Applied mathematics Books

Book SynopsisExamines the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and proves global regularity and modified scattering for suitably small and localized perturbations of a flat interface.Table of Contents Introduction Preliminaries Derivation of the main scalar equation Energy estimates I: high Sobolev estimates Energy estimates II: low frequencies Energy estimates III: weighted estimates for high frequencies Energy estimates IV: weighted estimates for low frequencies Decay estimates Proof of Lemma 8.6 Modified scattering Appendix A. Analysis of symbols Appendix B. The Dirichlet-Neumann operator Appendix C. Elliptic bounds Bibliography.

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Book SynopsisThis book has three main goals: to explore a selection of topics from the early period of the theory of relativity, focusing on particular aspects that are interesting or unusual; to provide an exposition of the differential geometry needed to understand these topics; to reflect on the historical development of the subject and its significance for our understanding of what reality is.Trade ReviewThis book is both pedagogical and humanistic in nature...in a historical setting, he gives a wealth of mathematical tools and many applications to astronomy, physics, and cosmology." — Alan S. McRae, Mathematical Reviews"Roger Cooke has successfully presented a wealth of fascinating ideas from the realm of physics, astronomy and cosmology while developing a range of powerful mathematical tools...This is an encyclopaedic discourse on relativity in a mathematical, philosophical and 'humanistic' setting...Being inexpert in this field myself, I was captivated by Roger Cooke's introduction to relativity. His book will appeal to a wide readership and it should provide the basis for a taught course at some suitable stage at the undergraduate level and beyond." — Peter Ruane, MAA ReviewsTable of Contents The special theory: Time, space, and space-time Relativistic mechanics Electromagnetic theory The general theory: Precession and deflection Concepts of curvature, 1700-1850 Concepts of curvature, 1850-1950 The geometrization fo gravity Historical and philosophical context: Experiments, chronology, metaphysics Bibliography Subject index Name index

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Book SynopsisTensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. Numerous open problems appropriate for graduate students and post-docs are included throughout.Table of Contents Basics: Motivation, first definitions and properties Tensors via linear algebra: Rank and border rank Tensor networks The asymptotic geometry of tensors: Detour into probability and information theory Strassen's laser method and spectral theory Quantum mechanics for quantum information theory Quantum information theory and the asymptotic geometry of tensors Moment maps and moment polytopes Hints and answers to selected exercises Bibliography Index.

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Book SynopsisTrade ReviewVillani writes with enthusiasm, and his approachable style is aided by pleasant typography. The exposition is far from rigid. ... As an introduction to an active and rapidly growing area of research, this book is greatly to be welcomed. Much of it is accessible to the novice research student possessing a solid background in real analysis, yet even experienced researchers will find it a stimulating source of novel applications, and a guide to the latest literature."" —Geoffrey Burton, Bulletin of the LMS""Cedric Villani's book is a lucid and very readable documentation of the tremendous recent analytic progress in 'optimal mass transportation' theory and of its diverse and unexpected applications in optimization, nonlinear PDE, geometry, and mathematical physics."" —Lawrence C. Evans, University of California at Berkeley""The book is clearly written and well organized and can be warmly recommended as an introductory text to this multidisciplinary area of research, both pure and applied - the mass transportation problem."" —Studia Universitatis Babes-BolyaiMathematica""This is a very interesting book: it is the first comprehensive introduction to the theory of mass transportation with its many - and sometimes unexpected - applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook."" —Olaf Ninnemann for Zentralblatt MATHTable of Contents Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric and Gaussian inequalities The metric side of optimal transportation A differential point of view on optimal transportation Entropy production and transportation inequalities Problems Bibliography Table of short statements Index

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Book SynopsisTrade Review“This book is a very welcome, untraditional, thorough and well-organized introduction to a young and quickly developing discipline on the crossroads between mathematics, computer science, and engineering.” —DMV NewsletterTable of Contents Computational geometric topology: Graphs Surfaces Complexes Computational algebraic topology: Homology Duality Morse functions Computational persistent topology: Persistence Stability Applications References Index

