Mathematical modelling Books

Book SynopsisThis volume examines the problems of predictability and nonlinear modelling by using tools from chaos theory and systems theory, adapted for the analysis of problems in the environmental sciences. Dependence on the initial state (chaos) and the parameters are analyzed using different methods.Table of Contents1: Geophysics. 2: Agriculture. 3: Population Biology. 4: Systems Sciences. 5: Environmental Sciences. 6: Economics.

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Book Synopsis1. LSM: The New Paradigm of Distributed Systems.- 2. Indoor GPS (iGPSTM).- 3. The Mobile Spatial coordinate Measuring System (MScMS).- 4. Positioning and Coverage of Distributed Devices.- 5. System Calibration.- 6. Self-Diagnostic Tools.- 7. Methodologies for Performance Enhancing.- 8. Evaluation of Measurement Uncertainty.Table of Contents1. LSM: The New Paradigm of Distributed Systems.- 2. Indoor GPS (iGPSTM).- 3. The Mobile Spatial coordinate Measuring System (MScMS).- 4. Positioning and Coverage of Distributed Devices.- 5. System Calibration.- 6. Self-Diagnostic Tools.- 7. Methodologies for Performance Enhancing.- 8. Evaluation of Measurement Uncertainty.

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Book SynopsisBringing together idiomatic Python programming, foundational numerical methods, and physics applications, this book covers linear algebra, differential equations, root-finding, interpolation, and integration. Fully implementing many numerical methods in Python and with 140 new problems, it is an ideal standalone textbook on computational physics.Trade Review'A fantastic addition as an introductory textbook for computational physics. The book is timely, and the author made thoughtful and, in my view, many wise choices. The book is comprehensive and yet accessible to undergraduate students.' Shiwei Zhang, the Flatiron Institute and the College of William & Mary'Gezerlis' book Numerical Methods in Physics with Python is a beautiful example of how an established subject can be brought to the next level by making it very accessible and by introducing several insightful and interdisciplinary applications. This second edition considerably extends the set of exercises, resulting in an extremely useful resource for both students and teachers. Strongly recommended!' Sonia Bacca, Johannes Gutenberg-Universität Mainz'This new edition of Numerical Methods… is another great example of Gezerlis' passion for teaching and for doing so carefully and precisely. Especially welcome, in my view, are the addition of problems at the end of each chapter and the discussion of singular value decomposition (SVD) and Bayesian methods. The SVD is one of the crown jewels of linear algebra which modern students interested in machine learning will surely find beneficial. To physics, computer science, or engineering students mesmerized by the fast Fourier transform, Gezerlis' excellent explanation of it in Chapter 6 is likely to shed some light on the underlying divide-and-conquer algorithm, which is an essential classic.' Joaquin Drut, University of North Carolina at Chapel HillPraise for the first edition: 'I enthusiastically recommend Numerical Methods in Physics with Python by Professor Gezerlis to any advanced undergraduate or graduate student who would like to acquire a solid understanding of the basic numerical methods used in physics. The methods are demonstrated with Python, a relatively compact, accessible computer language, allowing the reader to focus on understanding how the methods work rather than on how to program them. Each chapter offers a self-contained, clear, and engaging presentation of the relevant numerical methods, and captivates the reader with well-motivated physics examples and interesting physics projects. Written by a leading expert in computational physics, this outstanding textbook is unique in that it focuses on teaching basic numerical methods while also including a number of modern numerical techniques that are usually not covered in computational physics textbooks.' Yoram Alhassid, Yale University'In Numerical Methods in Physics with Python by Gezerlis, one finds a resource that has been sorely missing! As the usage of Python has become widespread, it is too often the case that students take libraries, functions, and codes and apply them without a solid understanding of what is truly being done 'under the hood' and why. Gezerlis' book fills this gap with clarity and rigor by covering a broad number of topics relevant for physics, describing the underlying techniques, and implementing them in detail. It should be an important resource for anyone applying numerical techniques to study physics.' Luis Lehner, Perimeter Institute for Theoretical Physics'Gezerlis' text takes a venerable subject - numerical techniques in physics - and brings it up to date and makes it accessible to modern undergraduate curricula through a popular, open-source programming language. Although the focus remains squarely on numerical techniques, each new lesson is motivated by topics commonly encountered in physics and concludes with a practical hands-on project to help cement the students' understanding. The net result is a textbook which fills an important and unique niche in pedagogy and scope, as well as a valuable reference for advanced students and practicing scientists.' Brian Metzger, Columbia UniversityTable of ContentsPreface; 1. Idiomatic Python; 2. Numbers; 3. Derivatives; 4. Matrices; 5. Zeroes and minima; 6. Approximation; 7. Integrals; 8. Differential equations; Appendix A. Installation and setup; Appendix B. Number representations; Appendix C. Math background; Bibliography; Index.

