Numerical analysis Books

Book SynopsisThis remarkable text by John R. Taylor has been a non-stop best-selling international hit since it was first published forty years ago. However, the two-plus decades since the second edition was released have seen two dramatic developments; the huge rise in popularity of Bayesian statistics, and the continued increase in the power and availability of computers and calculators. In response to the former, Taylor has added a full chapter dedicated to Bayesian thinking, introducing conditional probabilities and Bayes’ theorem. The several examples presented in the new third edition are intentionally very simple, designed to give readers a clear understanding of what Bayesian statistics is all about as their first step on a journey to become practicing Bayesians. In response to the second development, Taylor has added a number of chapter-ending problems that will encourage readers to learn how to solve problems using computers. While many of these can be solved using programs such as Matlab or Mathematica, almost all of them are stated to apply to commonly available spreadsheet programs like Microsoft Excel. These programs provide a convenient way to record and process data and to calculate quantities like standard deviations, correlation coefficients, and normal distributions; they also have the wonderful ability – if students construct their own spreadsheets and avoid the temptation to use built-in functions – to teach the meaning of these concepts.Trade ReviewThe new chapter on Bayesian statistics is extremely clear and well written, and is another one of John Taylor’s fabulous expositions. I enjoyed how Taylor develops the subject by using it to answer questions about the effectiveness of a vaccine. Before reading this chapter I wondered what assumptions are needed to derive a numerical value for a vaccine’s effectiveness, and I also wondered about the data needed and the methods used. Lo and behold, all my questions were answered in this chapter! I definitely will buy the new edition of Error Analysis and I look forward to delving into the Bayesian statistics. -- Mark Semon, Bates CollegeTable of ContentsPART I 1. Preliminary Description of Error Analysis 2. How to Report and Use Uncertainties 3. Propagation of Uncertainties 4. Statistical Analysis of Random Uncertainties 5. The Normal Distribution PART II 6. Rejection of Data 7. Weighted Averages 8. Least-Squares Fitting 9. Covariance and Correlation 10. The Binomial Distribution 11. The Poisson Distribution 12. The Chi-Squared Test for a Distribution 13. Bayesian Statistics APPENDICES A. Normal Error Integral, I B. Normal Error Integral, II C. Probabilities for Correlation Coefficients D. Probabilities for Chi Squared E. Two Proofs Concerning Sample Standard Deviations Answers to Quick Checks and Odd-Numbered Problems Index

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Book SynopsisThis is a practical guide to P-splines, a simple, flexible and powerful tool for smoothing. P-splines combine regression on B-splines with simple, discrete, roughness penalties. They were introduced by the authors in 1996 and have been used in many diverse applications. The regression basis makes it straightforward to handle non-normal data, like in generalized linear models. The authors demonstrate optimal smoothing, using mixed model technology and Bayesian estimation, in addition to classical tools like cross-validation and AIC, covering theory and applications with code in R. Going far beyond simple smoothing, they also show how to use P-splines for regression on signals, varying-coefficient models, quantile and expectile smoothing, and composite links for grouped data. Penalties are the crucial elements of P-splines; with proper modifications they can handle periodic and circular data as well as shape constraints. Combining penalties with tensor products of B-splines extends theseTrade Review'The title says it all. This is a practical book which shows how P-splines are used in an astonishingly wide range of settings. If you use P-splines already the book is indispensable; if you don't, then reading it will convince you it's time to start. Every example comes with an R-program available on the book's web-site, an important feature for the experienced user and novice alike.' Iain Currie, Heriot-Watt University'This book is an enlightening and at the same time extremely enjoyable read. It will serve the applied statistician who is looking for practical solutions but also the connoisseur in search of elegant concepts. The accompanying website offers reproducible code and invites to promptly enter the fascinating universe of P-splines.' Jutta Gampe, Max Planck Institute for Demographic Research'Everything you always wanted to know about P-splines, from the inventors themselves. Paul H.C. Eilers and Brian D. Marx make a compelling case for their claim that P-splines are the best practical smoother out there, providing intuition, methodology, applications, and R code that clearly demonstrate the power, flexibility, and wide applicability of this approach to smoothing.' Jeffrey Simonoff, New York University'This is the book that everyone working on smoothing models should keep handy. At last we have a manuscript that shows the real power of P-splines, their versatility, and the different perspectives you can take to use them. Chapters 1 to 3 will certainly appeal to those who want to start working in this field, and to researchers that need to deepen their knowledge of this technique. Scientists and practitioners from other areas will find chapters 4 to 8 very useful for the wide range of examples and applications. The companion package and the fact that all results (even figures) are reproducible is a real bonus. Thank you Paul and Brian for being truthful to your motto: 'show, don't tell'.' Maria Durbán, University Carlos III de MadridTable of Contents1. Introduction; 2. Bases, penalties, and likelihoods; 3. Optimal smoothing in action; 4. Multidimensional smoothing; 5. Smoothing of scale and shape; 6. Complex counts and composite links; 7. Signal regression; 8. Special subjects; A. P-splines for the impatient; B. P-splines and competitors; C. Computational details; D. Array algorithms; E. Mixed model equations; F. Standard errors in detail; G. The website.

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Book SynopsisHaskell is a purely functional language that allows programmers to rapidly develop clear, concise, and correct software. The language has grown in popularity in recent years, both in teaching and in industry. This book is based on the author''s experience of teaching Haskell for more than twenty years. All concepts are explained from first principles and no programming experience is required, making this book accessible to a broad spectrum of readers. While Part I focuses on basic concepts, Part II introduces the reader to more advanced topics. This new edition has been extensively updated and expanded to include recent and more advanced features of Haskell, new examples and exercises, selected solutions, and freely downloadable lecture slides and example code. The presentation is clean and simple, while also being fully compliant with the latest version of the language, including recent changes concerning applicative, monadic, foldable, and traversable types.Trade Review'The skills you acquire by studying this book will make you a much better programmer no matter what language you use to actually program in.' Erik Meijer, Facebook, from the ForewordReview of previous edition: 'The best introduction to Haskell available. There are many paths towards becoming comfortable and competent with the language but I think studying this book is the quickest path. I urge readers of this magazine to recommend Programming in Haskell to anyone who has been thinking about learning the language.' Duncan Coutts, The Monad.ReaderReview of previous edition: 'Where this book excels is in the order and style of its exposition … With its ripe selection of examples and its careful clarity of exposition, the book is a welcome addition to the introductory functional programming literature.' Journal of Functional ProgrammingTable of ContentsForeword; Preface; Part I. Basic Concepts: 1. Introduction; 2. First steps; 3. Types and classes; 4. Defining functions; 5. List comprehensions; 6. Recursive functions; 7. Higher-order functions; 8. Declaring types and classes; 9. The countdown problem; Part II. Going Further: 10. Interactive programming; 11. Unbeatable tic-tac-toe; 12. Monads and more; 13. Monadic parsing; 14. Foldables and friends; 15. Lazy evaluation; 16. Reasoning about programs; 17. Calculating compilers; Appendix A. Selected solutions; Appendix B. Standard prelude; Bibliography; Index.

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Book SynopsisProvides avenues for applying functional analysis to the practical study of natural sciences as well as mathematics. Contains worked problems on Hilbert space theory and on Banach spaces and emphasizes concepts, principles, methods and major applications of functional analysis.Table of ContentsMetric Spaces.Normed Spaces;Banach Spaces.Inner Product Spaces;Hilbert Spaces.Fundamental Theorems for Normed and Banach Spaces.Further Applications: Banach Fixed Point Theorem.Spectral Theory of Linear Operators in Normed Spaces.Compact Linear Operators on Normed Spaces and Their Spectrum.Spectral Theory of Bounded Self-Adjoint Linear Operators.Unbounded Linear Operators in Hilbert Space.Unbounded Linear Operators in Quantum Mechanics.Appendices.References.Index.

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Book SynopsisThis book addresses the state of the art of reduced order methods for modelling and computational reduction of complex parametrised systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in various fields.Consisting of four contributions presented at the CIME summer school, the book presents several points of view and techniques to solve demanding problems of increasing complexity. The focus is on theoretical investigation and applicative algorithm development for reduction in the complexity – the dimension, the degrees of freedom, the data – arising in these models.The book is addressed to graduate students, young researchers and people interested in the field. It is a good companion for graduate/doctoral classes.Table of Contents- 1. The Reduced Basis Method in Space and Time: Challenges, Limits and Perspectives. - 2. Inverse Problems: A Deterministic Approach Using Physics-Based Reduced Models. - 3. Model Order Reduction for Optimal Control Problems. - 4. Machine Learning Methods for Reduced Order Modeling.

