Calculus and mathematical analysis Books

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Book SynopsisThe problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in 19th century analysis and it has now been revived with the appearance of symbolic languages. In this book, the authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed in this book, first published in 2004, are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting, rather than the shortest, path to the results. Along the way, they illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This will be a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.Trade Review'I recommend this book highly as a source of rewarding projects for undergraduates (and others) to home their analytic skills and gain an appreciation for this area of mathematics. The authors clearly had great love for the material and their enthusiasm comes through in an infectious manner.' SIAM Review'The authors have managed to write a very readable account about integrals, accessible even to advanced undergraduates. Some of the topics of the book could be used for undergraduate reading and research projects. This way the book could serve as a 'springboard to many unexpected investigations and discoveries in mathematics.' Zentralblatt MATHTable of Contents1. Introduction; 2. Factorials and binomial coefficients; 3. The method of partial fractions; 4. A simple rational function; 5. A review of power series; 6. The exponential and logarithm functions; 7. The trigonometric functions and pi; 8. A quartic integral; 9. The normal integral; 10. Euler's constant; 11. Eulerian integrals: the Gamma and Beta functions; 12. The Riemann zeta function; 13. Logarithmic integrals; 14. A master formula; 15. Appendix: the revolutionary WZ method.

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Book SynopsisNonlinear dynamics has been successful in explaining complicated phenomena in well-defined low-dimensional systems. Now it is time to focus on real-life problems that are high-dimensional or ill-defined, for example, due to delay, spatial extent, stochasticity, or the limited nature of available data. How can one understand the dynamics of such systems? Written by international experts, Nonlinear Dynamics and Chaos: Where Do We Go from Here? assesses what the future holds for dynamics and chaos. The chapters address one or more of the broad and interconnected main themes: neural and biological systems, spatially extended systems, and experimentation in the physical sciences. The contributors offer suggestions as to what they see as the way forward, often in the form of open questions for future research.Trade Review"This handsome volume is the proceedings of a conference held in Bristol in 2001, which had the aim of charting new directions for the exploration of nonlinear dynamical systems. The editors must be commended for their work: the individual chapters have been given a clean, uniform style that reflects a serious effort to present the volume as a unified book rather than a recollection of articles, with several cross-references between the chapters. The book is also remarkably free of typographical errors. I heartily recommend this collection to students looking for some direction (as long as they don't think this is all of nonlinear dynamics!)." -UK Nonlinear News, May 2003 "This timely and important book is a record of papers presented at a conference in Bristol and is very well edited, and produced … The very richness of this book, in both theory and real-world applications, makes it difficult to summarize and even more difficult to put down." -Nonlinear Dynamics, Psychology and Life Sciences "The book is written by authors who are champions of their field. All researchers in nonlinear dynamics should have access to this book. It is a valuable resource of references and it contains a lot of ideas and open problems in various fields. One might think of it as a catalogue of problems in nonlinear dynamics. The introduction of the book is a 'must-read.' It presents the nature and the philosophy of the book (and the symposium). Reading the introduction, the editors clearly have done a great job of managing each of the invited lecturers to translate the philosophy of the symposium into their lectures … my impression is that all authors did a good job presenting the excitement of their research and addressing the interesting questions. This book in general is a valuable addition to the literature of the theory and practice of nonlinear dynamics and chaos." -Theo Tuwankotta, Institute of Technology,ITB, Bandung, IndonesiaTable of ContentsPreface. Bifurcation and Degenerate Decomposition in Multiple Time Scale Dynamical Systems. Many-body quantum mechanics. Unfolding Complexity: Hereditory Dynamical Systems-New Bifurcation Schemes and High Dimensional Chaos. Creating stability out of instability.Signal or Noise? A nonlinear dynamics approach to spatiotemporal communication. Outstanding problems in the theory of pattern formation. Is Chaos relevant to Fluid Mechanics?. Time-Reversed Acoustics and Chaos.Reduction methods applied to nonlocally coupled oscillator systems. A prime number of prime questions about vortex dynamcis in nonlinear media. Spontaneous pattern formation in primary visual cortex. Models for Pattern Formation in Development. Spatiotemporal nonlinear dynamics: a new beginning. Author index.

