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

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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 *

Read more less

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

Read more less

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

Read more less

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

Read more less

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

Read more less

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

Read more less

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

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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)

Read more less

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 $$\mathcal{G}\mathcal{L}\left( {3,\mathcal{R}} \right)$$.

Read more less

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.

Read more less

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.

Read more less

Book Synopsis

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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

Read more less

Book SynopsisLearning Data Science is the first book to cover foundational skills in both programming and statistics that encompass the entire data science lifecycle: the process of collecting, wrangling, analyzing, and drawing conclusions from data.

Read more less

Book SynopsisArising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Îx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.Trade Review'Damelin and Miller provide a very detailed and thorough treatment of all the important mathematics related to signal processing. This includes the required background information found in elementary mathematics courses, so their book is really self-contained. The style of writing is suitable not only for mathematicians, but also for practitioners from other areas. Indeed, Damelin and Miller managed to write their text in a form that is accessible to nonspecialists, without giving up mathematical rigor.' Kai Diethelm, Computing Reviews'In the last 20 years or so, many books on wavelets have been published; most of them deal with wavelets from either the engineering or the mathematics perspective, but few try to connect the two viewpoints. The book under review falls under the last category … Overall, the book is a good addition to the literature on engineering mathematics.' Ahmed I. Zayed, Mathematical ReviewsTable of Contents1. Introduction; 2. Normed vector spaces; 3. Analytic tools; 4. Fourier series; 5. Fourier transforms; 6. Compressive sensing; 7. Discrete transforms; 8. Linear filters; 9. Windowed Fourier transforms, continuous wavelets, frames; 10. Multiresolution analysis; 11. Discrete wavelet theory; 12. Biorthogonal filters and wavelets; 13. Parsimonious representation of data; Bibliography; Index.

Read more less

Book SynopsisProvides an introduction to numerical methods for students in engineering courses. This book covers the solution of equations, interpolation and data fitting, solution of differential equations, eigenvalue problems and optimisation. The algorithms are implemented in Python 3, a high-level programming language that rivals MATLAB® in readability and ease of use.Trade Review'… a practical introduction, pushing the theory as far in the background as possible.' The European Mathematical Society (euro-math-soc.eu)'This book is nicely focused on the most frequently encountered types of numerical problems that scientists and engineers usually face and the most common and robust algorithms for solving them. The text is just the right size for a semester-long course for upper-division undergraduates or first-year graduate students … this is a well-written text that is logically organized, attractively presented, and supported with challenging problems.' Anthony J. Duben, Computing ReviewsTable of Contents1. Introduction to Python; 2. Systems of linear algebraic equations; 3. Interpolation and curve fitting; 4. Roots of equations; 5. Numerical differentiation; 6. Numerical integration; 7. Initial value problems; 8. Two-point boundary value problems; 9. Symmetric matrix eigenvalue problems; 10. Introduction to optimization.

