevance in the areas of industrial economics, mathematical economics, market competition, game theory and epistemology of probability and statistics. Each of the seven authors reminds us of the force and modernity of Cournot's thought as a mathematician, historian of the sciences, philosopher and, not least, as an economist.Combining an epistemological perspective with a theoretical one, this book will be of great interest to researchers and students in the fields of economics, the history of economic thought, and epistemology.Trade Review'. . . readers, especially those with a methodological orientation, will find in this book useful material of a kind not so frequently available in more traditional HET books and journals.' -- Nicola Giocoli, Storia del Pensiero Economico'Augustin Cournot: Modelling Economics is not a biography, but rather a reflection on those ideas of Cournot that persist today and what we can still learn from this great thinker. One cannot help but wonder at the wide range of accomplishments detailed in this book, but the discussion of Cournot's missteps is an unexpected highlight. . . this book should appeal to those who would like to learn more about Cournot as well as the various settings in which his thoughts have been embraced or rejected.' -- Lauren E. Feiler, Journal of Economic Literature'This rich and fascinating collection of essays helps enormously to establish the reputation of Augustin Cournot as a diverse and powerful thinker, whose numerous contributions range far beyond his widely acknowledged model of oligopoly. Cournot is revealed not merely as a mathematician, but one who was engaged in philosophical debates concerning epistemology and the nature of science. Anyone with the preconception that the development of modern economics was confined to the Anglophone world - from Smith through Marshall to the Nobel Laureates of today - will be amazed by the details of Cournot's contribution revealed here.' -- Geoffrey M. Hodgson, University of Hertfordshire, UKTable of ContentsContents: Preface About the Series: Professor Robert Solow The Complete Works of Antoine Augustin Cournot Chronological Biography of Antoine Augustin Cournot Introduction Thierry Martin and Jean-Philippe Touffut 1. Cournot as Economist: 200 Years of Relevance Jean Magnan de Bornier 2. Cournot’s Probabilistic Epistemology Thierry Martin 3. The Functions of Economic Models Bernard Walliser 4. From Cournot’s Principle to Market Efficiency Glenn Shafer 5. War and Peace Robert J. Aumann 6. Cournot and the Social Income Robert M. Solow 7. Comparing the Incomparable: The Sociology of Statistics Alain Desrosières Index

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Book Synopsis"If Augustin Cournot had still been alive, he could have won the Nobel Memorial Prize in Economics on at least three different occasions", exclaimed Nobel Laureate Robert Aumann during the 2005 Cournot Centre conference.From his earliest publications, Cournot broke from tradition with his predecessors in applying mathematical modelling to the social sphere. Consequently, he was the first to affirm the mathematization of social phenomena as an essential principle. The fecundity of Cournot's works stems not only from this departure, but also from a richness that irrigated the social sciences of the twentieth century. In this collection, the contributors - including two Nobel laureates in economics - highlight Cournot's profound innovativeness and continued relevance in the areas of industrial economics, mathematical economics, market competition, game theory and epistemology of probability and statistics. Each of the seven authors reminds us of the force and modernity of Cournot's thought as a mathematician, historian of the sciences, philosopher and, not least, as an economist.Combining an epistemological perspective with a theoretical one, this book will be of great interest to researchers and students in the fields of economics, the history of economic thought, and epistemology.Trade Review'. . . readers, especially those with a methodological orientation, will find in this book useful material of a kind not so frequently available in more traditional HET books and journals.' -- Nicola Giocoli, Storia del Pensiero Economico'Augustin Cournot: Modelling Economics is not a biography, but rather a reflection on those ideas of Cournot that persist today and what we can still learn from this great