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Book SynopsisPresents an account of the development of laminar boundary layer theory as a historical study. This book includes a description of the application of the ideas of triple deck theory to flow past a plate, to separation from a cylinder and to flow in channels. It is intended to provide a graduate level teaching resource.Trade ReviewThis book provides various physical/engineering/historical insights on this topic. * EMS *Sobey includes recent work in a seamless manner ... a very readable book. * New Scientist *

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Book SynopsisSpectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more limited. More recently the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretisation procedures on unstructured meshes, which are also recognised as more efficient for solution of time-dependent oscillatory solutions over long time periods. Here Karniadakis and Sherwin present a much-updated and expanded version of their successful first edition covering the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing over 50% new material, including discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilisation and filtering techniques, this text aims to introduce a wider audience to the use oTrade ReviewThe book contains a large amount of material, including a number of exercises, examples and figures. The book will be helpful to specialists coming into contact with CFD, applied and numerical mathematicians, engineers, physicists and specialists in climate and ocean modeling. It can also be recommended for advanced students of these disciplines. * EMS Newsletter *This book will probably help popularize the spectral/hp element method. Not only should it be recommended to researchers working on spectral/hp methods but it should also be on the wish list of all those who are interested in computational fluid dynamics. * Jean-Luc Guermond, Mathematical Reviews *Table of ContentsIntroduction ; Fundamental concepts in one dimension ; Multi-dimensional expansion bases ; Multi-dimensional formulations ; Diffusion equation ; Advection and advection-diffusion ; Non-conforming elements ; Algorithms for incompressible flows ; Incompressible flow simulations:verification and validation ; Hyperbolic conservation laws ; Appendices ; Jacobi polynomials ; Gauss-Type integration ; Collocation differentiation ; Co discontinuous expansion bases ; Characteristic flux decomposition ; References ; Index

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Book SynopsisAlgorithms are the hidden methods that computers apply to process information and make decisions. Nowadays, our lives are run by algorithms. They determine what news we see. They influence which products we buy. They suggest our dating partners. They may even be determining the outcome of national elections. They are creating, and destroying, entire industries. Despite mounting concerns, few know what algorithms are, how they work, or who created them.Poems that Solve Puzzles tells the story of algorithms from their ancient origins to the present day and beyond. The book introduces readers to the inventors and inspirational events behind the genesis of the world''s most important algorithms. Professor Chris Bleakley recounts tales of ancient lost inscriptions, Victorian steam-driven contraptions, top secret military projects, penniless academics, hippy dreamers, tech billionaires, superhuman artificial intelligences, cryptocurrencies, and quantum computing. Along the way, the book explains, with the aid of clear examples and illustrations, how the most influential algorithms work.Compelling and impactful, Poems that Solve Puzzles tells the story of how algorithms came to revolutionise our world.Trade ReviewPoems that Solve Puzzles is a thorough investigation into the history of algorithms...It is an enjoyable read for anyone curious about how algorithms developed and were implemented throughout history.' * Notices of the American Mathematical Society *Poems that Solve Puzzles: The History and Science of Algorithms is an informative and entertaining book. It is appropriate for a wide swath of readers, from people who are interested in learning about what "blockchain" is without having to do any math to students and instructors in the mathematical sciences who need more examples of how these academic topics make important contributions to the technologically complex world we live in. * Ron Buckmire, Occidental College, Mathematical Association of America *Table of Contents0: Introduction 1: Ancient Algorithms 2: Ever Expanding Circles 3: Computer Dreams 4: Weather Forecasts 5: Artificial Intelligence Emerges 6: Needles in Haystacks 7: The Internet 8: Googling the Web 9: Facebook and Friends 10: America's Favourite Quiz Show 11: Mimicking the Brain 12: Superhuman Intelligence 13: Next Steps

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Book SynopsisCombining scientific computing methods and algorithms with modern data analysis techniques, including basic applications of compressive sensing and machine learning, this book develops techniques that allow for the integration of the dynamics of complex systems and big data. MATLAB is used throughout for mathematical solution strategies.Trade ReviewThe book allows methods for dealing with large data to be explained in a logical process suitable for both undergraduate and post-graduate students ... With sport performance analysis evolving into deal with big data, the book forms a key bridge between mathematics and sport science * John Francis, University of Worcester *Table of ContentsI BASIC COMPUTATIONS AND VISUALIZATION; II DIFFERENTIAL AND PARTIAL DIFFERENTIAL EQUATIONS; III COMPUTATIONAL METHODS FOR DATA ANALYSIS; IV SCIENTIFIC APPLICATIONS

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Book SynopsisThis textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection beTrade ReviewThe subject of the book is fascinating and written versions of the lecture series are nicley presented and preserve well the informal spirit of the lectures. This is a very useful book for graduate students and for mathematicians (or physicists) from other fields interested in the topic. * EMS *The lecturers cover an enormous amount of material, ranging from algeraic geometry and the theory of Riemann surfaces to loop groups, connections, Yang-Mills equations and twister theory. However despite this wide range, the book is surprisingly self-contained and readable. * Bulletin of the London Mathematical Society *Table of Contents1. Introduction ; 2. Riemann surfaces and integrable systems ; 3. Integrable systems and inverse scattering ; 4. Integrable systems and twistors ; Index

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Book SynopsisSelf-adaptive discretization methods are now an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. ChaptersTrade ReviewError control and adaptive solution algorithms for finite element approximation are a key concern of every practitioner. The present text, written by a leading authority in the field who has made many important contributions, will be valuable for theoreticians and practitioners alike. * Mark Ainsworth, Professor of Applied Mathematics, Brown University *Table of Contents1. A Simple Model Problem ; 2. Implementation ; 3. Auxiliary Results ; 4. Linear Elliptic Equations ; 5. Nonlinear Elliptic Equations ; 6. Parabolic Equations

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Book SynopsisMARY HART is a lecturer in Pure Mathematics at Sheffield University

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Book SynopsisDeveloped from the authorâs course on advanced mechanics of composite materials, Finite Element Analysis of Composite Materials with Abaqus shows how powerful finite element tools tackle practical problems in the structural analysis of composites. This Second Edition includes two new chapters on Fatigue and Abaqus Programmable Features as well as a major update of chapter 10 Delaminations and significant updates throughout the remaining chapters. Furthermore, it updates all examples, sample code, and problems to Abaqus 2020. Unlike other texts, this one takes theory to a hands-on level by actually solving problems. It explains the concepts involved in the detailed analysis of composites, the mechanics needed to translate those concepts into a mathematical representation of the physical reality, and the solution of the resulting boundary value problems using Abaqus. The reader can follow a process to recreate every example using Abaqus graphical user interfacTable of Contents1. Mechanics of Orthotropic Materials. 2. Introduction to Finite Element Analysis. 3. Elasticity and Strength of Laminates. 4. Buckling. 5. Free Edge Stresses. 6. Computational Micromechanics. 7. Viscoelasticity. 8. Continuum Damage Mechanics. 9. Discrete Damage Mechanics. 10. Delaminations. 11. Fatigue. 12. Abaqus Programmable Features.

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Book SynopsisOptimization is an important tool used in decision science and for the analysis of physical systems used in engineering. It begins with very simple ideas progressing through more complicated concepts, concentrating on methods for both unconstrained and constrained optimization.Trade ReviewMMOR Mathematical Methods of Operations Research, 2001: "The books looks very suitable to be used in an graduate-level course in optimization for students in mathematics, operations research, engineering, and others. Moreover, it seems to be very helpful to do some self-studies in optimization, to complete own knowledge and can be a source of new ideas... I recommend this excellent book to everyone who is interested in optimization problems."Table of ContentsPreface.-Preface to the Second Edition.-Introduction.-Fundamentals of Unconstrained Optimization.-Line Search Methods.-Trust-Region Methods.-Conjugate Gradient Methods.-Quasi-Newton Methods.-Large-Scale Unconstrained Optimization.-Calculating Derivatives.-Derivative-Free Optimization.-Least-Squares Problems.-Nonlinear Equations.-Theory of Constrained Optimization.-Linear Programming: The Simplex Method.-Linear Programming: Interior-Point Methods.-Fundamentals of Algorithms for Nonlinear Constrained Optimization.-Quadratic Programming.-Penalty and Augmented Lagrangian Methods.-Sequential Quadratic Programming.-Interior-Point Methods for Nonlinear Programming.-Background Material.- Regularization Procedure.