Read more less

Book SynopsisThis comprehensive, accessible textbook covers the basics of signal processing, building up from fundamental principles to practical applications. It uses engineering notation to make mathematical concepts easy to follow, includes numerous homework problems and is accompanied by an extensive Mathematica® companion and instructor solutions manual.Trade Review'This is a major book about a serious subject - the combination of engineering and mathematics that goes into modern signal processing: discrete time, continuous time, sampling, filtering, and compression. The theory is beautiful and the applications are so important and widespread.' Gil Strang, Massachusetts Institute of Technology'A refreshing new approach to teaching the fundamentals of signal processing. Starting from basic concepts in algebra and geometry, [the authors] bring the reader to deep understandings of modern signal processing. Truly a gem!' Rico Malvar, Microsoft Research'A wonderful book that connects together all the elements of modern signal processing … it's all here and seamlessly integrated, along with a summary of history and developments in the field. A real tour-de-force, and a must-have on every signal processor's shelf!' Robert D. Nowak, University of Wisconsin, Madison'Finally a wonderful and accessible book for teaching modern signal processing to undergraduate students.' Stéphane Mallat, École Normale Supérieure'Most introductory signal processing textbooks focus on classical transforms, and study how these can be used. Instead, Foundations of Signal Processing encourages readers to think of signals first. It develops a 'signal-centric' view, one that focuses on signals, their representation and approximation, through the introduction of signal spaces. Unlike most entry-level signal processing texts, this general view, which can be applied to many different signal classes, is introduced right at the beginning. From this, starting from basic concepts, and placing an emphasis on intuition, this book develops mathematical tools that give the readers gets a fresh perspective on classical results, while providing them with the tools to understand many state of the art signal representation techniques.' Antonio Ortega, University of Southern California'Foundations of Signal Processing … is a pleasure to read. Drawing on the authors' rich experience of research and teaching of signal processing and signal representations, it provides an intellectually cohesive and modern view of the subject from the geometric point of view of vector spaces. Emphasizing Hilbert spaces, where fine technicalities can be relegated to backstage, this textbook strikes an excellent balance between intuition and mathematical rigor, that will appeal to both undergraduate and graduate engineering students. The last two chapters, on sampling and interpolation, and on localization and uncertainty, take full advantage of the machinery developed in the previous chapters to present these two very important topics of modern signal processing, that previously were only found in specialized monographs. The explanations of advanced topics are exceptionally lucid, exposing the reader to the ideas and thought processes behind the results and their derivation. Students will learn … why things work, at a deep level, which will equip them for independent further reading and research. I look forward to using this text in my own teaching.' Yoram Bresler, University of Illinois, Urbana-ChampaignTable of Contents1. On rainbows and spectra; 2. From Euclid to Hilbert: 2.1 Introduction; 2.2 Vector spaces; 2.3 Hilbert spaces; 2.4 Approximations, projections, and decompositions; 2.5 Bases and frames; 2.6 Computational aspects; 2.A Elements of analysis and topology; 2.B Elements of linear algebra; 2.C Elements of probability; 2.D Basis concepts; Exercises with solutions; Exercises; 3. Sequences and discrete-time systems: 3.1 Introduction; 3.2 Sequences; 3.3 Systems; 3.4 Discrete-time Fourier Transform; 3.5 z-Transform; 3.6 Discrete Fourier Transform; 3.7 Multirate sequences and systems; 3.8 Stochastic processes and systems; 3.9 Computational aspects; 3.A Elements of analysis; 3.B Elements of algebra; Exercises with solutions; Exercises; 4. Functions and continuous-time systems: 4.1 Introduction; 4.2 Functions; 4.3 Systems; 4.4 Fourier Transform; 4.5 Fourier series; 4.6 Stochastic processes and systems; Exercises with solutions; Exercises; 5. Sampling and interpolation: 5.1 Introduction; 5.2 Finite-dimensional vectors; 5.3 Sequences; 5.4 Functions; 5.5 Periodic functions; 5.6 Computational aspects; Exercises with solutions; Exercises; 6. Approximation and compression: 6.1 Introduction; 6.2 Approximation of functions on finite intervals by polynomials; 6.3 Approximation of functions by splines; 6.4 Approximation of functions and sequences by series truncation; 6.5 Compression; 6.6 Computational aspects; Exercises with solutions; Exercises; 7. Localization and uncertainty: 7.1 Introduction; 7.2 Localization for functions; 7.3 Localization for sequences; 7.4 Tiling the time–frequency plane; 7.5 Examples of local Fourier and wavelet bases; 7.6 Recap and a glimpse forward; Exercises with solutions; Exercises.

