Maths for engineers Books

Book SynopsisAll students taking laboratory courses within the physical sciences and engineering will benefit from this book, whilst researchers will find it an invaluable reference. This concise, practical guide brings the reader up-to-speed on the proper handling and presentation of scientific data and its inaccuracies. It covers all the vital topics with practical guidelines, computer programs (in Python), and recipes for handling experimental errors and reporting experimental data. In addition to the essentials, it also provides further background material for advanced readers who want to understand how the methods work. Plenty of examples, exercises and solutions are provided to aid and test understanding, whilst useful data, tables and formulas are compiled in a handy section for easy reference.Trade Review"Overall, this would be a nice text or reference to accompany a short course in statistics for undergraduate science or engineering..also useful for researchers desiring a primer or review...Recommended." - CHOICETable of ContentsPart I. Data and Error Analysis: 1. Introduction; 2. The presentation of physical quantities with their inaccuracies; 3. Errors: classification and propagation; 4. Probability distributions; 5. Processing of experimental data; 6. Graphical handling of data with errors; 7. Fitting functions to data; 8. Back to Bayes: knowledge as a probability distribution; Answers to exercises; Part II. Appendices: A1. Combining uncertainties; A2. Systematic deviations due to random errors; A3. Characteristic function; A4. From binomial to normal distributions; A5. Central limit theorem; A6. Estimation of the varience; A7. Standard deviation of the mean; A8. Weight factors when variances are not equal; A9. Least squares fitting; Part III. Python Codes; Part IV. Scientific Data: Chi-squared distribution; F-distribution; Normal distribution; Physical constants; Probability distributions; Student's t-distribution; Units.

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Book SynopsisFourier transform theory is of central importance in a vast range of applications in physical science, engineering and applied mathematics. Providing a concise introduction to the theory and practice of Fourier transforms, this book is invaluable to students of physics, electrical and electronic engineering, and computer science. After a brief description of the basic ideas and theorems, the power of the technique is illustrated through applications in optics, spectroscopy, electronics and telecommunications. The rarely discussed but important field of multi-dimensional Fourier theory is covered, including a description of Computer Axial Tomography (CAT scanning). The book concludes by discussing digital methods, with particular attention to the Fast Fourier Transform and its implementation. This new edition has been revised to include new and interesting material, such as convolution with a sinusoid, coherence, the Michelson stellar interferometer and the van CittertâZernike theorem, Trade ReviewFrom previous editions: 'It is the wide range of topics that makes this book so appealing … I highly recommend this book for the advanced student … Even the expert who wants a deeper appreciation of the Fourier transform will find the book useful.' Computers in Physics'… this is an excellent book to initiate students who possess a reasonable mathematical background to the use of Fourier transforms …' Microscopy and AnalysisTable of Contents1. Physics and Fourier transforms; 2. Useful properties and theorems; 3. Applications 1: Fraunhofer diffraction; 4. Applications 2: signal analysis and communication theory; 5. Applications 3: spectroscopy and spectral line shapes; 6. Two-dimensional Fourier transforms; 7. Multi-dimensional Fourier transforms; 8. The formal complex Fourier transform; 9. Discrete and digital Fourier transforms; 10. Appendix; 11. Bibliography; 12. Index.

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Book SynopsisOriginally published in 1954, the purpose of this book was to provide a set of practical exercises for young engineers wishing to apply mathematical principles to problems confronting them in the workshop. The text was designed primarily for use in the Technical Secondary School, the County College, and the Works Training School. It will be of value to anyone with an interest in the development of engineering and educational practice.Table of ContentsPreface; Foreword; 1. Locomotives: lengths and weights; 2. Fractions: addition and subtraction; 3. Fractions: multiplication and division; 4. Fractions: miscellaneous problems; 5. Decimals: addition and subtraction; 6. Decimals: multiplication and division; 7. Decimals: conversion to engineering fraction; 8. Spacing of rivets and holes; 9. Algebra: generalized addition; 10. Algebra: generalized subtraction; 11. Algebra: generalized multiplication; 12. Algebra: generalized division; 13. Algebra: symbols, addition; 14. Algebra: symbols, subtraction; 15. Algebra: symbols, multiplication; 16. Algebra: polynomials and problems; 17. Algebra: symbols, division; 18. Algebra: simple equations; 19. Algebra: equations, fraction; 20. Algebra: equations, fractions brackets; 21. Algebra: harder equations; 22. Algebra: simple literal equations; 23. Algebra: problems on equations; 24. Algebra: substitution (positive numbers); 25. Algebra: substitution (negative numbers); 26. Formula manipulation and substitution; 27. Harder formula manipulation; 28. Square root: arithmetical; 29. Square root and squares from tables; 30. The theorem of Pythagoras: simple problems; 31. The theorem of Pythagoras: practical applications; 32. Logarithms: characteristics, use of tables; 33. Logarithms: multiplication and division (numbers greater than 1); 34. Logarithms: manipulation of negative characteristics; 35. Logarithms: multiplication and division (numbers less than 1); 36. Logarithms: powers and roots; 37. Logarithms: miscellaneous problems; 38. Proportion: speed, time and revolutions; 39. Mensuration: rectangle and square; 40. Mensuration: area of sections, weights per foot run; 41. Mensuration: rectangular solid; 42. Mensuration: triangle and triangular prism; 43. Mensuration: trapezium; 44. Mensuration: circle and annulus; 45. Mensuration: heating surface of tubes, volume and weight of bars and tubes; 46. Mensuration: capacity and rate of flow of water in pipes; 47. Mensuration: miscellaneous problems; 48. Mensuration: cone, frustum, sphere; 49. Mensuration: arcs, sectors, segments; 50. Trigonometry: sine, cosine and tangent; 51. Trigonometry: use of tables; 52. Trigonometry: simple problems; 53. Trigonometry: workshop problems; 54. Trigonometry: miscellaneous problems; Answers; Index.

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Book SynopsisThis 1994 book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. There are nearly 200 exercises, making the book ideal for both classroom use and self-study.Trade Review"...Darling's exegesis is clear and easy to understand, and his frequent use of examples is beneficial to the reader. There are many exercises that serve to reinforce the concepts." D.P. Turner, Choice"...easy on the eyes; some nice exercises..." American Mathematical Monthly"The exposition is clear and, in the American textbook style, has many exercises, both theoretical and computational. In summary, this text provides a worthwhile elementary introduction to anyone who wants to understand the basic mathematical ingredients of Differential Geometry and its interactions with Physics." F.E. Burstall, Contemporary Physics"...a good introduction to differential geometry and its applications to physics by using the calculus of differential forms...Nearly 200 exercises and many examples will help the reader's understanding...this book can be recommended as a good textbook for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering." Akira Asada, Mathematical ReviewsTable of ContentsPreface; 1. Exterior algebra; 2. Exterior calculus on Euclidean space; 3. Submanifolds of Euclidean spaces; 4. Surface theory using moving frames; 5. Differential manifolds; 6. Vector bundles; 7. Frame fields, forms and metrics; 8. Integration on oriented manifolds; 9. Connections on vector bundles; 10. Applications to gauge field theory; Bibliography; Index.

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Book SynopsisThis CD-ROM contains complete source code in C++ for the brand-new Numerical Recipes, Third Edition, plus source code from all earlier editions.Trade Review' … an essential component of any serious scientific or engineering library.' Computing Reviews' … an instant 'classic,' a book that should be purchased and read by anyone who uses numerical methods …' American Journal of Physics' … replete with the standard spectrum of mathematically pretreated and coded/numerical routines for linear equations, matrices and arrays, curves, splines, polynomials, functions, roots, series, integrals, eigenvectors, FFT and other transforms, distributions, statistics, and on to ODE's and PDE's … delightful.' Physics in Canada' … a must for anyone doing scientific computing.' Journal of the American Chemical Society'The authors are to be congratulated for providing the scientific community with a valuable resource.' The Scientist'If you were to have only a single book on numerical methods, this is the one I would recommend.' IEEE Computational Science & EngineeringTable of Contents1. Preliminaries; 2. Solution of linear algebraic equations; 3. Interpolation and extrapolation; 4. Integration of functions; 5. Evaluation of functions; 6. Special functions; 7. Random numbers; 8. Sorting and selection; 9. Root finding and nonlinear sets of equations; 10. Minimization or maximization of functions; 11. Eigensystems; 12. Fast Fourier transform; 13. Fourier and spectral applications; 14. Statistical description of data; 15. Modeling of data; 16. Classification and inference; 17. Integration of ordinary differential equations; 18. Two point boundary value problems; 19. Integral equations and inverse theory; 20. Partial differential equations; 21. Computational geometry; 22. Less-numerical algorithms; References.

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

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Book SynopsisThis first volume in a series provides a graduate-level introduction to the many facets of aperiodic order. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. Numerous illustrations and examples are included.Trade Review'Mathematicians add hypotheses to theorems either to bar known monsters or provisionally to enable proof, pending better ideas that lead to more general results … Monsters no more, aperiodic filings have joined mainstream mathematics, and undergraduates drawn here by beautiful graphics will find themselves initiated into algebraic number theory, Lie theory, ergodic theory, dynamical systems, finite-state automata, Fourier analysis, and more.' D. V. Feldman, University of New Hampshire'Aperiodic Order is a comprehensive introduction to this relatively new and multidisciplinary field. Sparked by Dan Shechtman's discovery of quasicrystals in 1982, which earned him the 2011 Nobel Prize in Chemistry, the field incorporates crystallography, discrete geometry, dynamical systems, harmonic analysis, mathematical diffraction theory, and more. Because the field spans such disparate fields, advances by one group often go unnoticed by the other. An important goal of this book is to remedy this by unifying and contextualizing results and providing a common language for researchers. … Readers who want to follow up on any details can certainly find a reference in the nearly 30 pages of bibliographic entries. Full of examples, construction techniques, and an array of analytic tools, this book is an outstanding resource for those hoping to enter the field, yet also contains plenty of useful information for seasoned experts.' Natalie Priebe Frank, Mathematical Association of AmericaTable of ContentsForeword Roger Penrose; Preface; 1. Introduction; 2. Preliminaries; 3. Lattices and crystals; 4. Symbolic substitutions and inflations; 5. Patterns and tilings; 6. Inflation tilings; 7. Projection method and model sets; 8. Fourier analysis and measures; 9. Diffraction; 10. Beyond model sets; 11. Random structures; A. The icosahedral group; Appendix B. The dynamical spectrum; References; Index.

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Book SynopsisThe first MATLAB-based numerical methods textbook specifically for bioengineers, including topics on hypothesis testing, plus numerous examples drawn exclusively from biomedical engineering applications. This is an ideal core text for one-semester undergraduate courses, and is also a valuable reference for anyone interested in the quantitative aspects of biology research.Trade Review'I think this book is a winner … [it] is really easy to read and places frameworks for numerical analysis into realistic bioengineering concepts that students will find familiar and relevant. This is most evident in the excellent boxed examples, but also in many of the homework problems. I also really liked the 'key points to consider' at the end of the chapters - these are useful reminders for the students. Finally, the book presents bioinformatics in a manageable fashion that should help demystify this subject for interested students.' K. Jane Grande-Allen, Rice UniversityTable of Contents1. Types and sources of numerical error; 2. Systems of linear equations; 3. Statistics and probability; 4. Hypothesis testing; 5. Root finding techniques for nonlinear equations; 6. Numerical quadrature; 7. Numerical integration of ordinary differential equations; 8. Nonlinear data regression and optimization; 9. Basic algorithms of bioinformatics; Appendix A. Introduction to MATLAB; Appendix B. Location of nodes for Gauss-Legendre quadrature.

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Book SynopsisAt present, most books on ecological modelling rely on very complex mathematics, resulting in students and researchers shying away from investigating the potential uses of ecological models and their methods of construction. This new book aims to open up this exciting area to a much wider audience.Trade Review"Teachers of courses on ecological modelling will find [this book] a useful source-book at a competitive price."Table of ContentsIntroduction: Themes Of Ecological Modelling. Probability Of Population Extinction. Looking For Cycles: The Dynamics Of Predators And Their Prey. Population Dynamics Of Species With Complex Life-Histories. Dynamics Of Ecological Communities. Spatial Models And Thresholds. Disease And Biological Control. Answers To Questions. Glossary Of Symbols And Terms. References. Index

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Book SynopsisIntroduces the distributed control of robotic networks. This book presents a set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity. It analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation.Trade Review"This book covers its subject very thoroughly. The framework the authors have established is very elegant and, if it catches on, this book could be the primary reference for this approach. I don't know of any other book that covers this set of topics."—Richard M. Murray, California Institute of Technology"The authors do an excellent job of clearly describing the problems and presenting rigorous, provably correct algorithms with complexity bounds for each problem. The authors also do a fantastic job of providing the mathematical insight necessary for such complex problems."—Ali Jadbabaie, University of Pennsylvania"The order of presentation makes much sense, and the book thoroughly covers what it sets out to cover. The algorithms and results are presented using a clear mathematical and computer science formalism, which allows a uniform presentation. The formalism used and the way of presenting the algorithms may be helpful for structuring the presentation of new algorithms in the future."—Vincent Blondel, Université catholique de LouvainTable of ContentsPreface ix Chapter 1. An introduction to distributed algorithms 1 1.1 Elementary concepts and notation 1 1.2 Matrix theory 6 1.3 Dynamical systems and stability theory 12 1.4 Graph theory 20 1.5 Distributed algorithms on synchronous networks 37 1.6 Linear distributed algorithms 52 1.7 Notes 66 1.8 Proofs 69 1.9 Exercises 85 Chapter 2. Geometric models and optimization 95 2.1 Basic geometric notions 95 2.2 Proximity graphs 104 2.3 Geometric optimization problems and multicenter functions 111 2.4 Notes 124 2.5 Proofs 125 2.6 Exercises 133 Chapter 3. Robotic network models and complexity notions 139 3.1 A model for synchronous robotic networks 139 3.2 Robotic networks with relative sensing 151 3.3 Coordination tasks and complexity notions 158 3.4 Complexity of direction agreement and equidistance 165 3.5 Notes 166 3.6 Proofs 169 3.7 Exercises 176 Chapter 4. Connectivity maintenance and rendezvous 179 4.1 Problem statement 180 4.2 Connectivity maintenance algorithms 182 4.3 Rendezvous algorithms 191 4.4 Simulation results 200 4.5 Notes 201 4.6 Proofs 204 4.7 Exercises 215 Chapter 5. Deployment 219 5.1 Problem statement 220 5.2 Deployment algorithms 222 5.3 Simulation results 233 5.4 Notes 237 5.5 Proofs 239 5.6 Exercises 245 Chapter 6. Boundary estimation and tracking 247 6.1 Event-driven asynchronous robotic networks 248 6.2 Problem statement 252 6.3 Estimate update and cyclic balancing law 256 6.4 Simulation results 266 6.5 Notes 268 6.6 Proofs 270 6.7 Exercises 275 Bibliography 279 Algorithm Index 305 Subject Index 307 Symbol Index 313

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Book SynopsisCo-published with Oxford University Press. This highly technical and thought-provoking book stresses the development of mathematical foundations for the application of the electromagnetic model to problems of research and technology.Table of ContentsPreface. Linear Analysis. The Green's Function Method. The Spectral Representation Method. Electromagnetic Sources. Electromagnetic Boundary Value Problems. Index.

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Book SynopsisSPATIAL ERROR ANALYSIS is an all-in-one sourcebook on error measurements in one-, two-, and three-dimensional spaces. This book features exhaustive, systematic coverage of error measurement relationships, techniques, and solutions used to solve general, correlated cases. It is packed with 62 figures and 24 tables. MATLAB-based M-files* for practical applications created especially for this volume are available on the Web at ftp://ftp.mathworks.com/pub/books/hsu. Solutions to two- and three-dimensional problems are presented without relying on equal standard deviations from each channel. They also make no assumption that the random variables of interest are independent or uncorrelated. * MATLAB (developed by MathWorks, Inc.) must be purchased separately. Sponsored by: IEEE Aerospace and Electronic Systems Society.Table of ContentsPreface. List of Figures. List of Tables. Introduction. Prameter Estimation from Samples. One-Dimensional Error Analysis. Two-Dimensional Error Analysis. Three-Dimensional Error Analysis. Maximum Likelihood Estimation of Radial Error PDF. Position Location Problems. Risk Analysis. Appendix A: Probability Density Functions. Appendix B: Method of Confidence Intervals. Appendix C: Function of N Random Variables. Appendix D: GPS Dilution of Precisions. Appendix E: Listing of Author-Generated M-files. Bibliography. Index. About the Author.

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Book SynopsisThis book is intended for mechanicians, engineering mathematicians, and, generally for theoretically inclined mechanical engineers. I did not think that the surface of the problem had even been scratched, so I joined de Pater, who had by then become Professor in the Engineering Mechanics Lab.Trade Review` This book clearly reflects such twofold remarkable expertise of the author and is recommended to all research workers in the title area. 'Table of Contents1 The Rolling Contact Problem.- 2 Review.- 3 The Simplified Theory of Contact.- 4 Variational and Numerical Theory of Contact.- 5 Results.- 6 Conclusion.- Appendix A The basic equations of the linear theory of elasticity.- Appendix B Some notions of mathematical programming.- Appendix C Numerical calculation of the elastic field in a half-space.- Appendix D Three-dimensional viscoelastic bodies in steady state frictional rolling contact with generalisation to contact perturbations.- Appendix E Tables.

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Book SynopsisThis book is an adventure into the computer analysis of three dimensional composite structures using the finite element method (FEM). Once the basic philosophy of the method is understood, the reader may expand its application and modify the computer programs to suit particular needs.Trade Review`The book is highly recommended as a reference text for advanced undergraduate students, as a graduate course on the FE analysis of composites, and as a reference work for both researchers in laboratories and practising engineers in industry.' Zentralblatt MATH, 906 Table of ContentsPreface. 1. Some Results from Continuum Mechanics. 2. A Brief History of FEM. 3. Natural Modes for Finite Elements. 4. Composites. 5. Composite Beam Element. 6. Composite Plate and Shell Element. 7. Computational Statistics. 8. Nonlinear Analysis of Anisotropic Shells. 9. Programming Aspects. Appendices: A. Geometry of the Bema Element in Space. B. Contents of the Floppy Disk. Bibliography. Index.