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Book SynopsisAn historical introduction.- Functional equations with two variables.- Functional equations with one variable.- Miscellaneous methods for functional equations.- Some closing heuristics.- Appendix: Hamel bases.- Hints and partial solutions to problems.Trade ReviewFrom the reviews: "This book is devoted to functional equations of a special type, namely to those appearing in competitions … . The book contains many solved examples and problems at the end of each chapter. … The book has 130 pages, 5 chapters and an appendix, a Hints/Solutions section, a short bibliography and an index. … The book will be valuable for instructors working with young gifted students in problem solving seminars." (EMS Newsletter, June, 2008)Table of ContentsAn historical introduction.- Functional equations with two variables.- Functional equations with one variable.- Miscellaneous methods for functional equations.- Some closing heuristics.- Appendix: Hamel bases.- Hints and partial solutions to problems.

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Book SynopsisThis book deals with several aspects of what is now called "explicit number theory." The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions.Trade ReviewFrom the reviews:"Cohen (Université Bordeaux I, France), an instant classic, uniquely bridges the gap between old-fashioned, naive treatments and the many modern books available that develop the tools just mentioned … . Summing Up: Recommended. … Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (5), January, 2008)"The book deals with aspects of ‘explicit number theory’. … The central theme … is the solution of Diophantine equations. … It combines an interesting ‘philosophy’ of the subject with an encyclopedic grasp of detail. The extension of the author’s reach via the contributed chapters is a good idea. Perhaps it is the start of a trend, as the subject grows more and more. … It will undoubtedly be mined by instructors for their graduate courses, particularly for the purpose of including some recently-proved content." (R. C. Baker, Mathematical Reviews, Issue 2008 e)“This is the second volume of a highly impressive two-volume textbook on Diophantine analysis. … readers are presented with an almost overwhelming amount of material. This … text book is bound to become an important reference for students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)Table of ContentsAnalytic Tools.- Bernoulli Polynomials and the Gamma Function.- Dirichlet Series and L-Functions.- p-adic Gamma and L-Functions.- Modern Tools.- Applications of Linear Forms in Logarithms.- Rational Points on Higher-Genus Curves.- The Super-Fermat Equation.- The Modular Approach to Diophantine Equations.- Catalan’s Equation.

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Book SynopsisIt can be used for a course that provides an introduction to basic functional analysis, approximation theory, and numerical analysis, while building upon and applying basic techniques of real variable theory.Trade ReviewSecond Edition S.C. Brenner and L.R. Scott The Mathematical Theory of Finite Element Methods "[This is] a well-written book. A great deal of material is covered, and students who have taken the trouble to master at least some of the advanced material in the later chapters would be well placed to embark on research in the area." ZENTRALBLATT MATH From the reviews of the third edition: "An excelent survey of the deep mathematical roots of finite element methods as well as of some of the newest and most formal results concerning these methods. … The approach remains very clear and precise … . A significant number of examples and exercises improve considerably the accessability of the text. The authors also point out different ways the book could be used in various courses. … valuable reference and source for researchers (mainly mathematicians) in the topic." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1135 (13), 2008)Table of ContentsPreface(3rdEd).- Preface(2ndEd).- Preface(1stED).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.

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Book SynopsisThis book presents a substantial part of matrix analysis that is functional analytic in spirit. Topics covered include the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, and perturbation of matrix functions and matrix inequalities.Trade ReviewR. Bhatia Matrix Analysis "A highly readable and attractive account of the subject. The book is a must for anyone working in matrix analysis; it can be recommended to graduate students as well as to specialists."—ZENTRALBLATT MATH "There is an ample selection of exercises carefully positioned throughout the text. In addition each chapter includes problems of varying difficulty in which themes from the main text are extended."—MATHEMATICAL REVIEWSTable of ContentsI A Review of Linear Algebra.- I.1 Vector Spaces and Inner Product Spaces.- I.2 Linear Operators and Matrices.- I.3 Direct Sums.- I.4 Tensor Products.- I.5 Symmetry Classes.- I.6 Problems.- I.7 Notes and References.- II Majorisation and Doubly Stochastic Matrices.- II.1 Basic Notions.- II. 2 Birkhoff’s Theorem.- II.3 Convex and Monotone Functions.- II.4 Binary Algebraic Operations and Majorisation.- II.5 Problems.- II.6 Notes and References.- III Variational Principles for Eigenvalues.- III.1 The Minimax Principle for Eigenvalues.- III.2 Weyl’s Inequalities.- III.3 Wielandt’s Minimax Principle.- III.4 Lidskii’s Theorems.- III. 5 Eigenvalues of Real Parts and Singular Values.- III.6 Problems.- III.7 Notes and References.- IV Symmetric Norms.- IV.l Norms on ?n.- IV.2 Unitarily Invariant Norms on Operators on ?n.- IV.3 Lidskii’s Theorem (Third Proof).- IV.4 Weakly Unitarily Invariant Norms.- IV.5 Problems.- IV.6 Notes and References.- V Operator Monotone and Operator Convex Functions.- V.1 Definitions and Simple Examples.- V.2 Some Characterisations.- V.3 Smoothness Properties.- V.4 Loewner’s Theorems.- V.5 Problems.- V.6 Notes and References.- VI Spectral Variation of Normal Matrices.- VI. 1 Continuity of Roots of Polynomials.- VI. 2 Hermitian and Skew-Hermitian Matrices.- VI. 3 Estimates in the Operator Norm.- VI. 4 Estimates in the Frobenius Norm.- VI. 5 Geometry and Spectral Variation: the Operator Norm.- VI. 6 Geometry and Spectral Variation: wui Norms.- VI. 7 Some Inequalities for the Determinant.- VI. 8 Problems.- VI. 9 Notes and References.- VII Perturbation of Spectral Subspaces of Normal Matrices.- VII. 1 Pairs of Subspaces.- VII. 2 The Equation AX — XB = Y.- VII. 3 Perturbation of Eigenspaces.- VII. 4 A Perturbation Bound for Eigenvalues.- VII. 5 Perturbation of the Polar Factors.- VII. 6 Appendix: Evaluating the (Fourier) constants.- VII. 7 Problems.- VII. 8 Notes and References.- VIII Spectral Variation of Nonnormal Matrices.- VIII. 1 General Spectral Variation Bounds.- VIII. 4 Matrices with Real Eigenvalues.- VIII. 5 Eigenvalues with Symmetries.- VIII. 6 Problems.- VIII. 7 Notes and References.- IX A Selection of Matrix Inequalities.- IX. 1 Some Basic Lemmas.- IX. 2 Products of Positive Matrices.- IX. 3 Inequalities for the Exponential Function.- IX. 4 Arithmetic-Geometric Mean Inequalities.- IX. 5 Schwarz Inequalities.- IX. 6 The Lieb Concavity Theorem.- IX. 7 Operator Approximation.- IX. 8 Problems.- IX. 9 Notes and References.- X Perturbation of Matrix Functions.- X. 1 Operator Monotone Functions.- X. 2 The Absolute Value.- X. 3 Local Perturbation Bounds.- X. 4 Appendix: Differential Calculus.- X. 5 Problems.- X. 6 Notes and References.- References.

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Book SynopsisComputational inference is based on an approach to statistical methods that uses modern computational power to simulate distributional properties of estimators and test statistics.Trade ReviewFrom the reviews:“This is a book that covers many of the computational issues that statisticians will encounter as part of their research and applied work. … The writing in the book is quite clear and the author has done a good job providing the essence of each topic. … Overall, I think this is an excellent book. … This book will give a graduate student a good overview of the field. There are exercises provided for each chapter together with some solutions.” (Michael J. Evans, Mathematical Reviews, Issue 2011 b)“This book is a superior treatment of the important subject of statistical computing. I strongly recommend this book to anyone who analyzes data using either a commercial statistical software package or statistical computer programs written by the user or someone else. Thus this book is important not only for data oriented statisticians but for econometricians, psychometricians, political methodologists and biometricians as well. … All terms in this work including computing terms are clearly defined.” (Melvin Hinich, Technometrics, Vol. 53 (1), February, 2011)“I greatly appreciated the author’s command of both numerical and statistical computing … . The book also contains many exercises that substantiate the concepts, with solutions and hints in the appendix, an extensive bibliography, and a link to further literature and notes. The target readership includes undergraduates, postgraduates in statistics and allied fields such as computer science and mathematics, scientific research workers, and practitioners of statistics and numerical techniques. … I strongly recommend it for all scientific libraries.” (Soubhik Chakraborty, ACM Computing Reviews, October, 2010)“This book has a very large scope in that … it covers the dual fields of computational statistics and of statistical computing. … must-read for all students and researchers engaging into any kind of serious statistical programming. … is well-written, in a lively and personal style. … a reference book that should appear in the shortlist of any computational statistics/statistical computing graduate course as well as on the shelves of any researchers supporting his or her statistical practice with a significant dose of computing backup.”­­­ (Christian P. Robert, Statistical and Computation, Vol. 21, 2011)Table of ContentsPreliminaries.- Mathematical and Statistical Preliminaries.- Statistical Computing.- Computer Storage and Arithmetic.- Algorithms and Programming.- Approximation of Functions and Numerical Quadrature.- Numerical Linear Algebra.- Solution of Nonlinear Equations and Optimization.- Generation of Random Numbers.- Methods of Computational Statistics.- Graphical Methods in Computational Statistics.- Tools for Identification of Structure in Data.- Estimation of Functions.- Monte Carlo Methods for Statistical Inference.- Data Randomization, Partitioning, and Augmentation.- Bootstrap Methods.- Exploring Data Density and Relationships.- Estimation of Probability Density Functions Using Parametric Models.- Nonparametric Estimation of Probability Density Functions.- Statistical Learning and Data Mining.- Statistical Models of Dependencies.