thinker. One cannot help but wonder at the wide range of accomplishments detailed in this book, but the discussion of Cournot's missteps is an unexpected highlight. . . this book should appeal to those who would like to learn more about Cournot as well as the various settings in which his thoughts have been embraced or rejected.' -- Lauren E. Feiler, Journal of Economic Literature'This rich and fascinating collection of essays helps enormously to establish the reputation of Augustin Cournot as a diverse and powerful thinker, whose numerous contributions range far beyond his widely acknowledged model of oligopoly. Cournot is revealed not merely as a mathematician, but one who was engaged in philosophical debates concerning epistemology and the nature of science. Anyone with the preconception that the development of modern economics was confined to the Anglophone world - from Smith through Marshall to the Nobel Laureates of today - will be amazed by the details of Cournot's contribution revealed here.' -- Geoffrey M. Hodgson, University of Hertfordshire, UKTable of ContentsContents: Preface About the Series: Professor Robert Solow The Complete Works of Antoine Augustin Cournot Chronological Biography of Antoine Augustin Cournot Introduction Thierry Martin and Jean-Philippe Touffut 1. Cournot as Economist: 200 Years of Relevance Jean Magnan de Bornier 2. Cournot’s Probabilistic Epistemology Thierry Martin 3. The Functions of Economic Models Bernard Walliser 4. From Cournot’s Principle to Market Efficiency Glenn Shafer 5. War and Peace Robert J. Aumann 6. Cournot and the Social Income Robert M. Solow 7. Comparing the Incomparable: The Sociology of Statistics Alain Desrosières Index

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Book SynopsisA transport network is typically a network of roads, streets, pipes, aqueducts, power lines, or nearly any structure that permits either vehicular movement or the flow of some commodity. Transport network analysis, a field of transport engineering that typically employs the use of graph theory, is used to determine the flow of vehicles, commodities, or people through these networks. It may combine different modes of transport - for example, walking and driving - to model multi-modal journeys. This edition is completely updated and contains two new chapters covering spatial analysis and urban management through graph theory simulation. Highly practical, the simulation approach allows readers to solve classic problems, such as placement of high-speed roads, the capacity of a network, pollution emission control, and more.

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Book SynopsisThis series of five volumes proposes an integrated description of physical processes modeling used by scientific disciplines from meteorology to coastal morphodynamics. Volume 1 describes the physical processes and identifies the main measurement devices used to measure the main parameters that are indispensable to implement all these simulation tools. Volume 2 presents the different theories in an integrated approach: mathematical models as well as conceptual models, used by all disciplines to represent these processes. Volume 3 identifies the main numerical methods used in all these scientific fields to translate mathematical models into numerical tools. Volume 4 is composed of a series of case studies, dedicated to practical applications of these tools in engineering problems. To complete this presentation, volume 5 identifies and describes the modeling software in each discipline.Trade Review"An inventory of ground measurement instruments, which provide necessary input data for the various modeling tools described in the book, is drawn up, and mathematical models describing each field within the overall subject area are detailed by a series of system equations. These are then solved by the use of numerical methods adapted to the particular characteristics of the application in question." (Environmental Expert, 19 April 2011)Table of ContentsIntroduction xix Jean-Michel TANGUY Chapter 1. Reminders on the Mechanical Properties of Fluids 1 Jacques GEORGE 1.1. Laws of conservation, principles and general theorems 1 1.2. Enthalpy, rotation, mixing, saturation 13 1.3. Thermodynamic relations, relations of state and laws of behavior 20 1.4. Turbulent flow 26 1.5. Dynamics of geophysical fluids 30 Chapter 2. 