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Book SynopsisThis textbook – a result of the author’s many years of research and teaching – brings together diverse concepts of the versatile tool of multibody dynamics, combining the efforts of many researchers in the field of mechanics. Trade Review"This textbook offers a comprehensive exposition of contemporary multibody dynamics elaborated by an expert in and an experienced teacher of the field. There are five features that distinguish this publication: comprehensivity, orientation towards computer implementation, utilization of matrix algebra, teaching-by-examples methodology, and embedding in the author's own research accomplishments.... [T]his is an excellent textbook for undergraduate, graduate, and doctoral students of mechanical engineering, industrial physics and robotics; it may also be recommended as a reference text for researchers in these areas." —Mathematical Reviews "This is a textbook intended for advanced undergraduate and beginning graduate students studying dynamics, physics, robotics, control, and biomechanics.... The book is well-written with good organization. It has good quality graphics. It clearly represents a considerable effort by the author to present a state-of-the-art exposition. The book should be of interest and use to practitioners and researches as well as students. It brings together methodologies from various fields into a single volume." —Zentralblatt MATH "Timely and forward looking, Fundamentals of Multibody Dynamics, has immense potential for use as a textbook with a strong computer orientation towards the subject for junior/senior undergraduates and first-year graduate engineering students taking a course in dynamics, physics, control, robotics, or biomechanics. The work may also be used as a self-study resource or research reference for practitioners in the above-mentioned fields; they will find the book a refreshing break from the usual textbook presentation...with the type of printing, layout and illustrations by way of figures, tables and mathematical equations that contribute to a helpful understanding of the material presented and instant location of what is required by the reader. The book is studded with useful and relevant references. Thus, we are stronly led to recommend this impressive volume to students and practicing engineers who are looking for a book that fills the gap between dynamics and engineering applications. We have nothing but admiration for what Farid Amirouche has done." —Current Engineering PracticeTable of ContentsPreface Particle Dynamics: The Principle fo Newton's Second Law Rigid-Body Kinematics Kinematics for General Multibody Systems Modeling of Forces in Multibody Systems Equations of Motion of Multibody Systems Hamilton–Lagrange and Gibbs–Appel Equations Handling of Constraints in Multibody Systems Dynamics Numerical Stability of Constrained Multibody Systems Linearization and Vibration Analysis of Multibody Systems Dynamics of Multibody Systems with Terminal Flexible Links Dynamic Analysis of Multiple Flexible-Body Systems Modeling of Flexibility Effects Using the Boundary-Element Method Appendix A: Multibody Dynamics Flowchart for the Construction of the Equations of Motion with Constraints Appendix B: Centroid Location and Area Moment of Inertia Appendix C: Center of Gravity and Mass Moment of Inertia of Homogeneous Solids Appendix D: Symbols Description Appendix E: Units and Conversion References Index

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Book Synopsisrii application of linear operators on a Hilbert space. We begin with a chapter on the geometry of Hilbert space and then proceed to the spectral theory of compact self adjoint operators;Table of ContentsIntroduction * I. Hilbert Spaces * II. Bounded Linear Operators on Hilbert Spaces * III. Spectral Theory of Compact Self Adjoint Operators * IV. Spectral Theory of Integral Operators * V. Oscillations of an Elastic String * VI. Operational Calculus with Applications * VII. Solving Linear Equations by Iterative Methods * VIII. Further Developments of the Spectral Theorem * IX. Banach Spaces * X. Linear Operators on a Banach Space * XI. Compact Operators on a Banach Space * XII. Non-Linear Operators * Appendix 1. Countable Sets and Separable Hilbert Spaces * Appendix 2. Lebesgue Integration and LP Spaces * Appendix 3. Proof of the Hahn-Banach Theorem * Appendix 4. Proof of the Closed Graph Theorem * Suggested Reading * References * Index

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Book SynopsisPresents articles originating from invited talks at an international conference held at The Fields Institute in Toronto celebrating the sixtieth birthday of the renowned mathematician, Vladimir Arnold. This work focuses on topics including: singularity theory, the theory of curves, symmetry groups, dynamical systems and mechanics.Table of ContentsFrom Hilbert's superposition problem to dynamical systems by V. I. Arnold Recollections by J. Moser Symplectization, complexification and mathematical trinities by V. I. Arnold Topological problems in wave propagation theory and topological economy principle in algebraic geometry by V. I. Arnold Geometry and control of three-wave interactions by M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins Standard basis along a Samuel stratum, and implicit differentiation by E. Bierstone and P. D. Milman A global weighted version of Bezout's theorem by J. Damon Real Enriques surfaces without real points and Enriques-Einstein-Hitchin 4-manifolds by A. Degtyarev and V. Kharlamov On the index of a vector field at an isolated singularity by W. Ebeling and S. M. Gusein-Zade The exponential map on $\mathcal{D}^s_\mu$ by D. G. Ebin and G. Misiolek Zeldovich's neutron star and the prediction of magnetic froth by M. H. Freedman Arnold conjecture and Gromov-Witten invariant for general symplectic manifolds by K. Fukaya and K. Ono Multiplicity of a zero of an analytic function on a trajectory of a vector field by A. Gabrielov Singularity theory and symplectic topology by A. B. Givental On enumeration of meromorphic functions on the line by V. V. Goryunov and S. K. Lando Pseudoholomorphic curves and dynamics by H. Hofer and E. Zehnder Bifurcation of planar and spatial polycycles: Arnold's program and its development by Yu. S. Ilyashenko and V. Yu. Kaloshin Singularity which has no $M$-smoothing by V. M. Kharlamov, S. Yu. Orevkov, and E. I. Shustin Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces by B. Khesin and A. Rosly Newton polyhedra, a new formula for mixed volume, product of roots of a system of equations by A. Khovanskii Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations by W. F. Langford and K. Zhan Solutions of the qKZB equation in tensor products of finite dimensional modules over the elliptic quantum group $E_{\tau,\eta}sl_2$ by E. Mukhin and A. Varchenko Schrodinger operators on graphs and symplectic geometry by S. P. Novikov On the dominant Fourier modes in the series associated with separatrix splitting for an a-priori stable, three degree-of-freedom Hamiltonian system by M. Rudnev and S. Wiggins Homology of $i$-connected graphs and invariants of knots, plane arrangements, etc. by V. A. Vassiliev On Arnold's variational principles in fluid mechanics by V. A. Vladimirov and K. I. Ilin On functions and curves defined by ordinary differential equations by S. Yakovenko Global finiteness properties of analytic families and algebra of their Taylor coefficients by Y. Yomdin.

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Book SynopsisAugmented with 1024 equations, 138 references and 82 figures and 69 problems, this book provides an introduction to and overview of signal detection and estimation.Table of ContentsProbability concepts; random processes; signal detection; parameter estimation; filtering; representation of signals; the generalized Gaussian problem; detection and parameter estimation.

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Book SynopsisThe second edition of this bestselling book provides an overview of the key topics in undergraduate mathematics, allowing beginning graduate students to fill in any gaps in their knowledge. With numerous examples, exercises and suggestions for further reading, it is a must-have for anyone looking to learn some serious mathematics quickly.Trade Review'Reading Garrity is like talking with your favorite uncle - he tells you the essential stories, in a clear and colorful way, and you get just what you need to explore further. The topics are well chosen (and there are more in this new edition). His points of view enrich the reader - not only do you learn what to know, but how to know it. I wish I had had this book when I started graduate school.' John McCleary, Vassar College'I admired one of the intentions behind the first edition of Garrity's All the Math You Missed: to give students the tools to appreciate the applications of mathematics without painting a simplistic picture of 'Applied Mathematics'. In this second edition, he takes this idea to the next level by introducing four additional chapters, dealing primarily with number theory and category theory.' Robert Kotiuga, Boston University'I felt like I was terribly underprepared for graduate school, and Garrity's book helped me fill in some of those gaps. But far more importantly, the welcoming tone made me see that I wasn't alone in feeling anxious, and it made grad school feel less intimidating.' Daniel Erman, University of Wisconsin, Madison'Incoming graduate students would find the book most useful … this book is designed to provide some useful guidance … The writing is clear and easy to read.' Bill Satzer, MAA ReviewsTable of ContentsOn the structure of mathematics; Brief summaries of topics; 1. Linear Algebra; 2. ε and δ real analysis; 3. Calculus for vector-valued functions; 4. Point set topology; 5. Classical Stokes' theorems; 6. Diff erential forms and Stokes' theorem; 7. Curvature for curves and surfaces; 8. Geometry; 9. Countability and the Axiom of Choice; 10. Elementary number theory; 11. Algebra; 12. Algebraic number theory; 13. Complex analysis; 14. Analytic number theory; 15. Lebesgue integration; 16. Fourier analysis; 17. Diff erential equations; 18. Combinatorics and probability theory; 19. Algorithms; 20. Category theory; Appendix A. Equivalence relations; References; Index.

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Book SynopsisAn example- and exercise-filled book for mathematical and scientific modelers with an introductory knowledge of category theory (e.g., readers of Cheng's 'Joy of Abstraction' or Fong & Spivak's 'Invitation to Applied Category Theory') interested in learning to apply the category of polynomial functors to real-world interacting dynamical systems.

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Book SynopsisAs more and more engineering departments and companies choose to use Python, this book provides an essential introduction to this open-source, free-to-use language. Expressly designed to support first-year engineering students, this book covers engineering and scientific calculations, Python basics, and structured programming.Based on extensive teaching experience, the text uses practical problem solving as a vehicle to teach Python as a programming language. By learning computing fundamentals in an engaging and hands-on manner, it enables the reader to apply engineering and scientific methods with Python, focusing this general language to the needs of engineers and the problems they are required to solve on a daily basis. Rather than inundating students with complex terminology, this book is designed with a leveling approach in mind, enabling students at all levels to gain experience and understanding of Python. It covers such topics as structured programming, graphics, matrTable of ContentsChapter 1 Engineering and Scientific CalculationsChapter 2 Computer-Based CalculationsChapter 3 Python BasicsChapter 4 Structured Programming with PythonChapter 5 Graphics—MatplotlibChapter 6 Array and Matrix OperationsChapter 7 Solving Single Algebraic EquationsChapter 8 Solving Sets of Algebraic EquationsChapter 9 Solving Differential EquationsChapter 10 Working with Data

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Book SynopsisTo properly operate a waterworks or wastewater treatment plant and to pass the examination for a waterworks/wastewater operator's license, it is necessary to know how to perform certain calculations. All operators, at all levels of licensure, need a basic understanding of arithmetic and problem-solving techniques to solve the problems they typically encounter in the workplace.Hailed on its first publication as a masterly account written in an engaging, highly readable, user-friendly style, the fully updated Mathematics Manual for Water and Wastewater Treatment Plant Operators: Basic Mathematics for Water and Wastewater Operators introduces and reviews fundamental concepts critical to qualified operators. It builds a strong foundation based on theoretical math concepts, which it then applies to solving practical problems for both water and wastewater operations.Features: Provides a strong foundation based on theoretical math concepts, which it then applieTable of Contents1. Introduction. 2. Basic Units of Measurement, Conversions. 3. Sequence of Operations. 4. Fractions, Decimals & Percent. 5. Rounding & Significant Digits. 6. Powers of Ten and Exponents. 7. Averages (Arithmetic Mean) & Median. 8. Solving for the Unknown. 9. Ratio/Proportion. 10. Electrical Calculations. 11. Circumference, Area & Volume. 12. Force, Pressure & Head, Velocity Calculations. 13. Mass Balance & Measuring Plant Performance. 14. Pumping Calculations. 15. Water Source & Storage Calculations. 16. Waste/Wastewater Laboratory Calculations. 17. Workbook Practice Problems.

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Book SynopsisTo properly operate a waterworks or wastewater treatment plant and to pass the examination for a waterworks/wastewater operator's license, it is necessary to know how to perform certain calculations. All operators, at all levels of licensure, need a basic understanding of arithmetic and problem-solving techniques to solve the problems they typically encounter in the workplace.Hailed on its first publication as a masterly account written in an engaging, highly readable, user-friendly style, the fully updated Mathematics Manual for Water and Wastewater Treatment Plant Operators: Water Treatment Operations covers all the necessary computations used in water treatment today. It presents math operations that progressively advance to higher, more practical applications, including math operations that operators at the highest level of licensure would be expected to know and perform.Features: Provides a strong foundation based on theoretical math concepts, whichTable of Contents1. Pumping Calculations. 2. Water Source & Storage Calculations. 3. Coagulation & Flocculation Calculations. 4. Sedimentation Calculations. 5. Filtration Calculations. 6. Water Chlorination Calculations. 7. Fluoridation. 8. Water Softening. 9. Water Treatment Practice Calculations.

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Book SynopsisWritten for senior undergraduates in all disciplines of physical science and engineering, the plain language style of this concise guide to numerical methods concentrates on developing computational skills and avoids potentially intimidating formal mathematical proofs. Including numerous worked examples and exercises, this textbook explains the practical realities of numerical techniques.Table of ContentsPreface; 1. Fitting functions to data; 2. Ordinary differential equations; 3. Two-point boundary conditions; 4. Partial differential equations; 5. Diffusion: parabolic PDEs; 6. Elliptic problems and iterative matrix solution; 7. Fluid dynamics and hyperbolic equations; 8. Boltzmann's equation and its solution; 9. Energy-resolved diffusive transport; 10. Atomistic and particle-in-cell simulation; 11. Monte Carlo techniques; 12. Monte Carlo radiation transport; 13. Next steps; Appendix A. Summary of matrix algebra; Index.

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Book SynopsisLinear Algebra offers a unified treatment of both matrix-oriented and theoretical approaches to the course, which will be useful for classes with a mix of mathematics, physics, engineering, and computer science students. Major topics include singular value decomposition, the spectral theorem, linear systems of equations, vector spaces, linear maps, matrices, eigenvalues and eigenvectors, linear independence, bases, coordinates, dimension, matrix factorizations, inner products, norms, and determinants.Trade Review'This is a book for anyone who wants to really understand linear algebra. Instead of mere cookbook recipes or dry proofs, it provides explanations, examples, pictures - and, yes, algorithms and proofs too, but only after the reader is able to understand them. And while it is aimed at beginners, even experts will have something to learn from this book.' John Baez, University of California, Riverside'This is an exciting and entertaining book. It keeps an informal tone, but without sacrificing accuracy or clarity. It takes care to address common difficulties (and the classroom testing shows), but without talking down to the reader. It uses the modern understanding of how to do linear algebra right, but remains accessible to first-time readers.' Tom Leinster, University of Edinburgh'Linear algebra is one of the most important topics in mathematics, as linearity is exploited throughout applied mathematics and engineering. Therefore, the tools from linear algebra are used in many fields. However, they are often not presented that way, which is a missed opportunity. The authors have written a linear algebra book that is useful for students from many fields (including mathematics). A great feature of this book is that it presents a formal linear algebra course that clearly makes (coordinate) matrices and vectors the fundamental tools for problem solving and computations.' Eric de Sturler, Virginia Polytechnic Institute and State University'It is a book well worth considering both for learning and teaching this important area of mathematics.' John Baylis, The Mathematical GazetteTable of Contents1. Linear systems and vector spaces; 2. Linear maps and matrices; 3. Linear independence, bases, and coordinates; 4. Inner products; 5. Singular value decomposition and the spectral theorem; 6. Determinants.

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Book SynopsisWritten for senior undergraduates in all disciplines of physical science and engineering, the plain language style of this concise guide to numerical methods concentrates on developing computational skills and avoids potentially intimidating formal mathematical proofs. Including numerous worked examples and exercises, this textbook explains the practical realities of numerical techniques.Table of ContentsPreface; 1. Fitting functions to data; 2. Ordinary differential equations; 3. Two-point boundary conditions; 4. Partial differential equations; 5. Diffusion: parabolic PDEs; 6. Elliptic problems and iterative matrix solution; 7. Fluid dynamics and hyperbolic equations; 8. Boltzmann's equation and its solution; 9. Energy-resolved diffusive transport; 10. Atomistic and particle-in-cell simulation; 11. Monte Carlo techniques; 12. Monte Carlo radiation transport; 13. Next steps; Appendix A. Summary of matrix algebra; Index.

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Book SynopsisAn introduction to the mathematical basis of finite element analysis as applied to vibrating systems. Finite element analysis is a technique that is very important in modeling the response of structures to dynamic loads and is widely used in aeronautical, civil, and mechanical engineering as well as naval architecture.Trade Review"The contents of this work are very well organized, and Petyt (Univ. of Southhamption, UK) gradually introduces important concepts, making it a very useful theoretical reference." X. Le, Wentworth Institute of Technology"The contents of this work are very well organized ... a very useful theoretical reference. ...Recommended." CHOICETable of Contents1. Formulation of the equations of motion; 2. Element energy functions; 3. Introduction to the finite element displacement method; 4. In-plane vibration of plates; 5. Vibration of solids; 6. Flexural vibration of plates; 7. Vibration of stiffened plates and folded plate structures; 8. Vibration of shells; 9. Vibration of laminated plates and shells; 10. Hierarchical finite element method; 11. Analysis of free vibration; 12. Forced response; 13. Forced response II; 14. Computer analysis technique.

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Book SynopsisThe study of complex variables is beautiful from a purely mathematical point of view, and very useful for solving a wide array of problems arising in applications. This introduction to complex variables, suitable as a text for a one-semester course, has been written for undergraduate students in applied mathematics, science, and engineering. Based on the authors'' extensive teaching experience, it covers topics of keen interest to these students, including ordinary differential equations, as well as Fourier and Laplace transform methods for solving partial differential equations arising in physical applications. Many worked examples, applications, and exercises are included. With this foundation, students can progress beyond the standard course and explore a range of additional topics, including generalized Cauchy theorem, Painlevé equations, computational methods, and conformal mapping with circular arcs. Advanced topics are labeled with an asterisk and can be included in the syllabus or form the basis for challenging student projects.Trade Review'… a stylish, well-written and up to date introduction to complex variable methods for undergraduate (or early graduate) students in applied mathematics, science and engineering … I thoroughly enjoyed reading this book and warmly commend it to anyone seeking a brisk, well-organised account of complex variables with a practical focus on applications and calculational aspects.' Nick Lord, The Mathematical GazetteTable of Contents1. Complex numbers and elementary functions; 2. Analytic functions and integration; 3. Sequences, series and singularities of complex functions; 4. Residue calculus and applications of contour integration; 5. Conformal mappings and applications; Appendix. Answers to selected odd-numbered exercises; References; Index.