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Book Synopsis* Novel approach to the mathematics of problem solving, in particular how to do logical calculations. * Many of the problems are well-known from (mathematical) puzzle books. * The solution method in the book is new and more relevant to the true nature of problem solving in the modern IT-dominated world.Table of ContentsPreface xi PART I Algorithmic Problem Solving 1 CHAPTER 1 – Introduction 3 1.1 Algorithms 3 1.2 Algorithmic Problem Solving 4 1.3 Overview 5 1.4 Bibliographic Remarks 6 CHAPTER 2 – Invariants 7 2.1 Chocolate Bars 10 2.1.1 The Solution 10 2.1.2 The Mathematical Solution 11 2.2 Empty Boxes 16 2.2.1 Review 19 2.3 The Tumbler Problem 22 2.3.1 Non-deterministic Choice 23 2.4 Tetrominoes 24 2.5 Summary 30 2.6 Bibliographic Remarks 34 CHAPTER 3 – Crossing a River 35 3.1 Problems 36 3.2 Brute Force 37 3.2.1 Goat, Cabbage and Wolf 37 3.2.2 State-Space Explosion 39 3.2.3 Abstraction 41 3.3 Nervous Couples 42 3.3.1 What Is the Problem? 42 3.3.2 Problem Structure 43 3.3.3 Denoting States and Transitions 44 3.3.4 Problem Decomposition 45 3.3.5 A Review 48 3.4 Rule of Sequential Composition 50 3.5 The Bridge Problem 54 3.6 Conditional Statements 63 3.7 Summary 65 3.8 Bibliographic Remarks 65 CHAPTER 4 – Games 67 4.1 Matchstick Games 67 4.2 Winning Strategies 69 4.2.1 Assumptions 69 4.2.2 Labelling Positions 70 4.2.3 Formulating Requirements 72 4.3 Subtraction-Set Games 74 4.4 Sums of Games 78 4.4.1 A Simple Sum Game 79 4.4.2 Maintain Symmetry! 81 4.4.3 More Simple Sums 82 4.4.4 Evaluating Positions 83 4.4.5 Using the Mex Function 87 4.5 Summary 91 4.6 Bibliographic Remarks 92 CHAPTER 5 – Knights and Knaves 95 5.1 Logic Puzzles 95 5.2 Calculational Logic 96 5.2.1 Propositions 96 5.2.2 Knights and Knaves 97 5.2.3 Boolean Equality 98 5.2.4 Hidden Treasures 100 5.2.5 Equals for Equals 101 5.3 Equivalence and Continued Equalities 102 5.3.1 Examples of the Associativity of Equivalence 104 5.3.2 On Natural Language 105 5.4 Negation 106 5.4.1 Contraposition 109 5.4.2 Handshake Problems 112 5.4.3 Inequivalence 113 5.5 Summary 117 5.6 Bibliographic Remarks 117 CHAPTER 6 – Induction 119 6.1 Example Problems 120 6.2 Cutting the Plane 123 6.3 Triominoes 126 6.4 Looking for Patterns 128 6.5 The Need for Proof 129 6.6 From Verification to Construction 130 6.7 Summary 134 6.8 Bibliographic Remarks 134 CHAPTER 7 – Fake-Coin Detection 137 7.1 Problem Formulation 137 7.2 Problem Solution 139 7.2.1 The Basis 139 7.2.2 Induction Step 139 7.2.3 The Marked-Coin Problem 140 7.2.4 The Complete Solution 141 7.3 Summary 146 7.4 Bibliographic Remarks 146 CHAPTER 8 – The Tower of Hanoi 147 8.1 Specification and Solution 147 8.1.1 The End of the World! 147 8.1.2 Iterative Solution 148 8.1.3 Why? 149 8.2 Inductive Solution 149 8.3 The Iterative Solution 153 8.4 Summary 156 8.5 Bibliographic Remarks 156 CHAPTER 9 – Principles of Algorithm Design 157 9.1 Iteration, Invariants and Making Progress 158 9.2 A Simple Sorting Problem 160 9.3 Binary Search 163 9.4 Sam Loyd’s Chicken-Chasing Problem 166 9.4.1 Cornering the Prey 170 9.4.2 Catching the Prey 174 9.4.3 Optimality 176 9.5 Projects 177 9.6 Summary 178 9.7 Bibliographic Remarks 180 CHAPTER 10 – The Bridge Problem 183 10.1 Lower and Upper Bounds 183 10.2 Outline Strategy 185 10.3 Regular Sequences 187 10.4 Sequencing Forward Trips 189 10.5 Choosing Settlers and Nomads 193 10.6 The Algorithm 196 10.7 Summary 199 10.8 Bibliographic Remarks 200 CHAPTER 11 – Knight’s Circuit 201 11.1 Straight-Move Circuits 202 11.2 Supersquares 206 11.3 Partitioning the Board 209 11.4 Summary 216 11.5 Bibliographic Remarks 218 PART II Mathematical Techniques 219 CHAPTER 12 – The Language of Mathematics 221 12.1 Variables, Expressions and Laws 222 12.2 Sets 224 12.2.1 The Membership Relation 224 12.2.2 The Empty Set 224 12.2.3 Types/Universes 224 12.2.4 Union and Intersection 225 12.2.5 Set Comprehension 225 12.2.6 Bags 227 12.3 Functions 227 12.3.1 Function Application 228 12.3.2 Binary Operators 230 12.3.3 Operator Precedence 230 12.4 Types and Type Checking 232 12.4.1 Cartesian Product and Disjoint Sum 233 12.4.2 Function Types 235 12.5 Algebraic Properties 236 12.5.1 Symmetry 237 12.5.2 Zero and Unit 238 12.5.3 Idempotence 239 12.5.4 Associativity 240 12.5.5 Distributivity/Factorisation 241 12.5.6 Algebras 243 12.6 Boolean Operators 244 12.7 Binary Relations 246 12.7.1 Reflexivity 247 12.7.2 Symmetry 248 12.7.3 Converse 249 12.7.4 Transitivity 249 12.7.5 Anti-symmetry 251 12.7.6 Orderings 252 12.7.7 Equality 255 12.7.8 Equivalence Relations 256 12.8 Calculations 257 12.8.1 Steps in a Calculation 259 12.8.2 Relations between Steps 260 12.8.3 ‘‘If’’ and ‘‘Only If’’ 262 12.9 Exercises 264 CHAPTER 13 – Boolean Algebra 267 13.1 Boolean Equality 267 13.2 Negation 269 13.3 Disjunction 270 13.4 Conjunction 271 13.5 Implication 274 13.5.1 Definitions and Basic Properties 275 13.5.2 Replacement Rules 276 13.6 Set Calculus 279 13.7 Exercises 281 CHAPTER 14 – Quantifiers 285 14.1 DotDotDot and Sigmas 285 14.2 Introducing Quantifier Notation 286 14.2.1 Summation 287 14.2.2 Free and Bound Variables 289 14.2.3 Properties of Summation 291 14.2.4 Warning 297 14.3 Universal and Existential Quantification 297 14.3.1 Universal Quantification 298 14.3.2 Existential Quantification 300 14.4 Quantifier Rules 301 14.4.1 The Notation 302 14.4.2 Free and Bound Variables 303 14.4.3 Dummies 303 14.4.4 Range Part 303 14.4.5 Trading 304 14.4.6 Term Part 304 14.4.7 Distributivity Properties 304 14.5 Exercises 306 CHAPTER 15 – Elements of Number Theory 309 15.1 Inequalities 309 15.2 Minimum and Maximum 312 15.3 The Divides Relation 315 15.4 Modular Arithmetic 316 15.4.1 Integer Division 316 15.4.2 Remainders and Modulo Arithmetic 320 15.5 Exercises 322 CHAPTER 16 – Relations, Graphs and Path Algebras 325 16.1 Paths in a Directed Graph 325 16.2 Graphs and Relations 328 16.2.1 Relation Composition 330 16.2.2 Union of Relations 332 16.2.3 Transitive Closure 334 16.2.4 Reflexive Transitive Closure 338 16.3 Functional and Total Relations 339 16.4 Path-Finding Problems 341 16.4.1 Counting Paths 341 16.4.2 Frequencies 343 16.4.3 Shortest Distances 344 16.4.4 All Paths 345 16.4.5 Semirings and Operations on Graphs 347 16.5 Matrices 351 16.6 Closure Operators 353 16.7 Acyclic Graphs 354 16.7.1 Topological Ordering 355 16.8 Combinatorics 357 16.8.1 Basic Laws 358 16.8.2 Counting Choices 359 16.8.3 Counting Paths 361 16.9 Exercises 366 Solutions to Exercises 369 References 405 Index 407