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Book SynopsisPresents the proceedings of the virtual workshop on Computational Aspects of Discrete Subgroups of Lie Groups, held in June 2021. The major theme deals with a novel domain of computational algebra: the design, implementation, and application of algorithms based on matrix representation of groups and their geometric properties.Table of Contents D. Gabai, R. Meyerhoff, N. Thurston, and A. Yarmola, Enumerating Kleinian groups W. A. de Graaf, Exploring Lie theory with GAP A. S. Detinko, D. L. Flannery, and A. Hulpke, Freeness and $S$-arithmeticity of rational Mobius groups J. Gilman, Computability models: Algebraic, topological and geometric algorithms W. M. Goldman, Compact components of planar surface group representations A. Hulpke, Proving infinite index for a subgroup of matrices M. Kapovich, Geometric algorithms for discreteness and faithfulness M. Kapovich, A. Detinko, and A. Kontorovich, List of problems on discrete subgroups of Lie groups and their computational aspects A. Mark, J. Paupert, and D. Polletta, Picard modular groups generated by complex reflections J. M. Riestenberg, Verifying the straight-and-spaced condition T. N. Venkataramana, Unipotent generators for arithmetic groups

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Book SynopsisContains the proceedings of the Virtual Conference on Noncommutative Rings and their Applications VII, held in July 2021, and the Virtual Conference on Quadratic Forms, Rings and Codes, held in July 2021. The articles cover topics in commutative and noncommutative algebra and applications to coding theory.Table of Contents R. R. de Araujo, A. L. M. Pereira, and C. Polcino Milies, A note on checkable codes over Frobenius and quasi-Frobenius rings N. Aydin, T. Guidotti, and P. Liu, Good classical and quantum codes from multi-twisted codes N. Bennenni and A. Leroy, Evaluation of iterated ore polynomials and skew Reed-Muller codes G. Blachar, L. H Rowen, and U. Vishne, Identities of the algebra $\mathbb{O}\otimes\mathbb{O}$ S. Breaz and Y. Zhou, When is every non central-unit a sum of two nilpotents? A. Chapman and K. McKinnie, Asymptotic Brauer $p$-dimension L. W. Christensen, S. Estrada, and P. Thompson, Five theorems on Gorenstein global dimensions S. T. Dougherty and S. Sahinkaya, Quasi-self-dual codes over a non-unital ring of order 4 S. T. Dougherty, S. Sahinkaya, and D. Ustun, Codes from the skew ring $M_2(\mathbb{F}_2\rtimes_\varphi G$ F. D'Este, The underlying vector spaces of certain endomorhism rings A. Facchini, Algebraic structures from the point of view of complete multiplicative lattices C. Fernandez-Cordoba and S. Szabo, $\mathbb{Z}_2\mathbb{Z}_4$-additive codes as codes over rings G. Lee, C. S. Roman, N. K. Tung, and X. Zhang, On Utumi rings and continuous regular Baer rings S. K. Jain and A. Leroy, Matrices representable as product of conjugates of a singular matrix F. Kaynarca, G. D'Este, and D. Keskin Tutuncu, Almost projective modules over non-hereditary algebras M. Salahuddin Khan, A. Abbasi, S. Ali, and M. Ayedh, On prime ideals with generalized derivations in rings with involution J. Krempa, On nil algebras and a problem of Passeman concerning nilpotent free algebras X. Mary, Rings with transitive chaining of idempotents J. K. Park and S. T. Rizvi, On continuous hulls of rings and modules

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Book SynopsisIntended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material.

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Book SynopsisProvides an overview of completion problems dealing with completions to different types of operators and can be considered as a natural extension of classical results concerned with matrix completions. The book assumes some basic familiarity with functional analysis and operator theory.