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Book SynopsisThis text is intended for the undergraduate course in math methods, with an audience of physics and engineering majors. As a required course in most departments, the text relies heavily on explained examples, real-world applications and student engagement.Trade Review"[Mathematical Methods in Engineering and Physics] is my book of choice for teaching undergraduates...I honestly never thought that I could be so enchanted by the heat equation before seeing how Felder and Felder effectively have students derive it as part of honing their intuition for how to think about partial differential equations." - Christine Aidala, PhD, Associate Professor of Physics at University of Michigan for the American Journal of Physics Table of ContentsPreface xi 1 Introduction to Ordinary Differential Equations 1 1.1 Motivating Exercise: The Simple Harmonic Oscillator 2 1.2 Overview of Differential Equations 3 1.3 Arbitrary Constants 15 1.4 Slope Fields and Equilibrium 25 1.5 Separation of Variables 34 1.6 Guess and Check, and Linear Superposition 39 1.7 Coupled Equations (see felderbooks.com) 1.8 Differential Equations on a Computer (see felderbooks.com) 1.9 Additional Problems (see felderbooks.com) 2 Taylor Series and Series Convergence 50 2.1 Motivating Exercise: Vibrations in a Crystal 51 2.2 Linear Approximations 52 2.3 Maclaurin Series 60 2.4 Taylor Series 70 2.5 Finding One Taylor Series from Another 76 2.6 Sequences and Series 80 2.7 Tests for Series Convergence 92 2.8 Asymptotic Expansions (see felderbooks.com) 2.9 Additional Problems (see felderbooks.com) 3 Complex Numbers 104 3.1 Motivating Exercise: The Underdamped Harmonic Oscillator 104 3.2 Complex Numbers 105 3.3 The Complex Plane 113 3.4 Euler’s Formula I—The Complex Exponential Function 117 3.5 Euler’s Formula II—Modeling Oscillations 126 3.6 Special Application: Electric Circuits (see felderbooks.com) 3.7 Additional Problems (see felderbooks.com) 4 Partial Derivatives 136 4.1 Motivating Exercise: The Wave Equation 136 4.2 Partial Derivatives 137 4.3 The Chain Rule 145 4.4 Implicit Differentiation 153 4.5 Directional Derivatives 158 4.6 The Gradient 163 4.7 Tangent Plane Approximations and Power Series (see felderbooks.com) 4.8 Optimization and the Gradient 172 4.9 Lagrange Multipliers 181 4.10 Special Application: Thermodynamics (see felderbooks.com) 4.11 Additional Problems (see felderbooks.com) 5 Integrals in Two or More Dimensions 188 5.1 Motivating Exercise: Newton’s Problem (or) The Gravitational Field of a Sphere 188 5.2 Setting Up Integrals 189 5.3 Cartesian Double Integrals over a Rectangular Region 204 5.4 Cartesian Double Integrals over a Non-Rectangular Region 211 5.5 Triple Integrals in Cartesian Coordinates 216 5.6 Double Integrals in Polar Coordinates 221 5.7 Cylindrical and Spherical Coordinates 229 5.8 Line Integrals 240 5.9 Parametrically Expressed Surfaces 249 5.10 Surface Integrals 253 5.11 Special Application: Gravitational Forces (see felderbooks.com) 5.12 Additional Problems (see felderbooks.com) 6 Linear Algebra I 266 6.1 The Motivating Example on which We’re Going to Base the Whole Chapter: The Three-Spring Problem 266 6.2 Matrices: The Easy Stuff 276 6.3 Matrix Times Column 280 6.4 Basis Vectors 286 6.5 Matrix Times Matrix 294 6.6 The Identity and Inverse Matrices 303 6.7 Linear Dependence and the Determinant 312 6.8 Eigenvectors and Eigenvalues 325 6.9 Putting It Together: Revisiting the Three-Spring Problem 336 6.10 Additional Problems (see felderbooks.com) 7 Linear Algebra II 346 7.1 Geometric Transformations 347 7.2 Tensors 358 7.3 Vector Spaces and Complex Vectors 369 7.4 Row Reduction (see felderbooks.com) 7.5 Linear Programming and the Simplex Method (see felderbooks.com) 7.6 Additional Problems (see felderbooks.com) 8 Vector Calculus 378 8.1 Motivating Exercise: Flowing Fluids 378 8.2 Scalar and Vector Fields 379 8.3 Potential in One Dimension 387 8.4 From Potential to Gradient 396 8.5 From Gradient to Potential: The Gradient Theorem 402 8.6 Divergence, Curl, and Laplacian 407 8.7 Divergence and Curl II—The Math Behind the Pictures 416 8.8 Vectors in Curvilinear Coordinates 419 8.9 The Divergence Theorem 426 8.10 Stokes’ Theorem 432 8.11 Conservative Vector Fields 437 8.12 Additional Problems (see felderbooks.com) 9 Fourier Series and Transforms 445 9.1 Motivating Exercise: Discovering Extrasolar Planets 445 9.2 Introduction to Fourier Series 447 9.3 Deriving the Formula for a Fourier Series 457 9.4 Different Periods and Finite Domains 459 9.5 Fourier Series with Complex Exponentials 467 9.6 Fourier Transforms 472 9.7 Discrete Fourier Transforms (see felderbooks.com) 9.8 Multivariate Fourier Series (see felderbooks.com) 9.9 Additional Problems (see felderbooks.com) 10 Methods of Solving Ordinary Differential Equations 484 10.1 Motivating Exercise: A Damped, Driven Oscillator 485 10.2 Guess and Check 485 10.3 Phase Portraits (see felderbooks.com) 10.4 Linear First-Order Differential Equations (see felderbooks.com) 10.5 Exact Differential Equations (see felderbooks.com) 10.6 Linearly Independent Solutions and the Wronskian (see felderbooks.com) 10.7 Variable Substitution 494 10.8 Three Special Cases of Variable Substitution 505 10.9 Reduction of Order and Variation of Parameters (see felderbooks.com) 10.10 Heaviside, Dirac, and Laplace 512 10.11 Using Laplace Transforms to Solve Differential Equations 522 10.12 Green’s Functions 531 10.13 Additional Problems (see felderbooks.com) 11 Partial Differential Equations 541 11.1 Motivating Exercise: The Heat Equation 542 11.2 Overview of Partial Differential Equations 544 11.3 Normal Modes 555 11.4 Separation of Variables—The Basic Method 567 11.5 Separation of Variables—More than Two Variables 580 11.6 Separation of Variables—Polar Coordinates and Bessel Functions 589 11.7 Separation of Variables—Spherical Coordinates and Legendre Polynomials 607 11.8 Inhomogeneous Boundary Conditions 616 11.9 The Method of Eigenfunction Expansion 623 11.10 The Method of Fourier Transforms 636 11.11 The Method of Laplace Transforms 646 11.12 Additional Problems (see felderbooks.com) 12 Special Functions and ODE Series Solutions 652 12.1 Motivating Exercise: The Circular Drum 652 12.2 Some Handy Summation Tricks 654 12.3 A Few Special Functions 658 12.4 Solving Differential Equations with Power Series 666 12.5 Legendre Polynomials 673 12.6 The Method of Frobenius 682 12.7 Bessel Functions 688 12.8 Sturm-Liouville Theory and Series Expansions 697 12.9 Proof of the Orthgonality of Sturm-Liouville Eigenfunctions (see felderbooks.com) 12.10 Special Application: The Quantum Harmonic Oscillator and Ladder Operators (see felderbooks.com) 12.11 Additional Problems (see felderbooks.com) 13 Calculus with Complex Numbers 708 13.1 Motivating Exercise: Laplace’s Equation 709 13.2 Functions of Complex Numbers 710 13.3 Derivatives, Analytic Functions, and Laplace’s Equation 716 13.4 Contour Integration 726 13.5 Some Uses of Contour Integration 733 13.6 Integrating Along Branch Cuts and Through Poles (see felderbooks.com) 13.7 Complex Power Series 742 13.8 Mapping Curves and Regions 747 13.9 Conformal Mapping and Laplace’s Equation 754 13.10 Special Application: Fluid Flow (see felderbooks.com) 13.11 Additional Problems (see felderbooks.com) Appendix A Different Types of Differential Equations 765 Appendix B Taylor Series 768 Appendix C Summary of Tests for Series Convergence 770 Appendix D Curvilinear Coordinates 772 Appendix E Matrices 774 Appendix F Vector Calculus 777 Appendix G Fourier Series and Transforms 779 Appendix H Laplace Transforms 782 Appendix I Summary: Which PDE Technique Do I Use? 787 Appendix J Some Common Differential Equations and Their Solutions 790 Appendix K Special Functions 798 Appendix L Answers to “Check Yourself” in Exercises 801 Appendix M Answers to Odd-Numbered Problems (see felderbooks.com) Index 805

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Book SynopsisAlgebraic Identification and Estimation Methods in Feedback Control Systems presents a model-based algebraic approach to online parameter and state estimation in uncertain dynamic feedback control systems. This approach evades the mathematical intricacies of the traditional stochastic approach, proposing a direct model-based scheme with several easy-to-implement computational advantages. The approach can be used with continuous and discrete, linear and nonlinear, mono-variable and multi-variable systems. The estimators based on this approach are not of asymptotic nature, and do not require any statistical knowledge of the corrupting noises to achieve good performance in a noisy environment. These estimators are fast, robust to structured perturbations, and easy to combine with classical or sophisticated control laws. This book uses module theory, differential algebra, and operational calculus in an easy-to-understand manner and also details how to apply these in the coTable of ContentsSeries Preface xiii Preface xv 1 Introduction 1 1.1 Feedback Control of Dynamic Systems 2 1.1.1 Feedback 2 1.1.2 Why Do We Need Feedback? 3 1.2 The Parameter Identification Problem 3 1.2.1 Identifying a System 4 1.3 A Brief Survey on Parameter Identification 4 1.4 The State Estimation Problem 5 1.4.1 Observers 6 1.4.2 Reconstructing the State via Time Derivative Estimation 7 1.5 Algebraic Methods in Control Theory: Differences from Existing Methodologies 8 1.6 Outline of the Book 9 References 12 2 Algebraic Parameter Identification in Linear Systems 15 2.1 Introduction 15 2.1.1 The Parameter-Estimation Problem in Linear Systems 16 2.2 Introductory Examples 17 2.2.1 Dragging an Unknown Mass in Open Loop 17 2.2.2 A Perturbed First-Order System 24 2.2.3 The Visual Servoing Problem 30 2.2.4 Balancing of the Plane Rotor 35 2.2.5 On the Control of the Linear Motor 38 2.2.6 Double-Bridge Buck Converter 42 2.2.7 Closed-Loop Behavior 43 2.2.8 Control of an unknown variable gain motor 47 2.2.9 Identifying Classical Controller Parameters 50 2.3 A Case Study Introducing a “Sentinel” Criterion 53 2.3.1 A Suspension System Model 54 2.4 Remarks 67 References 68 3 Algebraic Parameter Identification in Nonlinear Systems 71 3.1 Introduction 71 3.2 Algebraic Parameter Identification for Nonlinear Systems 72 3.2.1 Controlling an Uncertain Pendulum 74 3.2.2 A Block-Driving Problem 80 3.2.3 The Fully Actuated Rigid Body 84 3.2.4 Parameter Identification Under Sliding Motions 90 3.2.5 Control of an Uncertain Inverted Pendulum Driven by a DC Motor 92 3.2.6 Identification and Control of a Convey Crane 96 3.2.7 Identification of a Magnetic Levitation System 103 3.3 An Alternative Construction of the System of Linear Equations 105 3.3.1 Genesio–Tesi Chaotic System 107 3.3.2 The Ueda Oscillator 108 3.3.3 Identification and Control of an Uncertain Brushless DC Motor 112 3.3.4 Parameter Identification and Self-tuned Control for the Inertia Wheel Pendulum 119 3.3.5 Algebraic Parameter Identification for Induction Motors 128 3.3.6 A Criterion to Determine the Estimator Convergence: The Error Index 136 3.4 Remarks 141 References 141 4 Algebraic Parameter Identification in Discrete-Time Systems 145 4.1 Introduction 145 4.2 Algebraic Parameter Identification in Discrete-Time Systems 145 4.2.1 Main Purpose of the Chapter 146 4.2.2 Problem Formulation and Assumptions 147 4.2.3 An Introductory Example 148 4.2.4 Samuelson’s Model of the National Economy 150 4.2.5 Heating of a Slab from Two Boundary Points 155 4.2.6 An Exact Backward Shift Reconstructor 157 4.3 A Nonlinear Filtering Scheme 160 4.3.1 Hénon System 161 4.3.2 A Hard Disk Drive 164 4.3.3 The Visual Servo Tracking Problem 166 4.3.4 A Shape Control Problem in a Rolling Mill 170 4.3.5 Algebraic Frequency Identification of a Sinusoidal Signal by Means of Exact Discretization 175 4.4 Algebraic Identification in Fast-Sampled Linear Systems 178 4.4.1 The Delta-Operator Approach: A Theoretical Framework 179 4.4.2 Delta-Transform Properties 181 4.4.3 A DC Motor Example 181 4.5 Remarks 188 References 188 5 State and Parameter Estimation in Linear Systems 191 5.1 Introduction 191 5.1.1 Signal Time Derivation Through the “Algebraic Derivative Method” 192 5.1.2 Observability of Nonlinear Systems 192 5.2 Fast State Estimation 193 5.2.1 An Elementary Second-Order Example 193 5.2.2 An Elementary Third-Order Example 194 5.2.3 A Control System Example 198 5.2.4 Control of a Perturbed Third-Order System 201 5.2.5 A Sinusoid Estimation Problem 203 5.2.6 Identification of Gravitational Wave Parameters 205 5.2.7 A Power Electronics Example 210 5.2.8 A Hydraulic Press 213 5.2.9 Identification and Control of a Plotter 218 5.3 Recovering Chaotically Encrypted Signals 222 5.3.1 State Estimation for a Lorenz System 227 5.3.2 State Estimation for Chen’s System 229 5.3.3 State Estimation for Chua’s Circuit 231 5.3.4 State Estimation for Rossler’s System 232 5.3.5 State Estimation for the Hysteretic Circuit 234 5.3.6 Simultaneous Chaotic Encoding–Decoding with Singularity Avoidance 239 5.3.7 Discussion 240 5.4 Remarks 241 References 242 6 Control of Nonlinear Systems via Output Feedback 245 6.1 Introduction 245 6.2 Time-Derivative Calculations 246 6.2.1 An Introductory Example 247 6.2.2 Identifying a Switching Input 253 6.3 The Nonlinear Systems Case 255 6.3.1 Control of a Synchronous Generator 256 6.3.2 Control of a Multi-variable Nonlinear System 261 6.3.3 Experimental Results on a Mechanical System 267 6.4 Remarks 278 References 279 7 Miscellaneous Applications 281 7.1 Introduction 281 7.1.1 The Separately Excited DC Motor 282 7.1.2 Justification of the ETEDPOF Controller 285 7.1.3 A Sensorless Scheme Based on Fast Adaptive Observation 287 7.1.4 Control of the Boost Converter 292 7.2 Alternative Elimination of Initial Conditions 298 7.2.1 A Bounded Exponential Function 299 7.2.2 Correspondence in the Frequency Domain 300 7.2.3 A System of Second Order 301 7.3 Other Functions of Time for Parameter Estimation 304 7.3.1 A Mechanical System Example 304 7.3.2 A Derivative Approach to Demodulation 310 7.3.3 Time Derivatives via Parameter Identification 312 7.3.4 Example 314 7.4 An Algebraic Denoising Scheme 318 7.4.1 Example 321 7.4.2 Numerical Results 322 7.5 Remarks 325 References 326 Appendix A Parameter Identification in Linear Continuous Systems: A Module Approach 329 A.1 Generalities on Linear Systems Identification 329 A.1.1 Example 330 A.1.2 Some Definitions and Results 330 A.1.3 Linear Identifiability 331 A.1.4 Structured Perturbations 333 A.1.5 The Frequency Domain Alternative 337 References 338 Appendix B Parameter Identification in Linear Discrete Systems: A Module Approach 339 B.1 A Short Review of Module Theory over Principal Ideal Rings 339 B.1.1 Systems 340 B.1.2 Perturbations 340 B.1.3 Dynamics and Input–Output Systems 341 B.1.4 Transfer Matrices 341 B.1.5 Identifiability 342 B.1.6 An Algebraic Setting for Identifiability 342 B.1.7 Linear identifiability of transfer functions 344 B.1.8 Linear Identification of Perturbed Systems 345 B.1.9 Persistent Trajectories 347 References 348 Appendix C Simultaneous State and Parameter Estimation: An Algebraic Approach 349 C.1 Rings, Fields and Extensions 349 C.2 Nonlinear Systems 350 C.2.1 Differential Flatness 351 C.2.2 Observability and Identifiability 352 C.2.3 Observability 352 C.2.4 Identifiable Parameters 352 C.2.5 Determinable Variables 352 C.3 Numerical Differentiation 353 C.3.1 Polynomial Time Signals 353 C.3.2 Analytic Time Signals 353 C.3.3 Noisy Signals 354 References 354 Appendix D Generalized Proportional Integral Control 357 D.1 Generalities on GPI Control 357 D.2 Generalization to MIMO Linear Systems 365 References 368 Index 369

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Book SynopsisA solutions manual to accompany Statistics and Probability with Applications for Engineers and Scientists Unique among books of this kind, Statistics and Probability with Applications for Engineers and Scientists covers descriptive statistics first, then goes on to discuss the fundamentals of probability theory.Table of ContentsChapter 2 1 Practice Problems for Sections 2.1 and 2.2, 1 Practice Problems for Section 2.3, 1 Practice Problems for Section 2.4, 2 Practice Problems for Section 2.5, 8 Practice Problems for Section 2.6, 8 Practice Problems for Sections 2.7 and 2.8, 11 Practice Problems for Section 2.9, 12 Review Practice Problems, 13 Chapter 3 23 Practice Problems for Sections 3.2 and 3.3, 23 Practice Problems for Section 3.4, 25 Practice Problems for Sections 3.5 and 3.6, 26 Review Practice Problems, 28 Chapter 4 36 Practice Problems for Sections 4.1 and 4.2, 36 Practice Problems for Sections 4.3 and 4.4, 37 Practice Problems for Sections 4.5 and 4.6, 38 Practice Problems for Section 4.7, 39 Practice Problems for Section 4.8, 40 Practice Problems for Section 4.9, 41 Review Practice Problems, 42 Chapter 5 51 Practice Problems for Sections 5.1 and 5.2, 51 Practice Problems for Section 5.3, 53 Practice Problems for Section 5.4, 53 Practice Problems for Section 5.5, 54 Practice Problems for Section 5.6, 55 Practice Problems for Sections 5.7 and 5.8, 57 Practice Problems for Section 5.9.1, 58 Practice Problems for Section 5.9.2, 59 Practice Problems for Sections 5.9.3 and 5.9.4, 60 Review Practice Problems, 61 Chapter 6 70 Practice Problems for Section 6.2, 70 Practice Problems for Sections 6.3 and 6.4, 73 Review Practice Problems, 74 Chapter 7 80 Practice Problems for Section 7.1, 80 Practice Problems for Section 7.2, 80 Practice Problems for Section 7.3, 81 Practice Problems for Section 7.4, 81 Review Practice Problems, 83 Chapter 8 87 Practice Problems for Section 8.2, 87 Practice Problems for Section 8.3, 89 Practice Problems for Section 8.4, 90 Practice Problems for Sections 8.5 and 8.6, 92 Practice Problems for Section 8.7, 94 Practice Problems for Section 8.8, 95 Review Practice Problems, 96 Chapter 9 104 Practice Problems for Section 9.2, 104 Practice Problems for Section 9.3, 105 Practice Problems for Section 9.4, 106 Practice Problems for Section 9.5, 107 Practice Problems for Section 9.6, 108 Practice Problems for Section 9.7, 109 Practice Problems for Section 9.8, 110 Practice Problems for Section 9.9, 112 Practice Problems for Section 9.10, 113 Practice Problems for Sections 9.11 and 9.12, 114 Review Practice Problems, 116 Chapter 10 131 Practice Problems for Section 10.1, 131 Practice Problems for Section 10.2, 132 Practice Problems for Sections 10.3 and 10.4, 133 Review Practice Problems, 138 Chapter 11 145 Practice Problems for Sections 11.3 and 11.4, 145 Practice Problems for Section 11.5, 147 Practice Problems for Section 11.6, 151 Practice Problems for Section 11.7, 155 Review Practice Problems, 156 Chapter 12 165 Practice Problems for Section 12.2, 165 Practice Problems for Section 12.3, 165 Practice Problems for Section 12.4, 170 Practice Problems for Section 12.5, 174 Review Practice Problems, 178 Chapter 13 190 Practice Problems for Section 13.2, 190 Practice Problems for Section 13.3, 194 Practice Problems for Section 13.4, 197 Review Practice Problems, 199 Chapter 14 207 Practice Problems for Section 14.2, 207 Practice Problems for Section 14.3, 209 Practice Problems for Section 14.4, 211 Practice Problems for Section 14.5, 212 Review Practice Problems, 215 Chapter 15 219 Practice Problems for Section 15.2, 219 Practice Problems for Sections 15.3 and 15.4, 225 Practice Problems for Section 15.5, 230 Practice Problems for Section 15.6, 231 Practice Problems for Section 15.7, 233 Practice Problems for Section 15.8, 236 Practice Problems for Section 15.9, 239 Review Practice Problems, 240 Chapter 16 257 Practice Problems for Section 16.3, 257 Practice Problems for Section 16.4, 261 Practice Problems for Sections 16.6 and 16.7, 265 Practice Problems for Section 16.8, 269 Review Practice Problems, 270 Chapter 17 285 Practice Problems for Section 17.2, 285 Practice Problems for Section 17.3, 286 Practice Problems for Section 17.4, 289 Practice Problems for Section 17.5, 291 Practice Problems for Section 17.6, 294 Practice Problems for Section 17.7, 295 Review Practice Problems, 297 Chapter 18 308 Practice Problems for Section 18.2, 308 Practice Problems for Section 18.3, 309 Practice Problems for Section 18.4, 311 Practice Problems for Section 18.5, 314 Practice Problems for Section 18.6, 320 Review Practice Problems, 325 Chapter 19 345 Practice Problems for Section 19.2, 345 Practice Problems for Section 19.3, 348 Practice Problems for Section 19.4, 351 Review Practice Problems, 355