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Book SynopsisDesigned to present the chief results of the Lebesgue theory of integration, this textbook provides students with detailed coverage of the main ideas of the Lebesgue measure. The approach is particularly well suited for students of analysis, probability and statistics.Table of ContentsTHE ELEMENTS OF INTEGRATION. Measurable Functions. Measures. The Integral. Integrable Functions. The Lebesgue Spaces Lp. Modes of Convergence. Decomposition of Measures. Generation of Measures. Product Measures. THE ELEMENTS OF LEBESGUE MEASURE. Volumes of Cells and Intervals. The Outer Measure. Measurable Sets. Examples of Measurable Sets. Approximation of Measurable Sets. Additivity and Nonadditivity. Nonmeasurable and Non-Borel Sets. References. Index.

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Book SynopsisThis introduction to a wide range of theoretical studies in fluid dynamics, covers a great deal of material and offers updated information on topics such as stability and turbulence. It surveys nearly the entire field of classical fluid dynamics and discusses the various conceptual and analytical models of fluid flow.Trade Review"I know of no other modern book in theoretical fluid dynamics that covers so much material so well." (Physics Today, November 1998)Table of ContentsReview of Basic Concepts and Equations of Fluid Dynamics. Dynamics of Inviscid Incompressible Fluid Flows. Dynamics of Inviscid Compressible Fluid Flows. Dynamics of Viscous Fluid Flows. Hydrodynamic Stability. Dynamics of Turbulence. Bibliography. Index.

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Book SynopsisThis revised and updated volume describes methods fundamental to the theory and explanation of data analysis. This edition includes extensions and devices such as component and component-plus residual plots, cross-verification with a second sample and an index of required x-precision.Trade Review"...a grand historical document for industrial statistics in its glory days, as its selection for the Classics Library implies." --Technometrics Vol. 42, No. 4 May 2001 This book provides an excellent insight into the minds of two master craftsmen at work. I very much applaud the decision to include this in a "classics library" and would encourage more authors to produce statistics books in the same vein, i.e. focused on the practical application of the subject rather than methodology development. Anyone involved in the analysis of unbalanced multifactor dtaa will find this book an extremely useful source of practical advice. --The Statistician 50 (1) 2001.Table of ContentsAssumptions and Methods of Fitting Equations. One Independent Variable. Two or More Independent Variables. Fitting an Equation in Three Independent Variables. Selection of Independent Variables. Some Consequences of the Disposition of the Data Points. Selection of Variables in Nested Data. Nonlinear Least Squares, a Complex Example. Glossary. User's Manual. Bibliography. Index.

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Book SynopsisThis book provides an extensive introduction to the numerical solution of a large class of integral equations. The initial chapters provide a general framework for the numerical analysis of Fredholm integral equations of the second kind, covering degenerate kernel, projection and Nystrom methods. Additional discussions of multivariable integral equations and iteration methods update the reader on the present state of the art in this area. The final chapters focus on the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation, in both two and three dimensions. Two chapters are devoted to planar BIE problems, which include both existing methods and remaining questions. Practical problems for BIE such as the set up and solution of the discretised BIE are also discussed. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particTrade Review' This outstanding monograph ... represents a major milestone in the list of books on the numerical solution of integral equations ... deserves to be on the shelf of any researcher and graduate student interested in the numerical solution of elliptic boundary-value problems.' H. Brunner, Mathematics Abstracts 'It will become the standard reference in the area.' Zietschrift fur Angwandte Mathematik und PhysikTable of ContentsPreface; 1. A brief discussion of integral equations; 2. Degenerate kernel methods; 3. Projection methods; 4. The Nystrom method; 5. Solving multivariable integral equations; 6. Iteration methods; 7. Boundary integral equations on a smooth planar boundary; 8. Boundary integral equations on a piecewise smooth planar boundary; 9. Boundary integral equations in three dimensions; Discussion of the literature; Appendix; Bibliography; Index.

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Book SynopsisModern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and related topics such as modular arithmetic. The authors present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details.Trade Review'Very few books do justice to material that is suitable for both professional software engineers and graduate students. This book does just that, without losing its focus or stressing one audience over the other. As the authors make clear, this book is about algorithms for arithmetic (and not hardware considerations and implementations); this focus allows them to cover integer arithmetic, modular arithmetic, and floating-point arithmetic broadly and in detail. the notes and references at the end of each chapter guide readers to more details, and provide a historical backdrop for each major topic.' Marlin Thomas, Reviews.comTable of ContentsPreface; Acknowledgements; Notation; 1. Integer arithmetic; 2. Modular arithmetic and the FFT; 3. Floating-point arithmetic; 4. Newton's method and function evaluation; Appendix. Implementations and pointers; Bibliography; Index.

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Book SynopsisNumerical Recipes in Fortran 90 starts with a detailed introduction to the Fortran 90 language and then presents the basic concepts of parallel programming. All 350+ routines from the second edition of Numerical Recipes are presented in Fortran 90. Many are completely reworked algorithmically so as to be 'parallel-ready' and to utilise Fortran 90's advanced language features. Numerical Recipes in Fortran 90 emphasises general principles, but throughout there are also numerous hints and tips. This volume is intended for use with the original Numerical Recipes in Fortran, 2nd Edition (now called Numerical Recipes in Fortran 77) and does not discuss how the individual programs are used, or how the mathematical methods are used. An excellent guide for Fortran programmers interested in moving to Fortran 90, or C/C++ programmers interested in parallel programming.Trade Review'This new edition begins with three completely new chapters that provide a detailed introduction to the Fortran 90 language and then present the basic concepts of parallel programming, all with the same clarity and good cheer for which Numerical Recipes is famous.' L'Enseignement Mathématique' … certainly recommendable'. Eric de Sturler, ITW NieuwsTable of Contents1. Introduction to Fortran 90 language features; 2. Introduction to parallel programming; 3. Numerical recipes utility functions for Fortran 90; Part I. Fortran 90 Code chapters: 4. Preliminaries; 5. Solution of linear algebraic equations; 6. Interpolation and extrapolation; 7. Integration of functions; 8. Evaluation of functions; 9. Special functions; 10. Random numbers; 11. Sorting; 12. Root finding and nonlinear sets of equations; 13. Minimization or maximization of functions; 14. Eigensystems; 15. Fast Fourier transform; 16. Fourier and spectral applications; 17. Statistical description of Data; 18. Modelling of data; 19. Integration of ordinary differential equations; 20. Two point boundary value problems; 21. Integral equations and inverse theory; 22. Partial differential equations; 23 Less-numerical algorithms; Part II. Appendices: 24. Listing of utility modules (nrtype and nrutil); 25. Listing of explicit interfaces; 26. Index of programs and dependencies.

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Book SynopsisThe 2005 second edition of a highly successful graduate text giving a complete introduction to partial differential equations and numerical analysis. Revised to include new sections on finite volume methods, modified equation analysis, multigrid, and conjugate gradient methods.Trade Review' … attractive text … very clear and supported by many illuminating figures. Therefore, the book is suitable for a course for applied mathematicians or engineers at the advanced undergraduate level.' Math. Meth. Oper. Res.Table of Contents1. Introduction; 2. Parabolic equations in one space variable; 3. 2-D and 3-D parabolic equations; 4. Hyperbolic equations in one space dimension; 5. Consistency, convergence and stability; 6. Linear second order elliptic equations in two dimensions; 7. Iterative solution of linear algebraic equations; Bibliography; Index.