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Book SynopsisAn emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question ""How do we discretize differential geometry?"" arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful.Trade Review“This book gives new life to old concepts of classical differential geometry, and a beautiful introduction to new notions of discrete integrable systems. It should be of interest to researchers in several areas of mathematics (integrable systems, differential geometry, numerical approximation of special surfaces), but also to advanced students interested in a good introduction to several classical areas of mathematics. Parts of it could well be used for graduate or possibly advanced undergraduate courses in mathematics.” - Mathematical Reviews“It can serve as a very good introduction into contemporary research and it seems to be the first book devoted to the topic. ... The book is well and clearly written.” - EMS Newsletter Table of Contents Classical differential geometry Discretization principles. Multidimensional nets Discretization principles. Nets in quadrics Special classes of discrete surfaces Approximation Consistency as integrability Discrete complex analysis. Linear theory Discrete complex analysis. Integrable circle patterns Foundations Solutions of selected exercises Bibliography Notations Index

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Book SynopsisThis book provides mathematical analyses of scanning devices in optical and laser systems to yield results with higher accuracy than those obtained by geometrical imaging an object with a movable mirror or prism. Topics include the laws of reflection and refraction and the mathematical preliminaries of analytical raytracing; mirror-scanning devices with one axis of rotation (conic-section scanning) and with two axes of rotation (gimbaled mirror and galvanometric scanners in cascade for 2D scanning); and Risley-prism-based beam-steering systems. Readers should have a foundation in vector operation and calculus, and a reasonable knowledge of elementary optics and lasers.Table of Contents Introduction One-Mirror and One-Axis Scanning Devices Scan Field of Rotating Reflective Polygons Differential Geometry of the Ruled Surfaces Optically Produced by Mirror Scanning Devices Two-Mirror and Two-Axis Scanning Systems of Different Configurations Gimbaled Mirror for Two-Dimensional Beam-Steering Exact and Approximate Solutions for Risley-Prism-Based Beam-Steering Systems in Different Configurations Forward and Inverse Solutions for Two-Element Risley-Prism-Based Beam-Steering Systems in Different Configurations Inverse Solutions for Three-Element Risley-Prism-Based Beam-Steering Systems in Different Configurations Error Sources and Their Influence on the Performance of Risley-Prism-Based Beam Steering Systems

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Book SynopsisEssays in diatonic set theory, transformation theory, and neo-Riemannian theory -- the newest and most exciting fields in music theory today. The essays in Music Theory and Mathematics: Chords, Collections, and Transformations define the state of mathematically oriented music theory at the beginning of the twenty-first century. The volume includes essays in diatonic set theory, transformation theory, and neo-Riemannian theory -- the newest and most exciting fields in music theory today. The essays constitute a close-knit body of work -- a family in the sense of tracing their descentfrom a few key breakthroughs by John Clough, David Lewin, and Richard Cohn in the 1980s and 1990s. They are integrated by the ongoing dialogue they conduct with one another. The editors are Jack Douthett, a mathematician and music theorist who collaborated extensively with Clough; Martha M. Hyde, a distinguished scholar of twentieth-century music; and Charles J. Smith, a specialist in tonal theory. The contributors are all prominent scholars, teaching at institutions such as Harvard, Yale, Indiana University, and the University at Buffalo. Six of them (Clampitt, Clough, Cohn, Douthett, Hook, and Smith) have received the Society for Music Theory's prestigious PublicationAward, and one (Hyde) has received the ASCAP Deems Taylor Award. The collection includes the last paper written by Clough before his death, as well as the last paper written by David Lewin, an important music theorist also recently deceased. Contributors: David Clampitt, John Clough, Richard Cohn, Jack Douthett, Nora Engebretsen, Julian Hook, Martha Hyde, Timothy Johnson, Jon Kochavi, David Lewin, Charles J. Smith, and Stephen Soderberg.Trade ReviewThese essays, by leading American music theorists, continue the development of some of the most important research of the last twenty years into mathematical models of basic musical structures. These models are elegant in the abstract, but they are also shown to have many practical applications in explaining a wide range of art music. Several of the contributions are bound to be classics in this literature. --John Roeder, Professor of Music Theory, University of British Columbia * . *Music Theory and Mathematics is a fitting memorial to John Clough, one of music theory's great pioneers. Clough was among the first scholars to introduce non-trivial mathematics into what has emerged as diatonic set theory or scale theory. This volume consists of essays by important theorists on a variety of topics ranging from scale and Riemannian theory to analysis of works by Bartók and Schoenberg. Building on Clough's research, Music Theory and Mathematics poses new questions and approaches to what are perhaps the most exciting directions in music theory today. --Robert Morris, Professor of Composition, Eastman School of Music, University of Rochester * . *Table of ContentsIntroduction by Normal Carey, Jack Douthett, and Martha M. Hyde Preface by Charles J. Smith "Cardinality Equals Variety for Chords" in Well-Formed Scales, with a Note on the Twin Primes Conjecture - David Clampitt Flip-Flop Circles and Their Groups - John Clough Pitch-Time Analogies and Transformations in Bartók's Sonata for Two Pianos and Percussion - Richard Cohn Filtered Point-Symmetry and Dynamical Voice-Leading - Jack Douthett The "Over-Determined" Triad as a Source of Discord: Nascent Groups and the Emergent Chromatic Tonality in Nineteenth-Century German Harmonic Theory Nineteenth-Century German Harmonic Theory - Nora Engebretsen Signature Transformations - Julian Hook Some Pedagogical Implications of Diatonic and Neo-Riemannian Theory - Timothy Johnson A Parsimony Metric for Diatonic Sequences - Jonathan Kochavi Transformational Considerations in Schoenberg's Opus 23, Number 3 - David Lewin Transformational Etudes: Basic Principles and Applications of Interval String Theory - Stephen Soderberg