Read more less

Book SynopsisAn informative look at the theory, computer implementation, and application of the scaled boundary finite element method This reliable resource, complete with MATLAB, is an easy-to-understand introduction to the fundamental principles of the scaled boundary finite element method. It establishes the theory of the scaled boundary finite element method systematically as a general numerical procedure, providing the reader with a sound knowledge to expand the applications of this method to a broader scope. The book also presents the applications of the scaled boundary finite element to illustrate its salient features and potentials. The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation covers the static and dynamic stress analysis of solids in two and three dimensions. The relevant concepts, theory and modelling issues of the scaled boundary finite element method are discussed and the unique features of the method are highlightedTable of ContentsPreface xv Acknowledgements xix About the Companion Website xxi 1 Introduction 1 1.1 Numerical Modelling 1 1.2 Overview of the Scaled Boundary Finite Element Method 6 1.3 Features and Example Applications of the Scaled Boundary Finite Element Method 10 1.3.1 Linear Elastic Fracture Mechanics: Crack Terminating at Material Interface 11 1.3.2 Automatic Mesh Generation Based on Quadtree/Octree 13 1.3.3 Treatment of Non-matching Meshes 14 1.3.4 Crack Propagation 17 1.3.5 Adaptive Analysis 17 1.3.6 TransientWave Scattering in an Alluvial Basin 19 1.3.7 Automatic Image-based Analysis 19 1.3.7.1 Two-dimensional Elastoplastic Analysis of Cast Iron 20 1.3.7.2 Three-dimensional Concrete Specimen 22 1.3.8 Automatic Analysis of STL Models 24 1.4 Summary 26 Part I Basic Concepts and MATLAB Implementation of the Scaled Boundary Finite Element Method in Two Dimensions 27 2 Basic Formulations of the Scaled Boundary Finite Element Method 31 2.1 Introduction 31 2.2 Modelling of Geometry in Scaled Boundary Coordinates 31 2.2.1 S-domains: Scaling Requirement on Geometry, Scaling Centre and Scaling of Boundary 31 2.2.2 S-elements: Boundary Discretization of S-domains 37 2.2.3 Scaled Boundary Transformation 40 2.2.3.1 Scaled Boundary Coordinates 40 2.2.3.2 Coordinate Transformation of Partial Derivatives 42 2.2.3.3 Geometrical Properties in Scaled Boundary Coordinates 44 2.3 Governing Equations of Linear Elasticity in Scaled Boundary Coordinates 50 2.4 Semi-analytical Representation of Displacement and Strain Fields 51 2.5 Derivation of the Scaled Boundary Finite Element Equation by the Virtual Work Principle 53 2.5.1 Virtual Displacement and Strain Fields in Scaled Boundary Coordinates 54 2.5.2 Nodal Force Functions 54 2.5.3 The Scaled Boundary Finite Element Equation 55 2.6 Computer Program Platypus: Coefficient Matrices of an S-element 63 2.6.1 Element Coefficient Matrices of a 2-node Line Element 63 2.6.2 Assembly of Coefficient Matrices of an S-element 67 3 Solution of the Scaled Boundary Finite Element Equation by Eigenvalue Decomposition 73 3.1 Solution Procedure for the Scaled Boundary Finite Element Equations in Displacement 73 3.2 Pre-conditioning of Eigenvalue Problems 77 3.3 Computer Program