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Book SynopsisThe book combines topics in mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation). The motivation for these topics is the difficulty, both conceptually and in the technical execution, of combining elements of combinatorial and of numerical algorithms. Mesh generation is a topic where a meaningful combination of these different approaches to problem solving is inevitable. The book develops methods from both areas that are amenable to combination, and explains breakthrough solutions to meshing that fit into this category. This book emphasizes topics that are elementary, attractive, useful, interesting, and lend themselves to teaching, making it an ideal graduate text for courses on mesh generation.Trade Review'… a very readable exposition …'. Monatshefte für Mathematik'… well organised … We recommend the book to graduate students and researchers in computational geometry.' János Kincses, Acta Sci. Math.Table of Contents1. Delaunay triangulations; 2. Triangle meshes; 3. Combinatorial topology; 4. Surface simplification; 5. Delaunay tetrahedrizations; 6. Tetrahedron meshes; 7. Open problems.

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Book SynopsisThis text introduces numerical methods and shows how to develop, analyse, and use them. Complete MATLAB programs are now available at www.cambridge.org/Moin, with more than 30 exercises. This thorough and practical book is a first course in numerical analysis for new graduate students in engineering and physical science.Trade Review'… thorough and practical …' Mathematical ReviewsTable of Contents1. Interpolation; 2. Numerical differentiation - finite differences; 3. Numerical integration; 4. Numerical solution of ordinary differential equations; 5. Numerical solution of partial differential equations; 6. Discrete transform methods; Appendix. A review of linear algebra.

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Book SynopsisAccurately predicting the behaviour of multiphase flows is a problem of immense industrial and scientific interest. Modern computers can now study the dynamics in great detail and these simulations yield unprecedented insight. This book provides a comprehensive introduction to direct numerical simulations of multiphase flows for researchers and graduate students.Trade Review"This book provides a comprehensive introduction to direct numerical simulations of multiphase flows. It is useful for researchers and graduate students in computational engineering science who are interested in the development and application of numerical simulation methods for multiphase incompressible flows." Arnold Reusken, Mathematical ReviewsTable of ContentsPreface; 1. Introduction; 2. Fluid mechanics with interfaces; 3. Numerical solutions of the Navier–Stokes equations; 4. Advecting a fluid interface; 5. The volume-of-fluid method; 6. Advecting marker points - front tracking; 7. Surface tension; 8. Disperse bubbly flows; 9. Atomization and breakup; 10. Droplet collision, impact and splashing; 11. Extensions; Appendix A. Interfaces: description and definitions; Appendix B. Distributions on the interface; Appendix C. Cube-chopping; Appendix D. Dynamics of liquid sheets; Bibliography; Index.

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Book SynopsisA comprehensive look at the Schwarz-Christoffel transformation, including its many applications.Trade Review'Altogether an excellent book written by the masters of the SC mapping who command both theory and numerics.' Dieter Gaier, Zbl. MATHTable of ContentsPreface; 1. Introduction; 2. Essentials; 3. Numerical methods; 4. Variations; 5. Applications; Using the SC toolbox.

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Book SynopsisThis outrageous graphic novel investigates key concepts in mathematics by taking readers on a voyage of forensic discovery, exploring some of the most fundamental ideas in mathematics within a thrilling murder mystery.Trade Review"Prime Suspects will appeal to a variety of readers in a variety of venues . . . . For the mathematician who eats, sleeps, and drinks numbers, start on page one and just enjoy the story . . . the book is fun, and interesting, and a challenge on many levels."---Judith Reveal, New York Journal of Books"Prime Suspects blends together the worlds of mathematics and forensic science to give readers both an interesting mystery and an education in numbers."---Anelise Farris, Rogues Portal"A total one-off."---Matthew Reisz, Times Educational Supplement"This is really a great book with so many references to mathematical ideas, that you can read and reread and find every time new details hidden in the graphics."---Adhemar Bultheel, European Mathematical Society"What a spectacular book! I am rather blown away by it."---Jonathan Shock, MathemAfrica"Granville and Granville have performed something of a feat. They've written a graphic detective novel that is both interesting to read and yet simultaneously teaches its readers some deep mathematics . . . . It's very difficult to write a book on an advanced topic in mathematics that's accessible to math students and enthusiasts yet touches on contemporary research that is of interest to a broad swath of practicing mathematicians. Prime Suspects is such a book. And it's entertaining to boot. I recommend it in the strongest terms."---Benjamin Linowitz, MAA Reviews"Bringing in elements from film noir, TV police shows and famous movies, coupled with some amazing art work, subtlemathematical humour and corny science jokes, and what you have is a one-of-a-kind creation – indeed, Prime Suspects has it all from minus to plus infinity."---David Appell, Physics World"Renowned number-theorist Andrew Granville here explores the graphic novel as a format to popularize mathematical discoveries. This absolutely brilliantly illustrated book arose from an earlier play and an accompanying commissioned musical piece. The mathematics too is astonishing—but explained understandably." * Mathematics Magazine *"If you are a mathematician, I definitely recommend reading (and listening to) Prime Suspects. If you are not a mathematician, you might miss a number of references, and some passages might be too mysterious or technical, but still it makes for interesting reading; it might be good for you, too."---Marco Abate, The Mathematical Intelligencer"The artwork and presentation are both well up to the standard of a mainstream graphic novel."---Andrew Ruddle, Mathematics Today"Prime Suspects is a work of art which anyone with a passion for maths will appreciate and enjoy."---Lennie Wells, Mathematical Gazette"I definitely recommend."---Marco Abate, Mathematical Intelligencer