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Book SynopsisDiscrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics.This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis.Table of Contents List of Figures Notation Preface Chapter 1: Introduction to the Central Concepts Chapter 2: Convex Functions with Combinatorial Structures Chapter 3: Convex Analysis, Linear Programming, and Integrality Chapter 4: M-Convex Sets and Submodular Set Functions Chapter 5: L-Convex Sets and Distance Functions Chapter 6: M-Convex Functions Chapter 7: L-Convex Functions Chapter 8: Conjugacy and Duality Chapter 9: Network Flows Chapter 10: Algorithms Chapter 11: Application to Mathematical Economics Chapter 12: Application to Systems Analysis by Mixed Matrices Bibliography Index.

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Book SynopsisEmphasizes mathematical techniques and ideas arising across the spectrum of medical imaging modalities and explains important concepts concerning inversion, stability, incomplete data effects, the role of interior information, and other issues critical to all medical imaging methods.

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Book SynopsisSystems of linear equations are ubiquitous in numerical analysis and scientific computing. and iterative methods are indispensable for the numerical treatment of such systems. This book offers a rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning. The book supplements standard texts on numerical mathematics for first-year graduate and advanced undergraduate courses and is suitable for advanced graduate classes covering numerical linear algebra and Krylov subspace and multigrid iterative methods. It will be useful to researchers interested in numerical linear algebra and engineers who use iterative methods for solving large algebraic systems.Table of ContentsList of figures; List of algorithms; Preface; 1. Krylov subspace methods; 2. Toeplitz matrices and preconditioners; 3. Multigrid preconditioners; 4. Preconditioners by space decomposition; 5. Some applications; Bibliography; Index.

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Book SynopsisSolving Transcendental Equations is unique in that it is the first book to describe the Chebyshev-proxy rootfinder, which is the most reliable way to find all zeros of a smooth function on the interval, and the very reliable spectrally enhanced Weyl bisection/marching triangles method for bivariate rootfinding.

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Book SynopsisThis informal introduction to computational fluid dynamics and practical guide to numerical simulation of transport phenomena covers the derivation of the governing equations, construction of finite element approximations, and qualitative properties of numerical solutions, among other topics.

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Book SynopsisThe first comprehensive book on the AIMD algorithm. The authors offer a new approach that is based on positive switched linear systems. It is used to develop most of the main results found in the book, and fundamental results on stochastic switched nonnegative and consensus systems are derived to obtain these results.