Platypus: Solution of the Scaled Boundary Finite Element Equation of a Bounded S-element by the Eigenvalue Method 78 3.4 Assembly of S-elements and Solution of Global System of Equations 84 3.4.1 Assembly of S-elements 84 3.4.2 Surface Tractions 85 3.4.3 Enforcing Displacement Boundary Conditions 87 3.5 Computer Program Platypus: Assembly and Solution 87 3.5.1 Assembly of Global Stiffness Matrix 87 3.5.2 Assembly of Load Vector 95 3.5.3 Solution of Global System of Equations 96 3.5.4 Utility Functions 97 3.6 Examples of Static Analysis Using Platypus 102 3.7 Evaluation of Internal Displacements and Stresses of an S-element 111 3.7.1 Integration Constants and Internal Displacements 111 3.7.2 Strain/Stress Modes and Strain/Stress Fields 112 3.7.3 Shape Functions of Polygon Elements Modelled as S-elements 114 3.8 Computer Program Platypus: Internal Displacements and Strains 114 3.9 Body Loads 132 3.10 Dynamics and Vibration Analysis 135 3.10.1 Mass Matrix and Equation of Motion 135 3.10.2 Natural Frequencies and Mode Shapes 140 3.10.3 Response History Analysis Using the Newmark Method 143 4 Automatic Polygon Mesh Generation for Scaled Boundary Finite Element Analysis 149 4.1 Introduction 149 4.2 Basics of Geometrical Representation by Signed Distance Functions 150 4.3 Computer Program Platypus: Generation of Polygon S-elementMesh 154 4.3.1 Mesh Data Structure 157 4.3.2 Centroid of a Polygon 165 4.3.3 Converting a TriangularMesh to an S-elementMesh 166 4.3.4 Use of Polygon Meshes Generated by PolyMesher in a Scaled Boundary Finite Element Analysis 171 4.3.5 Dividing Edges of Polygons into Multiple Elements 172 4.4 Examples of Scaled Boundary Finite Element Analysis Using Platypus 175 4.4.1 A Deep Beam 178 4.4.1.1 Static Analysis 186 4.4.1.2 Modal Analysis 189 4.4.1.3 Response History Analysis 190 4.4.1.4 Pure Bending of a Beam: 2 Line Elements on an Edge of Polygons 190 4.4.2 A Circular Hole in an Infinite Plane Under Remote Uniaxial Tension 193 4.4.3 An L-shaped Panel 197 4.4.3.1 Static Analysis 203 4.4.3.2 Modal Analysis 204 4.4.3.3 Response History Analysis 207 5 Modelling Considerations in the Scaled Boundary Finite Element Analysis 209 5.1 Effect of Location of Scaling Centre on Accuracy 209 5.2 Mesh Transition 212 5.2.1 Local Mesh Refinement 212 5.2.2 Rapid Mesh Transition 214 5.2.3 Effect of Nonuniformity of Line Element Length on the Boundary of S-elements 216 5.3 Connecting Non-matching Meshes of Multiple Domains 218 5.3.1 Computer Program Platypus: Combining Two Non-matching Meshes 220 5.3.2 Computer Program Platypus: Modelling of a Problem by Multiple Domains with Non-matching Meshes 223 5.3.3 Examples 225 5.4 Modelling of Stress Singularities 234 Part II Theory and Applications of the Scaled Boundary Finite Element Method 237 6 Derivation of the Scaled Boundary Finite Element Equation in Three Dimensions 239 6.1 Introduction 239 6.2 Scaling of Boundary 239 6.3 Boundary Discretization of an S-domain 242 6.3.1 Isoparametric Quadrilateral Elements 243 6.3.1.1 Four-node Quadrilateral Element 243 6.3.1.2 Quadrilateral Element of Variable Number of Nodes 245 6.3.2 Isoparametric Triangular Elements 246 6.3.2.1 Three-node Triangular Elements 247 6.3.2.2 Six-node