3D Navier-Stokes Equations 35 Véronique DUCROCQ 2.1. The continuity hypothesis 35 2.2. Lagrangian description/Eulerian description 36 2.3. The continuity equation 37 2.4. The movement quantity assessment equation 38 2.5. The energy balance equation 41 2.6. The equation of state 41 2.7. Navier-Stokes equations for a fluid in rotation 41 Chapter 3. Models of the Atmosphere 43 Jean COIFFIER 3.1. Introduction 43 3.2. The various simplifications and corresponding models 44 3.3. The equations with various systems of coordinates 56 3.4. Some typical conformal projections 61 3.5. The operational models 67 3.6. Bibliography 69 Chapter 4. Hydrogeologic Models 71 Dominique THIÉRY 4.1. Equation of fluid mechanics 71 4.2. Continuity equation in porous media 72 4.3. Navier-Stokes’ equations 74 4.4. Darcy’s law 76 4.5. Calculating mass storage from the equations of state 80 4.6. General equation of hydrodynamics in porous media 82 4.7. Flows in unsaturated media 84 4.8. Bibliography 91 Chapter 5. Fluvial and Maritime Currentology Models 93 Jean-Michel TANGUY 5.1. 3D hydrostatic model 99 5.2. 2D horizontal model for shallow water 107 5.3. 1D models of fluvial flows 119 5.4. Putting 1D models into real time 131 5.5. Bibliography 151 Chapter 6. Urban Hydrology Models 155 Bernard CHOCAT 6.1. Global models and detailed models used in surface flows 156 6.2. Rainfall representation and rainfall-flow transformation 161 6.3. Modeling of the losses into the ground 164 6.4. Transfer function 169 6.5. Modeling of the hydraulic operating conditions of the networks 177 6.6. Production and transport of polluting agents 189 6.7. Conclusion 205 6.8. Bibliography 206 Chapter 7. Tidal Model and Tide Streams 213 Bernard SIMON 7.1. Tidal coefficient 214 7.2. Non-harmonic methods 215 7.3. Compatibilities 216 7.4. Tidal coefficient 222 7.5. Modeling 223 7.6. Tidal currents 226 Chapter 8. Wave Generation and Coastal Current Models 235 Jean-Michel TANGUY, Jean-Michel LEFÈVRE and Philippe SERGENT 8.1. Types of swell models 235 8.2. Spectral approach in high waters 242 8.3. Wave generation models 246 8.4. Wave propagation models 260 8.5. Agitating models within the harbors 266 8.6. Non-linear wave model: Boussinesq model 298 8.7. Coastal current models influenced or created by the swell 320 8.8. Bibliography 325 Chapter 9. Solid Transport Models and Evolution of the Seabed 335 Benoît LE GUENNEC and Jean-Michel TANGUY 9.1. Transport due to the overthrust effect 338 9.2. Total load 344 9.3. Bed forms and roughness 344 9.4. Suspension transport 346 9.5. Evolution model of movable beds 357 9.6. Conclusion 364 9.7. Bibliography 364 Chapter 10. Oil Spill Models 371 Pierre DANIEL 10.1. Behavior of hydrocarbons in marine environment 371 10.2. Oil spill drift models 372 10.3. Example: the MOTHY model 375 10.4. Calculation algorithm of the path of polluting particles 378 10.5. Example of a drift prediction map 379 10.6. Bibliography 379 Chapter 11. Conceptual, Empirical and Other Models 381 Christelle ALOT and Florence HABETS 11.1. Evapotranspiration 382 11.2. Bibliography 394 Chapter 12. Reservoir Models in Hydrology 397 Patrick FOURMIGUÉ and Patrick ARNAUD 12.1. Background 397 12.2. Main principles 399 12.3. Mathematical tools 401 12.4. Forecasting 403 12.5. Integration of the spatial information 405 12.6. Modeling limits 406 12.7. Bibliography 406 Chapter 13. Reservoir Models in Hydrogeology 409 Dominique THIÉRY 13.1. Principles and objectives 409 13.2. Catchment basin 410 13.3. Setting the model up 411 13.4. Data and parameters 412 13.5. Application domains 412 Chapter 14. Artificial Neural Network Models 419 Anne JOHANNET 14.1. Neural networks: a rapidly changing domain 420 14.2. Neuron and architecture models 422 14.3. How to take into account the non-linearity 429 14.4. Case study: identification of the rainfall-runoff relation of a karst 434 14.5. Acknowledgments 441 14.6. Bibliography 441 Chapter 15. Model Coupling 445 Rachid ABABOU, Denis DARTUS and Jean-Michel TANGUY 15.1. Model coupling 446 15.2. Bibliography 488 Chapter 16. A Set of Hydrological Models 493 Charles PERRIN, Claude MICHEL and Vasken ANDRÉASSIAN 16.1. Introduction 493 16.2. Description of the annual GR1A rainfall-runoff model 495 16.3. Description of the monthly GR2M rainfall-runoff model 496 16.4. Description of the daily GR4J rainfall-runoff model 500 16.5. Applications of the models 505 16.6. Conclusions and future work 506 16.7. Bibliography 507 List of Authors 511 Index 515 General Index of Authors 517 Summary of the Other Volumes in the Series 519