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Book SynopsisAddresses the problem of finding an ellipsoid to represent a large set of points in high-dimensional space, which has applications in computational geometry, data representations, and optimal design in statistics. The book covers the formulation of this and related problems, theoretical properties of their optimal solutions, and algorithms for their solution.

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Book SynopsisOptimization is of critical importance in engineering. Engineers constantly strive for the best possible solutions, the most economical use of limited resources, and the greatest efficiency. As system complexity increases, these goals mandate the use of state-of-the-art optimization techniques.In recent years the theory and methodology of optimization have seen revolutionary improvements. Moreover, the exponential growth in computational power, along with the availability of multicore computing with virtually unlimited memory and storage capacity, has fundamentally changed what engineers can do to optimize their designs. This is a two-way process: engineers benefit from developments in optimization methodology, and challenging new classes of optimization problems arise from novel engineering applications.Advances and Trends in Optimization with Engineering Applications reviews 10 major areas of optimization and related engineering applications in a distinct part, providing a broad summary of state-of-the-art optimization techniques most important to engineering practice. Each part provides a clear overview of a specific area, followed by chapters detailing applications to a wide range of real-world problems.The book provides a solid foundation for engineers and mathematical optimizers alike who want to understand not only the importance of optimization methods to engineering but also the capabilities of current methods.

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Book SynopsisThe primary goal of this book is to provide a self-contained, comprehensive study of the main ?rst-order methods that are frequently used in solving large-scale problems. First-order methods exploit information on values and gradients/subgradients (but not Hessians) of the functions composing the model under consideration. With the increase in the number of applications that can be modeled as large or even huge-scale optimization problems, there has been a revived interest in using simple methods that require low iteration cost as well as low memory storage.The author has gathered, reorganized, and synthesized (in a unified manner) many results that are currently scattered throughout the literature, many of which cannot be typically found in optimization books.First-Order Methods in Optimization offers comprehensive study of first-order methods with the theoretical foundations; provides plentiful examples and illustrations; emphasizes rates of convergence and complexity analysis of the main first-order methods used to solve large-scale problems; and covers both variables and functional decomposition methods.Table of Contents Preface; Chapter 1: Vector Spaces; Chapter 2: Extended Real-Value Functions; Chapter 3: Subgradients; Chapter 4: Conjugate Functions; Chapter 5: Smoothness and Strong Convexity; Chapter 6: The Proximal Operator; Chapter 7: Spectral Functions; Chapter 8: Primal and Dual Projected Subgradient Methods; Chapter 9: Mirror Descent; Chapter 10: The Proximal Gradient Method; Chapter 11: The Block Proximal Gradient Method; Chapter 12: Dual-Based Proximal Gradient Methods; Chapter 13: The Generalized Conditional Gradient Method; Chapter 14: Alternating Minimization; Chapter 15: ADMM; Appendix A: Strong Duality and Optimality Conditions; Appendix B: Tables; Appendix C: Symbols and Notation; Appendix D: Bibliographic Notes; Bibliography; Index.

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Book Synopsis“If mathematical modeling is the process of turning real phenomena into mathematical abstractions, then numerical computation is largely about the transformation from abstract mathematics to concrete reality. Many science and engineering disciplines have long benefited from the tremendous value of the correspondence between quantitative information and mathematical manipulation.” -from the PrefaceFundamentals of Numerical Computation is an advanced undergraduate-level introduction to the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. The book is organized with simpler methods in the first half and more advanced methods in the second half, allowing use for either a single course or a sequence of two courses. The authors take readers from basic to advanced methods, illustrating them with over 200 self-contained MATLAB functions and examples designed for those with no prior MATLAB experience. Although the text provides many examples, exercises, and illustrations, the aim of the authors is not to provide a cookbook per se, but rather an exploration of the principles of cooking.Professors Driscoll and Braun have developed an online resource that includes well-tested materials related to every chapter. Among these materials are lecture-related slides and videos, ideas for student projects, laboratory exercises, computational examples and scripts, and all the functions presented in the book.