Triangular Elements 248 6.4 Scaled Boundary Transformation of Geometry 249 6.5 Geometrical Properties in Scaled Boundary Coordinates 253 6.6 Governing Equations of Elastodynamics with Geometry in Scaled Boundary Coordinates 257 6.7 Derivation of the Scaled Boundary Finite Element Equation by the Galerkin’s Weighted Residual Technique 259 6.7.1 Displacement, Strain Fields and Nodal Force Functions in Scaled Boundary Coordinates 259 6.7.2 The Scaled Boundary Finite Element Equation 262 6.8 Unified Formulations in Two andThree Dimensions 267 6.9 Formulation of the Scaled Boundary Finite Element Equation as a System of First-order Differential Equations 268 6.10 Properties of Coefficient Matrices 269 6.10.1 Coefficient Matrices [E0] and [M0] 270 6.10.2 Coefficient Matrix [E2] 270 6.10.3 Matrix [Zp] 271 6.11 Linear Completeness of the Scaled Boundary Finite Element Solution 272 6.11.1 Constant Displacement Field 272 6.11.2 Linear Displacement Field 273 6.12 Scaled Boundary Finite Element Equation in Stiffness 278 7 Solution of the Scaled Boundary Finite Element Equation in Statics by Schur Decomposition 281 7.1 Introduction 281 7.2 Basics of Matrix Exponential Function 283 7.3 Schur Decomposition 287 7.3.1 Introduction 287 7.3.2 Treatment of the Diagonal Block of Eigenvalues of 0 288 7.4 Solution Procedure for a Bounded S-element by Schur Decomposition 291 7.4.1 Transformation of the Scaled Boundary Finite Element Equation 291 7.4.2 Enforcing the Boundary Condition at the Scaling Centre 292 7.4.3 Determining the Solution for Displacement and Nodal Force Functions 294 7.4.4 Determining the Static Stiffness Matrix 295 7.5 Solution of Displacement and Stress Fields of an S-element 295 7.5.1 Integration Constants 295 7.5.2 Stress Modes and Stresses on the Boundary 296 7.6 Block-diagonal Schur Decomposition 297 7.7 Solution Procedure by Block-diagonal Schur Decomposition 303 7.7.1 General Solution of the Scaled Boundary Finite Element Equation 303 7.7.1.1 [Zp] Having No Eigenvalues of Zero 304 7.7.1.2 [Zp] Having Eigenvalues of Zero 304 7.7.2 Solution for Bounded S-elements 305 7.7.3 Solution for Unbounded S-elements 307 7.7.3.1 [Zp] Having No Eigenvalues of Zero 307 7.7.3.2 [Zp] Having Eigenvalues of Zero 308 7.8 Displacements and Stresses of an S-element by Block-diagonal Schur Decomposition 310 7.8.1 Integration Constants and Displacement Fields 310 7.8.2 Stress Modes and Stress Fields 311 7.8.3 Shape Functions of Polytope Elements 312 7.9 Body Loads 313 7.10 Mass Matrix 315 7.11 Remarks 317 7.12 Examples 319 7.12.1 Circular Cavity in Full-plane 319 7.12.2 Bi-materialWedge 322 7.12.3 Interface Crack in Anisotropic Bi-material Full-plane 325 7.13 Summary 327 8 High-order Elements 329 8.1 Lagrange Interpolation 330 8.2 One-dimensional Spectral Elements 333 8.2.1 Shape Functions 334 8.2.2 Numerical Integration of Element Coefficient Matrices 337 8.2.2.1 Gauss-Legendre Quadrature 337 8.2.2.2 Gauss-Lobatto-Legendre Quadrature 338 8.3 Two-dimensional Quadrilateral Spectral Elements 341 8.3.1 Shape Functions 341 8.3.2 Integration of Element Coefficient Matrices by Gauss-Lobatto-Legendre Quadrature 342 8.4 Examples 344 8.4.1 A Cantilever Beam Subject to End Loading 345 8.4.2 