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Book SynopsisTheory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the model most suitable for the problems they are interested in. The book is targeted at graduate students and researchers who plan to investigate multiphase flows using LBMs. Throughout the text most of the popular multiphase LBMs are analyzed both theoretically and through numerical simulation. The authors present many of the mathematical derivations of the models in greater detail tTable of ContentsPreface xi About the companion website xiii 1 Introduction 1 1.1 History of the Lattice Boltzmann method 2 1.2 The Lattice Boltzmann method 3 1.3 Multiphase LBM 6 1.3.1 Color-gradient model 7 1.3.2 Shan–Chen model 7 1.3.3 Free-energy model 8 1.3.4 Interface tracking model 9 1.4 Comparison of models 9 1.5 Units in this book and parameter conversion 11 1.6 Appendix: Einstein summation convention 14 1.6.1 Kronecker δ function 15 1.6.2 Lattice tensors 15 1.7 Use of the Fortran code in the book 16 2 Single-component multiphase Shan–Chen-type model 18 2.1 Introduction 18 2.1.1 "Equilibrium" velocity in the SC model 20 2.1.2 Inter-particle forces in the SC SCMP LBM 20 2.2 Typical equations of state 21 2.2.1 Parameters in EOS 27 2.3 Thermodynamic consistency 28 2.3.1 The SCMP LBM EOS 29 2.3.2 Incorporating other EOS into the SC model 31 2.4 Analytical surface tension 32 2.4.1 Inter-particle Force Model A 32 2.4.2 Inter-particle Force Model B 33 2.5 Contact angle 34 2.6 Capillary rise 36 2.7 Parallel flow and relative permeabilities 39 2.8 Forcing term in the SC model 40 2.8.1 Schemes to incorporate the body force 42 2.8.2 Scheme overview 44 2.8.3 Theoretical analysis 44 2.8.4 Numerical results and discussion 46 2.9 Multirange pseudopotential (Inter-particle Force Model B) 55 2.10 Conclusions 58 2.11 Appendix A: Analytical solution for layered multiphase flow in a channel 58 2.12 Appendix B: FORTRAN code to simulate single component multiphase droplet contacting a wall as shown in Figure 2.7(c) 60 3 Shan and Chen-type multi-component multiphase models 71 3.1 Multi-component multiphase SC LBM 71 3.1.1 Fluid–fluid cohesion and fluid–solid adhesion 73 3.2 Derivation of the pressure 73 3.2.1 Pressure in popular papers (2D) 74 3.2.2 Pressure in popular papers (3D) 75 3.3 Determining Gc and the surface tension 76 3.4 Contact angle 78 3.4.1 Application of Young's equation to MCMP LBM 79 3.4.2 Contact angle measurement 79 3.4.3 Verification of proposed equation 80 3.5 Flow through capillary tubes 83 3.6 Layered two-phase flow in a 2D channel 85 3.7 Pressure or velocity boundary conditions 87 3.7.1 Boundary conditions for 2D simulations 87 3.7.2 Boundary conditions for 3D simulations 89 3.8 Displacement in a 3D porous medium 91 4 Rothman–Keller multiphase Lattice Boltzmann model 94 4.1 Introduction 94 4.2 RK color-gradient model 96 4.3 Theoretical analysis (Chapman–Enskog expansion) 99 4.3.1 Discussion of above formulae 103 4.4 Layered two-phase flow in a 2D channel 103 4.4.1 Cases of two fluids with identical densities 104 4.4.2 Cases of two fluids with different densities 106 4.5 Interfacial tension and isotropy of the RK model 110 4.5.1 Interfacial tension 110 4.5.2 Isotropy 110 4.6 Drainage and capillary filling 111 4.7 MRT RK model 113 4.8 Contact angle 114 4.8.1 Spurious currents 115 4.9 Tests of inlet/outlet boundary conditions 117 4.10 Immiscible displacements in porous media 118 4.11 Appendix A 121 4.12 Appendix B 122 5 Free-energy-based multiphase Lattice Boltzmann model 136 5.1 Swift free-energy based single-component multiphase LBM 136 5.1.1 Derivation of the coefficients in the equilibrium distribution function 138 5.2 Chapman–Enskog expansion 143 5.3 Issue of Galilean invariance 146 5.4 Phase separation 149 5.5 Contact angle 154 5.5.1 How to specify a desired contact angle 154 5.5.2 Numerical verification 155 5.6 Swift free-energy-based multi-component multiphase LBM 158 5.7 Appendix 158 6 Inamuro's multiphase Lattice Boltzmann model 167 6.1 Introduction 167 6.1.1 Inamuro's method 167 6.1.2 Comment on the presentation 169 6.1.3 Chapman–Enskog expansion analysis 170 6.1.4 Cahn–Hilliard equation (equation for order parameter) 173 6.1.5 Poisson equation 174 6.2 Droplet collision 175 6.3 Appendix 178 7 He–Chen–Zhang multiphase Lattice Boltzmann model 196 7.1 Introduction 196 7.2 HCZ model 196 7.3 Chapman–Enskog analysis 199 7.3.1 N–S equations 199 7.3.2 CH equation 202 7.4 Surface tension and phase separation 202 7.5 Layered two-phase flow in a channel 204 7.6 Rayleigh–Taylor instability 205 7.7 Contact angle 210 7.8 Capillary rise 213 7.9 Geometric scheme to specify the contact angle and its hysteresis 215 7.9.1 Examples of droplet slipping in shear flows 218 7.10 Oscillation of an initially ellipsoidal droplet 219 7.11 Appendix A 222 7.12 Appendix B: 2D code 223 7.13 Appendix C: 3D code 238 8 Axisymmetric multiphase HCZ model 253 8.1 Introduction 253 8.2 Methods 253 8.2.1 Macroscopic governing equations 253 8.2.2 Axisymmetric HCZ LBM (Premnath and Abraham 2005a) 255 8.2.3 MRT version of the axisymmetric LBM (McCracken and Abraham 2005) 256 8.2.4 Axisymmetric boundary conditions 258 8.3 The Laplace law 258 8.4 Oscillation of an initially ellipsoidal droplet 259 8.5 Cylindrical liquid column break 263 8.6 Droplet collision 265 8.6.1 Effect of gradient and Laplacian calculation 267 8.6.2 Effect of BGK and MRT 274 8.7 A revised axisymmetric HCZ model (Huang et al. 2014) 276 8.7.1 MRT collision 276 8.7.2 Calculation of the surface tension 277 8.7.3 Mass correction 278 8.8 Bubble rise 279 8.8.1 Numerical validation 281 8.8.2 Surface-tension calculation effect 283 8.8.3 Terminal bubble shape 284 8.8.4 Wake behind the bubble 284 8.9 Conclusion 286 8.10 Appendix A: Chapman–Enskog analysis 288 8.10.1 Preparation for derivation 288 8.10.2 Mass conservation 289 8.10.3 Momentum conservation 289 8.10.4 CH equation 291 9 Extensions of the HCZ model for high-density ratio two-phase flows 292 9.1 Introduction 292 9.2 Model I (Lee and Lin 2005) 293 9.2.1 Stress and potential form of intermolecular forcing terms 293 9.2.2 Model description 294 9.2.3 Implementation 297 9.2.4 Directional derivative 298 9.2.5 Droplet splashing on a thin liquid film 299 9.3 Model II (Amaya-Bower and Lee 2010) 301 9.3.1 Implementation 302 9.4 Model III (Lee and Liu 2010) 304 9.5 Model IV 305 9.6 Numerical tests for different models 306 9.6.1 A drop inside a box with periodic boundary conditions 306 9.6.2 Layered two-phase flows in a channel 311 9.6.3 Galilean invariance 313 9.7 Conclusions 316 9.8 Appendix A: Analytical solutions for layered two-phase flow in a channel 317 9.9 Appendix B: 2D code based on Amaya-Bower and Lee (2010) 319 10 Axisymmetric high-density ratio two-phase LBMs (extension of the HCZ model) 334 10.1 Introduction 334 10.2 The model based on Lee and Lin (2005) 334 10.2.1 The equilibrium distribution functions I 336 10.2.2 The equilibrium distribution functions II 336 10.2.3 Source terms 337 10.2.4 Stress and potential form of intermolecular forcing terms 337 10.2.5 Chapman–Enskog analysis 338 10.2.6 Implementation 340 10.2.7 Droplet splashing on a thin liquid film 342 10.2.8 Head-on droplet collision 342 10.3 Axisymmetric model based on Lee and Liu (2010) 345 10.3.1 Implementation 347 10.3.2 Head-on droplet collision 348 10.3.3 Bubble rise 353 Index 371

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Book SynopsisAn introduction to mathematical techniques used in engineering with an emphasis on applications in linear circuits and systems This book provides an integrated approach to learning the necessary mathematical tools specifically used for linear circuits and systems.Table of ContentsPreface xiii Notation and Bibliography xvii About the Companion Website xix 1 Overview and Background 1 1.1 Introduction 1 1.2 Mathematical Models 3 1.3 Frequency Content 12 1.4 Functions and Properties 16 1.5 Derivatives and Integrals 22 1.6 Sine, Cosine, and 𝜋 33 1.7 Napier’s Constant e and Logarithms 38 PART I CIRCUITS, MATRICES, AND COMPLEX NUMBERS 51 2 Circuits and Mechanical Systems 53 2.1 Introduction 53 2.2 Voltage, Current, and Power 54 2.3 Circuit Elements 60 2.4 Basic Circuit Laws 67 2.4.1 Mesh-Current and Node-Voltage Analysis 69 2.4.2 Equivalent Resistive Circuits 71 2.4.3 RC and RL Circuits 75 2.4.4 Series RLC Circuit 78 2.4.5 Diode Circuits 82 2.5 Mechanical Systems 85 2.5.1 Simple Pendulum 86 2.5.2 Mass on a Spring 92 2.5.3 Electrical and Mechanical Analogs 95 3 Linear Equations and Matrices 105 3.1 Introduction 105 3.2 Vector Spaces 106 3.3 System of Linear Equations 108 3.4 Matrix Properties and Special Matrices 113 3.5 Determinant 122 3.6 Matrix Subspaces 128 3.7 Gaussian Elimination 135 3.7.1 LU and LDU Decompositions 146 3.7.2 Basis Vectors 148 3.7.3 General Solution of 𝐀𝐲 = 𝐱 151 3.8 Eigendecomposition 152 3.9 MATLAB Functions 156 4 Complex Numbers and Functions 163 4.1 Introduction 163 4.2 Imaginary Numbers 165 4.3 Complex Numbers 167 4.4 Two Coordinates 169 4.5 Polar Coordinates 171 4.6 Euler’s Formula 175 4.7 Matrix Representation 182 4.8 Complex Exponential Rotation 183 4.9 Constant Angular Velocity 189 4.10 Quaternions 192 PART II SIGNALS, SYSTEMS, AND TRANSFORMS 203 5 Signals, Generalized Functions, and Fourier Series 205 5.1 Introduction 205 5.2 Energy and Power Signals 206 5.3 Step and Ramp Functions 208 5.4 Rectangle and Triangle Functions 211 5.5 Exponential Function 214 5.6 Sinusoidal Functions 217 5.7 Dirac Delta Function 220 5.8 Generalized Functions 223 5.9 Unit Doublet 233 5.10 Complex Functions and Singularities 240 5.11 Cauchy Principal Value 242 5.12 Even and Odd Functions 245 5.13 Correlation Functions 248 5.14 Fourier Series 251 5.15 Phasor Representation 261 5.16 Phasors and Linear Circuits 265 6 Differential Equation Models for Linear Systems 275 6.1 Introduction 275 6.2 Differential Equations 276 6.3 General Forms of The Solution 278 6.4 First-Order Linear ODE 280 6.4.1 Homogeneous Solution 283 6.4.2 Nonhomogeneous Solution 285 6.4.3 Step Response 287 6.4.4 Exponential Input 287 6.4.5 Sinusoidal Input 289 6.4.6 Impulse Response 290 6.5 Second-Order Linear ODE 294 6.5.1 Homogeneous Solution 296 6.5.2 Damping Ratio 304 6.5.3 Initial Conditions 306 6.5.4 Nonhomogeneous Solution 307 6.6 Second-Order ODE Responses 311 6.6.1 Step Response 311 6.6.2 Step Response (Alternative Method) 313 6.6.3 Impulse Response 319 6.7 Convolution 319 6.8 System of ODEs 323 7 Laplace Transforms and Linear Systems 335 7.1 Introduction 335 7.2 Solving ODEs Using Phasors 336 7.3 Eigenfunctions 339 7.4 Laplace Transform 340 7.5 Laplace Transforms and Generalized Functions 347 7.6 Laplace Transform Properties 352 7.7 Initial and Final Value Theorems 364 7.8 Poles and Zeros 367 7.9 Laplace Transform Pairs 372 7.9.1 Constant Function 372 7.9.2 Rectangle Function 373 7.9.3 Triangle Function 374 7.9.4 Ramped Exponential Function 376 7.9.5 Sinusoidal Functions 376 7.10 Transforms and Polynomials 377 7.11 Solving Linear ODEs 380 7.12 Impulse Response and Transfer Function 382 7.13 Partial Fraction Expansion 387 7.13.1 Distinct Real Poles 388 7.13.2 Distinct Complex Poles 391 7.13.3 Repeated Real Poles 396 7.13.4 Repeated Complex Poles 402 7.14 Laplace Transforms and Linear Circuits 409 8 Fourier Transforms and Frequency Responses 423 8.1 Introduction 423 8.2 Fourier Transform 425 8.3 Magnitude and Phase 435 8.4 Fourier Transforms and Generalized Functions 437 8.5 Fourier Transform Properties 442 8.6 Amplitude Modulation 449 8.7 Frequency Response 453 8.7.1 First-Order Low-Pass Filter 455 8.7.2 First-Order High-Pass Filter 459 8.7.3 Second-Order Band-Pass Filter 460 8.7.4 Second-Order Band-Reject Filter 463 8.8 Frequency Response of Second-Order Filters 466 8.9 Frequency Response of Series RLC Circuit 475 8.10 Butterworth Filters 478 8.10.1 Low-Pass Filter 481 8.10.2 High-Pass Filter 484 8.10.3 Band-Pass Filter 487 8.10.4 Band-Reject Filter 490 APPENDICES 499 Introduction to Appendices 500 A Extended Summaries of Functions and Transforms 501 A.1 Functions and Notation 501 A.2 Laplace Transform 502 A.3 Fourier Transform 504 A.4 Magnitude and Phase 506 A.5 Impulsive Functions 511 A.5.1 Dirac Delta Function (Shifted) 511 A.5.2 Unit Doublet (Shifted) 514 A.6 Piecewise Linear Functions 514 A.6.1 Unit Step Function 514 A.6.2 Signum Function 517 A.6.3 Constant Function (Two-Sided) 517 A.6.4 Ramp Function 521 A.6.5 Absolute Value Function (Two-Sided Ramp) 523 A.6.6 Rectangle Function 524 A.6.7 Triangle Function 528 A.7 Exponential Functions 529 A.7.1 Exponential Function (Right-Sided) 529 A.7.2 Exponential Function (Ramped) 531 A.7.3 Exponential Function (Two-Sided) 533 A.7.4 Gaussian Function 537 A.8 Sinusoidal Functions 539 A.8.1 Cosine Function (Two-Sided) 539 A.8.2 Cosine Function (Right-Sided) 541 A.8.3 Cosine Function (ExponentiallyWeighted) 544 A.8.4 Cosine Function (ExponentiallyWeighted and Ramped) 544 A.8.5 Sine Function (Two-Sided) 549 A.8.6 Sine Function (Right-Sided) 550 A.8.7 Sine Function (ExponentiallyWeighted) 553 A.8.8 Sine Function (ExponentiallyWeighted and Ramped) 554 B Inverse Laplace Transforms 559 B.1 Improper Rational Function 559 B.2 Unbounded System 562 B.3 Double Integrator and Feedback 563 C Identities, Derivatives, and Integrals 565 C.1 Trigonometric Identities 565 C.2 Summations 566 C.3 Miscellaneous 567 C.4 Completing the Square 567 C.5 Quadratic and Cubic Formulas 568 C.6 Derivatives 571 C.7 Indefinite Integrals 573 C.8 Definite Integrals 574 D Set Theory 577 D.1 Sets and Subsets 577 D.2 Set Operations 579 E Series Expansions 583 E.1 Taylor Series 583 E.2 Maclaurin Series 585 E.3 Laurent Series 588 F Lambert W-Function 593 F.1 LambertW-Function 593 F.2 Nonlinear Diode Circuit 597 F.3 System of Nonlinear Equations 598 Glossary 601 Bibliography 609 Index 615