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Book SynopsisHighly oscillatory phenomena range across numerous areas in science and engineering and their computation represents a difficult challenge. A case in point is integrals of rapidly oscillating functions in one or more variables. The quadrature of such integrals has been historically considered very demanding. Research in the past 15 years (in which the authors played a major role) resulted in a range of very effective and affordable algorithms for highly oscillatory quadrature. This is the only monograph bringing together the new body of ideas in this area in its entirety.The starting point is that approximations need to be analyzed using asymptotic methods rather than by more standard polynomial expansions. As often happens in computational mathematics, once a phenomenon is understood from a mathematical standpoint, effective algorithms follow. As reviewed in this monograph, we now have at our disposal a number of very effective quadrature methods for highly oscillatory integrals—Filon-type and Levin-type methods, methods based on steepest descent, and complex-valued Gaussian quadrature. Their understanding calls for a fairly varied mathematical toolbox—from classical numerical analysis, approximation theory, and theory of orthogonal polynomials all the way to asymptotic analysis—yet this understanding is the cornerstone of efficient algorithms.

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Book SynopsisThe mathematical challenges coming from the social and behavioral sciences differ significantly from typical applied mathematical concerns. ""Change,"" for instance, is ubiquitous, but without knowing the fundamental driving force, standard differential and iterative methods are not appropriate. Although differing forms of aggregation are widely used, a general mathematical assessment of potential pitfalls is missing. These realities provide opportunities to create new mathematical approaches.These themes are described in an introductory, expository, and accessible manner by exploring new ways to handle dynamics and evolutionary game theory, to identify subtleties of decision and voting methods, to recognize unexpected modeling concerns, and to introduce new approaches with which to examine game theory. Applications range from avoiding undesired consequences when designing policy to identifying unanticipated voting (where the ""wrong"" person could win), nonparametric statistical, and economic ""supply and demand"" properties.Table of Contents Preface Chapter 1: Evolutionary game theory Chapter 2: All those puzzling voting mysteries! Chapter 3: Voting theory applied elsewhere Chapter 4: Voting: Symmetry and decompositions Chapter 5: Game theory: A decomposition Chapter 6: The reductionist approach Bibliography Index

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Book SynopsisEigenvalue computations are ubiquitous in science and engineering. John Francis’s implicitly shifted QR algorithm has been the method of choice for small to medium sized eigenvalue problems since its invention in 1959. This book presents a new view of this classical algorithm. While Francis’s original procedure chases bulges, the new version chases core transformations, which allows the development of fast algorithms for eigenvalue problems with a variety of special structures. This also leads to a fast and backward stable algorithm for computing the roots of a polynomial by solving the companion matrix eigenvalue problem. The authors received a SIAM Outstanding Paper prize for this work.This book will be of interest to researchers in numerical linear algebra and their students.

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Book SynopsisThis book develops the theory of probability and mathematical statistics with the goal of analyzing real-world data. Throughout the text, the R package is used to compute probabilities, check analytically computed answers, simulate probability distributions, illustrate answers with appropriate graphics, and help students develop intuition surrounding probability and statistics. Examples, demonstrations, and exercises in the R programming language serve to reinforce ideas and facilitate understanding and confidence. The book’s Chapter Highlights provide a summary of key concepts, while the examples utilizing R within the chapters are instructive and practical. Exercises that focus on real-world applications without sacrificing mathematical rigor are included, along with more than 200 figures that help clarify both concepts and applications. In addition, the book features two helpful appendices: annotated solutions to 700 exercises and a Review of Useful Math.