A Circular Hole in an Infinite Plate 347 8.4.3 An L-shaped Panel 349 8.4.4 A 3D Cantilever Beam Subject to End-shear Loading 351 8.4.5 A Pressurized Hollow Sphere 352 9 Quadtree/Octree Algorithm of Mesh Generation for Scaled Boundary Finite Element Analysis 355 9.1 Introduction 355 9.1.1 Mesh Generation 355 9.1.2 The Quadtree/Octree Algorithm 357 9.2 Data Structure of S-element Meshes 360 9.3 Quadtree/Octree Mesh Generation of Digital Images 361 9.3.1 Illustration of Quadtree Decomposition of Two-dimensional Images by an Example 361 9.3.2 Octree Decomposition 366 9.4 Solutions of S-elements with the Same Pattern of Node Configuration 370 9.4.1 Two-dimensional S-elements 370 9.4.2 Three-dimensional S-elements 372 9.5 Examples of Image-based Analysis 374 9.5.1 A 2D Concrete Specimen 374 9.5.2 A 3D Concrete Specimen 376 9.6 Quadtree/Octree Mesh Generation for CAD Models 378 9.6.1 Quadtree/Octree Grid 380 9.6.2 Trimming of Boundary Cells 381 9.7 Examples Using Quadtree/Octree Meshes of CAD Models 383 9.7.1 Square Body with Multiple Holes 384 9.7.2 An Evolving Void in a Square Body 385 9.7.3 Adaptive Analysis of an L-shaped Panel 386 9.7.4 A Mechanical Part 387 9.7.5 STL Models 389 9.8 Remarks 394 10 Linear Elastic Fracture Mechanics 395 10.1 Introduction 395 10.2 Basics of Fracture Analysis: Asymptotic Solutions, Stress Intensity Factors, and the T-stress 397 10.2.1 Crack in Homogeneous Isotropic Material 397 10.2.2 Interfacial Cracks between Two Isotropic Materials 401 10.2.3 Interfacial Cracks between Two AnisotropicMaterials 402 10.2.4 Multi-materialWedges 405 10.3 Modelling of Singular Stress Fields by the Scaled Boundary Finite Element Method 406 10.4 Stress Intensity Factors and the T-stress of a Cracked Homogeneous Body 407 10.5 Definition and Evaluation of Generalized Stress Intensity Factors 416 10.6 Examples of Highly Accurate Stress Intensity Factors and T-stress 432 10.6.1 A Single Edge-cracked Rectangular Body Under Tension 433 10.6.2 A Single Edge-cracked Rectangular Body Under Bending 435 10.6.3 A Centre-cracked Rectangular Body Under Tension 437 10.6.4 A Double Edge-cracked Rectangular Body Under Tension 438 10.6.5 A Single Edge-cracked Rectangular Body Under End Shearing 439 10.7 Modelling of Crack Propagation 440 10.7.1 Modelling of Crack Paths by Polygon Meshes 442 10.7.2 Modelling of Crack Paths by Quadtree Meshes 443 10.7.3 Examples of Crack PropagationModelling 444 10.7.3.1 Fatigue Crack Propagation Using Polygon Mesh 444 10.7.3.2 Crack Propagation in a Beam with Three Holes 447 Appendix A Governing Equations of Linear Elasticity 449 A.1 Three-dimensional Problems 449 A.1.1 Strain 449 A.1.2 Stress and Equilibrium Equation 450 A.1.3 Stress-strain Relationship and Material Elasticity Matrix 451 A.1.4 Boundary Conditions 453 A.2 Two-dimensional Problems 454 A.2.1 Elasticity Matrix in Plane Stress 455 A.2.2 Elasticity Matrix in Plane Strain 456 A.3 Unified Expressions of Governing Equations 457 Appendix B Matrix Power Function 459 B.1 Definition of Matrix Power Function 459 B.2 Application to Solution of System of Ordinary Differential Equations 460 B.3 Computation of Matrix Power Function by Eigenvalue Method 461 Bibliography 463 Index 475