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Book SynopsisA comprehensive guide to using energy principles and variational methods for solving problems in solid mechanics This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. It begins with a review of the basic equations of mechanics, the concepts of work and energy, and key topics from variational calculus. It presents virtual work and energy principles, energy methods of solid and structural mechanics, Hamilton's principle for dynamical systems, and classical variational methods of approximation. And it takes a more unified approach than that found in most solid mechanics books, to introduce the finite element method. Featuring more than 200 illustrations and tables, this Third Edition has been extensively reorganized and contains much new Table of ContentsAbout the Author xvii About the Companion Website xix Preface to the Third Edition xxi Preface to the Second Edition xxiii Preface to the First Edition xxv 1. Introduction and Mathematical Preliminaries 1 1.1 Introduction 1 1.1.1 Preliminary Comments 1 1.1.2 The Role of Energy Methods and Variational Principles 1 1.1.3 A Brief Review of Historical Developments 2 1.1.4 Preview 4 1.2 Vectors 5 1.2.1 Introduction 5 1.2.2 Definition of a Vector 6 1.2.3 Scalar and Vector Products 8 1.2.4 Components of a Vector 12 1.2.5 Summation Convention 13 1.2.6 Vector Calculus 17 1.2.7 Gradient, Divergence, and Curl Theorems 22 1.3 Tensors 26 1.3.1 Second-Order Tensors 26 1.3.2 General Properties of a Dyadic 29 1.3.3 Nonion Form and Matrix Representation of a Dyad 30 1.3.4 Eigenvectors Associated with Dyads 34 1.4 Summary 39 Problems 40 2. Review of Equations of Solid Mechanics 47 2.1 Introduction 47 2.1.1 Classification of Equations 47 2.1.2 Descriptions of Motion 48 2.2 Balance of Linear and Angular Momenta 50 2.2.1 Equations of Motion 50 2.2.2 Symmetry of Stress Tensors 54 2.3 Kinematics of Deformation 56 2.3.1 Green-Lagrange Strain Tensor 56 2.3.2 Strain Compatibility Equations 62 2.4 Constitutive Equations 65 2.4.1 Introduction 65 2.4.2 Generalized Hooke's Law 66 2.4.3 Plane Stress-Reduced Constitutive Relations 68 2.4.4 Thermoelastic Constitutive Relations 70 2.5 Theories of Straight Beams 71 2.5.1 Introduction 71 2.5.2 The Bernoulli-Euler Beam Theory 73 2.5.3 The Timoshenko Beam Theory 76 2.5.4 The von Ka’rma’n Theory of Beams 81 2.5.4.1 Preliminary Discussion 81 2.5.4.2 The Bernoulli-Euler Beam Theory 82 2.5.4.3 The Timoshenko Beam Theory 84 2.6 Summary 85 Problems 88 3. Work, Energy, and Variational Calculus 97 3.1 Concepts of Work and Energy 97 3.1.1 Preliminary Comments 97 3.1.2 External and Internal Work Done 98 3.2 Strain Energy and Complementary Strain Energy 102 3.2.1 General Development 102 3.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids 107 3.2.2.1 Stain energy density 107 3.2.2.2 Complementary stain energy density 108 3.2.3 Strain Energy and Complementary Strain Energy for Trusses 109 3.2.4 Strain Energy and Complementary Strain Energy for Torsional Members 114 3.2.5 Strain Energy and Complementary Strain Energy for Beams 117 3.2.5.1 The Bernoulli-Euler Beam Theory 117 3.2.5.2 The Timoshenko Beam Theory 119 3.3 Total Potential Energy and Total Complementary Energy 123 3.3.1 Introduction 123 3.3.2 Total Potential Energy of Beams 124 3.3.3 Total Complementary Energy of Beams 125 3.4 Virtual Work 126 3.4.1 Virtual Displacements 126 3.4.2 Virtual Forces 131 3.5 Calculus of Variations 135 3.5.1 The Variational Operator 135 3.5.2 Functionals 138 3.5.3 The First Variation of a Functional 139 3.5.4 Fundamental Lemma of Variational Calculus 140 3.5.5 Extremum of a Functional 141 3.5.6 The Euler Equations 143 3.5.7 Natural and Essential Boundary Conditions 146 3.5.8 Minimization of Functionals with Equality Constraints 151 3.5.8.1 The Lagrange Multiplier Method 151 3.5.8.2 The Penalty Function Method 153 3.6 Summary 156 Problems 159 4. Virtual Work and Energy Principles of Mechanics 167 4.1 Introduction 167 4.2 The Principle of Virtual Displacements 167 4.2.1 Rigid Bodies 167 4.2.2 Deformable Solids 168 4.2.3 Unit Dummy-Displacement Method 172 4.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I 179 4.3.1 The Principle of Minimum Total Potential Energy179 4.3.2 Castigliano's Theorem I 188 4.4 The Principle of Virtual Forces 196 4.4.1 Deformable Solids 196 4.4.2 Unit Dummy-Load Method 198 4.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II 204 4.5.1 The Principle of the Minimum total Complementary Potential Energy 204 4.5.2 Castigliano's Theorem II 206 4.6 Clapeyron's, Betti's, and Maxwell's Theorems 217 4.6.1 Principle of Superposition for Linear Problems 217 4.6.2 Clapeyron's Theorem 220 4.6.3 Types of Elasticity Problems and Uniqueness of Solutions 224 4.6.4 Betti's Reciprocity Theorem 226 4.6.5 Maxwell's Reciprocity Theorem 230 4.7 Summary 232 Problems 235 5. Dynamical Systems: Hamilton's Principle 243 5.1 Introduction 243 5.2 Hamilton's Principle for Discrete Systems 243 5.3 Hamilton's Principle for a Continuum 249 5.4 Hamilton's Principle for Constrained Systems 255 5.5 Rayleigh's Method 260 5.6 Summary 262 Problems 263 6. Direct Variational Methods 269 6.1 Introduction 269 6.2 Concepts from Functional Analysis 270 6.2.1 General Introduction 270 6.2.2 Linear Vector Spaces 271 6.2.3 Normed and Inner Product Spaces 276 6.2.3.1 Norm 276 6.2.3.2 Inner product 279 6.2.3.3 Orthogonality 280 6.2.4 Transformations, and Linear and Bilinear Forms 281 6.2.5 Minimum of a Quadratic Functional 282 6.3 The Ritz Method 287 6.3.1 Introduction 287 6.3.2 Description of the Method 288 6.3.3 Properties of Approximation Functions 293 6.3.3.1 Preliminary Comments 293 6.3.3.2 Boundary Conditions 293 6.3.3.3 Convergence 294 6.3.3.4 Completeness 294 6.3.3.5 Requirements on ɸ0 and ɸi 295 6.3.4 General Features of the Ritz Method 299 6.3.5 Examples 300 6.3.6 The Ritz Method for General Boundary-Value Problems 323 6.3.6.1 Preliminary Comments 323 6.3.6.2 Weak Forms 323 6.3.6.3 Model Equation 1 324 6.3.6.4 Model Equation 2 328 6.3.6.5 Model Equation 3 330 6.3.6.6 Ritz Approximations 332 6.4 Weighted-Residual Methods 337 6.4.1 Introduction 337 6.4.2 The General Method of Weighted Residuals 339 6.4.3 The Galerkin Method 44 6.4.4 The Least-Squares Method 349 6.4.5 The Collocation Method 356 6.4.6 The Subdomain Method 359 6.4.7 Eigenvalue and Time-Dependent Problems 361 6.4.7.1 Eigenvalue Problems 361 6.4.7.2 Time-Dependent Problems 362 6.5 Summary 381 Problems 383 7. Theory and Analysis of Plates 391 7.1 Introduction 391 7.1.1 General Comments 391 7.1.2 An Overview of Plate Theories 393 7.1.2.1 The Classical Plate Theory 394 7.1.2.2 The First-Order Plate Theory 395 7.1.2.3 The Third-Order Plate Theory 396 7.1.2.4 Stress-Based Theories 397 7.2 The Classical Plate Theory 398 7.2.1 Governing Equations of Circular Plates 398 7.2.2 Analysis of Circular Plates 405 7.2.2.1 Analytical Solutions For Bending 405 7.2.2.2 Analytical Solutions For Buckling 411 7.2.2.3 Variational Solutions 414 7.2.3 Governing Equations in Rectangular Coordinates 427 7.2.4 Navier Solutions of Rectangular Plates 435 7.2.4.1 Bending 438 7.2.4.2 Natural Vibration 443 7.2.4.3 Buckling Analysis 445 7.2.4.4 Transient Analysis 447 7.2.5 Lévy Solutions of Rectangular Plates 449 7.2.6 Variational Solutions: Bending 454 7.2.7 Variational Solutions: Natural Vibration 470 7.2.8 Variational Solutions: Buckling 475 7.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides 475 7.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions 478 7.3 The First-Order Shear Deformation Plate Theory 486 7.3.1 Equations of Circular Plates 486 7.3.2 Exact Solutions of Axisymmetric Circular Plates 488 7.3.3 Equations of Plates in Rectangular Coordinates 492 7.3.4 Exact Solutions of Rectangular Plates 496 7.3.4.1 Bending Analysis 498 7.3.4.2 Natural Vibration 501 7.3.4.3 Buckling Analysis 502 7.3.5 Variational Solutions of Circular and Rectangular Plates 503 7.3.5.1 Axisymmetric Circular Plates 503 7.3.5.2 Rectangular Plates 505 7.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories 507 7.4.1 Beams 507 7.4.1.1 Governing Equations 508 7.4.1.2 Relationships Between BET and TBT 508 7.4.2 Circular Plates 512 7.4.3 Rectangular Plates 516 7.5 Summary 521 Problems 521 8. The Finite Element Method 527 8.1 Introduction 527 8.2 Finite Element Analysis of Straight Bars 529 8.2.1 Governing Equation 529 8.2.2 Representation of the Domain by Finite Elements 530 8.2.3 Weak Form over an Element 531 8.2.4 Approximation over an Element 532 8.2.5 Finite Element Equations 537 8.2.5.1 Linear Element 538 8.2.5.2 Quadratic Element 539 8.2.6 Assembly (Connectivity) of Elements 539 8.2.7 Imposition of Boundary Conditions 542 8.2.8 Postprocessing 543 8.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory 549 8.3.1 Governing Equation 549 8.3.2 Weak Form over an Element 549 8.3.3 Derivation of the Approximation Functions 550 8.3.4 Finite Element Model 552 8.3.5 Assembly of Element Equations 553 8.3.6 Imposition of Boundary Conditions 555 8.4 Finite Element Analysis of the Timoshenko Beam Theory 558 8.4.1 Governing Equations 558 8.4.2 Weak Forms 558 8.4.3 Finite Element Models 559 8.4.4 Reduced Integration Element (RIE) 559 8.4.5 Consistent Interpolation Element (CIE) 561 8.4.6 Superconvergent Element (SCE) 562 8.5 Finite Element Analysis of the Classical Plate Theory 565 8.5.1 Introduction 565 8.5.2 General Formulation 566 8.5.3 Conforming and Nonconforming Plate Elements 568 8.5.4 Fully Discretized Finite Element Models 569 8.5.4.1 Static Bending 569 8.5.4.2 Buckling 569 8.5.4.3 Natural Vibration 570 8.5.4.4 Transient Response 570 8.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory 574 8.6.1 Governing Equations and Weak Forms 574 8.6.2 Finite Element Approximations 576 8.6.3 Finite Element Model 577 8.6.4 Numerical Integration 579 8.6.5 Numerical Examples 582 8.6.5.1 Isotropic Plates 582 8.6.5.2 Laminated Plates 584 8.7 Summary 587 Problems 588 9. Mixed Variational and Finite Element Formulations 595 9.1 Introduction 595 9.1.1 General Comments 595 9.1.2 Mixed Variational Principles 595 9.1.3 Extremum and Stationary Behavior of Functionals 597 9.2 Stationary Variational Principles 599 9.2.1 Minimum Total Potential Energy 599 9.2.2 The Hellinger-Reissner Variational Principle 601 9.2.3 The Reissner Variational Principle 605 9.3 Variational Solutions Based on Mixed Formulations 606 9.4 Mixed Finite Element Models of Beams 610 9.4.1 The Bernoulli-Euler Beam Theory 610 9.4.1.1 Governing Equations And Weak Forms 610 9.4.1.2 Weak-Form Mixed Finite Element Model 610 9.4.1.3 Weighted-Residual Finite Element Models 613 9.4.2 The Timoshenko Beam Theory 615 9.4.2.1 Governing Equations 615 9.4.2.2 General Finite Element Model 615 9.4.2.3 ASD-LLCC Element 617 9.4.2.4 ASD-QLCC Element 617 9.4.2.5 ASD-HQLC Element 618 9.5 Mixed Finite Element Analysis of the Classical Plate Theory 620 9.5.1 Preliminary Comments 620 9.5.2 Mixed Model I 620 9.5.2.1 Governing Equations 620 9.5.2.2 Weak Forms 621 9.5.2.3 Finite Element Model 622 9.5.3 Mixed Model II 625 9.5.3.1 Governing Equations 625 9.5.3.2 Weak Forms 625 9.5.3.3 Finite Element Model 626 9.6 Summary 630 Problems 631 10. Analysis of Functionally Graded Beams and Plates 635 10.1 Introduction 635 10.2 Functionally Graded Beams 638 10.2.1 The Bernoulli-Euler Beam Theory 638 10.2.1.1 Displacement and strain fields 638 10.2.1.2 Equations of motion and boundary conditions 638 10.2.2 The Timoshenko Beam Theory 639 10.2.2.1 Displacement and strain fields 639 10.2.2.2 Equations of motion and boundary conditions 640 10.2.3 Equations of Motion in terms of Generalized Displacements 641 10.2.3.1 Constitutive Equations 641 10.2.3.2 Stress Resultants of BET 641 10.2.3.3 Stress Resultants of TBT 642 10.2.3.4 Equations of Motion of the BET 642 10.2.3.5 Equations of Motion of the TBT 642 10.2.4 Stiffiness Coefficients643 10.3 Functionally Graded Circular Plates 645 10.3.1 Introduction 645 10.3.2 Classical Plate Theory 646 10.3.2.1 Displacement and Strain Fields 646 10.3.2.2 Equations of Motion 646 10.3.3 First-Order Shear Deformation Theory 647 10.3.3.1 Displacement and Strain Fields 647 10.3.3.2 Equations of Motion 648 10.3.4 Plate Constitutive Relations 649 10.3.4.1 Classical Plate Theory 649 10.3.4.2 First-Order Plate Theory 649 10.4 A General Third-Order Plate Theory 650 10.4.1 Introduction 650 10.4.2 Displacements and Strains 651 10.4.3 Equations of Motion 653 10.4.4 Constitutive Relations 657 10.4.5 Specialization to Other Theories 658 10.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces 658 10.4.5.2 The Reddy Third-Order Plate Theory 661 10.4.5.3 The First-Order Plate Theory 663 10.4.5.4 The Classical Plate Theory 664 10.5 Navier's Solutions 664 10.5.1 Preliminary Comments 664 10.5.2 Analysis of Beams 665 10.5.2.1 Bernoulli-Euler Beams 665 10.5.2.2 Timoshenko Beams 667 10.5.2.3 Numerical Results 669 10.5.3 Analysis of Plates 671 10.5.3.1 Boundary Conditions 672 10.5.3.2 Expansions of Generalized Displacements 672 10.5.3.3 Bending Analysis 673 10.5.3.4 Free Vibration Analysis 676 10.5.3.5 Buckling Analysis 677 10.5.3.6 Numerical Results 679 10.6 Finite Element Models 681 10.6.1 Bending of Beams 681 10.6.1.1 Bernoulli-Euler Beam Theory 681 10.6.1.2 Timoshenko Beam Theory 683 10.6.2 Axisymmetric Bending of Circular Plates 684 10.6.2.1 Classical Plate Theory 681 10.6.2.2 First-Order Shear Deformation Plate Theory 686 10.6.3 Solution of Nonlinear Equations 688 10.6.3.1 Times approximation 688 10.6.3.2 Newton's Iteration Approach 688 10.6.3.3 Tangent Stiffiness Coefficients for the BET 690 10.6.3.4 Tangent Stiffiness Coefficients for the TBT 692 10.6.3.5 Tangent Stiffiness Coefficients for the CPT 693 10.6.3.6 Tangent Stiffiness Coefficients for the FSDT 693 10.6.4 Numerical Results for Beams and Circular Plates 694 10.6.4.1 Beams 694 10.6.4.2 Circular Plates 697 10.7 Summary 699 Problems 700 References 701 Answers to Most Problems 711 Index 723