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Book SynopsisPhysical phenomena can be modeled at varying degrees of complexity and at different scales. Multiscale modeling provides a framework, based on fundamental principles, for constructing mathematical and computational models of such phenomena, by examining the connection between models at different scales. This book, by a leading contributor to the field, is the first to provide a unified treatment of the subject, covering, in a systematic way, the general principles of multiscale models, algorithms and analysis. After discussing the basic techniques and introducing the fundamental physical models, the author focuses on the two most typical applications of multiscale modeling: capturing macroscale behavior and resolving local events. The treatment is complemented by chapters that deal with more specific problems. Throughout, the author strikes a balance between precision and accessibility, providing sufficient detail to enable the reader to understand the underlying principles without allTrade Review'[This] book can be considered as the standard work in this research field and is a rich source for this topic … It is a valuable book and serves as a useful tool for newcomers and researchers working on these problems … highly recommended.' Willi-Hans Steeb, Zentralblatt MATHTable of ContentsPreface; 1. Introduction; 2. Analytical methods; 3. Classical multiscale algorithms; 4. The hierarchy of physical models; 5. Examples of multi-physics models; 6. Capturing the macroscale behavior; 7. Resolving local events or singularities; 8. Elliptic equations with multiscale coefficients; 9. Problems with multiple time scales; 10. Rare events; 11. Some perspectives; Index.

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Book SynopsisPerfect for senior undergraduates and first-year graduate students in geophysics, physics, mathematics, geology and engineering, this book is devoted exclusively to seismic wave theory. The result is an invaluable teaching tool, with its detailed derivations of formulas, clear explanations of topics, exercises along with selected answers, and an additional set of exercises with derived answers on the book''s website. Some highlights of the text include: a review of vector calculus and Fourier transforms and an introduction to tensors, which prepare readers for the chapters to come; and a detailed discussion on computing reflection and transmission coefficients, a topic of wide interest in the field; a discussion in later chapters of plane waves in anisotropic and anelastic media, which serves as a useful introduction to these two areas of current research in geophysics. Students will learn to understand seismic wave theory through the book''s clear and concise pedagogy.Table of Contents1. Vectors, tensors, and Fourier transforms; 2. Stress, strain, and seismic waves; 3. Reflection and transmission of plane waves; 4. Surface waves, head waves, and normal modes; 5. Waves in heterogeneous media; 6. Data transformations; 7. Synthetic seismograms; 8. Seismic migration; 9. Plane waves in anisotropic media; 10. Plane waves in anelastic media; Answers to selected exercises; References; Index.

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Book SynopsisJust like humans, animals and plants suffer from infectious diseases, which can critically threaten biodiversity. This book describes key studies that have driven our understanding of the ecology and evolution of wildlife diseases. Each chapter introduces the host and disease, and explains how that system has aided our general understanding of the evolution and spread of wildlife diseases, through the development and testing of important epidemiological and evolutionary theories. Questions addressed include: How do hosts and parasites co-evolve? What determines how fast a disease spreads through a population? How do co-infecting parasites interact? Why do hosts vary in parasite burden? Which factors determine parasite virulence and host resistance? How do parasites influence the spread of invasive species? How do we control infectious diseases in wildlife? This book will provide a valuable introduction to students new to the topic, and novel insights to researchers, professionals and pTrade Review'Overall, this is a fascinating collection of studies that showcases why wildlife diseases are worthy of study and how combining field observations, experiments, mathematical models and the latest in genomic and molecular research provides not only research insight, but also contributes to effective conservation and management efforts.' Rob Robinson, British Trust for Ornithology'Advances in modeling, epidemiologic techniques, and genetics have been crucial in some examples treated by contributors, and the importance of long-term field studies, essential for understanding dynamic systems, is emphasized throughout the volume. Some studies are observational, some experimental, and some largely theoretical. All contributions are extensively referenced and effectively illustrated.' M. Gochfeld, Choice'Overall, this well-written book is, in my opinion, a valuable contribution that will encourage further collecting and analysing long-term data in the study of wildlife diseases. It also gives hope. The advances in our understanding of wildlife disease dynamics enable better planning of conservation and management efforts, as shown in the case of wild and farmed salmon or the bighorn sheep pneumonia. As such, it is undoubtedly of high value for researchers and managers working in the field of wildlife disease ecology, but also for advanced undergraduate students or academic lecturers who would like to broaden their knowledge. The book was a great company during the coronavirus lockdown and a fascinating journey through the realm of wildlife diseases. I highly recommend it!' Agata Mrugala, Basic and Applied Ecology'This book comes to fill an important niche in disease ecology: synthesizing the state of knowledge about wildlife disease ecology while integrating theoretical models with a wide variety empirical case studies … this book presents an invaluable synthesis of our knowledge of disease ecology in wildlife hosts.' Miguel A. Acevedo, The Quarterly Review of BiologyTable of ContentsPreface: wildlife disease ecology; Glossary of terms; Part I. Understanding Within-Host Processes: 1. Pollinator diseases: the Bombus-Crithidia system; 2. Genetic diversity and disease spread: epidemiological models and empirical studies of a snail-trematode system; 3. Wild rodents as a natural model to study within-host parasite interactions; 4. From population to individual host scale and back again: testing theories of infection and defence in the Soay sheep of St Kilda; 5. The causes and consequences of parasite interactions: African buffalo as a case study; 6. Effects of host lifespan on the evolution of age-specific resistance: a case study of anther-smut disease on wild carnations; 7. Sexually transmitted infections in natural populations: what have we learnt from beetles and beyond?; Part II. Understanding Between-Host Processes: 8. Using insect baculoviruses to understand how population structure affects disease spread; 9. Infection and invasion: study cases from aquatic communities; 10. Parasite mediated selection in red grouse – consequences for population dynamics and mate choice; 11. Emergence, transmission and evolution of an uncommon enemy: Tasmanian devil facial tumour disease; 12. Bovine tuberculosis in badgers: sociality, infection and demography in a social mammal; 13. Mycoplasma ovipneumoniae in bighorn sheep: from exploration to action; 14. Manipulating parasites in an Arctic herbivore: gastrointestinal nematodes and the population regulation of Svalbard reindeer; Part III. Understanding Wildlife Disease Ecology at the Community and Landscape Level: 15. The ecological and evolutionary trajectory of oak powdery mildew in Europe; 16. Healthy herds or predator spreaders? Insights from the plankton into how predators suppress and spread disease; 17. Multi-trophic interactions and migration behaviour determine the ecology and evolution of parasite infection in monarch butterflies; 18. When chytrid fungus invades: integrating theory and data to understand disease- induced amphibian declines; 19. Ecology of a marine ectoparasite in farmed and wild salmon; 20. Mycoplasmal conjunctivitis in house finches: the study of an emerging disease; 21. Heterogeneities in infection and transmission in a parasite-rabbit system: key issues for understanding disease dynamics and persistence; 22. Sylvatic plague in Central Asia: a case study of abundance thresholds.