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Book SynopsisWhenever two or more objects or entities—be they bubbles, vortices, black holes, magnets, colloidal particles, microorganisms, swimming bacteria, Brownian random walkers, airfoils, turbine blades, electrified drops, magnetized particles, dislocations, cracks, or heterogeneities in an elastic solid—interact in some ambient medium, they make holes in that medium. Such holey regions with interacting entities are called multiply connected.This book describes a novel mathematical framework for solving problems in two-dimensional, multiply connected regions. The framework is built on a central theoretical concept: the prime function, whose significance for the applied sciences, especially for solving problems in multiply connected domains, has been missed until recent work by the author.This monograph is a one-of-a-kind treatise on the prime function associated with multiply connected domains and how to use it in applications. The book contains many results familiar in the simply connected, or single-entity, case that are generalized naturally to any number of entities, in many instances for the first time.Solving Problems in Multiply Connected Domains is aimed at applied and pure mathematicians, engineers, physicists, and other natural scientists; the framework it describes finds application in a diverse array of contexts. The book provides a rich source of project material for undergraduate and graduate courses in the applied sciences and could serve as a complement to standard texts on advanced calculus, potential theory, partial differential equations and complex analysis, and as a supplement to texts on applied mathematical methods in engineering and science.

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Book SynopsisThis book addresses an important class of mathematical problems (the Riemann problem) for first-order hyperbolic partial differential equations (PDEs), which arise when modeling wave propagation in applications such as fluid dynamics, traffic flow, acoustics, and elasticity.It covers the fundamental ideas related to classical Riemann solutions, including their special structure and the types of waves that arise, as well as the ideas behind fast approximate solvers for the Riemann problem.The emphasis is on the general ideas, but each chapter delves into a particular application. The book is available in electronic form as a collection of Jupyter notebooks that contain executable computer code and interactive figures and animations.

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Book SynopsisThe Mathematical Elasticity set contains three self-contained volumes that together provide the only modern treatise on elasticity. They introduce contemporary research on three-dimensional elasticity, the theory of plates, and the theory of shells. Each volume contains proofs, detailed surveys of all mathematical prerequisites, and many problems for teaching and self-study. An extended preface and extensive bibliography have been added to each volume to highlight the progress that has been made since the original publication.The first book, Three-Dimensional Elasticity, covers the modeling and mathematical analysis of nonlinear three-dimensional elasticity. In volume two, Theory of Plates, asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear plate and shallow shell theories. The objective of Theory of Shells, the final volume, is to show how asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear shell theories: membrane, generalized membrane, and flexural.These classic textbooks are for advanced undergraduates, first-year graduate students, and researchers in pure or applied mathematics or continuum mechanics. They are appropriate for courses in mathematical elasticity, theory of plates and shells, continuum mechanics, computational mechanics, and applied mathematics in general.

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Book SynopsisCollecting a set of classical and emerging methods that otherwise would not be available in a single treatment, Foundations of Computational Imaging: A Model-Based Approach is the first book to define a common foundation for the mathematical and statistical methods used in computational imaging. The book is designed to bring together an eclectic group of researchers with a wide variety of applications and disciplines including applied math, physics, chemistry, optics, and signal processing, to address a collection of problems that can benefit from a common set of methods. Inside, readers will find: Basic techniques of model-based image processing. A comprehensive treatment of Bayesian and regularized image reconstruction methods. An integrated treatment of advanced reconstruction techniques such as majorization, constrained optimization, ADMM, and Plug-and-Play methods for model integration. Foundations of Computational Imaging can be used in courses on Model-Based or Computational Imaging, Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory. It is also for researchers or practitioners in medical imaging, scientific imaging, commercial imaging, or industrial imaging.

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Book SynopsisThis expansive volume describes the history of numerical methods proposed to solve linear algebra problems,, from antiquity to the present day. The authors focus on methods for solving linear systems of equations and eigenvalue problems and describe the interplay between numerical methods and the computing tools available at the time. The second part of the book consists of 78 biographies of the main important contributors to the field.A Journey through the History of Numerical Linear Algebra will be of special interest to applied mathematicians, especially researchers in numerical linear algebra, and to applied mathematiciansas well as to and historians of mathematics as well.

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Book SynopsisReduced order modeling is an important, growing field in computational science and engineering, and this is the first book to address the subject in relation to computational fluid dynamics. It focuses on complex parametrization of shapes for their optimization and includes recent developments in advanced topics such as turbulence, stability of flows, inverse problems, optimization, and flow control, as well as applications.This book will be of interest to researchers and graduate students in the field of reduced order modeling.