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Book SynopsisProvides a one-stop resource for engineers learning biostatistics using MATLAB and WinBUGS Through its scope and depth of coverage, this book addresses the needs of the vibrant and rapidly growing bio-oriented engineering fields while implementing software packages that are familiar to engineers. The book is heavily oriented to computation and hands-on approaches so readers understand each step of the programming. Another dimension of this book is in parallel coverage of both Bayesian and frequentist approaches to statistical inference. It avoids taking sides on the classical vs. Bayesian paradigms, and many examples in this book are solved using both methods. The results are then compared and commented upon. Readers have the choice of MATLAB for classical data analysis and WinBUGS/OpenBUGS for Bayesian data analysis. Every chapter starts with a box highlighting what is covered in that chapter and ends with exercises, a list of software scripts, datasets, and referenceTable of ContentsPreface v 1 Introduction 1 Chapter References 7 2 The Sample and Its Properties 9 2.1 Introduction 9 2.2 A MATLAB Session on Univariate Descriptive Statistics 10 2.3 Location Measures 12 2.4 Variability Measures 15 2.4.1 Ranks 24 2.5 Displaying Data 25 2.6 Multidimensional Samples: Fisher’s Iris Data and Body Fat Data 29 2.7 Multivariate Samples and Their Summaries 35 2.8 Principal Components of Data 40 2.9 Visualizing Multivariate Data 45 2.10 Observations as Time Series 49 2.11 About Data Types 52 2.12 Big Data Paradigm 53 2.13 Exercises 55 Chapter References 70 3 Probability, Conditional Probability, and Bayes’ Rule 73 3.1 Introduction 73 3.2 Events and Probability 74 3.3 Odds 85 3.4 Venn Diagrams 86 3.5 Counting Principles 88 3.6 Conditional Probability and Independence 92 3.6.1 Pairwise and Global Independence 97 3.7 Total Probability 97 3.8 Reassesing Probabilities: Bayes’ Rule 100 3.9 Bayesian Networks 105 3.10 Exercises 111 Chapter References 130 4 Sensitivity, Specificity, and Relatives 133 4.1 Introduction 133 4.2 Notation 134 4.2.1 Conditional Probability Notation 138 4.3 Combining Two or More Tests 141 4.4 ROC Curves 144 4.5 Exercises 149 Chapter References 157 5 Random Variables 159 5.1 Introduction 159 5.2 Discrete Random Variables 161 5.2.1 Jointly Distributed Discrete Random Variables 166 5.3 Some Standard Discrete Distributions 169 5.3.1 Discrete Uniform Distribution 169 5.3.2 Bernoulli and Binomial Distributions 170 5.3.3 Hypergeometric Distribution 174 5.3.4 Poisson Distribution 177 5.3.5 Geometric Distribution 180 5.3.6 Negative Binomial Distribution 183 5.3.7 Multinomial Distribution 184 5.3.8 Quantiles 186 5.4 Continuous Random Variables 187 5.4.1 Joint Distribution of Two Continuous Random Variables 192 5.4.2 Conditional Expectation 193 5.5 Some Standard Continuous Distributions 195 5.5.1 Uniform Distribution 196 5.5.2 Exponential Distribution 198 5.5.3 Normal Distribution 200 5.5.4 Gamma Distribution 201 5.5.5 Inverse Gamma Distribution 203 5.5.6 Beta Distribution 203 5.5.7 Double Exponential Distribution 205 5.5.8 Logistic Distribution 206 5.5.9 Weibull Distribution 207 5.5.10 Pareto Distribution 208 5.5.11 Dirichlet Distribution 209 5.6 Random Numbers and Probability Tables 210 5.7 Transformations of Random Variables 211 5.8 Mixtures 214 5.9 Markov Chains 215 5.10 Exercises 219 Chapter References 232 6 Normal Distribution 235 6.1 Introduction 235 6.2 Normal Distribution 236 6.2.1 Sigma Rules 240 6.2.2 Bivariate Normal Distribution 241 6.3 Examples with a Normal Distribution 243 6.4 Combining Normal Random Variables 246 6.5 Central Limit Theorem 249 6.6 Distributions Related to Normal 253 6.6.1 Chi-square Distribution 254 6.6.2 t-Distribution 258 6.6.3 Cauchy Distribution 259 6.6.4 F-Distribution 260 6.6.5 Noncentral χ2, t, and F Distributions 262 6.6.6 Lognormal Distribution 263 6.7 Delta Method and Variance Stabilizing Transformations 265 6.8 Exercises 268 Chapter References 274 7 Point and Interval Estimators 277 7.1 Introduction 277 7.2 Moment Matching and Maximum Likelihood Estimators 278 7.2.1 Unbiasedness and Consistency of Estimators 285 7.3 Estimation of a Mean, Variance, and Proportion 288 7.3.1 Point Estimation of Mean 288 7.3.2 Point Estimation of Variance 290 7.3.3 Point Estimation of Population Proportion 294 7.4 Confidence Intervals 295 7.4.1 Confidence Intervals for the Normal Mean 296 7.4.2 Confidence Interval for the Normal Variance 299 7.4.3 Confidence Intervals for the Population Proportion . . . 302 7.4.4 Confidence Intervals for Proportions When X = 0 306 7.4.5 Designing the Sample Size with Confidence Intervals 307 7.5 Prediction and Tolerance Intervals 309 7.6 Confidence Intervals for Quantiles 311 7.7 Confidence Intervals for the Poisson Rate 312 7.8 Exercises 315 Chapter References 328 8 Bayesian Approach to Inference 331 8.1 Introduction 331 8.2 Ingredients for Bayesian Inference 334 8.3 Conjugate Priors 338 8.4 Point Estimation 340 8.4.1 Normal-Inverse Gamma Conjugate Analysis 343 8.5 Prior Elicitation 345 8.6 Bayesian Computation and Use of WinBUGS 348 8.6.1 Zero Tricks in WinBUGS 351 8.7 Bayesian Interval Estimation: Credible Sets 353 8.8 Learning by Bayes’ Theorem 357 8.9 Bayesian Prediction 358 8.10 Consensus Means 362 8.11 Exercises 365 Chapter References 372 9 Testing Statistical Hypotheses 375 9.1 Introduction 375 9.2 Classical Testing Problem 377 9.2.1 Choice of Null Hypothesis 377 9.2.2 Test Statistic, Rejection Regions, Decisions, and Errors in Testing 379 9.2.3 Power of the Test 380 9.2.4 Fisherian Approach: p-Values 381 9.3 Bayesian Approach to Testing 382 9.3.1 Criticism and Calibration of p-Values 386 9.4 Testing the Normal Mean 388 9.4.1 z-Test 389 9.4.2 Power Analysis of a z-Test 389 9.4.3 Testing a Normal Mean When the Variance Is Not Known: t-Test 391 9.4.4 Power Analysis of t-Test 394 9.5 Testing Multivariate Mean: T-Square Test∗ 397 9.5.1 T-Square Test 397 9.5.2 Test for Symmetry 401 9.6 Testing the Normal Variances 402 9.7 Testing the Proportion 404 9.7.1 Exact Test for Population Proportions 406 9.7.2 Bayesian Test for Population Proportions 409 9.8 Multiplicity in Testing, Bonferroni Correction, and False Discovery Rate 412 9.9 Exercises 415 Chapter References 425 10 Two Samples 427 10.1 Introduction 427 10.2 Means and Variances in Two Independent Normal Populations 428 10.2.1 Confidence Interval for the Difference of Means 433 10.2.2 Power Analysis for Testing Two Means 434 10.2.3 More Complex Two-Sample Designs 438 10.2.4 A Bayesian Test for Two Normal Means 439 10.3 Testing the Equality of Normal Means When Samples Are Paired 443 10.3.1 Sample Size in Paired t-Test 448 10.3.2 Difference-in-Differences (DiD) Tests 449 10.4 Two Multivariate Normal Means 451 10.4.1 Confidence Intervals for Arbitrary Linear Combinations of Mean Differences 453 10.4.2 Profile Analysis With Two Independent Groups 454 10.4.3 Paired Multivariate Samples 456 10.5 Two Normal Variances 459 10.6 Comparing Two Proportions 463 10.6.1 The Sample Size 465 10.7 Risk Differences, Risk Ratios, and Odds Ratios 466 10.7.1 Risk Differences 466 10.7.2 Risk Ratio 467 10.7.3 Odds Ratios 469 10.7.4 Two Proportions from a Single Sample 473 10.8 Two Poisson Rates 476 10.9 Equivalence Tests 479 10.10 Exercises 483 Chapter References 500 11 ANOVA and Elements of Experimental Design 503 11.1 Introduction 503 11.2 One-Way ANOVA 504 11.2.1 ANOVA Table and Rationale for F-Test 506 11.2.2 Testing Assumption of Equal Population Variances . . . 509 11.2.3 The Null Hypothesis Is Rejected. What Next? 511 11.2.4 Bayesian Solution 516 11.2.5 Fixed- and Random-Effect ANOVA 518 11.3 Welch’s ANOVA 518 11.4 Two-Way ANOVA and Factorial Designs 521 11.4.1 Two-way ANOVA: One Observation Per Cell 527 11.5 Blocking 529 11.6 Repeated Measures Design 531 11.6.1 Sphericity Tests 534 11.7 Nested Designs 535 11.8 Power Analysis in ANOVA 539 11.9 Functional ANOVA 545 11.10 Analysis of Means (ANOM) 548 11.11 Gauge R&R ANOVA 550 11.12 Testing Equality of Several Proportions 556 11.13 Testing the Equality of Several Poisson Means 557 11.14 Exercises 559 Chapter References 582 12 Models for Tables 585 12.1 Introduction 586 12.2 Contingency Tables: Testing for Independence 586 12.2.1 Measuring Association in Contingency Tables 591 12.2.2 Power Analysis for Contingency Tables 593 12.2.3 Cohen’s Kappa 594 12.3 Three-Way Tables 596 12.4 Fisher’s Exact Test 600 12.5 Stratified Tables: Mantel–Haenszel Test 603 12.5.1 Testing Conditional Independence or Homogeneity . . . 604 12.5.2 Odds Ratio from Stratified Tables 607 12.6 Paired Tables: McNemar’s Test 608 12.7 Risk Differences, Risk Ratios, and Odds Ratios for Paired Tables 610 12.7.1 Risk Differences 610 12.7.2 Risk Ratios 611 12.7.3 Odds Ratios 612 12.7.4 Liddell’s Procedure 617 12.7.5 Garth Test 619 12.7.6 Stuart–Maxwell Test 620 12.7.7 Cochran’s Q Test∗ 626 12.8 Exercises 628 Chapter References 643 13 Correlation 647 13.1 Introduction 647 13.2 The Pearson Coefficient of Correlation 648 13.2.1 Inference About ρ 650 13.2.2 Bayesian Inference for Correlation Coefficients 663 13.3 Spearman’s Coefficient of Correlation 665 13.4 Kendall’s Tau 667 13.5 Cum hoc ergo propter hoc 670 13.6 Exercises 671 Chapter References 677 14 Regression 679 14.1 Introduction 679 14.2 Simple Linear Regression 680 14.2.1 Inference in Simple Linear Regression 688 14.3 Calibration 697 14.4 Testing the Equality of Two Slopes 699 14.5 Multiple Regression 702 14.5.1 Matrix Notation 703 14.5.2 Residual Analysis, Influential Observations, Multicollinearity, and Variable Selection 709 14.6 Sample Size in Regression 720 14.7 Linear Regression That Is Nonlinear in Predictors 720 14.8 Errors-In-Variables Linear Regression 723 14.9 Analysis of Covariance 724 14.9.1 Sample Size in ANCOVA 728 14.9.2 Bayesian Approach to ANCOVA 729 14.10 Exercises 731 Chapter References 748 15 Regression for Binary and Count Data 751 15.1 Introduction 751 15.2 Logistic Regression 752 15.2.1 Fitting Logistic Regression 753 15.2.2 Assessing the Logistic Regression Fit 758 15.2.3 Probit and Complementary Log-Log Links 769 15.3 Poisson Regression 773 15.4 Log-linear Models 779 15.5 Exercises 783 Chapter References 798 16 Inference for Censored Data and Survival Analysis 801 16.1 Introduction 801 16.2 Definitions 802 16.3 Inference with Censored Observations 807 16.3.1 Parametric Approach 807 16.3.2 Nonparametric Approach: Kaplan–Meier or Product–Limit Estimator 809 16.3.3 Comparing Survival Curves 815 16.4 The Cox Proportional Hazards Model 818 16.5 Bayesian Approach 822 16.5.1 Survival Analysis in WinBUGS 823 16.6 Exercises 829 Chapter References 835 17 Goodness of Fit Tests 837 17.1 Introduction 837 17.2 Probability Plots 838 17.2.1 Q–Q Plots 838 17.2.2 P–P Plots 841 17.2.3 Poissonness Plots 842 17.3 Pearson’s Chi-Square Test 843 17.4 Kolmogorov–Smirnov Tests 852 17.4.1 Kolmogorov’s Test 852 17.4.2 Smirnov’s Test to Compare Two Distributions 854 17.5 Cramér-von Mises and Watson’s Tests 858 17.5.1 Rosenblatt’s Test 860 17.6 Moran’s Test 862 17.7 Departures from Normality 863 17.7.1 Ellimination of Unknown Parameters by Transformations 866 17.8 Exercises 867 Chapter References 876 18 Distribution-Free Methods 879 18.1 Introduction 879 18.2 Sign Test 880 18.3 Wilcoxon Signed-Rank Test 884 18.4 Wilcoxon Sum Rank Test and Mann–Whitney Test 887 18.5 Kruskal–Wallis Test 890 18.6 Friedman’s Test 894 18.7 Resampling Methods 898 18.7.1 The Jackknife 898 18.7.2 Bootstrap 901 18.7.3 Bootstrap Versions of Some Popular Tests 908 18.7.4 Randomization and Permutation Tests 916 18.7.5 Discussion 919 18.8 Exercises 919 Chapter References 929 19 Bayesian Inference Using Gibbs Sampling – BUGS Project 931 19.1 Introduction 931 19.2 Step-by-Step Session 932 19.3 Built-in Functions and Common Distributions in WinBUGS 937 19.4 MATBUGS: A MATLAB Interface to WinBUGS 938 19.5 Exercises 942 Chapter References 943 Index 945

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Book SynopsisCovers multivariable calculus, starting from the basics and leading up to the three theorems of Green, Gauss, and Stokes, but always with an eye on practical applications. Written for a wide spectrum of undergraduate students by an experienced author, this book provides a very practical approach to advanced calculusstarting from the basics and leading up to the theorems of Green, Gauss, and Stokes. It explains, clearly and concisely, partial differentiation, multiple integration, vectors and vector calculus, and provides end-of-chapter exercises along with their solutions to aid the readers' understanding. Written in an approachable style and filled with numerous illustrative examples throughout, Two and Three Dimensional Calculus: with Applications in Science and Engineering assumes no prior knowledge of partial differentiation or vectors and explains difficult concepts with easy to follow examples. Rather than concentrating on mathematical structures, tTable of ContentsPreface xi 1 Revision of One-Dimensional Calculus 1 1.1 Limits and Convergence 1 1.2 Differentiation 3 1.2.1 Rules for Differentiation 5 1.2.2 Mean Value Theorem 7 1.2.3 Taylor’s Series 8 1.2.4 Maxima and Minima 12 1.2.5 Numerical Differentiation 13 1.3 Integration 16 Exercises 22 2 Partial Differentiation 25 2.1 Introduction 25 2.2 Differentials 29 2.2.1 Small Errors 30 2.3 Total Derivative 33 2.4 Chain Rule 36 2.4.1 Leibniz Rule 39 2.4.2 Chain Rule in n Dimensions 41 2.4.3 Implicit Functions 42 2.5 Jacobian 43 2.6 Higher Derivatives 46 2.6.1 Higher Differentials 49 2.7 Taylor’sTheorem 50 2.8 Conjugate Functions 52 2.9 Case Study:Thermodynamics 54 Exercises 58 3 Maxima and Minima 61 3.1 Introduction 61 3.2 Maxima, Minima and Saddle Points 63 3.3 Lagrange Multipliers 74 3.3.1 Generalisations 77 3.4 Optimisation 81 3.4.1 Hill Climbing Techniques 81 Exercises 85 4 Vector Algebra 89 4.1 Introduction 89 4.2 Vector Addition 90 4.3 Components 92 4.4 Scalar Product 94 4.5 Vector Product 97 4.5.1 Scalar Triple Product 102 4.5.2 Vector Triple Product 105 Exercises 106 5 Vector Differentiation 109 5.1 Introduction 109 5.2 Differential Geometry 111 5.2.1 Space Curves 112 5.2.2 Surfaces 120 5.3 Mechanics 129 Exercises 135 6 Gradient, Divergence, and Curl 139 6.1 Introduction 139 6.2 Gradient 139 6.3 Divergence 143 6.4 Curl 145 6.5 Vector Identities 146 6.6 Conjugate Functions 151 Exercises 154 7 Curvilinear Co-ordinates 157 7.1 Introduction 157 7.2 Curved Axes and Scale Factors 157 7.3 Curvilinear Gradient, Divergence, and Curl 161 7.3.1 Gradient 161 7.3.2 Divergence 163 7.3.3 Curl 165 7.4 Further Results and Tensors 166 7.4.1 Tensor Notation 166 7.4.2 Covariance and Contravariance 168 Exercises 171 8 PathIntegrals 173 8.1 Introduction 173 8.2 Integration Along a Curve 173 8.3 Practical Applications 181 Exercises 186 9 Multiple Integrals 191 9.1 Introduction 191 9.2 The Double Integral 191 9.2.1 Rotation and Translation 199 9.2.2 Change of Order of Integration 201 9.2.3 Plane Polar Co-ordinates 203 9.2.4 Applications of Double Integration 208 9.3 Triple Integration 213 9.3.1 Cylindrical and Spherical Polar Co-ordinates 219 9.3.2 Applications of Triple Integration 227 Exercises 233 10 Surface Integrals 241 10.1 Introduction 241 10.2 Green’s Theorem in the Plane 242 10.3 Integration over a Curved Surface 246 10.4 Applications of Surface Integration 253 Exercises 256 11 Integral Theorems 259 11.1 Introduction 259 11.2 Stokes’ Theorem 260 11.3 Gauss’ DivergenceTheorem 268 11.3.1 Green’s Second Identity 275 11.4 Co-ordinate-Free Definitions 277 11.5 Applications of Integral Theorems 279 11.5.1 Electromagnetic Theory 279 11.5.1.1 Maxwell’s Equations 279 11.5.2 Fluid Mechanics 283 11.5.3 ElasticityTheory 287 11.5.4 Heat Transfer 297 Exercises 298 12 Solutions and Answers to Exercises 301 References 375 Index 377

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Book SynopsisExplore the diverse electrical engineering application of polymer composite materials with this in-depth collection edited by leaders in the field Polymer Composites for Electrical Engineering delivers a comprehensive exploration of the fundamental principles, state-of-the-art research, and future challenges of polymer composites. Written from the perspective of electrical engineering applications, like electrical and thermal energy storage, high temperature applications, fire retardance, power cables, electric stress control, and others, the book covers all major application branches of these widely used materials. Rather than focus on polymer composite materials themselves, the distinguished editors have chosen to collect contributions from industry leaders in the area of real and practical electrical engineering applications of polymer composites. The book?s relevance will only increase as advanced polymer composites receive more attention and interest iTable of ContentsAbout the Author xi Preface to the Fourth Edition xiii 1 Introduction 1 The Case for Imprecision 2 A Historical Perspective 4 The Utility of Fuzzy Systems 7 Limitations of Fuzzy Systems 9 The Illusion: Ignoring Uncertainty and Accuracy 11 Uncertainty and Information 13 Fuzzy Sets and Membership 14 Chance versus Fuzziness 17 Intuition of Uncertainty: Fuzzy versus Probability 19 Sets as Points in Hypercubes 21 Summary 23 References 23 Problems 24 2 Classical Sets and Fuzzy Sets 27 Classical Sets 28 Fuzzy Sets 36 Summary 45 References 46 Problems 46 3 Classical Relations and Fuzzy Relations 51 Cartesian Product 52 Crisp Relations 53 Fuzzy Relations 58 Tolerance and Equivalence Relations 67 Fuzzy Tolerance and Equivalence Relations 70 Value Assignments 72 Other Forms of the Composition Operation 76 Summary 77 References 77 Problems 77 4 Properties of Membership Functions, Fuzzification, and Defuzzification 84 Features of the Membership Function 85 Various Forms 87 Fuzzification 88 Defuzzification to Crisp Sets 90 λ-Cuts for Fuzzy Relations 92 Defuzzification to Scalars 93 Summary 102 References 103 Problems 104 5 Logic and Fuzzy Systems 107 Part I: Logic 107 Classical Logic 108 Fuzzy Logic 122 Part II: Fuzzy Systems 132 Summary 151 References 153 Problems 154 6 Historical Methods of Developing Membership Functions 163 Membership Value Assignments 164 Intuition 164 Inference 165 Rank Ordering 167 Neural Networks 168 Genetic Algorithms 179 Inductive Reasoning 188 Summary 195 References 196 Problems 197 7 Automated Methods for Fuzzy Systems 201 Definitions 202 Batch Least Squares Algorithm 205 Recursive Least Squares Algorithm 210 Gradient Method 213 Clustering Method 218 Learning from Examples 221 Modified Learning from Examples 224 Summary 233 References 235 Problems 235 8 Fuzzy Systems Simulation 237 Fuzzy Relational Equations 242 Nonlinear Simulation Using Fuzzy Systems 243 Fuzzy Associative Memories (FAMs) 246 Summary 257 References 258 Problems 259 9 Decision Making with Fuzzy Information 265 Fuzzy Synthetic Evaluation 267 Fuzzy Ordering 269 Nontransitive Ranking 272 Preference and Consensus 275 Multiobjective Decision Making 279 Fuzzy Bayesian Decision Method 285 Decision Making under Fuzzy States and Fuzzy Actions 295 Summary 309 References 310 Problems 311 10 Fuzzy Classification and Pattern Recognition 323 Fuzzy Classification 324 Classification by Equivalence Relations 324 Cluster Analysis 332 Cluster Validity 332 c-Means Clustering 333 Hard c-Means (HCM) 333 Fuzzy c-Means (FCM) 343 Classification Metric 351 Hardening the Fuzzy c-Partition 354 Similarity Relations from Clustering 356 Fuzzy Pattern Recognition 357 Single-Sample Identification 357 Multifeature Pattern Recognition 365 Summary 378 References 379 Problems 380 11 Fuzzy Control Systems 388 Control System Design Problem 390 Examples of Fuzzy Control System Design 393 Fuzzy Engineering Process Control 404 Fuzzy Statistical Process Control 417 Industrial Applications 431 Summary 434 References 437 Problems 438 12 Applications of Fuzzy Systems Using Miscellaneous Models 455 Fuzzy Optimization 455 Fuzzy Cognitive Mapping 462 Agent-Based Models 477 Fuzzy Arithmetic and the Extension Principle 481 Fuzzy Algebra 487 Data Fusion 491 Summary 498 References 498 Problems 500 13 Monotone Measures: Belief, Plausibility, Probability, and Possibility 505 Monotone Measures 506 Belief and Plausibility 507 Evidence Theory 512 Probability Measures 515 Possibility and Necessity Measures 517 Possibility Distributions as Fuzzy Sets 525 Possibility Distributions Derived from Empirical Intervals 528 Summary 548 References 549 Problems 550 Index 554