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Book SynopsisProvides an exploration of standard numerical analysis topics, as well as non-traditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. This textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering.Trade Review"Distinguishing features are the inclusion of many recent applications of numerical methods and the extensive discussion of methods based on Chebyshev interpolation. This book would be suitable for use in courses aimed at advanced undergraduate students in mathematics, the sciences, and engineering."--Choice "An instructor could assemble several different one-semester courses using this book--numerical linear algebra and interpolation, or numerical solutions of differential equations--or perhaps a two-semester sequence. This is a charming book, well worth consideration for the next numerical analysis course."--William J. Satzer, MAA FocusTable of ContentsPreface xiii Chapter 1: MATHEMATICAL MODELING 1 1.1 Modeling in Computer Animation 2 1.1.1 A Model Robe 2 1.2 Modeling in Physics: Radiation Transport 4 1.3 Modeling in Sports 6 1.4 Ecological Models 8 1.5 Modeling a Web Surfer and Google 11 1.5.1 The Vector Space Model 11 1.5.2 Google's PageRank 13 1.6 Chapter 1 Exercises 14 Chapter 2: BASIC OPERATIONS WITH MATLAB 19 2.1 Launching MATLAB 19 2.2 Vectors 20 2.3 Getting Help 22 2.4 Matrices 23 2.5 Creating and Running .m Files 24 2.6 Comments 25 2.7 Plotting 25 2.8 Creating Your Own Functions 27 2.9 Printing 28 2.10 More Loops and Conditionals 29 2.11 Clearing Variables 31 2.12 Logging Your Session 31 2.13 More Advanced Commands 31 2.14 Chapter 2 Exercises 32 Chapter 3: MONTE CARLO METHODS 41 3.1 A Mathematical Game of Cards 41 3.1.1 The Odds in Texas Holdem 42 3.2 Basic Statistics 46 3.2.1 Discrete Random Variables 48 3.2.2 Continuous Random Variables 51 3.2.3 The Central Limit Theorem 53 3.3 Monte Carlo Integration 56 3.3.1 Buffon's Needle 56 3.3.2 Estimating pi 58 3.3.3 Another Example of Monte Carlo Integration 60 3.4 Monte Carlo Simulation of Web Surfing 64 3.5 Chapter 3 Exercises 67 Chapter 4: SOLUTION OF A SINGLE NONLINEAR EQUATION IN ONE UNKNOWN 71 4.1 Bisection 75 4.2 Taylor's Theorem 80 4.3 Newton's Method 83 4.4 Quasi-Newton Methods 89 4.4.1 Avoiding Derivatives 89 4.4.2 Constant Slope Method 89 4.4.3 Secant Method 90 4.5 Analysis of Fixed Point Methods 93 4.6 Fractals, Julia Sets, and Mandelbrot Sets 98 4.7 Chapter 4 Exercises 102 Chapter 5: FLOATING-POINT ARITHMETIC 107 5.1 Costly Disasters Caused by Rounding Errors 108 5.2 Binary Representation and Base 2 Arithmetic 110 5.3 Floating-Point Representation 112 5.4 IEEE Floating-Point Arithmetic 114 5.5 Rounding 116 5.6 Correctly Rounded Floating-Point Operations 118 5.7 Exceptions 119 5.8 Chapter 5 Exercises 120 Chapter 6: CONDITIONING OF PROBLEMS; STABILITY OF ALGORITHMS 124 6.1 Conditioning of Problems 125 6.2 Stability of Algorithms 126 6.3 Chapter 6 Exercises 129 Chapter 7: DIRECT METHODS FOR SOLVING LINEAR SYSTEMS AND LEAST SQUARES PROBLEMS 131 7.1 Review of Matrix Multiplication 132 7.2 Gaussian Elimination 133 7.2.1 Operation Counts 137 7.2.2 LU Factorization 139 7.2.3 Pivoting 141 7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required 144 7.2.5 Implementation Considerations for High Performance 148 7.3 Other Methods for Solving Ax = b 151 7.4 Conditioning of Linear Systems 154 7.4.1 Norms 154 7.4.2 Sensitivity of Solutions of Linear Systems 158 7.5 Stability of Gaussian Elimination with Partial Pivoting 164 7.6 Least Squares Problems 166 7.6.1 The Normal Equations 167 7.6.2 QR Decomposition 168 7.6.3 Fitting Polynomials to Data 171 7.7 Chapter 7 Exercises 175 Chapter 8: POLYNOMIAL AND PIECEWISE POLYNOMIAL INTERPOLATION 181 8.1 The Vandermonde System 181 8.2 The Lagrange Form of the Interpolation Polynomial 181 8.3 The Newton Form of the Interpolation Polynomial 185 8.3.1 Divided Differences 187 8.4 The Error in Polynomial Interpolation 190 8.5 Interpolation at Chebyshev Points and chebfun 192 8.6 Piecewise Polynomial Interpolation 197 8.6.1 Piecewise Cubic Hermite Interpolation 200 8.6.2 Cubic Spline Interpolation 201 8.7 Some Applications 204 8.8 Chapter 8 Exercises 206 Chapter 9: NUMERICAL DIFFERENTIATION AND RICHARDSON EXTRAPOLATION 212 9.1 Numerical Differentiation 213 9.2 Richardson Extrapolation 221 9.3 Chapter 9 Exercises 225 Chapter 10: NUMERICAL INTEGRATION 227 10.1 Newton-Cotes Formulas 227 10.2 Formulas Based on Piecewise Polynomial Interpolation 232 10.3 Gauss Quadrature 234 10.3.1 Orthogonal Polynomials 236 10.4 Clenshaw-Curtis Quadrature 240 10.5 Romberg Integration 242 10.6 Periodic Functions and the Euler-Maclaurin Formula 243 10.7 Singularities 247 10.8 Chapter 10 Exercises 248 Chapter 11: NUMERICAL SOLUTION OF THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 251 11.1 Existence and Uniqueness of Solutions 253 11.2 One-Step Methods 257 11.2.1 Euler's Method 257 11.2.2 Higher-Order Methods Based on Taylor Series 262 11.2.3 Midpoint Method 262 11.2.4 Methods Based on Quadrature Formulas 264 11.2.5 Classical Fourth-Order Runge-Kutta and Runge-Kutta-Fehlberg Methods 265 11.2.6 An Example Using MATLAB's ODE Solver 267 11.2.7 Analysis of One-Step Methods 270 11.2.8 Practical Implementation Considerations 272 11.2.9 Systems of Equations 274 11.3 Multistep Methods 275 11.3.1 Adams-Bashforth and Adams-Moulton Methods 275 11.3.2 General Linear m-Step Methods 277 11.3.3 Linear Difference Equations 280 11.3.4 The Dahlquist Equivalence Theorem 283 11.4 Stiff Equations 284 11.4.1 Absolute Stability 285 11.4.2 Backward Differentiation Formulas (BDF Methods) 289 11.4.3 Implicit Runge-Kutta (IRK) Methods 290 11.5 Solving Systems of Nonlinear Equations in Implicit Methods 291 11.5.1 Fixed Point Iteration 292 11.5.2 Newton's Method 293 11.6 Chapter 11 Exercises 295 Chapter 12: MORE NUMERICAL LINEAR ALGEBRA: EIGENVALUES AND ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS 300 12.1 Eigenvalue Problems 300 12.1.1 The Power Method for Computing the Largest Eigenpair 310 12.1.2 Inverse Iteration 313 12.1.3 Rayleigh Quotient Iteration 315 12.1.4 The QR Algorithm 316 12.1.5 Google's PageRank 320 12.2 Iterative Methods for Solving Linear Systems 327 12.2.1 Basic Iterative Methods for Solving Linear Systems 327 12.2.2 Simple Iteration 328 12.2.3 Analysis of Convergence 332 12.2.4 The Conjugate Gradient Algorithm 336 12.2.5 Methods for Nonsymmetric Linear Systems 334 12.3 Chapter 12 Exercises 345 Chapter 13: NUMERICAL SOLUTION OF TWO-POINT BOUNDARY VALUE PROBLEMS 350 13.1 An Application: Steady-State Temperature Distribution 350 13.2 Finite Difference Methods 352 13.2.1 Accuracy 354 13.2.2 More General Equations and Boundary Conditions 360 13.3 Finite Element Methods 365 13.3.1 Accuracy 372 13.4 Spectral Methods 374 13.5 Chapter 13 Exercises 376 Chapter 14: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 379 14.1 Elliptic Equations 381 14.1.1 Finite Difference Methods 381 14.1.2 Finite Element Methods 386 14.2 Parabolic Equations 388 14.2.1 Semidiscretization and the Method of Lines 389 14.2.2 Discretization in Time 389 14.3 Separation of Variables 396 14.3.1 Separation of Variables for Difference Equations 400 14.4 Hyperbolic Equations 402 14.4.1 Characteristics 402 14.4.2 Systems of Hyperbolic Equations 403 14.4.3 Boundary Conditions 404 14.4.4 Finite Difference Methods 404 14.5 Fast Methods for Poisson's Equation 409 14.5.1 The Fast Fourier Transform 411 14.6 Multigrid Methods 414 14.7 Chapter 14 Exercises 418 APPENDIX A REVIEW OF LINEAR ALGEBRA 421 A.1 Vectors and Vector Spaces 421 A.2 Linear Independence and Dependence 422 A.3 Span of a Set of Vectors; Bases and Coordinates; Dimension of a Vector Space 423 A.4 The Dot Product; Orthogonal and Orthonormal Sets; the Gram-Schmidt Algorithm 423 A.5 Matrices and Linear Equations 425 A.6 Existence and Uniqueness of Solutions; the Inverse; Conditions for Invertibility 427 A.7 Linear Transformations; the Matrix of a Linear Transformation 431 A.8 Similarity Transformations; Eigenvalues and Eigenvectors 432 APPENDIX B TAYLOR'S THEOREM IN MULTIDIMENSIONS 436 References 439 Index 445

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Book SynopsisPacked with illustrations, amusing facts, puzzles, brainteasers and anecdotes - an enthralling and thought-provoking numerical voyage through the history of mathematics. A must-have for trivia addicts, maths-lovers and even numberphobes.Trade Review'This book is a complete joy. It made me smile. A lot' Carol Vorderman'There's more to maths than just numbers - but, as this entertaining and engaging book amply demonstrates, the depth and variety of mathematical ideas that appear when you start with 1, 2, 3 and keep going is astonishing. Once you start reading 'Number Freak' it's just like the number system itself - impossible to stop' Ian Stewart, author of Professor Stewart's Cabinet of Mathematical Curiosities'A great maths book for geeks and non-geeks alike' Johnny Ball'A fun book... definitely challenging' Vanity Fair'A fascinating parade of diverse numerical characters... An entertaining mix of numerical fun and theory' Booklist'All sorts of fascinating mathematical minutiae' Time Out (Chicago)