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Book SynopsisPreface to the Third Edition.- Preface to the Second Edition.- Preface to the First Edition.- The Dirac Delta Function and Delta Sequences.- The Schwartz-Sobolev Theory of Distributions.- Additional Properties of Distributions.- Distributions Defined by Divergent Integrals.- Distributional Derivatives of Functions with Jump Discontinuities.- Tempered Distributions and the Fourier Transforms.- Direct Products and Convolutions of Distributions.- The Laplace Transform.- Applications to Ordinary Differential Equations.- Applications to Partial Differential Equations.- Applications to Boundary Value Problems.- Applications to Wave Propagation.- Interplay between Generalized Functions and the Theory of Moments.- Linear Systems.- Miscellaneous Topics.- References.- Index.Trade Review"This book on generalized functions is suitable for physicists, engineers and applied mathematicians. The author presents the notion of generalized functions, their properties and their applications for solving ordinary differential equations and partial differential equations. ... The author demonstrates through various examples that familiarity with generalized functions is very helpful for students in mathematics, physical sciences and technology. The proposed exercises are very good for better understanding of notions and properties presented in the chapters. The book contains new topics and important features." —Mathematica "The advantage of this text is in carefully gathered examples explaining how to use corresponding properties.... Even the standard material connecting with partial and ordinary differential equations is rewritten in modern terminology." —Zentralblatt (Review of a previous edition) "The author has done an excellent job in presenting examples and in displaying the calculational techniques associated with distributions and the applications. Throughout the book there are a wealth of examples concerning the distributional topics and caluclations introduced and concering the applications, and the examples are presented in detail." ---Zentralblatt (Review of the 1st edition) "The collaboration of physicists or engineers and mathematics, which is more and more popular and necessary in modern investigations, requires…a common language. The book under review provides this language…. [It] is a well written book, most of the material is accessible to senior undergraduate and graduate students in mathematical, physical and engineering sciences…. [The] book will [also] be useful…for specialists in ODEs, PDEs, functional analysis, [and] physicists, engineers, and lecturers." —Acta. Sci. Math. (Review of a previous edition) "An exceptionally clear exposition... The exercises at the end of each chapter are well-chosen." —The American Mathematical Monthly (Review of a previous edition) "This fully revised edition of well-received book expands the treatment of fundamental concepts and theoretical background material delineates connections to a variety of applications in mathematical physics, elasticity, wave propagation, magnetohydrodynamics, linear systems, probability and statistics, optical control problems in economics, and more. It has many new topics and [features] driven by additional examples and exercises. . . It presents a wealth of applications that connot be found in any other single source. the book will be important reading for graduate students in physics and engineering." --- Educational Book ReviewTable of ContentsPreface to the Third Edition * Preface to the Second Edition * Preface to the First Edition * Chapter 1. The Dirac Delta Function and Delta Sequences * 1.1 The Heaviside Function * 1.2 The Dirac Delta Function * 1.3 The Delta Sequences * 1.4 A Unit Dipole * 1.5 The Heaviside Sequences * Exercises * Chapter 2. The Schwartz-Sobolev Theory of Distributions * 2.1 Some Introductory Definitions * 2.2 Test Functions * 2.3 Linear Functionals and the Schwartz–Sobolev Theory of Distributions * 2.4 Examples * 2.5 Algebraic Operations on Distributions * 2.6 Analytic Operations on Distributions * 2.7 Examples * 2.8 The Support and Singular Support of a Distribution Exercises * Chapter 3. Additional Properties of Distributions * 3.1 Transformation Properties of the Delta Distributions * 3.2 Convergence of Distributions * 3.3 Delta Sequences with Parametric Dependence * 3.4 Fourier Series * 3.5 Examples * 3.6 The Delta Function as a Stieltjes Integral Exercises * Chapter 4. Distributions Defined by Divergent Integrals * 4.1 Introduction * 4.2 The Pseudofunction H(x)/x n , n = 1, 2,3, * 4.3 Functions with Algebraic Singularity of Order m * 4.4 Examples * Exercises * Chapter 5. Distributional Derivatives of Functions with Jump Discontinuities * 5.1 Distributional Derivatives in R 1 * 5.2 Moving Surfaces of Discontinuity in R n , n 2 * 5.3 Surface Distributions * 5.4 Various Other Representations * 5.5 First-Order Distributional Derivatives * 5.6 Second Order Distributional Derivatives * 5.7 Higher-Order Distributional Derivatives * 5.8 The Two-Dimensional Case * 5.9 Examples * 5.10 The Function Pf ( l/r ) and its Derivatives * Chapter 6. Tempered Distributions and the Fourier Transforms * 6.1 Preliminary Concepts * 6.2 Distributions of Slow Growth (Tempered Distributions) * 6.3 The Fourier Transform * 6.4 Examples * Exercises * Chapter 7. Direct Products and Convolutions of Distributions * 7.1 Definition of the Direct Product * 7.2 The Direct Product of Tempered Distributions * 7.3 The Fourier Transform of the Direct Product of Tempered Distributions * 7.4 The Convolution * 7.5 The Role of Convolution in the Regularization of the Distributions * 7.6 The Dual Spaces E and E' * 7.7 Examples * 7.8 The Fourier Transform of the Convolution * 7.9 Distributional Solutions of Integral Equations * Exercises * Chapter 8. The Laplace Transform * 8.1 A Brief Discussion of the Classical Results * 8.2 The Laplace Transform of the Distributions * 8.3 The Laplace Transform of the Distributional Derivatives and Vice Versa * 8.4 Examples * Exercises * Chapter 9. Applications to Ordinary Differential Equations * 9.1 Ordinary Differential Operators * 9.2 Homogeneous Differential Equations * 9.3 Inhomogeneous Differentational Equations: The Integral of a Distribution * 9.4 Examples * 9.5 Fundamental Solutions and Green's Functions * 9.6 Second Order Differential Equations with Constant Coefficients * 9.7 Eigenvalue Problems * 9.8 Second Order Differential Equations with Variable Coefficients * 9.9 Fourth Order Differential Equations * 9.10 Differential Equations of n th Order * 9.11 Ordinary Differential Equations with Singular Coefficients * Exercises * Chapter 10. Applications to Partial Differential Equations * 10.1 Introduction * 10.2 Classical and Generalized Solutions * 10.3 Fundamental Solutions * 10.4 The Cauchy–Riemann Operator * 10.5 The Transport Operator * 10.6 The Laplace Operator * 10.7 The Heat Operator * 10.8 The Schroedinger Operator * 10.9 The Helmholtz Operator * 10.10 The Wave Operator * 10.11 The Inhomogeneous Wave Equation * 10.12 The Klein–Gordon Operator * Exercises * Chapter 11. Applications to Boundary Value Problems * 11.1 Poisson's Equation * 11.2 Dumbbell-Shaped Bodies * 11.3 Uniform Axial Distributions * 11.4 Linear Axial Distributions * 11.5 Parabolic Axial Distributions * 11.6 The Four-Order Polynomial Distribution, n = 7; Spheroidal Cavities * 11.7 The Polarization Tensor for a Spheroid * 11.8 The Virtual Mass Tensor for a Spheroid * 11.9 The Electric and Magnetic Polarizability Tensors * 11.10 The Distributional Approach to Scattering Theory * 11.11 Stokes Flow * 11.12 Displacement-Type Boundary Value Problems in Elastostatics * 11.13 The Extension to Elastodynamics * 11.14 Distributions on Arbitrary Lines * 11.15 Distributions on Plane Curves * 11.16 Distributions on a Circular Disk * Chapter 12. Applications to Wave Propagation * 12.1 Introduction * 12.2 The Wave Equation * 12.3 First-Order Hyperbolic Systems * 12.4 Aerodynamic Sound Generation * 12.5 The Rankine–Hugoniot Conditions * 12.6 Wave Fronts That Carry Infinite Singularities * 12.7 Kinematics of Wave Fronts * 12.8 Derivation of the Transport Theorems for Wave Fronts * 12.9 Propagation of Wave Fronts Carrying Multilayer Densities * 12.10 Generalized Functions with Support on the Light Cone * 12.11 Examples * Chapter 13. Interplay Between Generalized Functions and the Theory of Moments * 13.1 The Theory of Moments * 13.2 Asymptotic Approximation of Integrals * 13.3 Applications to the Singular Perturbation Theory * 13.4 Applications to Number Theory * 13.5 Distributional Weight Functions for Orthogonal Polynomials * 13.6 Convolution Type Integral Equations Revisited * 13.7 Further Applications * Chapter 14. Linear Systems * 14.1 Operators * 14.2 The Step Response * 14.3 The Impulse Response * 14.4 The Response to an Arbitrary Input * 14.5 Generalized Functions as Impulse Response Functions * 14.6 The Transfer Function * 14.7 Discrete-Time Systems * 14.8 The Sampling Theorem * Chapter 15. Miscellaneous Topics * 15.1 Applications to Probability and Statistics * 15.2 Applications to Mathematical Economics * 15.3 Periodic Generalized Functions * 15.4 Microlocal Theory * References * Index

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Book SynopsisShows the effectiveness of abstract analysis for solving fundamental problems of stochastic theory, specifically the use of functional analytic methods for elucidating stochastic processes.Trade Review"More than 20 original papers reflect Rao's broad scientific interests in probability, stochastic processes, Banach space theory, measure theory and (stochastic) differential equations. …The volume is completed with a biography and bibliography of M. M. Rao, a remarkable collection of personal reminiscences (written by his former students) adds a human dimension to this fine book."-EMS Newsletter, June 2005Table of ContentsBiography of M. M. Rao, Published Writings of M. M. Rao, Ph.D. Theses Completed Under the Direction of M. M. Rao, Contributors, For M. M. Rao, An Appreciation of My Teacher, M. M. Rao, 1001 Words About Rao, A Guide to Life, Mathematical and Otherwise, Rao and the Early Riverside Years, On M. M. Rao, Reflections on M. M. Rao, 1: Stochastic Analysis and Function Spaces, 2: Applications of Sinkhorn Balancing to Counting Problems, 3: Zakai Equation of Nonlinear Filtering with Ornstein-Uhlenbeck Noise: Existence and Uniqueness, 4: Hyperfunctionals and Generalized Distributions, 5: Process-Measures and Their Stochastic Integral, 6: Invariant Sets for Nonlinear Operators, 7: The Immigration-Emigration with Catastrophe Model, 8: Approximating Scale Mixtures, 9: Cyclostationary Arrays: Their Unitary Operators and Representations, 10: Operator Theoretic Review for Information Channels, 11: Pseudoergodicity in Information Channels, 12: Connections Between Birth-Death Processes, 13: Integrated Gaussian Processes and Their Reproducing Kernel Hilbert Spaces, 14: Moving Average Representation and Prediction for Multidimensional Harmonizable Processes, 15: Double-Level Averaging on a Stratified Space, 16: The Problem of Optimal Asset Allocation with Stable Distributed Returns, 17: Computations for Nonsquare Constants of Orlicz Spaces, 18: Asymptotically Stationary and Related Processes, 19: Superlinearity and Weighted Sobolev Spaces, 20: Doubly Stochastic Operators and the History of Birkhoff s Problem 111, 21: Classes of Harmonizable Isotropic Random Fields, 22: On Geographically-Uniform Coevolution: Local Adaptation in Non-fluctuating Spatial Patterns, 23: Approximating the Time Delay in Coupled van der Pol Oscillators with Delay Coupling