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Book SynopsisA problem-solving approach to statistical signal processing for practicing engineers, technicians, and graduate students This book takes a pragmatic approach in solving a set of common problems engineers and technicians encounter when processing signals.Table of ContentsList of Figures xvii List of Tables xxiii Preface xxv List of Abbreviations xxix How to Use the Book xxxi About the Companion Website xxxiii Prerequisites xxxv Why are there so many matrixes in this book? xxxvii 1 Manipulations on Matrixes 1 1.1 Matrix Properties 1 1.1.1 Elementary Operations 2 1.2 Eigen-Decomposition 6 1.3 Eigenvectors in Everyday Life 9 1.3.1 Conversations in a Noisy Restaurant 9 1.3.2 Power Control in a Cellular System 12 1.3.3 Price Equilibrium in the Economy 14 1.4 Derivative Rules 15 1.4.1 Derivative with respect to x 16 1.4.2 Derivative with respect to x 17 1.4.3 Derivative with respect to the Matrix X 18 1.5 Quadratic Forms 19 1.6 Diagonalization of a Quadratic Form 20 1.7 Rayleigh Quotient 21 1.8 Basics of Optimization 22 1.8.1 Quadratic Function with Simple Linear Constraint (M=1) 23 1.8.2 Quadratic Function with Multiple Linear Constraints 23 Appendix A: Arithmetic vs. Geometric Mean 24 2 Linear Algebraic Systems 27 2.1 Problem Definition and Vector Spaces 27 2.1.1 Vector Spaces in Tomographic Radiometric Inversion 29 2.2 Rotations 31 2.3 Projection Matrixes and Data-Filtering 33 2.3.1 Projections and Commercial FM Radio 34 2.4 Singular Value Decomposition (SVD) and Subspaces 34 2.4.1 How to Choose the Rank of Afor Gaussian Model? 35 2.5 QR and Cholesky Factorization 36 2.6 Power Method for Leading Eigenvectors 38 2.7 Least Squares Solution of Overdetermined Linear Equations 39 2.8 Efficient Implementation of the LS Solution 41 2.9 Iterative Methods 42 3 Random Variables in Brief 45 3.1 Probability Density Function (pdf), Moments, and Other Useful Properties 45 3.2 Convexity and Jensen Inequality 49 3.3 Uncorrelatedness and Statistical Independence 49 3.4 Real-Valued Gaussian Random Variables 51 3.5 Conditional pdf for Real-Valued Gaussian Random Variables 54 3.6 Conditional pdf in Additive Noise Model 56 3.7 Complex Gaussian Random Variables 56 3.7.1 Single Complex Gaussian Random Variable 56 3.7.2 Circular Complex Gaussian Random Variable 57 3.7.3 Multivariate Complex Gaussian Random Variables 58 3.8 Sum of Square of Gaussians: Chi-Square 59 3.9 Order Statistics for N rvs 60 4 Random Processes and Linear Systems 63 4.1 Moment Characterizations and Stationarity 64 4.2 Random Processes and Linear Systems 66 4.3 Complex-Valued Random Processes 68 4.4 Pole-Zero and Rational Spectra (Discrete-Time) 69 4.4.1 Stability of LTI Systems 70 4.4.2 Rational PSD 71 4.4.3 Paley–Wiener Theorem 72 4.5 Gaussian Random Process (Discrete-Time) 73 4.6 Measuring Moments in Stochastic Processes 75 Appendix A: Transforms for Continuous-Time Signals 76 Appendix B: Transforms for Discrete-Time Signals 79 5 Models and Applications 83 5.1 Linear Regression Model 84 5.2 Linear Filtering Model 86 5.2.1 Block-Wise Circular Convolution 88 5.2.2 Discrete Fourier Transform and Circular Convolution Matrixes 89 5.2.3 Identification and Deconvolution 90 5.3 MIMO systems and Interference Models 91 5.3.1 DSL System 92 5.3.2 MIMO in Wireless Communication 92 5.4 Sinusoidal Signal 97 5.5 Irregular Sampling and Interpolation 97 5.5.1 Sampling With Jitter 100 5.6 Wavefield Sensing System 101 6 Estimation Theory 105 6.1 Historical Notes 105 6.2 Non-Bayesian vs. Bayesian 106 6.3 Performance Metrics and Bounds 107 6.3.1 Bias 107 6.3.2 Mean Square Error (MSE) 108 6.3.3 Performance Bounds 109 6.4 Statistics and Sufficient Statistics 110 6.5 MVU and BLU Estimators 111 6.6 BLUE for Linear Models 112 6.7 Example: BLUE of the Mean Value of Gaussian rvs 114 7 Parameter Estimation 117 7.1 Maximum Likelihood Estimation (MLE) 117 7.2 MLE for Gaussian Model 119 7.2.1 Additive Noise Model with 119 7.2.2 Additive Noise Model with 120 7.2.3 Additive Noise Model with Multiple Observations with Known 121 7.2.3.1 Linear Model 121 7.2.3.2 Model 122 7.2.3.3 Model 123 7.2.4 Model 123 7.2.5 Additive Noise Model with Multiple Observations with Unknown 124 7.3 Other Noise Models 125 7.4 MLE and Nuisance Parameters 126 7.5 MLE for Continuous-Time Signals 128 7.5.1 Example: Amplitude Estimation 129 7.5.2 MLE for Correlated Noise 130 7.6 MLE for Circular Complex Gaussian 131 7.7 Estimation in Phase/Frequency Modulations 131 7.7.1 MLE Phase Estimation 132 7.7.2 Phase Locked Loops 133 7.8 Least Square (LS) Estimation 135 7.8.1 Weighted LS with 136 7.8.2 LS Estimation and Linear Models 137 7.8.3 Under or Over-Parameterizing? 138 7.8.4 Constrained LS Estimation 139 7.9 Robust Estimation 140 8 Cramér–Rao Bound 143 8.1 Cramér–Rao Bound and Fisher Information Matrix 143 8.1.1 CRB for Scalar Problem (P=1) 143 8.1.2 CRB and Local Curvature of Log-Likelihood 144 8.1.3 CRB for Multiple Parameters (p 1) 144 8.2 Interpretation of CRB and Remarks 146 8.2.1 Variance of Each Parameter 146 8.2.2 Compactness of the Estimates 146 8.2.3 FIM for Known Parameters 147 8.2.4 Approximation of the Inverse of FIM 148 8.2.5 Estimation Decoupled From FIM 148 8.2.6 CRB and Nuisance Parameters 149 8.2.7 CRB for Non-Gaussian rv and Gaussian Bound 149 8.3 CRB and Variable Transformations 150 8.4 FIM for Gaussian Parametric Model 151 8.4.1 FIM for with 151 8.4.2 FIM for Continuous-Time Signals in Additive White Gaussian Noise 152 8.4.3 FIM for Circular Complex Model 152 Appendix A: Proof of CRB 154 Appendix B: FIM for Gaussian Model 156 Appendix C: Some Derivatives for MLE and CRB Computations 157 9 MLE and CRB for Some Selected Cases 159 9.1 Linear Regressions 159 9.2 Frequency Estimation 162 9.3 Estimation of Complex Sinusoid 164 9.3.1 Proper, Improper, and Non-Circular Signals 165 9.4 Time of Delay Estimation 166 9.5 Estimation of Max for Uniform pdf 170 9.6 Estimation of Occurrence Probability for Binary pdf 172 9.7 How to Optimize Histograms? 173 9.8 Logistic Regression 176 10 Numerical Analysis and Montecarlo Simulations 179 10.1 System Identification and Channel Estimation 181 10.1.1 Matlab Code and Results 184 10.2 Frequency Estimation 184 10.2.1 Variable (Coarse/Fine) Sampling 187 10.2.2 Local Parabolic Regression 189 10.2.3 Matlab Code and Results 190 10.3 Time of Delay Estimation 192 10.3.1 Granularity of Sampling in ToD Estimation 193 10.3.2 Matlab Code and Results 194 10.4 Doppler-Radar System by Frequency Estimation 196 10.4.1 EM Method 197 10.4.2 Matlab Code and Results 199 11 Bayesian Estimation 201 11.1 Additive Linear Model with Gaussian Noise 203 11.1.1 Gaussian A-priori: 204 11.1.2 Non-Gaussian A-Priori 206 11.1.3 Binary Signals: MMSE vs. MAP Estimators 207 11.1.4 Example: Impulse Noise Mitigation 210 11.2 Bayesian Estimation in Gaussian Settings 212 11.2.1 MMSE Estimator 213 11.2.2 MMSE Estimator for Linear Models 213 11.3 LMMSE Estimation and Orthogonality 215 11.4 Bayesian CRB 218 11.5 Mixing Bayesian and Non-Bayesian 220 11.5.1 Linear Model with Mixed Random/Deterministic Parameters 220 11.5.2 Hybrid CRB 222 11.6 Expectation-Maximization (EM) 223 11.6.1 EM of the Sum of Signals in Gaussian Noise 224 11.6.2 EM Method for the Time of Delay Estimation of Multiple Waveforms 227 11.6.3 Remarks 228 Appendix A: Gaussian Mixture pdf 229 12 Optimal Filtering 231 12.1 Wiener Filter 231 12.2 MMSE Deconvolution (or Equalization) 233 12.3 Linear Prediction 234 12.3.1 Yule–Walker Equations 235 12.4 LS Linear Prediction 237 12.5 Linear Prediction and AR Processes 239 12.6 Levinson Recursion and Lattice Predictors 241 13 Bayesian Tracking and Kalman Filter 245 13.1 Bayesian Tracking of State in Dynamic Systems 246 13.1.1 Evolution of the A-posteriori pdf 247 13.2 Kalman Filter (KF) 249 13.2.1 KF Equations 251 13.2.2 Remarks 253 13.3 Identification of Time-Varying Filters in Wireless Communication 255 13.4 Extended Kalman Filter (EKF) for Non-Linear Dynamic Systems 257 13.5 Position Tracking by Multi-Lateration 258 13.5.1 Positioning and Noise 260 13.5.2 Example of Position Tracking 263 13.6 Non-Gaussian Pdf and Particle Filters264 14 Spectral Analysis 267 14.1 Periodogram 268 14.1.1 Bias of the Periodogram 268 14.1.2 Variance of the Periodogram 271 14.1.3 Filterbank Interpretation 273 14.1.4 Pdf of the Periodogram (White Gaussian Process) 274 14.1.5 Bias and Resolution 275 14.1.6 Variance Reduction and WOSA 278 14.1.7 Numerical Example: Bandlimited Process and (Small) Sinusoid 280 14.2 Parametric Spectral Analysis 282 14.2.1 MLE and CRB 284 14.2.2 General Model for AR, MA, ARMA Spectral Analysis 285 14.3 AR Spectral Analysis 286 14.3.1 MLE and CRB 286 14.3.2 A Good Reason to Avoid Over-Parametrization in AR 289 14.3.3 Cramér–Rao Bound of Poles in AR Spectral Analysis 291 14.3.4 Example: Frequency Estimation by AR Spectral Analysis 293 14.4 MA Spectral Analysis 296 14.5 ARMA Spectral Analysis 298 14.5.1 Cramér–Rao Bound for ARMA Spectral Analysis 300 Appendix A: Which Sample Estimate of the Autocorrelation to Use? 302 Appendix B: Eigenvectors and Eigenvalues of Correlation Matrix 303 Appendix C: Property of Monic Polynomial 306 Appendix D: Variance of Pole in AR(1) 307 15 Adaptive Filtering 309 15.1 Adaptive Interference Cancellation 311 15.2 Adaptive Equalization in Communication Systems 313 15.2.1 Wireless Communication Systems in Brief 313 15.2.2 Adaptive Equalization 315 15.3 Steepest Descent MSE Minimization 317 15.3.1 Convergence Analysis and Step-Size 318 15.3.2 An Intuitive View of Convergence Conditions 320 15.4 From Iterative to Adaptive Filters 323 15.5 LMS Algorithm and Stochastic Gradient 324 15.6 Convergence Analysis of LMS Algorithm 325 15.6.1 Convergence in the Mean 326 15.6.2 Convergence in the Mean Square 326 15.6.3 Excess MSE 329 15.7 Learning Curve of LMS 331 15.7.1 Optimization of the Step-Size 332 15.8 NLMS Updating and Non-Stationarity 333 15.9 Numerical Example: Adaptive Identification 334 15.10 RLS Algorithm 338 15.10.1 Convergence Analysis 339 15.10.2 Learning Curve of RLS 341 15.11 Exponentially-Weighted RLS 342 15.12 LMS vs. RLS 344 Appendix A: Convergence in Mean Square 344 16 Line Spectrum Analysis 347 16.1 Model Definition 349 16.1.1 Deterministic Signals 350 16.1.2 Random Signals 350 16.1.3 Properties of Structured Covariance 351 16.2 Maximum Likelihood and Cramér–Rao Bounds 352 16.2.1 Conditional ML 353 16.2.2 Cramér–Rao Bound for Conditional Model 354 16.2.3 Unconditional ML 356 16.2.4 Cramér–Rao Bound for Unconditional Model 356 16.2.5 Conditional vs. Unconditional Model & Bounds 357 16.3 High-Resolution Methods 357 16.3.1 Iterative Quadratic ML (IQML) 358 16.3.2 Prony Method 360 16.3.3 MUSIC 360 16.3.4 ESPRIT 363 16.3.5 Model Order 365 17 Equalization in Communication Engineering 367 17.1 Linear Equalization 369 17.1.1 Zero Forcing (ZF) Equalizer 370 17.1.2 Minimum Mean Square Error (MMSE) Equalizer 371 17.1.3 Finite-Length/Finite-Block Equalizer 371 17.2 Non-Linear Equalization 372 17.2.1 ZF-DFE 373 17.2.2 MMSE–DFE 374 17.2.3 Finite-Length MMSE–DFE 375 17.2.4 Asymptotic Performance for Infinite-Length Equalizers 376 17.3 MIMO Linear Equalization 377 17.3.1 ZF MIMO Equalization 377 17.3.2 MMSE MIMO Equalization 379 17.4 MIMO–DFE Equalization 379 17.4.1 Cholesky Factorization and Min/Max Phase Decomposition 379 17.4.2 MIMO–DFE 380 18 2D Signals and Physical Filters 383 18.1 2D Sinusoids 384 18.1.1 Moiré Pattern 386 18.2 2D Filtering 388 18.2.1 2D Random Fields 390 18.2.2 Wiener Filtering 391 18.2.3 Image Acquisition and Restoration 392 18.3 Diffusion Filtering 394 18.3.1 Evolution vs. Time: Fourier Method 394 18.3.2 Extrapolation of the Density 395 18.3.3 Effect of Phase-Shift 396 18.4 Laplace Equation and Exponential Filtering 398 18.5 Wavefield Propagation 400 18.5.1 Propagation/Backpropagation 400 18.5.2 Wavefield Extrapolation and Focusing 402 18.5.3 Exploding Reflector Model 402 18.5.4 Wavefield Extrapolation 404 18.5.5 Wavefield Focusing (or Migration) 406 Appendix A: Properties of 2D Signals 406 Appendix B: Properties of 2D Fourier Transform 410 Appendix C: Finite Difference Method for PDE-Diffusion 412 19 Array Processing 415 19.1 Narrowband Model 415 19.1.1 Multiple DoAs and Multiple Sources 419 19.1.2 Sensor Spacing Design 420 19.1.3 Spatial Resolution and Array Aperture 421 19.2 Beamforming and Signal Estimation 422 19.2.1 Conventional Beamforming 425 19.2.2 Capon Beamforming (MVDR) 426 19.2.3 Multiple-Constraint Beamforming 429 19.2.4 Max-SNR Beamforming 431 19.3 DoA Estimation 432 19.3.1 ML Estimation and CRB 433 19.3.2 Beamforming and Root-MVDR 434 20 Multichannel Time of Delay Estimation 435 20.1 Model Definition for ToD 440 20.2 High Resolution Method for ToD (L=1) 441 20.2.1 ToD in the Fourier Transformed Domain 441 20.2.2 CRB and Resolution 444 20.3 Difference of ToD (DToD) Estimation 445 20.3.1 Correlation Method for DToD 445 20.3.2 Generalized Correlation Method 448 20.4 Numerical Performance Analysis of DToD 452 20.5 Wavefront Estimation: Non-Parametric Method (L=1) 454 20.5.1 Wavefront Estimation in Remote Sensing and Geophysics 456 20.5.2 Narrowband Waveforms and 2D Phase Unwrapping 457 20.5.3 2D Phase Unwrapping in Regular Grid Spacing 458 20.6 Parametric ToD Estimation and Wideband Beamforming 460 20.6.1 Delay and Sum Beamforming 462 20.6.2 Wideband Beamforming After Fourier Transform 464 20.7 Appendix A: Properties of the Sample Correlations 465 20.8 Appendix B: How to Delay a Discrete-Time Signal? 466 20.9 Appendix C: Wavefront Estimation for 2D Arrays 467 21 Tomography 467 21.1 X-ray Tomography 471 21.1.1 Discrete Model 471 21.1.2 Maximum Likelihood 473 21.1.3 Emission Tomography 473 21.2 Algebraic Reconstruction Tomography (ART) 475 21.3 Reconstruction From Projections: Fourier Method 475 21.3.1 Backprojection Algorithm 476 21.3.2 How Many Projections to Use? 479 21.4 Traveltime Tomography 480 21.5 Internet (Network) Tomography 483 21.5.1 Latency Tomography 484 21.5.2 Packet-Loss Tomography 484 22 Cooperative Estimation 487 22.1 Consensus and Cooperation 490 22.1.1 Vox Populi: The Wisdom of Crowds 490 22.1.2 Cooperative Estimation as Simple Information Consensus 490 22.1.3 Weighted Cooperative Estimation ( ) 493 22.1.4 Distributed MLE ( ) 495 22.2 Distributed Estimation for Arbitrary Linear Models (p>1) 496 22.2.1 Centralized MLE 497 22.2.2 Distributed Weighted LS 498 22.2.3 Distributed MLE 500 22.2.4 Distributed Estimation for Under-Determined Systems 501 22.2.5 Stochastic Regressor Model 503 22.2.6 Cooperative Estimation in the Internet of Things (IoT) 503 22.2.7 Example: Iterative Distributed Estimation 505 22.3 Distributed Synchronization 506 22.3.1 Synchrony-States for Analog and Discrete-Time Clocks 507 22.3.2 Coupled Clocks 510 22.3.3 Internet Synchronization and the Network Time Protocol (NTP) 512 Appendix A: Basics of Undirected Graphs 515 23 Classification and Clustering 521 23.1 Historical Notes 522 23.2 Classification 523 23.2.1 Binary Detection Theory 523 23.2.2 Binary Classification of Gaussian Distributions 528 23.3 Classification of Signals in Additive Gaussian Noise 529 23.3.1 Detection of Known Signal 531 23.3.2 Classification of Multiple Signals 532 23.3.3 Generalized Likelihood Ratio Test (GLRT) 533 23.3.4 Detection of Random Signals 535 23.4 Bayesian Classification 536 23.4.1 To Classify or Not to Classify? 537 23.4.2 Bayes Risk 537 23.5 Pattern Recognition and Machine Learning 538 23.5.1 Linear Discriminant 539 23.5.2 Least Squares Classification 540 23.5.3 Support Vectors Principle 541 23.6 Clustering 543 23.6.1 K-Means Clustering 544 23.6.2 EM Clustering 545 References 549 Index 557