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Book Synopsis.- Preliminaries..- First Look at QPLEX..- Part 1 QPLEX Modeling and Calculus..- Introduction to QPLEX Modeling and Calculus..- Simple Transition Dynamics..- Models with Simple Transition Dynamics..- Advanced Transition Dynamics..- Models with Advanced Transition Dynamics..- Conditional and Joint Probabilities..- Part 2 Graphical QPLEX Calculus..- Introduction to Graphical QPLEX Calculus..- Subsystem QPLEX Calculus..- Conditional Independence..- Information Structure..- Graphical QPLEX Calculus with Distributional Programs..- Efficient Calculation for Distributional Programs..- Part 3 Foundations..- Introduction to Foundations..- Optimality of QPLEX Iterates..- Exactness Results.

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Book SynopsisPreface.- The Root of Mind and Mathematics.- Equality, Similarity, and Transformations.- Mind and Mathematics in an Event-Centered Approach.- Symmetry, the Unconscious, and Imagination.- Imagination, Mathematics, and Mysticism.- Epilogue.- Index.

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Book SynopsisPreface.- The Starting Point: Lakoff and Núñez.- Image Schema Theory.- Related Processes.- Learning, Diagrams, and AI.- Index.

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Book SynopsisThis concise and clear introduction to the topic requires only basic knowledge of calculus and linear algebra - all other concepts and ideas are developed in the course of the book. Lucidly written so as to appeal to undergraduates and practitioners alike, it enables readers to set up simple mathematical models on their own and to interpret their results and those of others critically. To achieve this, many examples have been chosen from various fields, such as biology, ecology, economics, medicine, agricultural, chemical, electrical, mechanical and process engineering, which are subsequently discussed in detail. Based on the author`s modeling and simulation experience in science and engineering and as a consultant, the book answers such basic questions as: What is a mathematical model? What types of models do exist? Which model is appropriate for a particular problem? What are simulation, parameter estimation, and validation? The book relies exclusively upon open-source software which is available to everybody free of charge. The entire book software - including 3D CFD and structural mechanics simulation software - can be used based on a free CAELinux-Live-DVD that is available in the Internet (works on most machines and operating systems).Trade Review"Very solid introductory text at the undergraduate level aimed at wide audience. Perfectly fits introductory modeling courses at colleges and universities that prefer to use open-source software rather than commercial one, and is an enjoyable reading in the first place. Highly recommended both as a main text and a supplementary one. (...) This delightful book has two unbeatable features that should absolutely win the audience (...) First of all, it illuminates many important conceptual ideas of mathematical modelling (...) Second, (...) this book enthusiastically promotes open-source software that works on most computers and operating systems and is freely available on the web. (...) Professor Velten suggests an elegant approach to mathematical modeling, carefully going through all important steps from identification of a problem, definition of the associated system under study and analysis of the system's properties to design of a mathematical model for the system, its numerical simulation and validation." (Yuri V. Rogovchenko, Zentralblatt MATH, European Mathematical Society) "The book is certainly a reference for those, beginners or professional, who search for a complete and easy to follow step-by-step guide in the amazing world of modeling and simulation (...) it is shown that mathematical models and simulation, if adequately used, help to reduce experimental costs by a better exploration of the information content of experimental data (...) it is