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Book SynopsisThis series of five volumes proposes an integrated description of physical processes modeling used by scientific disciplines from meteorology to coastal morphodynamics. Volume 1 describes the physical processes and identifies the main measurement devices used to measure the main parameters that are indispensable to implement all these simulation tools. Volume 2 presents the different theories in an integrated approach: mathematical models as well as conceptual models, used by all disciplines to represent these processes. Volume 3 identifies the main numerical methods used in all these scientific fields to translate mathematical models into numerical tools. Volume 4 is composed of a series of case studies, dedicated to practical applications of these tools in engineering problems. To complete this presentation, volume 5 identifies and describes the modeling software in each discipline.Trade Review"An inventory of ground measurement instruments, which provide necessary input data for the various modeling tools described in the book, is drawn up, and mathematical models describing each field within the overall subject area are detailed by a series of system equations." (Live-PR (EN), 19 April 2011)Table of ContentsIntroduction xiii PART 1. GENERAL CONSIDERATIONS CONCERNING NUMERICAL TOOLS 1 Chapter 1. Feedback on the Notion of a Model and the Need for Calibration 3 Denis DARTUS 1.1. “Static” and “dynamic” calibrations of a model 6 1.2. “Dynamic” calibration of a model or data assimilation 10 1.3. Bibliography 10 Chapter 2. Engineering Model and Real-Time Model 11 Jean-Michel TANGUY 2.1. Categories of modeling tools 11 2.2. Weather forecasting at Météo France 12 2.3. Flood forecasting 18 2.4. Characteristics of real-time models 23 2.5. Environment of real-time platforms 25 2.6. Interpretation of hydrological forecasting by those responsible for civil protection 27 2.7. Conclusion 29 2.8. Bibliography 30 Chapter 3. From Mathematical Model to Numerical Model 31 Jean-Michel TANGUY 3.1. Classification of the systems of differential equations 32 3.3. Discrete systems and continuous systems 40 3.4. Equilibrium and propagation problems 41 3.5. Linear and non-linear systems 43 3.6. Conclusion 57 3.7. Bibliography 57 PART 2. DISCRETIZATION METHODS 59 Chapter 4. Problematic Issues Encountered 61 Marie-Madeleine MAUBOURGUET 4.1. Examples of unstable problems 62 4.2. Loss of material 63 4.3. Unsuitable scheme 66 4.4. Bibliography 69 Chapter 5. General Presentation of Numerical Methods 71 Serge PIPERNO and Alexandre ERN 5.1. Introduction 71 5.2. Finite difference method 72 5.3. Finite volume method 77 5.4. Finite element method 78 5.5. Comparison of the different methods on a convection/diffusion problem 92 5.6. Bibliography 93 Chapter 6. Finite Differences 95 Marie-Madeleine MAUBOURGUET and Jean-Michel TANGUY 6.1. General principles of the finite difference method 95 6.2. Discretization of initial and boundary conditions 102 6.3. Resolution on a 2D domain 105 Chapter 7. Introduction to the Finite Element Method 109 Jean-Michel TANGUY 7.1. Elementary FEM concepts and presentation of the section 109 7.2. Method of approximation by finite elements 111 7.3. Geometric transformation 114 7.4. Transformation of derivation and integration operators 121 7.5. Geometric definition of the elements 125 7.6. Method of weighted residuals 128 7.7. Transformation of integral forms 130 7.8. Matrix presentation of the finite element method 133 7.9. Integral form of We on the reference element 140 7.10. Introduction of the Dirichlet-type boundary conditions 148 7.11. Summary: implementation of the finite element method 151 7.12. Application example: wave propagation 151 7.13. Bibliography 158 Chapter 8. Presentation of the Finite Volume Method 161 Alexandre ERN and Serge PIPERNO, section 8.6 written by Dominique THIÉRY 8.1. 