Read more less

Book SynopsisApproaches computational engineering sciences from the perspective of engineering applications Uniting theory with hands-on computer practice, this book gives readers a firm appreciation of the error mechanisms and control that underlie discrete approximation implementations in the engineering sciences. Key features: Illustrative examples include heat conduction, structural mechanics, mechanical vibrations, heat transfer with convection and radiation, fluid mechanics and heat and mass transport Takes a cross-discipline continuum mechanics viewpoint Includes Matlab toolbox and .m data files on a companion website, immediately enabling hands-on computing in all covered disciplines Website also features eight topical lectures from the author's own academic courses It provides a holistic view of the topic from covering the different engineering problems that can be solved using finite element to how each pTable of ContentsPreface viii Notation xi 1 COMPUTATIONAL ENGINEERING SCIENCE 1 1.1 Engineering simulation 1 1.2 A problem solving environment 2 1.3 Problem statements in engineering 4 1.4 Decisions on forming WSN 6 1.5 Discrete approximate WSh implementation 8 1.6 Chapter summary 9 1.7 Chapter references 10 2 PROBLEM STATEMENTS 11 2.1 Engineering simulation 11 2.2 Continuum mechanics viewpoint 12 2.3 Continuum conservation law forms 12 2.4 Constitutive closure for conservation law PDEs 14 2.5 Engineering science continuum mechanics 18 2.6 Chapter references 20 3 SOME INTRODUCTORY MATERIAL 21 3.1 Introduction 21 3.2 Multi-dimensional PDEs, separation of variables 22 3.3 Theoretical foundations, GWSh 27 3.4 A legacy FD construction 28 3.5 An FD approximate solution 30 3.6 Lagrange interpolation polynomials 31 3.7 Chapter summary 32 3.8 Exercises 34 3.9 Chapter references 34 4 HEAT CONDUCTION35 4.1 A steady heat conduction example 35 4.2 Weak form approximation, error minimization 37 4.3 GWSN discrete implementation, FE basis38 4.4 Finite element matrix statement 41 4.5 Assembly of {WS}e to form algebraic GWSh 43 4.6 Solution accuracy, error distribution 45 4.7 Convergence, boundary heat flux 47 4.8 Chapter summary 47 4.9 Exercises 48 4.10 Chapter reference 48 5 STEADY HEAT TRANSFER, n =149 5.1 Introduction 49 5.2 Steady heat transfer, n = 1 50 5.3 FE k = 1 trial space basis matrix library 52 5.4 Object-oriented GWSh programming 55 5.5 Higher completeness degree trial space bases58 5.6 Global theory, asymptotic error estimate 62 5.7 Non-smooth data, theory generalization 66 5.8 Temperature dependent conductivity, non-linearity 69 5.9 Static condensation, p-elements 72 5.10 Chapter summary 75 5.11 Exercises 76 5.12 Computer labs 77 5.13 Chapter references 78 6 ENGINEERING SCIENCES, n =1 79 6.1 Introduction 79 6.2 The Euler-Bernoulli beam equation 80 6.3 Euler-Bernoulli beam theory GWSh reformulation 85 6.4 The Timoshenko beam theory 92 6.5 Mechanical vibrations of a beam 99 6.6 Fluid mechanics, potential flow 106 6.7 Electromagnetic plane wave propagation110 6.8 Convective-radiative finned cylinder heat transfer 112 6.9 Chapter summary 120 6.10 Exercises122 6.10 Computer labs 123 6.11 Chapter references 124 7 STEADY HEAT TRANSFER, n > 1 125 7.1 Introduction 125 7.2 Multi-dimensional FE bases and DOF 126 7.3 Multi-dimensional FE operations 129 7.4 The NC k = 1,2 basis FE matrix library 132 7.5 NC basis {WS}e template, accuracy, convergence 136 7.6 The tensor product basis element family 139 7.7 Gauss numerical quadrature, k = 1 TP basis library 141 7.8 Convection-radiation BC GWSh implementation 146 7.9 Linear basis GWSh template unification 150 7.10 Accuracy, convergence revisited 152 7.11 Chapter summary 153 7.12 Exercises155 7.13 Computer labs 155 7.14 Chapter references 156 8 FINITE DIFFERENCES OF OPINION 159 8.1 The FD-FE correlation159 8.2 The FV-FE correlation162 8.3 Chapter summary 167 8.4 Exercises168 9 CONVECTION-DIFFUSION, n = 1 169 9.1 Introduction169 9.2 The Galerkin weak statement 170 9.3 GWSh completion for time dependence172 9.4 GWSh + qTS algorithm templates 173 9.5 GWSh + qTS algorithm asymptotic error estimates 175 9.6 Performance verification test cases 177 9.7 Dispersive error characterization 180 9.8 A modified Galerkin weak statement 184 9.9 Verification problem statements revisited 187 9.10 Unsteady heat conduction 190 9.11 Chapter summary 193 9.12 Exercises 193 9.13 Computer labs 194 9.14 Chapter references 195 10 CONVECTION-DIFFUSION, n > 1 197 10.1 The problem statement 197 10.2 GWSh + qTS formulation reprise 198 10.3 Matrix library additions, templates 200 10.4 mPDE Galerkin weak forms, theoretical analyses 202 10.5 Verification, benchmarking and validation 207 10.6 Mass transport, the rotating cone verification 208 10.7 The gaussian plume benchmark 211 10.8 The steady n-D Peclet problem verification 213 10.9 Mass transport, a validated n = 3 experiment 215 10.10 Numerical linear algebra, matrix iteration 222 10.11 Newton and AF TP jacobian templates 227 10.12 Chapter summary 229 10.13 Exercises231 10.14 Computer labs 231 10.15 Chapter references232 11 ENGINEERING SCIENCES, n > 1 235 11.1 Introduction 235 11.2 Structural mechanics236 11.3 Structural mechanics, virtual work FE form 240 11.4 Plane stress/strain, GWSh implementation 242 11.5 Elasticity computer lab 246 11.6 Fluid mechanics, incompressible-thermal flow 251 11.7 Vorticity-streamfunction GWSh + qTS algorithm 254 11.8 An isothermal INS validation experiment 258 11.9 Multi-mode convection heat transfer262 11.10 Mechanical vibrations, normal mode GWSh 267 11.11 Normal modes of a vibrating membrane270 11.12 Multi-physics solid-fluid mass transport 276 11.13 Chapter summary 280 11.14 Exercises 282 11.15 Computer labs283 11.14 Chapter references 284 12 CONCLUSION 287 Index 289