explained how to analyze a real problem arising from science or engineering and how to best describe it through a mathematical model. A number of examples help the reader to follow step by step the basics of modelling." (Marcello Vasta, Meccanica: International Journal of Theoretical and Applied Mechanics, Vol. 44(3), 2009) "The broad subject area covered in this book reflects the background of the author, an experienced mathematical consultant and academic (...) This book differs from almost all other available modeling books in that the author addresses both mechanistic and statistical models as well as "hybrid" models. Since many problems coming out of industrial and medical applications in recent years require hybrid models, this text is timely. The modeling range is enormous (...) In this single chapter ("Phenomenological Models") he manages to cover almost all the material one would expect to find in an undergraduate statistics program. (...) Parameter sensitivity and overfitting problems are discussed in a very simple context - very nice! (...) The author points out that, by translating a real-world problem into a mathematical form, one brings to bear on that problem the vast knowledge and powerful and free software tools available within the "mathematical universe", and his aim is to enable the reader to source this information. (...) I believe the author has succeeded in providing access to the available tools and an understanding of how to go about using these tools to solve real-world problems." Neville Fowkes (University of Western Australia) in: SIAM Rev. 53(2), 2011, pp. 387-388 (Society of Industrial and Applied Mathematics, Philadelphia, USA)Table of ContentsPreface xiii 1 Principles of Mathematical Modeling 1 1.1 A Complex World Needs Models 1 1.2 Systems, Models, Simulations 3 1.2.1 Teleological Nature of Modeling and Simulation 4 1.2.2 Modeling and Simulation Scheme 4 1.2.3 Simulation 7 1.2.4 System 7 1.2.5 Conceptual and Physical Models 8 1.3 Mathematics as a Natural Modeling Language 9 1.3.1 Input–Output Systems 9 1.3.2 General Form of Experimental Data 10 1.3.3 Distinguished Role of Numerical Data 10 1.4 Definition of Mathematical Models 11 1.5 Examples and Some More Definitions 13 1.5.1 State Variables and System Parameters 15 1.5.2 Using Computer Algebra Software 18 1.5.3 The Problem Solving Scheme 19 1.5.4 Strategies to Set up Simple Models 20 1.5.4.1 Mixture Problem 24 1.5.4.2 Tank Labeling Problem 27 1.5.5 Linear Programming 30 1.5.6 Modeling a Black Box System 31 1.6 Even More Definitions 34 1.6.1 Phenomenological and Mechanistic Models 34 1.6.2 Stationary and Instationary models 38 1.6.3 Distributed and Lumped models 38 1.7 Classification of Mathematical Models 39 1.7.1 From Black to White Box Models 40 1.7.2 SQM Space Classification: S Axis 41 1.7.3 SQM Space Classification: Q Axis 42 1.7.4 SQM Space Classification: M Axis 43 1.8 Everything Looks Like a Nail? 45 2 Phenomenological Models 47 2.1 Elementary Statistics 48 2.1.1 Descriptive Statistics 48 2.1.1.1 Using Calc 49 2.1.1.2 Using the R Commander 51 2.1.2 Random Processes and Probability 52 2.1.2.1 Random Variables 53 2.1.2.2 Probability 53 2.1.2.3 Densities and Distributions 55 2.1.2.4 The Uniform Distribution 57 2.1.2.5 The Normal Distribution 57 2.1.2.6 Expected Value and Standard Deviation 58 2.1.2.7 More on Distributions 60 2.1.3 Inferential Statistics 60 2.1.3.1 Is Crop A’s Yield Really Higher? 61 2.1.3.2 Structure of a Hypothesis Test 61 2.1.3.3 The t test 62 2.1.3.4 Testing Regression Parameters 63 2.1.3.5 Analysis of Variance 63 2.2 Linear Regression 65 2.2.1 The Linear Regression Problem 65 2.2.2 Solution Using Software 66 2.2.3 The Coefficient of Determination 68 2.2.4 Interpretation of the Regression Coefficients 70 2.2.5 Understanding LinRegEx1.r 70 2.2.6 Nonlinear Linear Regression 72 2.3 Multiple Linear Regression 74 2.3.1 The Multiple Linear Regression Problem 74 2.3.2 Solution Using Software 76 2.3.3 Cross-Validation 78 2.4 Nonlinear Regression 80 2.4.1 The Nonlinear Regression Problem 80 2.4.2 Solution