1D conservation equations 162 8.2. Classical, weak and entropic solutions 170 8.3. Numerical solution of a conservation law 175 8.4. Numerical solution of hyperbolic systems 183 8.5. High-order, finite volume methods 194 8.6. Application of the finite volume method to the flow development of groundwater 195 8.7. Bibliography 210 Chapter 9. Spectral Methods in Meteorology 213 Jean COIFFIER 9.1. Introduction 213 9.2. Using finite series expansion of functions 214 9.3. The spectral method on the sphere 216 9.4. The spectral method on a biperiodic domain 227 9.5. Bibliography 232 Chapter 10. Numerical-Scheme Study 235 Jean-Michel TANGUY 10.1. Reminder of the notion of the numerical scheme 235 10.2. Time discretization 236 10.3. Space discretization 240 10.4. Scheme study: notions of consistency, stability and convergence 241 10.5. Bibliography 264 Chapter 11. Resolution Methods 267 Marie-Madeleine MAUBOURGUET 11.1. Temporal integration methods 268 11.2. Linearization methods for non-linear systems 270 11.3. Methods for solving linear systems AX = B 271 11.4. Bibliography 272 PART 3. INTRODUCTION TO DATA ASSIMILATION 273 Chapter 12. Data Assimilation 275 Jean PAILLEUX, Denis DARTUS, Xijun LAI, Jérôme MONNIER and Marc HONNORAT 12.1. Several examples of the application of data assimilation 277 12.2. Data assimilation in hydraulics with the Dassflow model 284 12.3. Bibliography 290 Chapter 13. Data Assimilation Methodology 295 Hélène BESSIÈRE, Hélène ROUX, François-Xavier LE DIMET and Denis DARTUS 13.1. Representation of the system 295 13.2. Taking errors into account 296 13.3. Simplified approach to optimum static estimation theory 297 13.4. Generalization in the multidimensional case 300 13.5. The different data assimilation techniques 303 13.6. Sequential assimilation method: the Kalman filter 304 13.7. Extension to non-linear models: the extended Kalman filter 307 13.8. Assessment of the Kalman filter 308 13.9. Variational methods 312 13.10. Discreet formulation of the cost function: the 3D-VAR 313 13.11. General variational formalism: the 4D-VAR 314 13.12. Continuous formulation of the cost function 314 13.13. Principle of automatic differentiation 322 13.14. Summary of variational methods 322 13.15. A complete application example: the Burgers equation 324 13.16. Feedback on the notion of a model and the need for calibration 335 13.17. Bibliography 343 List of Authors 349 Index 351 General Index of Authors 353 Summary of the Other Volumes in the Series . . . 355

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Book SynopsisThis book contains a selection of the best theoretical and applied papers from the inaugural International Choice Modelling Conference. The conference was organised by the Institute for Transport Studies at the University of Leeds and held in Harrogate, North Yorkshire on 30 March to 1 April 2009. The conference brought together leading researchers and practitioners from across the many different areas in which choice modelling is a key technique for understanding behaviour and evaluating policy. The diversity of the field was reflected in presentations by both academics and practitioners, coming from six continents and a variety of fields including transport and economics. Key contributions include papers from Professor Daniel McFadden, from the University of California, Berkeley - Nobel Prize laureate in Economics and chief architect of random utility modelling. The conference also included keynote presentations by five other leading choice modellers, namely Professor Moshe Ben-Akiva, Professor Chandra Bhat, Professor Michel Bierlaire, Professor David Hensher, and Professor Riccardo Scarpa.Trade ReviewAn important collection of papers by many of the world's leading researchers in choice modeling. With topics that range from the conceptual basis of choice to the pragmatics of data collection, this volume provides a vivid snapshot of the current state of the field.A" Kenneth Train, University of California, Berkeley.Table of ContentsPART I: Guest Speaker Papers 1. Sociality, Rationality, and the Ecology of Choice - Daniel McFadden 2. Planning and Action in a Model of Choice - Moshe Ben-Akiva 3. Attribute Processing, Heuristics, and Preference Construction in Choice Analysis - David Hensher 4. The Multiple Discrete-Continuous Extreme Value (MDCEV) Model: Formulation and Applications - Chandra R. Bhat & Naveen Eluru 5. Capturing Human Perception of Facial Expressions by Discrete Choice Modelling - Matteo Sorci,Thomas Robin, Javier Cruz, Michel Bierlaire, J.