Read more less

Book SynopsisBasic R Programming.- Random Variable Generation.- Monte Carlo Integration.- Controlling and Accelerating Convergence.- Monte Carlo Optimization.- Metropolis#x2013;Hastings Algorithms.- Gibbs Samplers.- Convergence Monitoring and Adaptation for MCMC Algorithms.Trade ReviewFrom the reviews:“Robert and Casella’s new book uses the programming language R, a favorite amongst (Bayesian) statisticians to introduce in eight chapters both basic and advanced Monte Carlo techniques … . The book could be used as the basic textbook for a semester long course on computational statistics with emphasis on Monte Carlo tools … . useful for (and should be next to the computer of) a large body of hands on graduate students, researchers, instructors and practitioners … .” (Hedibert Freitas Lopes, Journal of the American Statistical Association, Vol. 106 (493), March, 2011)“Chapters focuses on MCMC methods the Metropolis–Hastings algorithm, Gibbs sampling, and monitoring and adaptation for MCMC algorithms. … There are exercises within and at the end of all chapters … . Overall, the level of the book makes it suitable for graduate students and researchers. Others who wish to implement Monte Carlo methods, particularly MCMC methods for Bayesian analysis will also find it useful.” (David Scott, International Statistical Review, Vol. 78 (3), 2010)“The primary audience is graduate students in statistics, biostatistics, engineering, etc. who need to know how to utilize Monte Carlo simulation methods to analyze their experiments and/or datasets. … this text does an effective job of including a selection of Monte Carlo methods and their application to a broad array of simulation problems. … Anyone who is an avid R user and has need to integrate and/or optimize complex functions will find this text to be a necessary addition to his or her personal library.” (Dean V. Neubauer, Technometrics, Vol. 53 (2), May, 2011)Table of ContentsBasic R Programming.- Random Variable Generation.- Monte Carlo Integration.- Controlling and Accelerating Convergence.- Monte Carlo Optimization.- Metropolis#x2013;Hastings Algorithms.- Gibbs Samplers.- Convergence Monitoring and Adaptation for MCMC Algorithms.

Read more less

Book SynopsisAs well as providing an overview of the current state of science in the analysis and synthesis of non-rigid shapes, the authors include everyday examples to explain concepts. Practice problems follow at the end of each chapter, along with detailed solutions.Trade ReviewFrom the reviews: "This book provides an introduction to this geometry. … Overall, the book … does explain relevant mathematical notions, such as Gromov’s metric geometry ideas, in a very understandable and entertaining way, with numerous images and exercises. … I highly recommend it to both computer scientists interested in learning more about the latest advances in computational geometry and to geometers looking for applications. This unique book can serve as an excellent textbook for many related courses, for self-study, or as a reference." (V. Kreinovich, ACM Computing Reviews, May, 2009) “Numerical geometry of non-rigid shapes by A. Bronstein, M. Bronstein, and R. Kimmel combines the beauty of modern mathematics … with the interesting field of computer vision and pattern recognition. … The book is developed at an intermediate-advanced level. Students will find the material clear and easy to understand, and will benefit from its good presentation.” (Stefan Henn, Mathematical Reviews, Issue 2010 b)Table of ContentsA Taste of Geometry.- Discrete Geometry.- Shortest Paths and Fast Marching Methods.- Numerical Optimization.- In the Rigid Kingdom.- Multidimensional Scaling.- Spectral Embedding.- Non Euclidean Embedding.- Isometry Invariant Similarity.- Partial Similarity.- Non rigid Correspondence and Calculus of Shapes.- Three dimensional Face Recognition.- Epilogue.

Read more less