Using Software 81 2.4.3 Multiple Nonlinear Regression 83 2.4.4 Implicit and Vector-Valued Problems 86 2.5 Neural Networks 87 2.5.1 General Idea 87 2.5.2 Feed-Forward Neural Networks 89 2.5.3 Solution Using Software 91 2.5.4 Interpretation of the Results 92 2.5.5 Generalization and Overfitting 95 2.5.6 Several Inputs Example 97 2.6 Design of Experiments 99 2.6.1 Completely Randomized Design 100 2.6.2 Randomized Complete Block Design 103 2.6.3 Latin Square and More Advanced Designs 104 2.6.4 Factorial Designs 106 2.6.5 Optimal Sample Size 108 2.7 Other Phenomenological Modeling Approaches 109 2.7.1 Soft Computing 109 2.7.1.1 Fuzzy Model of a Washing Machine 110 2.7.2 Discrete Event Simulation 111 2.7.3 Signal Processing 113 3 Mechanistic Models I: ODEs 117 3.1 Distinguished Role of Differential Equations 117 3.2 Introductory Examples 118 3.2.1 Archaeology Analogy 118 3.2.2 Body Temperature 120 3.2.2.1 Phenomenological Model 120 3.2.2.2 Application 121 3.2.3 Alarm Clock 122 3.2.3.1 Need for a Mechanistic Model 122 3.2.3.2 Applying the Modeling and Simulation Scheme 123 3.2.3.3 Setting Up the Equations 125 3.2.3.4 Comparing Model and Data 126 3.2.3.5 Validation Fails – What Now? 127 3.2.3.6 A Different Way to Explain the Temperature Memory 128 3.2.3.7 Limitations of the Model 129 3.3 General Idea of ODE’s 130 3.3.1 Intrinsic Meaning of π 130 3.3.2 E X Solves An Ode 130 3.3.3 Infinitely Many Degrees of Freedom 131 3.3.4 Intrinsic Meaning of the Exponential Function 132 3.3.5 ODEs as a Function Generator 134 3.4 Setting Up ODE Models 135 3.4.1 Body Temperature Example 135 3.4.1.1 Formulation of an ODE Model 135 3.4.1.2 ODE Reveals the Mechanism 136 3.4.1.3 ODE’s Connect Data and Theory 137 3.4.1.4 Three Ways to Set up ODEs 138 3.4.2 Alarm Clock Example 139 3.4.2.1 A System of Two ODEs 139 3.4.2.2 Parameter Values Based on A priori Information 140 3.4.2.3 Result of a Hand-fit 141 3.4.2.4 A Look into the Black Box 142 3.5 Some Theory You Should Know 143 3.5.1 Basic Concepts 143 3.5.2 First-order ODEs 145 3.5.3 Autonomous, Implicit, and Explicit ODEs 146 3.5.4 The Initial Value Problem 146 3.5.5 Boundary Value Problems 147 3.5.6 Example of Nonuniqueness 149 3.5.7 ODE Systems 150 3.5.8 Linear versus Nonlinear 152 3.6 Solution of ODE’s: Overview 153 3.6.1 Toward the Limits of Your Patience 153 3.6.2 Closed Form versus Numerical Solutions 154 3.7 Closed Form Solutions 156 3.7.1 Right-hand Side Independent of the Independent Variable 156 3.7.1.1 General and Particular Solutions 156 3.7.1.2 Solution by Integration 157 3.7.1.3 Using Computer Algebra Software 158 3.7.1.4 Imposing Initial Conditions 160 3.7.2 Separation of Variables 161 3.7.2.1 Application to the Body Temperature Model 164 3.7.2.2 Solution Using Maxima and Mathematica 165 3.7.3 Variation of Constants 166 3.7.3.1 Application to the Body Temperature Model 167 3.7.3.2 Using Computer Algebra Software 169 3.7.3.3 Application to the Alarm Clock Model 170 3.7.3.4 Interpretation of the Result 171 3.7.4 Dust Particles in the ODE Universe 173 3.8 Numerical Solutions 174 3.8.1 Algorithms 175 3.8.1.1 The Euler Method 175 3.8.1.2 Example Application 176 3.8.1.3 Order of Convergence 178 3.8.1.4 Stiffness 179 3.8.2 Solving ODE’s Using Maxima 180 3.8.2.1 Heuristic Error Control 181 3.8.2.2 ODE Systems 182 3.8.3 Solving ODEs Using R 184 3.8.3.1 Defining the ODE 184 3.8.3.2 Defining Model and Program Control Parameters 186 3.8.3.3 Local Error Control in lsoda 186 3.8.3.4 Effect of the Local Error Tolerances 187 3.8.3.5 A Rule of Thumb to Set the Tolerances 188 3.8.3.6 The Call of lsoda 189 3.8.3.7 Example Applications 190 3.9 Fitting ODE’s to Data 194 3.9.1 Parameter Estimation in the Alarm Clock Model 194 3.9.1.1 Coupling lsoda with nls 195 3.9.1.2 Estimating One Parameter 197 3.9.1.3 Estimating Two Parameters 198 3.9.1.4 Estimating Initial Values 199 3.9.1.5 Sensitivity of the Parameter Estimates 200 3.9.2 The General Parameter Estimation Problem 201 3.9.2.1 One State Variable Characterized by Data 202 3.9.2.2 Several State Variables Characterized by Data 203 3.9.3 Indirect