-P. Thiran and Gianluca Antonini Part II: Data Collection 6. Serial Choice Conjoint Analysis for Estimating Discrete Choice Models - Michiel C.J. Bliemer and John M. Rose 7. Observed Efficiency of a D-Optimal Design in an Interactive Agency Choice Experiment - Sean M. Puckett and John M. Rose 8. Effects of Stated Choice Design Dimensions on Model Estimates - Phani Kumar Chintakayala, Stephane Hess, John M. Rose and Mark Wardman 9. Stated Choice Experimental Designs for Scheduling Models - Paul Koster and Yin-Yen Tseng Part III: Concepts & Methodology 10. Systematically Heterogeneous Covariance in Network GEV Models - Jeffrey P. Newman 11. On Estimation of Hybrid Choice Models - Denis Bolduc and Ricardo Alvarez-Daziano 12. A Model of Travel Happiness and Mode Switching - Maya Abou-Zeid and Moshe Ben-Akiva 13. On Path Generation Algorithms for Route Choice Models - Emma Frejinger and Michel Bierlaire Part IV: Endogeneity and Heterogeneity 14. Mode Choice Endogeneity in Value of Travel Time Estimation - Stefan L. Mabit and Mogens Fosgerau 15. Accommodating Coefficient Outliers in Discrete Choice Modelling: A Comparison of Discrete and Continuous Mixing Approaches - Danny Campbell, Stephane Hess, Riccardo Scarpa and John M. Rose 16. Addressing Endogeneity in Discrete Choice Models: Assessing Control-Function and Latent-Variable Methods - Cristian Angelo Guevara and Moshe Ben-Akiva 17. Latent Class and Mixed Logit Models with Endogenous Choice Set Formation Based on Compensatory Screening Rules - Matthieu de Lapparent Part V: Transport Matters 18. Transport Welfare Benefits in the Presence of an Income Effect - James Laird 19. Which Commuters will Car Share? An Examination of Alternative Approaches to Identifying Market Segments - Jon Crockett, Gerard Andrew Whelan, Caroline Louise Sinclair and Hugh Gillies 20. Modelling Choice in a Changing Environment: Assessing the Shock Effects of a New Transport System - Maria Francisca Yanez and Juan de Dios Ortuzar 21. What do We Really Know About Travellers' Response to Unreliability? - Yaron Hollander Part VI: Beyond Transport 22. Optimizing Product Portfolios Using Discrete Choice Modeling and TURF - Thomas J. Adler, Colin Smith and Jeffrey Dumont 23. Preference Stability: Modelling how Consumer Preferences Shift after Receiving New Product Information - Harmen Oppewal, Mark Morrison, Paul Wang and David Waller 24. Investigating Willingness to Pay -Willingness to Accept Asymmetry in Choice Experiments - Bruno Lanz, Allan Provins, Ian J. Bateman, Riccardo Scarpa, Ken Willis and Ece Ozdermiroglu 25. Clustering Ranked Preference Data Using Sociodemographic Covariates - Isobel Claire Gormley and Thomas Brendan Murphy 26. Continuous versus Discrete Representation of Investing Firm Heterogeneity in Modelling FDI Location Decisions - Simona Rasciute and Eric J. Pentecost 27. Development of Integrated Choice and Latent Variable (ICLV) Models for the Residential Relocation Decision in Island Areas - Eleni Kitrinou, Amalia Polydoropoulou and Denis Bolduc

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Book SynopsisIn this book on mathematical programming, the postulate spacial-time certainty of economic process at uncertainty conditions in finite-dimensional vector space and the principle piecewise-linear homogeneity of economic process at uncertainty conditions in finite-dimensional vector space are first suggested. A special theory on constructing piecewise-linear economic-mathematical models was developed, and a criterion of multivariate prediction of economic process and their control at uncertainty conditions in a finite-dimensional vector space was suggested. A packet of numerical programs for computer simulation in constructing and multivariate prediction of economic state with the help of n-element piecewise-linear economic-mathematical models with regard to the uncertainty factors effect in m-dimensional vector space is also suggested.This book is intended for students of economic and administrative specialties as well as for research associates in the sphere of economic-mathematical methods, management, and banking.Table of ContentsBrief Information on Finite-Dimensional Vector Space and its Application in Economics. Bases of Piecewise-Linear Economic-Mathematical Models with Regard to Influence of Unaccounted Factors in Finite-Dimensional Vector Space. Piecewise Linear Economic-Mathematical Models with Regard to Unaccounted Factors Influence in Three-Dimensional Vector Space. Piecewise-Linear Economic-Mathematical Models with Regard to Unaccounted Factors Influence on a Plane. Bases of Software for Computer Simulation and Multivariant Prediction of Economic Even at Uncertainty Conditions on the Base of N-Component Piecewise-Linear Economic-Mathematical Models in M-Dimensional Vector Space. Index.