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Book SynopsisOften it is more instructive to know ''what can go wrong'' and to understand ''why a result fails'' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling''s other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).Trade Review'This book is an admirable counterpart, both to the first author's well-known text Measures, Integrals and Martingales (Cambridge, 2005/2017), and to the books on counter-examples in analysis (Gelbaum and Olmsted), topology (Steen and Seebach) and probability (Stoyanov). To paraphrase the authors' preface: in a good theory, it is valuable and instructive to probe the limits of what can be said by investigating what cannot be said. The task is thus well-conceived, and the execution is up to the standards one would expect from the books of the first author and of their papers. I recommend it warmly.' N. H. Bingham, Imperial College'… an excellent reference text and companion reader for anyone interested in deepening their understanding of measure theory.' John Ross, MAA Reviews'… the unique nature of the book makes it an essential acquisition for any university with a doctoral program in pure mathematics … Essential.' M. Bona, Choice Connect'The book is well written, the demonstrations are clear and the bibliographic references are competent. We appreciate this work as extremely useful for those interested in measure theory and integration, starting with beginners and extending even to advanced researchers in the field.' Liviu Constantin Florescu, Mathematical Reviews/MathSciNet'Counterexamples in Measure and Integration is an ideal companion to help better understand canonically problematic examples in analysis … This collection of counterexamples is an excellent resource to researchers who rely on measure and integration theory. It would be helpful for students studying for their analysis qualifying exam as it draws on common misconceptions and enables readers to build intuition about why a given counterexample works and how conditions can be changed to make a particular statement hold.' Katelynn Kochalski, Notices of the AMS'This is a remarkable book covering Measure and Integration, perhaps one of the most important parts of Mathematics. It is written in a master style by following the best traditions in writing this kind of books. The authors are passionate about the topic. Look at the great care with which each of the counterexamples is presented. It is done in a way to help maximally the reader. The names of the counterexamples are chosen very carefully. Any name can be considered as a 'door' behind which is a treasure!' Jordan M. Stoyanov, zbMATH'… compendia of counterexamples remain a useful and thought-provoking resource, and this new text is a high-quality example in an analytic direction.' Dominic Yeo, The Mathematical GazetteTable of ContentsPreface; User's guide; List of topics and phenomena; 1. A panorama of Lebesgue integration; 2. A refresher of topology and ordinal numbers; 3. Riemann is not enough; 4. Families of sets; 5. Set functions and measures; 6. Range and support of a measure; 7. Measurable and non-measurable sets; 8. Measurable maps and functions; 9. Inner and outer measure; 10. Integrable functions; 11. Modes of convergence; 12. Convergence theorems; 13. Continuity and a.e. continuity; 14. Integration and differentiation; 15. Measurability on product spaces; 16. Product measures; 17. Radon–Nikodým and related results; 18. Function spaces; 19. Convergence of measures; References; Index.