Measurements Using Parameter Estimation 204 3.10 More Examples 205 3.10.1 Predator–Prey Interaction 205 3.10.1.1 Lotka–Volterra Model 205 3.10.1.2 General Dynamical Behavior 207 3.10.1.3 Nondimensionalization 208 3.10.1.4 Phase Plane Plots 209 3.10.2 Wine Fermentation 211 3.10.2.1 Setting Up a Mathematical Model 212 3.10.2.2 Yeast 213 3.10.2.3 Ethanol and Sugar 215 3.10.2.4 Nitrogen 216 3.10.2.5 Using a Hand-fit to Estimate N 0 217 3.10.2.6 Parameter Estimation 219 3.10.2.7 Problems with Nonautonomous Models 220 3.10.2.8 Converting Data into a Function 222 3.10.2.9 Using Weighting Factors 222 3.10.3 Pharmacokinetics 223 3.10.4 Plant Growth 226 4 Mechanistic Models II: PDEs 229 4.1 Introduction 229 4.1.1 Limitations of ODE Models 229 4.1.2 Overview: Strange Animals, Sounds, and Smells 230 4.1.3 Two Problems You Should Be Able to Solve 231 4.2 The Heat Equation 233 4.2.1 Fourier’s Law 234 4.2.2 Conservation of Energy 235 4.2.3 Heat Equation = Fourier’s Law + Energy Conservation 236 4.2.4 Heat Equation in Multidimensions 238 4.2.5 Anisotropic Case 238 4.2.6 Understanding Off-diagonal Conductivities 239 4.3 Some Theory You Should Know 241 4.3.1 Partial Differential Equations 241 4.3.1.1 First-order PDEs 242 4.3.1.2 Second-order PDEs 243 4.3.1.3 Linear versus Nonlinear 243 4.3.1.4 Elliptic, Parabolic, and Hyperbolic Equations 244 4.3.2 Initial and Boundary Conditions 245 4.3.2.1 Well Posedness 246 4.3.2.2 A Rule of Thumb 246 4.3.2.3 Dirichlet and Neumann Conditions 247 4.3.3 Symmetry and Dimensionality 248 4.3.3.1 1D Example 249 4.3.3.2 2D Example 251 4.3.3.3 3D Example 252 4.3.3.4 Rotational Symmetry 252 4.3.3.5 Mirror Symmetry 253 4.3.3.6 Symmetry and Periodic Boundary Conditions 253 4.4 Closed Form Solutions 254 4.4.1 Problem 1 255 4.4.2 Separation of Variables 255 4.4.3 A Particular Solution for Validation 257 4.5 Numerical Solution of PDE’s 257 4.6 The Finite Difference Method 258 4.6.1 Replacing Derivatives with Finite Differences 258 4.6.2 Formulating an Algorithm 259 4.6.3 Implementation in R 260 4.6.4 Error and Stability Issues 262 4.6.5 Explicit and Implicit Schemes 263 4.6.6 Computing Electrostatic Potentials 264 4.6.7 Iterative Methods for the Linear Equations 264 4.6.8 Billions of Unknowns 265 4.7 The Finite-Element Method 266 4.7.1 Weak Formulation of PDEs 267 4.7.2 Approximation of the Weak Formulation 269 4.7.3 Appropriate Choice of the Basis Functions 270 4.7.4 Generalization to Multidimensions 271 4.7.5 Summary of the Main Steps 272 4.8 Finite-element Software 274 4.9 A Sample Session Using Salome-Meca 276 4.9.1 Geometry Definition Step 277 4.9.1.1 Organization of the GUI 277 4.9.1.2 Constructing the Geometrical Primitives 278 4.9.1.3 Excising the Sphere 279 4.9.1.4 Defining the Boundaries 281 4.9.2 Mesh Generation Step 281 4.9.3 Problem Definition and Solution Step 283 4.9.4 Postprocessing Step 285 4.10 A Look Beyond the Heat Equation 286 4.10.1 Diffusion and Convection 288 4.10.2 Flow in Porous Media 290 4.10.2.1 Impregnation Processes 291 4.10.2.2 Two-phase Flow 293 4.10.2.3 Water Retention and Relative Permeability 293 4.10.2.4 Asparagus Drip Irrigation 295 4.10.2.5 Multiphase Flow and Poroelasticity 296 4.10.3 Computational Fluid Dynamics (CFD) 296 4.10.3.1 Navier–Stokes Equations 296 4.10.3.2 Backward Facing Step Problem 298 4.10.3.3 Solution Using Code-Saturne 299 4.10.3.4 Postprocessing Using Salome-Meca 301 4.10.3.5 Coupled Problems 302 4.10.4 Structural Mechanics 303 4.10.4.1 Linear Static Elasticity 303 4.10.4.2 Example: Eye Tonometry 306 4.11 Other Mechanistic Modeling Approaches 309 4.11.1 Difference Equations 309 4.11.2 Cellular Automata 310 4.11.3 Optimal Control Problems 312 4.11.4 Differential-algebraic Problems 314 4.11.5 Inverse Problems 314 A CAELinux and the Book Software 317 B R (Programming Language and Software Environment) 321 B.1 Using R in a Konsole Window 321 B.1.1 Batch Mode 321 B.1.2 Command Mode 322 B.2 R Commander 322 C Maxima 323 C. 1 Using Maxima in a Konsole Window 323 C.1. 1 Batch Mode 323 C.1. 2 Command Mode 323 C. 2 wxMaxima 324 References 325 Index 335