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Book SynopsisThis book provides qualitative and quantitative methods to analyze and better understand phenomena that change in space and time. An innovative approach is to incorporate ideas and methods from dynamical systems and equivariant bifurcation theory to model, analyze and predict the behavior of mathematical models. In addition, real-life data is incorporated in the derivation of certain models. For instance, the model for a fluxgate magnetometer includes experiments in support of the model. The book is intended for interdisciplinary scientists in STEM fields, who might be interested in learning the skills to derive a mathematical representation for explaining the evolution of a real system. Overall, the book could be adapted in undergraduate- and postgraduate-level courses, with students from various STEM fields, including: mathematics, physics, engineering and biology.Table of ContentsIntroduction- Algebraic Models.- Discrete Models.- Continuous Models.- Bifurcation Theory.- Network-Based Modeling.- Delay Models.- Spatial-Temporal Models.- Stochastic Models.- Model Reduction and Simplification.

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Book SynopsisFree thinking, unconstrained by facts The book is based on the thesis that we live in a world of abundance, full of natural riches, and cultural artifacts, full of human intellect and powerful technologies. Our thinking, however, is dominated by the opposite, the notion of scarcity. The limits of nature act as an inevitable necessity. In his book, David Schildberger adopts a novel approach to the subject of resources, with the help of intelligent instruments that introduce new foods, such as chocolate made from cocoa cell cultures, and even a fruit-bearing vine raised far from a vineyard. With his imagined scenarios, the author invites the reader to dare stretch their intellectual imaginations and ultimately presents nature as a contingent. Conceptual models on the subject of nature and alternative ways of producing food Recommended reading for architectural IT specialists New volume in the Applied Virtuality Book Series

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Book Synopsis„Was ist Mathematik?” – auf diese Frage gibt dieses dicke Buch zahllose Antworten. Mathematik ist eben viel mehr als ein Schul- und Studienfach oder Rechnen: Es ist Teil der menschlichen Kultur, ein riesiges aktives Forschungsgebiet und ein nützlicher Werkzeugkasten. „Was ist Mathematik?” – statt einer einzelnen Antwort zeichnen die Autoren ein Panorama, bunt und vielfältig. Da geht es um Philosophie, Beweise, große und kleine Probleme, fundamentale Konzepte, Teilgebiete, Forschungspraxis, Anwendungen der Mathematik. Und um Geschichten aus der Geschichte. Das Buch richtet sich an alle, die wissen und darüber nachdenken wollen, was Mathematik ist, insbesondere auch an Studierende der Mathematik. Es begleitet eine Vorlesung, die an der Freien Universität Berlin jährlich vor allem für Lehramtsstudierende angeboten wird.Table of ContentsWas ist Mathematik?- Mathematische Forschung.- Beweise.- Formeln, Zeichnungen und Bilder.- Philosophie der Mathematik.- Primzahlen.- Zahlenbereiche.- Unendlichkeit.- Dimensionen.- Zufall – Wahrscheinlichkeiten – Statistik.- Funktionen.- Anwendungen.- Rechnen.- Algorithmen und Komplexität.- Mathematik in der Öffentlichkeit.

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Book SynopsisStarting from a specific industrial problem related to the propellant flow within a liquid propellant rocket engine, this book develops a numerical method for simulating three-dimensional, generic barotropic flows in rotating frames. It proposes a novel finite volume compressible approach for unstructured grids, suitably preconditioned for accurately dealing with nearly-incompressible flows.