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Book SynopsisCarl Friedrich Gauss, the foremost of mathematicians, was a land surveyor. Measuring and calculating geodetic networks on the curved Earth was the inspiration for some of his greatest mathematical discoveries. This is just one example of how mathematics and geodesy, the science and art of measuring and mapping our world, have evolved together throughout history.This text is for students and professionals in geodesy, land surveying, and geospatial science who need to understand the mathematics of describing the Earth and capturing her in maps and geospatial data: the discipline known as mathematical geodesy. Map of the World: An Introduction to Mathematical Geodesy aims to provide an accessible introduction to this area, presenting and developing the mathematics relating to maps, mapping, and the production of geospatial data. Described are the theory and its fundamental concepts, its application for processing, analyzing, transforming, and projecting geospatial data, and how these are used in producing charts and atlases. Also touched upon are the multitude of cross-overs into other sciences sharing in the adventure of discovering what our world really looks like.FEATURESâ Written in a fluid and accessible style, replete with exercises; adaptable for courses on different levels.â Suitable for students and professionals in the mapping sciences, but also for lovers of maps and map making.Trade Review"Map of the World: An Introduction to Mathematical Geodesy is organized, written and presented in an impressively accessible style that is replete with exercises -- making it highly adaptable textbook for curriculum courses on different levels. Especially and unreservedly recommended for students and professionals in the mapping sciences, Map of the World will prove to be an ideal and instructive source for non-specialist readers with an interest in maps and map making. While a critically important addition to college and university library collections, it should be noted for personal reading lists that Maps of the World is also available in a digital book format."—Midwest Book Review"This is a textbook covering mathematics applied to geodesy: the measuring and mapping of our ellipsoid spheroid earth that includes an overview of earth measurement and mapping back to remote times. The mathematics of describing the Earth through maps and geospatial data is covered from underpinnings to application. [. . .] This textbook, including some exercises (without solutions), is aimed at students and practitioners in geodesy, land surveying, and geospatial science. It is easy to see this as a reference work. [. . .] this is a concise review of the theory and development of coordinate reference systems."—Tom Schulte, MAA Reviews ". . .(T)his text, by a geodesist (Vermeer) and a mathematician (Rasila), focuses primarily on the mathematics enabling map projections, coordinate systems, and transformation of three-dimensional coordinate representations, ranging from Euclidean to Reimannian geometries. Although the geometry is beyond what most geography students would need to address, the detailed mathematics offers a bridge for integration of collaborative teaching appropriate for upper-level mathematics and physics students, with applications to both cartography and geophysics. Each chapter concludes with exercises that provide an opportunity for learning the explicit mathematics behind the calculation presented. Interesting historical anecdotes about mathematicians and the evolution of geodesy are also included throughout. Students and readers of mathematics and geophysics as well as scientists working in the interdisciplinary area of geodesy will appreciate this book."– Choice Review, C. A. Badurek, SUNY Cortland"Map of the World: An Introduction to Mathematical Geodesy is organized, written and presented in an impressively accessible style that is replete with exercises -- making it highly adaptable textbook for curriculum courses on different levels. Especially and unreservedly recommended for students and professionals in the mapping sciences, Map of the World will prove to be an ideal and instructive source for non-specialist readers with an interest in maps and map making. While a critically important addition to college and university library collections, it should be noted for personal reading lists that Maps of the World is also available in a digital book format."—Midwest Book Review"This is a textbook covering mathematics applied to geodesy: the measuring and mapping of our ellipsoid spheroid earth that includes an overview of earth measurement and mapping back to remote times. The mathematics of describing the Earth through maps and geospatial data is covered from underpinnings to application. [. . .] This textbook, including some exercises (without solutions), is aimed at students and practitioners in geodesy, land surveying, and geospatial science. It is easy to see this as a reference work. [. . .] this is a concise review of the theory and development of coordinate reference systems."—Tom Schulte, MAA Reviews Table of Contents1. A Brief History of Mapping. 2. Popular Conformal Map Projections. 3. The Complex Plane and Conformal Mappings. 4. Complex Analysis. 5. Conformal Mappings. 6. Transversal Mercator Projections. 7. Sperical Trigonometry. 8. The Geometry of the Ellipsoid of Revolution. 9. Three-dimensional Co-ordinates and Transformations. 10. Co-ordinate Reference Systems. 11. Co-ordinates of Heaven and Earth. 12. The Orbital Motion of Satellites. 13. The Surface Theory of Gauss. 14. Riemann Surfaces and Charts. 15. Map Projections in the Light of Surface Theory. 16. Appendices