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Book SynopsisThis book is an overview of recent mathematical models developed to examine stages of cancer growth and development, from single mutant cell to metastatic spread of the disease.Table of ContentsPreface * Biology/Pathology of Cancer * Modeling the Immune Response to Cancer * Simple Models of Tumor Growth * Modeling Avascular, Multicellular Spheroid Growth * Pre-pattern Models of Tumor Growth * Modeling Tumor-induced Angiogenesis * Modeling Invasion and Metastasis * Mechanical Models of Growth * Index

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Book SynopsisFocusing on modeling applications, this outstanding reference provides a step-by-step, non-mathematical approach to constructing and using realistic workable groundwater models on a daily basis. Extensive detailed drawings, case studies, practical examples, and sample models illustrate important concepts.Table of ContentsUnderstanding Groundwater Modeling. Reviewing the Mathematical Model for Flow. Reviewing the Mathematical Model for Transport. Using Analytical Models for Idealized Flow and Transport Problems. Understanding Numerical Flow Models. Understanding Numerical Transport Models. Dimensionality of Flow and Transport Problems. Applying Numerical Models. Special Topics. Appendices. Glossary. References. Index.

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Book SynopsisThis contributed volume contains a collection of articles on state-of-the-art developments on the construction of theoretical integral techniques and their application to specific problems in science and engineering. Chapters in this book are based on talks given at the Symposium on the Theory and Applications of Integral Methods in Science and Engineering, held virtually in July 2021, and are written by internationally recognized researchers. This collection will be of interest to researchers in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines and other professionals for whom integration is an essential tool.Table of ContentsApproximate Solution for One-Dimensional Compressible Two-Phase Immiscible Flow in Porous Media for Variable Boundary Conditions (P. Pires).- On Pseudo-Cross Sections for Neutron Escape from a Domain by a Physical Monte Carlo Simulation (D.G. Benvenutti).- From a Unitary Symmetry Hypothesis to Dynamical Structures in Quantum Mechanics Models (B.E.J. Bodmann).- The Traction Boundary Value Problem for Thin Elastic Structures (C. Constanda).- Mapping Properties of Potential Operators Related to the 2D Compressible Stokes System in Weighted Sobolev Spaces (C. Fresneda-Portillo).- Stochastic effects of the meander on the dispersion of pollutants in the planetary boundary layer under low wind conditions (D. Buske).- Asymptotics for the spectrum of a Floquet-parametric family of homogenization problems associated with a Dirichlet waveguide (Pérez-Martínez).- The Wavelet-Based Integral Formula for the Solutions of the Wave Equation in an inhomogeneous Medium: Convergence of Integrals (Perel).- Modelling The Spread of a Disease in an Epidemic Through a Country Divided into Geographcal Regions (P.J. Harris).- Computing Elastic Interior Transmission Eigenvalues (A. Kleefeld).- A Novel Solution of the Multi-Group Neutron Diffusion Equation by the Hankel Transform Formalism (J.C.L. Fernandes).- A Simple Numerical Scheme to Obtain Reflectivity and Transmissivity of an Isotropically Scattering Slab (C.A. Ladeia).- A Unified Integral Equation Formulation for Linear and Geometrically Nonlinear Analysis of Thick Plates: Derivation of Equations (R.J. Marczak).- On viscous fluid flow in curvilinear coordinate systems (Meneghetti).- Impact Loading of Interface Cracks: Effects of Cracks Closure and Friction (Menshykova).- Periodic Solutions in Rn for Stationary Anisotropic Stokes and Navier-Stokes Systems (S.E. Mikhailov).- Null-solutions of elliptic partial differential equations with power growth (D. Mitrea).- On the Use of the Adjoint Technique to the Estimation of Neutron Source Distributions in the Context of Subcritical Nuclear Reactors (R.C. Barros).- The Nodal LTSN Solution and a New Approach to Determine the Outgoing Angular Flux at the Boundary in a Retangular Domain (A.R. Parigi).- A numerical study of the convergence of two hybrid convolution quadrature schemes for broadband wave problems (D.J. Chappell).- Analytical Reconstruction of the Nonlinear Transfer Function for a Wiener-HammersteinModel (J. Schmith).- Variation of Zero-Net Liquid Holdup in Gas-Liquid Cylindrical Cyclone (GLCC c) (O. Shoham).- On the Mono-Energetic Neutron Space Kinetics Equation in Cartesian Geometry: An Analytic Solution by a Spectral Method (F. Tumelero).

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Book SynopsisThis volume addresses recent developments in mathematical modeling in three areas of optical science: diffractive optics, photonic band gap structures, and waveguides. Particular emphasis is on the formulation of mathematical models and the design and analysis of new computational approaches. The book contains cutting-edge discourses on emerging technology in optics that provides significant challenges and opportunities for applied mathematicians, researchers, and engineers. Each of the three topics is presented through a series of survey papers to provide a broad overview focusing on the mathematical models. Chapters present model problems, physical principles, mathematical and computational approaches, and engineering applications corresponding to each of the three areas. Although some of the subject matter is classical, the topics presented are new and represent the latest developments in their respective fields.

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Book SynopsisAn accessible textbook that explains, discusses, and applies both the frequentist and Bayesian theoretical frameworks to fit the different types of statistical models that allow an analysis of the types of data most commonly gathered by life scientists.Table of ContentsPart 1: The Conceptual Basis For Fitting Statistical Models 1: General introduction 2: Statistical modeling: a short historical background 3: Estimating parameters: the main purpose of statistical inference Part II: Applying The Generalized Linear Model to Varied Data Types 4: The General Linear Model I: numerical explanatory variables 5: The General Linear Model II: categorical explanatory variables 6: The General Linear Model III: interactions between explanatory variables 7: Model selection: one, two, and more models fitted to the data 8: The Generalized Linear Model 9: When the response variable is binary 10: When the response variables are counts, often with many zeros 11: Further issues involved in the modeling of counts 12: Models for positive real-valued response variables: proportions and others Part III: Incorporating Experimental and Survey Design Using Mixed Models 13: Accounting for structure in mixed/hierachical structures 14: Experimental design in the life sciences - the basics 15: Mixed-hierachical models and experimental design data Afterword R packages used in the book Appendix 1: Using R and RStudio: the basics (only available online at www.oup.com/companion/InchaustiSMWR) Appendix 2: Exploring and describing the evidence in graphics (only available online at www.oup.com/companion/InchaustiSMWR)