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Book SynopsisThis book presents original papers reflecting topics featured at the international symposium entitled “Fusion of Mathematics and Biology” and organized by the editor of the book. The symposium, held in October 2020 at Osaka University in Japan, was the core event for the final year of the research project entitled “Establishing International Research Networks of Mathematical Oncology.” The project had been carried out since April 2015 as part of the Core-to-Core Program of Japan Society for the Promotion of Science (JSPS). In this book, the editor presents collaborative research from prestigious organizations in France, the UK, and the USA. By utilizing their individual strengths and realizing the fusion of life science and mathematical science, the project achieved a combination of mathematical analysis, verification by biomedical experiments, and statistical analysis of chemical databases.Mathematics is sometimes regarded as a universal language. It is a valuable property that everyone can understand beyond the boundaries of culture, religion, and language. This unifying force of mathematics also applies to the various fields of science. Mathematical oncology has two aspects, i.e., data science and mathematical modeling, and definitely helps in the prediction and control of biological phenomena observed in cancer evolution.The topics addressed in this book represent several methods of applying mathematical modeling to scientific problems in the natural sciences. Furthermore, novel reviews are included that may motivate many mathematicians to become interested in biological research.Table of ContentsPART 1: Mathematical Modeling: D. Guan, X. Luo, and H. Gao, Constitutive Modelling of Soft Biological Tissue from Ex Vivo to In Vivo: Myocardium as an Example.- T. Colin, T. Michel, and C. Poignard, Mathematical Modeling of Gastro-intestinal Metastasis Resistance to Tyrosine Kinase Inhibitors.- Y. Tanaka and T. Yasugi, Mathematical Modeling and Experimental Verification of the Proneural Wave.- D. Kumakura and S. Nakaoka, Exploring Similarity between Embedding Dimension of Time-series Data and Flows of an Ecological Population Model.- T. Hayashi, Mathematical Modeling for Angiogenesis.- S. Collin, Corridore and C. Poignard, Floating Potential Boundary Condition in Smooth Domains in an Electroporation Context.- N. L. Othman and T. Suzuki, Free Boundary Problem of Cell Deformation and Invasion.- L. Preziosi and M. Scianna, Multi-level Mathematical Models for Cell Migration in Confined Environments.- S. Magi, Mathematical Modeling of Cancer Signaling Addressing Tumor Heterogeneity.- N. Sfakianakis and Mark A.J. Chaplain, Mathematical Modelling of Cancer Invasion: A Review.- T. Williams, A. Wilson, and N. Sfakianakis, The First Step towards the Mathematical Understanding of the Role of Matrix Metalloproteinase-8 in Cancer Invasion.- PART II: Biological Prediction: T. Ito, T. Suzuki, and Y. Murakami, Mathematical Modeling of the Dimerization of EGFR and ErbB3 in Lung Adenocarcinoma.- H. Kubota, Selective Regulation of the Insulin-Akt Pathway by Simultaneous Processing of Blood Insulin Pattern in the Liver.- D. Oikawa, N. Hatanaka, T. Suzuki, and F. Tokunaga, Mathematical Simulation of Linear Ubiquitination in T Cell Receptor-mediated NF-κB Activation Pathway.- Y. Ito, D. Minerva, S. Tasaki, M. Yoshida, T. Suzuki, and A. Goto, Time Changes in the VEGF-A Concentration Gradient Lead Neovasculature to Engage in Stair-like Growth.- N. Hatanaka, M. Futakuchi, and T. Suzuki, Mathematical Modeling of Tumor Malignancy in Bone Microenvironment.- M. Yamamoto and Jun-ichiro Inoue, Signaling Networks Involved in the Malignant Transformation of Breast Cancer.- PART III: Data Science: R. Morishita, H. Takahashi, and T. Sawasaki, Cell-free Based Protein Array Technology.- Y. Nojima and Y. Takeda, Omics Data Analysis Tools for Biomarker Discovery and the Tutorial.- M. Oyama and H. Kozuka-Hata, Integrative Network Analysis of Cancer Cell Signaling by High-resolution Proteomics.- N. Nakamura and R. Yamada, Distance-matrix-based Extraction of Motility Features from Functionally Heterogeneous Cell Populations.- S. Kawasaki, H. Hayashi, and Y. Tominaga, Data Analytic Study of Genetic Mechanism of Ovarian Carcinoma from Single Cell RNA-seq Data.

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