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Book SynopsisCovering fractional order theory, simulation and experiments, this book explains how fractional order modelling and fractional order controller design compares favourably with traditional velocity and position control systems. The authors systematically compare the two approaches using applied fractional calculus.Table of ContentsAcronyms xix Foreword xxiii Preface xxv Acknowledgments xxix PART I FUNDAMENTALS OF FRACTIONAL CONTROLS 1 Introduction 3 1.1 Fractional Calculus 3 1.2 Fractional Order Controls 9 1.3 Fractional Order Motion Controls 20 1.4 Contributions 22 1.5 Organization 22 PART II FRACTIONAL ORDER VELOCITY SERVO 2 Fractional Order PI Controller Designs for Velocity Servo Systems 25 2.1 Introduction 25 2.2 FOPTD Systems and Three Controllers Considered 27 2.3 Design Specifications 27 2.4 Fractional Order PI and [PI] Controller Designs 28 2.5 Simulation 38 2.6 Chapter Summary 39 3 Tuning Fractional Order PI Controllers for Fractional Order Velocity Systems with Experimental Validation 41 3.1 Introduction 41 3.2 Three Controllers to Be Designed and Tuning Specifications 42 3.3 Tuning Three Controllers for FOVS 42 3.4 Illustrative Examples and Design Procedure Summaries 43 3.5 Simulation Illustration 45 3.6 Experimental Validation 49 3.7 Chapter Summary 54 4 Relay Feedback Tuning of Robust PID Controllers 59 4.1 Introduction 59 4.2 Slope Adjustment of the Phase Bode Plot 62 4.3 The New PID Controller Design Formulae 65 4.4 Phase and Magnitude Measurement Via Relay Feedback Tests 66 4.5 Illustrative Examples 67 4.6 Chapter Summary 72 5 Auto-Tuning of Fractional Order Controllers with Iso-Damping 73 5.1 Introduction 73 5.2 FOPI and FO[PI] Controllers Design Formulae 75 5.3 Measurements for Auto-Tuning 80 5.4 Simulation Illustration 80 5.5 Chapter Summary 87 PART III FRACTIONAL ORDER POSITION SERVO 6 Fractional Order PD Controller Tuning for Position Systems 91 6.1 Introduction 91 6.2 Fractional Order PD Controller Design for Position Servos 92 6.3 Design Procedures 94 6.4 Simulation Example 95 6.5 Experiments 99 6.6 Chapter Summary 101 7 Fractional Order [PD] Controller Synthesis for Position Servo Systems 105 7.1 Introduction 105 7.2 Position Control Plants and Design Specifications 106 7.3 Fractional Order [PD] Controller Design 106 7.4 Parameter Design Examples and Bode Plot Validations 108 7.5 Implementation of Two Fractional Order Operators 110 7.6 Simulation 111 7.7 Experiment 120 7.8 Chapter Summary 122 8 Time-Constant Robust Analysis and Design of Fractional Order [PD] Controller 123 8.1 Introduction 123 8.2 Problem Statement 124 8.3 FO[PD] Tuning Specifications and Rules 125 8.4 The Solution Existence Range and An Online Computation Method 127 8.5 Experiment 135 8.6 Chapter Summary 136 9 Experimental Study of Fractional OrderPDController Synthesis for Fractional Order Position Servo Systems 139 9.1 Introduction 139 9.2 Fractional Order Systems and Fractional Order Controller Considered 140 9.3 FOPD Controller Design Procedure for the Fractional Order Position Servo Systems 141 9.4 Simulation Illustration 144 9.5 Experimental Study 148 9.6 Chapter Summary 153 10 Fractional Order [PD] Controller Design and Comparison for Fractional Order Position Servo Systems 155 10.1 Introduction 155 10.2 Fractional Order Position Servo Systems and Fractional Order Controllers 156 10.3 Fractional Order [PD] Controller Design 156 10.4 Integer Order PID Controller and Fractional Order PD Controller Designs 159 10.5 Simulation Comparisons 160 10.6 Chapter Summary 162 PART IV STABILITY AND FEASIBILITY FOR FOPID DESIGN 11 Stability and Design Feasibility of Robust PID Controllers for FOPTD Systems 165 11.1 Introduction 165 11.2 Stability Region and Flat Phase Tuning Rule for the Robust PID Controller Design 168 11.3 PID Controller Design with Pre-Specifications on Ám and !c 171 11.4 Simulation Illustration 180 11.5 Chapter Summary 185 12 Stability and Design Feasibility of Robust FOPI Controllers for FOPTD Systems 187 12.1 Introduction 187 12.2 Stabilizing and Robust FOPI Controller Design for FOPTD Systems 188 12.3 Design Procedures Summary with An Illustrative Example 194 12.4 Complete Information Collection for Achievable Region of wc and Φm 197 12.5 Simulation Illustration 201 12.6 Chapter Summary 207 PART V FRACTIONAL ORDER DISTURBANCE COMPENSATORS 13 Fractional Order Disturbance Observer 211 13.1 Introduction 211 13.2 Disturbance Observer (DOB) 212 13.3 Actual Design Parameters In DOB and Their Effects 213 13.4 Loss of The Phase Margin With DOB 215 13.5 Solution One: Rule-Based Switched Low Pass Filtering With Varying Relative Degree 216 13.6 The Proposed Solution: Guaranteed Phase Margin Method Using Fractional Order Low Pass Filtering 216 13.7 Implementation Issues: Stable Minimum-Phase Frequency Domain Fitting 218 13.8 Chapter Summary 222 14 Fractional Order Adaptive Feed-forward Cancellation 223 14.1 Introduction 223 14.2 Fractional Order Adaptive Feed-forward Cancellation 225 14.3 Equivalence Between Fractional Order Internal Model Principle and Fractional Order Adaptive Feed-Forward Cancellation 229 14.4 Frequency-domain analysis of the FOAFC performance for the periodic disturbance 231 14.5 Simulation Illustration 233 14.6 Experiment Validation 237 14.7 Chapter Summary 241 15 Fractional Order Robust Control for Cogging Effect 243 15.1 Introduction 243 15.2 Fractional Order Robust Control of Cogging Effect Compensation 244 15.3 Simulation Illustration 252 15.4 Experiments on A Lab Testbed - Dynamometer 258 15.5 Chapter Summary 264 16 Fractional Order Periodic Adaptive Learning Compensation 275 16.1 Introduction 275 16.2 Fractional Order Periodic Adaptive learning Compensation for the State-dependent Periodic Disturbance 276 16.3 Simulation Illustrations 282 16.4 Experimental Validation 284 16.5 Chapter Summary 288 PART VI EFFECTS OF FRACTIONAL ORDER CONTROLS ON NONLINEARITIES 17 Fractional Order PID Control of A DC-Motor with Elastic Shaft 293 17.1 Introduction 293 17.2 The Benchmark Position Servo System 294 17.3 A Modified Approximate Realization Method 295 17.4 Comparative Simulations 297 17.5 Chapter Summary 305 18 Fractional Order Ultra Low-Speed Position Servo 313 18.1 Introduction 313 18.2 Ultra Low-Speed Position Tracking using Designed FOPD and Optimized IOPI 314 18.3 Static and Dynamic Models of Friction and DescribingFunctions for Friction Models 316 18.4 Simulation Analysis with IOPI and FOPD Controllers Using Describing Function 321 18.5 Extended Experimental Demonstration 324 18.6 Chapter Summary 325 19 Optimized Fractional Order Conditional Integrator 329 19.1 Introduction 329 19.2 Clegg Conditional Integrator 330 19.3 Intelligent Conditional Integrator 331 19.4 The Optimized Fractional Order Conditional Integrator 332 19.5 Simulation Validation 340 19.6 Chapter Summary 342 PART VII FRACTIONAL ORDER CONTROL APPLICATIONS 20 Lateral Directional Fractional Order Control of A Small Fixed-Wing UAV 345 20.1 Introduction 345 20.2 Flight Control System of Small Fixed-Wing UAV 346 20.3 Integer/Fractional Order Controller Designs 351 20.4 Modified Ziegler-Nichols PI Controller Design 352 20.5 Fractional Order (PI)¸ Controller Design 353 20.6 Fractional Order PI Controller Design 355 20.7 Integer Order PID Controller Design 356 20.8 Simulation Illustration 357 20.9 Flight Experiments 363 20.10 Chapter Summary 367 21 Fractional Order PD Controller Synthesis and Implementation for HDD Servo System 369 21.1 Introduction 369 21.2 Fractional Order Controller Design with “Flat Phase” 370 21.3 Implementation of the Fractional Order Controller 372 21.4 Readjustment for the Designed FOPD Controller 377 21.5 Experiment 380 21.6 Chapter Summary 383 References 385 Index 403

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Book SynopsisThis book describes the basics and developments of the new XFEM approach to fracture analysis of structures and materials, providing state of the art techniques and algorithms for fracture analysis of structures.Table of ContentsPreface xiii Nomenclature xvii 1 Introduction 1 1.1 Composite Structures 1 1.2 Failures of Composites 2 1.2.1 Matrix Cracking 2 1.2.2 Delamination 2 1.2.3 Fibre/Matrix Debonding 2 1.2.4 Fibre Breakage 3 1.2.5 Macro Models of Cracking in Composites 3 1.3 Crack Analysis 3 1.3.1 Local and Non-Local Formulations 3 1.3.2 Theoretical Methods for Failure Analysis 5 1.4 Analytical Solutions for Composites 6 1.4.1 Continuum Models 6 1.4.2 Fracture Mechanics of Composites 6 1.5 Numerical Techniques 8 1.5.1 Boundary Element Method 8 1.5.2 Finite Element Method 8 1.5.3 Adaptive Finite/Discrete Element Method 10 1.5.4 Meshless Methods 10 1.5.5 Extended Finite Element Method 11 1.5.6 Extended Isogeometric Analysis 12 1.5.7 Multiscale Analysis 13 1.6 Scope of the Book 13 2 Fracture Mechanics, A Review 17 2.1 Introduction 17 2.2 Basics of Elasticity 20 2.2.1 Stress–Strain Relations 20 2.2.2 Airy Stress Function 22 2.2.3 Complex Stress Functions 22 2.3 Basics of LEFM 23 2.3.1 Fracture Mechanics 23 2.3.2 Infinite Tensile Plate with a Circular Hole 24 2.3.3 Infinite Tensile Plate with an Elliptical Hole 26 2.3.4 Westergaard Analysis of a Line Crack 28 2.3.5 Williams Solution of a Wedge Corner 29 2.4 Stress Intensity Factor, K 30 2.4.1 Definition of the Stress Intensity Factor 30 2.4.2 Examples of Stress Intensity Factors for LEFM 33 2.4.3 Griffith Energy Theories 35 2.4.4 Mixed Mode Crack Propagation 38 2.5 Classical Solution Procedures for K and G 41 2.5.1 Displacement Extrapolation/Correlation Method 41 2.5.2 Mode I Energy Release Rate 41 2.5.3 Mode I Stiffness Derivative/Virtual Crack Model 42 2.5.4 Two Virtual Crack Extensions for Mixed Mode Cases 42 2.5.5 Single Virtual Crack Extension Based on Displacement Decomposition 43 2.6 Quarter Point Singular Elements 44 2.7 J Integral 47 2.7.1 Generalization of J 48 2.7.2 Effect of Crack Surface Traction 48 2.7.3 Effect of Body Force 49 2.7.4 Equivalent Domain Integral (EDI) Method 49 2.7.5 Interaction Integral Method 49 2.8 Elastoplastic Fracture Mechanics (EPFM) 51 2.8.1 Plastic Zone 51 2.8.2 Crack-Tip Opening Displacements (CTOD) 53 2.8.3 J Integral for EPFM 55 3 Extended Finite Element Method 57 3.1 Introduction 57 3.2 Historic Development of XFEM 58 3.2.1 A Review of XFEM Development 58 3.2.2 A Review of XFEM Composite Analysis 62 3.3 Enriched Approximations 62 3.3.1 Partition of Unity 62 3.3.2 Intrinsic and Extrinsic Enrichments 63 3.3.3 Partition of Unity Finite Element Method 66 3.3.4 MLS Enrichment 66 3.3.5 Generalized Finite Element Method 67 3.3.6 Extended Finite Element Method 67 3.3.7 Generalized PU Enrichment 67 3.4 XFEM Formulation 67 3.4.1 Basic XFEM Approximation 68 3.4.2 Signed Distance Function 69 3.4.3 Modelling the Crack 70 3.4.4 Governing Equation 71 3.4.5 XFEM Discretization 72 3.4.6 Evaluation of Derivatives of Enrichment Functions 73 3.4.7 Selection of Nodes for Discontinuity Enrichment 75 3.4.8 Numerical Integration 77 3.5 XFEM Strong Discontinuity Enrichments 79 3.5.1 A Modified FE Shape Function 79 3.5.2 The Heaviside Function 81 3.5.3 The Sign Function 84 3.5.4 Strong Tangential Discontinuity 85 3.5.5 Crack Intersection 85 3.6 XFEM Weak Discontinuity Enrichments 86 3.7 XFEM Crack-Tip Enrichments 87 3.7.1 Isotropic Enrichment 87 3.7.2 Orthotropic Enrichment Functions 88 3.7.3 Bimaterial Enrichments 88 3.7.4 Orthotropic Bimaterial Enrichments 89 3.7.5 Dynamic Enrichment 89 3.7.6 Orthotropic Dynamic Enrichments for Moving Cracks 90 3.7.7 Bending Plates 91 3.7.8 Crack-Tip Enrichments in Shells 91 3.7.9 Electro-Mechanical Enrichment 92 3.7.10 Dislocation Enrichment 93 3.7.11 Hydraulic Fracture Enrichment 94 3.7.12 Plastic Enrichment 94 3.7.13 Viscoelastic Enrichment 95 3.7.14 Contact Corner Enrichment 96 3.7.15 Modification for Large Deformation Problems 97 3.7.16 Automatic Enrichment 99 3.8 Transition from Standard to Enriched Approximation 99 3.8.1 Linear Blending 100 3.8.2 Hierarchical Transition Domain 100 3.9 Tracking Moving Boundaries 103 3.9.1 Level Set Method 103 3.9.2 Alternative Methods 106 3.10 Numerical Simulations 107 3.10.1 A Central Crack in an Infinite Tensile Plate 107 3.10.2 An Edge Crack in a Finite Plate 109 3.10.3 Tensile Plate with a Central Inclined Crack 110 3.10.4 A Bending Plate in Fracture Mode III 111 3.10.5 Crack Propagation in a Shell 112 3.10.6 Shear Band Simulation 115 3.10.7 Fault Simulation 116 3.10.8 Sliding Contact Stress Singularity by PUFEM 119 3.10.9 Hydraulic Fracture 122 3.10.10 Dislocation Dynamics 126 4 Static Fracture Analysis of Composites 131 4.1 Introduction 131 4.2 Anisotropic Elasticity 134 4.2.1 Elasticity Solution 134 4.2.2 Anisotropic Stress Functions 136 4.3 Analytical Solutions for Near Crack Tip 137 4.3.1 The General Solution 137 4.3.2 Special Solutions for Different Types of Composites 140 4.4 Orthotropic Mixed Mode Fracture 142 4.4.1 Energy Release Rate for Anisotropic Materials 142 4.4.2 Anisotropic Singular Elements 142 4.4.3 SIF Calculation by Interaction Integral 143 4.4.4 Orthotropic Crack Propagation Criteria 147 4.5 Anisotropic XFEM 149 4.5.1 Governing Equation 149 4.5.2 XFEM Discretization 150 4.5.3 Orthotropic Enrichment Functions 151 4.6 Numerical Simulations 152 4.6.1 Plate with a Crack Parallel to the Material Axis of Orthotropy 152 4.6.2 Edge Crack with Several Orientations of the Axes of Orthotropy 155 4.6.3 Inclined Edge Notched Tensile Specimen 156 4.6.4 Central Slanted Crack 160 4.6.5 An Inclined Centre Crack in a Disk Subjected to Point Loads 164 4.6.6 Crack Propagation in an Orthotropic Beam 166 5 Dynamic Fracture Analysis of Composites 169 5.1 Introduction 169 5.1.1 Dynamic Fracture Mechanics 169 5.1.2 Dynamic Fracture Mechanics of Composites 170 5.1.3 Dynamic Fracture by XFEM 172 5.2 Analytical Solutions for Near Crack Tips in Dynamic States 173 5.2.1 Analytical Solution for a Propagating Crack in Isotropic Material 174 5.2.2 Asymptotic Solution for a Stationary Crack in Orthotropic Media 175 5.2.3 Analytical Solution for Near Crack Tip of a Propagating Crack in Orthotropic Material 176 5.3 Dynamic Stress Intensity Factors 178 5.3.1 Stationary and Moving Crack Dynamic Stress Intensity Factors 178 5.3.2 Dynamic Fracture Criteria 179 5.3.3 J Integral for Dynamic Problems 180 5.3.4 Domain Integral for Orthotropic Media 181 5.3.5 Interaction Integral 182 5.3.6 Crack-Axis Component of the Dynamic J Integral 183 5.3.7 Field Decomposition Technique 185 5.4 Dynamic XFEM 185 5.4.1 Dynamic Equations of Motion 185 5.4.2 XFEM Discretization 185 5.4.3 XFEM Enrichment Functions 187 5.4.4 Time Integration Schemes 191 5.5 Numerical Simulations 195 5.5.1 Plate with a Stationary Central Crack 195 5.5.2 Mode I Plate with an Edge Crack 196 5.5.3 Mixed Mode Edge Crack in Composite Plates 199 5.5.4 A Composite Plate with Double Edge Cracks under Impulsive Loading 210 5.5.5 Pre-Cracked Three Point Bending Beam under Impact Loading 213 5.5.6 Propagating Central Inclined Crack in a Circular Orthotropic Plate 217 6 Fracture Analysis of Functionally Graded Materials (FGMs) 225 6.1 Introduction 225 6.2 Analytical Solution for Near a Crack Tip 227 6.2.1 Average Material Properties 227 6.2.2 Mode I Near Tip Fields in FGM Composites 228 6.2.3 Stress and Displacement Field (Similar to Homogeneous Orthotropic Composites) 233 6.3 Stress Intensity Factor 235 6.3.1 J Integral 235 6.3.2 Interaction Integral 236 6.3.3 FGM Auxillary Fields 236 6.3.4 Isoparametric FGM 240 6.4 Crack Propagation in FGM Composites 240 6.5 Inhomogeneous XFEM 241 6.5.1 Governing Equation 241 6.5.2 XFEM Approximation 241 6.5.3 XFEM Discretization 243 6.6 Numerical Examples 244 6.6.1 Plate with a Centre Crack Parallel to the Material Gradient 244 6.6.2 Proportional FGM Plate with an Inclined Central Crack 247 6.6.3 Non-Proportional FGM Plate with a Fixed Inclined Central Crack 250 6.6.4 Rectangular Plate with an Inclined Crack (Non-Proportional Distribution) 251 6.6.5 Crack Propagation in a Four-Point FGM Beam 253 7 Delamination/Interlaminar Crack Analysis 261 7.1 Introduction 261 7.2 Fracture Mechanics for Bimaterial Interface Cracks 264 7.2.1 Isotropic Bimaterial Interfaces 265 7.2.2 Orthotropic Bimaterial Interface Cracks 266 7.2.3 Stress Contours for a Crack between Two Dissimilar Orthotropic Materials 270 7.3 Stress Intensity Factors for Interlaminar Cracks 271 7.4 Delamination Propagation 273 7.4.1 Fracture Energy-Based Criteria 273 7.4.2 Stress-Based Criteria 273 7.4.3 Contact-Based Criteria 274 7.5 Bimaterial XFEM 275 7.5.1 Governing Equation 275 7.5.2 XFEM Discretization 276 7.5.3 XFEM Enrichment Functions for Bimaterial Problems 278 7.5.4 Discretization and Integration 280 7.6 Numerical Examples 280 7.6.1 Central Crack in an Infinite Bimaterial Plate 280 7.6.2 Isotropic-Orthotropic Bimaterial Crack 289 7.6.3 Orthotropic Double Cantilever Beam 291 7.6.4 Concrete Beams Strengthened with Fully Bonded GFRP 294 7.6.5 FRP Reinforced Concrete Cantilever Beam Subjected to Edge Loadings 295 7.6.6 Delamination of Metallic I Beams Strengthened by FRP Strips 298 7.6.7 Variable Section Beam Reinforced by FRP 300 8 New Orthotropic Frontiers 303 8.1 Introduction 303 8.2 Orthotropic XIGA 303 8.2.1 NURBS Basis Function 304 8.2.2 Extended Isogeometric Analysis 305 8.2.3 XIGA Simulations 313 8.3 Orthotropic Dislocation Dynamics 321 8.3.1 Straight Dislocations in Anisotropic Materials 321 8.3.2 Edge Dislocations in Anisotropic Materials 322 8.3.3 Curve Dislocations in Anisotropic Materials 324 8.3.4 Anisotropic Dislocation XFEM 324 8.3.5 Plane Strain Anisotropic Solution 329 8.3.6 Individual Sliding Systems s1 and s2 in an Infinite Domain 330 8.3.7 Simultaneous Sliding Systems in an Infinite Domain 330 8.4 Other Anisotropic Applications 333 8.4.1 Biomechanics 333 8.4.2 Piezoelectric 335 References 339 Index 363

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