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Book SynopsisExtremum Seeking through Delays and PDEs, the first book on the topic, expands the scope of applicability of the extremum seeking method, from static and finite-dimensional systems to infinite-dimensional systems. Readers will find: Numerous algorithms for model-free real-time optimization are developed and their convergence guaranteed. Extensions from single-player optimization to noncooperative games, under delays and PDEs, are provided. The delays and pdes are compensated in the control designs using the PDE backstepping approach, and stability is ensured using infinite-dimensional versions of averaging theory. Accessible and powerful tools for analysis. This book is intended for control engineers in all disciplines (electrical, mechanical, aerospace, chemical), mathematicians, physicists, biologists, and economists. It is appropriate for graduate students, researchers, and industrial users.

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Book SynopsisOptimization is presented in most multivariable calculus courses as an application of the gradient, and while this treatment makes sense for a calculus course, there is much more to the theory of optimization. Optimization problems are generated constantly, and the theory of optimization has grown and developed in response to the challenges presented by these problems. This textbook aims to show readers how optimization is done in practice and help them to develop an appreciation for the richness of the theory behind the practice.Exercises, problems (including modeling and computational problems), and implementations are incorporated throughout the text to help students learn by doing. Python notes are inserted strategically to help readers complete computational problems and implementations.The Basics of Practical Optimization, Second Edition is intended for undergraduates who have completed multivariable calculus, as well as anyone interested in optimization. The book is appropriate for a course that complements or replaces a standard linear programming course.

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Book SynopsisInverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers—a subset of transmission eigenvalues—using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles.This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.

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Book SynopsisComposites have been studied for more than 150 years, and interest in their properties has been growing. This classic volume provides the foundations for understanding a broad range of composite properties, including electrical, magnetic, electromagnetic, elastic and viscoelastic, piezoelectric, thermal, fluid flow through porous materials, thermoelectric, pyroelectric, magnetoelectric, and conduction in the presence of a magnetic field (Hall effect). Exact solutions of the PDEs in model geometries provide one avenue of understanding composites; other avenues include microstructure-independent exact relations satisfied by effective moduli, for which the general theory is reviewed; approximation formulae for effective moduli; and series expansions for the fields and effective moduli that are the basis of numerical methods for computing these fields and moduli. The range of properties that composites can exhibit can be explored either through the model geometries or through microstructure-independent bounds on the properties. These bounds are obtained through variational principles, analytic methods, and Hilbert space approaches. Most interesting is when the properties of the composite are unlike those of the constituent materials, and there has been an explosion of interest in such composites, now known as metamaterials. The Theory of Composites surveys these aspects, among others, and complements the new body of literature that has emerged since the book was written. It remains relevant today by providing historical background, a compendium of numerous results, and through elucidating many of the tools still used today in the analysis of composite properties. This book is intended for applied mathematicians, physicists, and electrical and mechanical engineers. It will also be of interest to graduate students.

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Book SynopsisRounding Errors in Algebraic Processes was the first book to give systematic analyses of the effects of rounding errors on a variety of key computations involving polynomials and matrices.A detailed analysis is given of the rounding errors made in the elementary arithmetic operations and inner products, for both floating-point arithmetic and fixed-point arithmetic. The results are then applied in the error analyses of a variety of computations involving polynomials as well as the solution of linear systems, matrix inversion, and eigenvalue computations.The conditioning of these problems is investigated. The aim was to provide a unified method of treatment, and emphasis is placed on the underlying concepts.This book is intended for mathematicians, computer scientists, those interested in the historical development of numerical analysis, and students in numerical analysis and numerical linear algebra.Trade Review[This book] combines a rigorous mathematical analysis with a practicality that stems from an obvious first-hand contact with the actual numerical computation. The well-chosen examples alone show vividly both the importance of the study of rounding errors and the perils of its neglect. A. A. Grau, SIAM Review (1966)

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