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Book SynopsisA concise, accessible introduction to the principal ideas, methods, and underlying statistical concepts for understanding and applying evolutionary quantitative genetics. It includes a broad taxonomic range of examples - human, animal, and plant; both model organisms and wild populations.Table of ContentsIntroduction 1: Selection on a Single Trait 2: Selection on Multiple Traits 3: The Selection Surface and Adaptive Landscape for a Single Trait 4: The Selection Surface and Adaptive Landscape for Multiple Traits 5: Inheritance of a Single Trait 6: Inheritance of Multiple Traits 7: Modularity, Performance, and Functional Complexes 8: Drift of a Single, Neutral Trait 9: Drift of Multiple, Neutral Traits 10: Response of a Single Trait to Selection 11: Response of Multiple Traits to Selection 12: Evolution of a Single Trait on a Stationary Adaptive Landscape 13: Evolution of Multiple Traits on a Stationary Adaptive Landscape 14: Trait Evolution on Dynamic Adaptive Landscapes 15: Evolution of Genetic Variance 16: Evolution of the G-Matrix on a Stationary Adaptive Landscape 17: Evolution of the G-Matrix on Dynamic Adaptive Landscapes 18: Evolution Along Selective Lines of Least Resistance 19: Speciation and Extinction of Lineages 20: Coevolution of Species with Trait-Based Interactions 21: Coevolution of Species with Density-Dependent Interactions 22: From Evolutionary Process to Pattern: A Synthesis

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Book SynopsisAn accessible and interdisciplinary introduction to the applications of statistical mechanics across the sciences. The book contains a discussion of the methods of statistical physics and includes mathematical explanations alongside guidance to enable the reader to translate theory into practice.Trade ReviewThis book has a good mix of interesting topics and shows the breadth of application of the statistical mechanics concepts. * Robert M. Ziff, University of Michigan *The book's subject is one which is of great interest and impacts many areas both within and outside physics. I am not aware of any other textbook which includes engaging mathematical content alongside a wide range of accessible applications, so this text has the potential to appeal to both the lay person and the technical expert. * Peter Richmond, Trinity College Dublin *Explores statistical mechanics at its glorious best, in the form of practical applications of collective behaviors found in the real world. Schulman is refreshingly honest in his approach, helping to stake out the frontiers of the field, posing problems that will inspire and direct future generations of scientists. * Daniel Sheehan, University of San Diego *I think that the book's collection of topics and its unique style make it a useful addition to the more standard textbook offering. Moreover, given the more colloquial style of the book, I imagine that it may be suitable for an audience that is interested in the physics of emergence and complexity that goes beyond the popular science literature. * Stefan Kirchner, Zhejiang University *I expect that anyone interested in complex systems and who has the requisite knowledge of elementary calculus and linear algebra will find When Things Grow Many to be a rewarding read. * Robert Deegan, University of Michigan, Physics Today *I enjoyed reading When Things Grow Many and learned something new from each chapter. Schulman writes in a conversational style, and he peppers the book with jokes and opinions. Even though he intimates that he doesn't have all the answers, his fun, inviting tone will make readers want to find out if he does. * Robert Deegan, Physics Today *This book ensures that all readers can grasp the fundamental principles and applications of physics, making it an excellent educational tool for a wide range of students. * Miguel A. F. Sanjuán, Contemporary Physics *This book ensures that all readers can grasp the fundamental principles and applications of physics, making it an excellent educational tool for a wide range of students. * Miguel A. F. Sanjuán, Contemporary Physics *Table of Contents1: Introduction 2: Ideal Gas 3: Rubber Bands 4: Percolitis 5: Ferromagnetism 6: Maximum Entropy Methods 7: Power Laws 8: Universality, Renormalization and Critical Phenomena 9: Social Sciences 10: Biological Sciences 11: Physical Sciences Free

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Book SynopsisEncompasses the full range of computational science and engineering from modelling to solution, both analytical and numerical. It develops a framework for the equations and numerical methods of applied mathematics. Gilbert Strang has taught this material to thousands of engineers and scientists (and many more on MIT's OpenCourseWare 18.085-6). His experience is seen in his clear explanations, wide range of examples, and teaching method. The book is solution-based and not formula-based: it integrates analysis and algorithms and MATLAB codes to explain each topic as effectively as possible. The topics include applied linear algebra and fast solvers, differential equations with finite differences and finite elements, Fourier analysis and optimization. This book also serves as a reference for the whole community of computational scientists and engineers. Supporting resources, including MATLAB codes, problem solutions and video lectures from Gilbert Strang's 18.085 courses at MIT, are provi

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Book SynopsisThe Earth and environmental sciences are becoming progressively more quantitative due to the increased use of mathematical models and new data analysis techniques. This accessible introduction presents an overview of the mathematical methods essential for understanding Earth processes, providing an invaluable resource for students and early career researchers who may have missed (or forgotten) the mathematics they need to succeed as scientists. Topics build gently from basic methods such as calculus to more advanced techniques including linear algebra and differential equations. The practical applications of the mathematical methods to a variety of topics are discussed, ranging from atmospheric science and oceanography to biogeochemistry and geophysics. Including over 530 exercises and end-of-chapter problems, as well as additional computer codes in Python and MATLAB, this book supports readers in applying appropriate analytical or computational methods to solving real research questioTable of ContentsPreface; 1. Estimation and dimensional analysis; 2. Derivatives and integrals; 3. Series and summations; 4. Scalars, vectors, and matrices; 5. Probability; 6. Ordinary differential equations; 7. Vectors and calculus; 8. Special functions; 9. Fourier series and integral transforms; 10. Partial differential equations; 11. Tensors; Appendix A. Units and dimensions; Appendix B. Tables of useful formulae; Appendix C. Complex numbers; Notes; References; Index.

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