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Book SynopsisThe problem of developing a systematic approach to the design of feed­ back strategies capable of shaping the response of complicated dynamical control systems illustrates the integration of a wide variety of mathemat­ ical disciplines typical of the modern theory of systems and control.Table of ContentsSimultaneous Stabilization of Linear Time Varying Systems by Linear Time Varying Compensation.- Robust Feedback Stabilization of Nonlinear Systems.- Feedback Design from the Zero Dynamics Point of View.- Two Examples of Stabilizable Second Order Systems.- Orthogonality — Conventional and Unconventional — in Numerical Analysis.- Discrete Observability of Parabolic Initial Boundary Value Problems.- Numerical Optimal Control via Smooth Penalty Functions.- Observability and Inverse Problems Arising in Electrocardiography.- Eigenvalue Approximations on the Entire Real Line.- Prediction Bands for Ill-Posed Problems.- Controllability, Approximations and Stabilization.- Interval Mathematics Techniques for Control Theory Computations.- Accuracy and Conditioning in the Inversion of the Heat Equation.- On the Recovery of Surface Temperature and Heat Flux via Convolutions.- Observability, Interpolation and Related Topics.- Constructing Polynomials over Finite Fields.- A Collocative Variation of the Sinc-Galerkin Method for Second Order Boundary Value Problems.- A Sinc-Collocation Method for Weakly Singular Volterra Integral Equations.- Tuning Natural Frequencies by Output Feedback.- Efficient Numerical Solution of Fourth-Order Problems in the Modeling of Flexible Structures.- Explicit Approximate Methods for Computational Control Theory.- Sinc Approximate Solution of Quasilinear Equations of Conservation Law Type.- Systems with Fast Chaotic Components.- Bifurcation and Persistance of Minima in Nonlinear Parametric Programming.- Numerical Solution of an Ill-Posed Coefficient Identification Problem.- Observability, Predictability and Chaos.- Geometric Inverse Eigenvalue Problem.- Observability and Group Representation Theory.- Highly-Accurate Difference Schemes for Solving Hyperbolic Problems.- A Finite Spectrum Unmixing Set for $$\mathcal{G}\mathcal{L}\left( {3,\mathcal{R}} \right)$$.

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Book SynopsisThis text presents a complete treatment of variational problems on discrete sets with an overall behavior driven by surface energies. Covering both applications and perspectives, it can be used as an advanced graduate course text, as well as a reference for mathematical analysts and applied mathematicians working in related fields.Table of Contents1. Introduction; 2. Preliminaries; 3. Homogenization of pairwise systems with positive coefficients; 4. Compactness and integral representation; 5. Random lattices; 6. Extensions; 7. Frustrated systems; 8. Perspectives towards dense graphs; A. Multiscale analysis; B. Spin systems as limits of elastic interactions; References; Index.

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Book SynopsisThis concise introduction covers inverse problems and data assimilation, before exploring their inter-relations. Suitable for both classroom teaching and self-guided study, it is aimed at advanced undergraduates and beginning graduate students in mathematical sciences, together with researchers in science and engineering.Table of ContentsIntroduction; Part I. Inverse Problems: 1. Bayesian inverse problems and well-posedness; 2. The linear-Gaussian setting; 3. Optimization perspective; 4. Gaussian approximation; 5. Monte Carlo sampling and importance sampling; 6. Markov chain Monte Carlo; Exercises for Part I; Part II. Data Assimilation: 7. Filtering and smoothing problems and well-posedness; 8. The Kalman filter and smoother; 9. Optimization for filtering and smoothing: 3DVAR and 4DVAR; 10. The extended and ensemble Kalman filters; 11. Particle filter; 12. Optimal particle filter; Exercises for Part II; Part III. Kalman Inversion: 13. Blending inverse problems and data assimilation; References; Index.

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Book SynopsisThis textbook offers a guided tutorial that reviews the theoretical fundamentals while going through the practical examples used for constructing the computational frame, applied to various real-life models.Computational Optimization: Success in Practice will lead the readers through the entire process. They will start with the simple calculus examples of fitting data and basics of optimal control methods and end up constructing a multi-component framework for running PDE-constrained optimization. This framework will be assembled piece by piece; the readers may apply this process at the levels of complexity matching their current projects or research needs.By connecting examples with the theory and discussing the proper communication between them, the readers will learn the process of creating a big house. Moreover, they can use the framework exemplified in the book as the template for their research or course problems they will know how to change theTable of ContentsChapter 1. Introduction to Optimization. Chapter 2. Minimization Approaches for Functions of One Variable. Chapter 3. Generalized Optimization Framework. Chapter 4. Exploring Optimization Algorithms. Chapter 5. Line Search Algorithms. Chapter 6. Choosing Optimal Step Size. Chapter 7. Trust Region and Derivative-Free Methods. Chapter 8. Large-Scale and Constrained Optimization. Chapter 9. ODE-based Optimization. Chapter 10. Implementing Regularization Techniques. Chapter 11. Moving to PDE-based Optimization. Chapter 12. Sharing Multiple Software Environments.

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Book SynopsisThis textbook is a comprehensive introduction to computational mathematics and scientific computing suitable for undergraduate and postgraduate courses. It presents both practical and theoretical aspects of the subject, as well as advantages and pitfalls of classical numerical methods alongside with computer code and experiments in Python. Each chapter closes with modern applications in physics, engineering, and computer science.Features: No previous experience in Python is required. Includes simplified computer code for fast-paced learning and transferable skills development. Includes practical problems ideal for project assignments and distance learning. Presents both intuitive and rigorous faces of modern scientific computing. Provides an introduction to neural networks and machine learning. Table of Contents1. Introduction to Python. 2. Matrices and Python. 3. Scientific computing. 4. Calculus facts. 5. Roots of equations. 6. Interpolation and approximation. 7. Numerical integration. 8. Numerical differentiation and applications to differential equations. 9. Numerical linear algebra. 10. Best approximations. 11. Unconstrained optimization and neural networks. 12. Eigenvalue problems.

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Book SynopsisNumerical methods are vital to the practice of chemical engineering, allowing for the solution of real-world problems. Written in a concise and practical format, this textbook introduces readers to the numerical methods required in the discipline of chemical engineering and enables them to validate their solutions using both Python and Simulink. Introduces numerical methods, followed by the solution of linear and nonlinear algebraic equations. Deals with the numerical integration of a definite function and solves initial and boundary value ordinary differential equations with different orders. Weaves in examples of various numerical methods and validates solutions to each with Python and Simulink graphical programming. Features appendices on how to use Python and Simulink. Aimed at advanced undergraduate and graduate chemical engineering students, as well as practicing chemical engineers, thiTable of Contents1. Introduction. 2. Numerical Solutions of Linear Systems. 3. Bracketing Numerical Methods for Solving Systems of Nonlinear Equations. 4. Open Numerical Methods for Solving Systems of Nonlinear Equations. 5. Initial Value Problem Differential Equations. 6. Numerical Integration of Definite Functions. 7. Numerical Solution of Ordinary Differential Equations. 8. Simultaneous Systems of Differential Equations. 9. Boundary Value Problems Ordinary Differential Equations. Appendix A. Appendix B.

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Book SynopsisThis book investigates in detail the emerging deep learning (DL) technique in computational physics, assessing its promising potential to substitute conventional numerical solvers for calculating the fields in real-time. After good training, the proposed architecture can resolve both the forward computing and the inverse retrieve problems.Pursuing a holistic perspective, the book includes the following areas. The first chapter discusses the basic DL frameworks. Then, the steady heat conduction problem is solved by the classical U-net in Chapter 2, involving both the passive and active cases. Afterwards, the sophisticated heat flux on a curved surface is reconstructed by the presented Conv-LSTM, exhibiting high accuracy and efficiency. Additionally, a physics-informed DL structure along with a nonlinear mapping module are employed to obtain the space/temperature/time-related thermal conductivity via the transient temperature in Chapter 4. Finally, in Chapter 5, a series of thTable of Contents1. Deep Learning Framework and Paradigm in Computational Physics 2. Application of U-net in 3D Steady Heat Conduction Solver 3. Inversion of complex surface heat flux based on ConvLSTM 4. Time-domain electromagnetic inverse scattering based on deep learning 5. Reconstruction of thermophysical parameters based on deep learning 6. Advanced Deep Learning Techniques in Computational Physics

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Book SynopsisWhat sets Numerical Methods and Analysis with Mathematical Modelling apart are the modelling aspects utilizing numerical analysis (methods) to obtain solutions. The authors cover first the basic numerical analysis methods with simple examples to illustrate the techniques and discuss possible errors. The modelling prospective reveals the practical relevance of the numerical methods in context to real-world problems.At the core of this text are the real-world modelling projects. Chapters are introduced and techniques are discussed with common examples. A modelling scenario is introduced that will be solved with these techniques later in the chapter. Often, the modelling problems require more than one previously covered technique presented in the book.Fundamental exercises to practice the techniques are included. Multiple modelling scenarios per numerical methods illustrate the applications of the techniques introduced. Each chapter has several modelling

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