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Book SynopsisPartial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners' course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor. This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The teaching-by-examples approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book's level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses. Highlights: Table of Contents Introduction Basic definitions Examples First-order equations Linear first-order equations General solution Initial condition Quasilinear first-order equations Characteristic curves Examples Second-order equations Classification of second-order equations Canonical forms Hyperbolic equations Elliptic equations Parabolic equations The Sturm-Liouville Problem General consideration Examples of Sturm-Liouville Problems One-Dimensional Hyperbolic Equations Wave Equation Boundary and Initial Conditions Longitudinal Vibrations of a Rod and Electrical Oscillations Rod oscillations: Equations and boundary conditions Electrical Oscillations in a Circuit Traveling Waves: D'Alembert Method Cauchy problem for nonhomogeneous wave equation D'Alembert's formula The Green's function Well-posedness of the Cauchy problem Finite intervals: The Fourier Method for Homogeneous Equations The Fourier Method for Nonhomogeneous Equations The Laplace Transform Method: simple cases Equations with Nonhomogeneous Boundary Conditions The Consistency Conditions and Generalized Solutions Energy in the Harmonics Dispersion of waves Cauchy problem in an infinite region Propagation of a wave train One-Dimensional Parabolic Equations Heat Conduction and Diffusion: Boundary Value Problems Heat conduction Diffusion equation One-dimensional parabolic equations and initial and boundary conditions The Fourier Method for Homogeneous Equations Nonhomogeneous Equations The Green's function and Duhamel's principle The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions Large time behavior of solutions Maximum principle The heat equation in an infinite region Elliptic equations Elliptic differential equations and related physical problems Harmonic functions Boundary conditions Example of an ill-posed problem Well-posed boundary value problems Maximum principle and its consequences Laplace equation in polar coordinates Laplace equation and interior BVP for circular domain Laplace equation and exterior BVP for circular domain Poisson equation: general notes and a simple case Poisson Integral Application of Bessel functions for the solution of Poisson equations in a circle Three-dimensional Laplace equation for a cylinder Three-dimensional Laplace equation for a ball Axisymmetric case Non-axisymmetric case BVP for Laplace Equation in a Rectangular Domain The Poisson Equation with Homogeneous Boundary Conditions Green's function for Poisson equations Homogeneous boundary conditions Nonhomogeneous boundary conditions Some other important equations Helmholtz equation Schrӧdinger equation Two Dimensional Hyperbolic Equations Derivation of the Equations of Motion Boundary and Initial Conditions Oscillations of a Rectangular Membrane The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions Small Transverse Oscillations of a Circular Membrane The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions Axisymmetric Oscillations of a Membrane The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions Forced Axisymmetric Oscillations The Fourier Method for Equations with Nonhomogeneous Boundary Conditions Two-Dimensional Parabolic Equations Heat Conduction within a Finite Rectangular Domain The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange) The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary conditions Heat Conduction within a Circular Domain The Fourier Method for the Homogeneous Heat Equation The Fourier Method for the Nonhomogeneous Heat Equation Heat conduction in an Infinite Medium Heat Conduction in a Semi-Infinite Medium Nonlinear equations Burgers equation Kink solution Symmetries of the Burgers equation General solution of the Cauchy problem. Interaction of kinks Korteweg-de Vries equation Symmetry properties of the KdV equation Cnoidal waves Solitons Bilinear formulation of the KdV equation Hirota's method Multisoliton solutions Nonlinear Schrӧdinger equation Symmetry properties of NSE Solitary waves Appendix A. Fourier Series, Fourier and Laplace Transforms Appendix B. Bessel and Legendre Functions Appendix C. Sturm-Liouville problem and auxiliary functions for one and two dimensions Appendix D. D1. The Sturm-Liouville problem for a circle D2. The Sturm-Liouville problem for the rectangle Appendix E. E1. The Laplace and Poisson equations for a rectangular domain with nonhomogeneous boundary conditions. E2. The heat conduction equations with nonhomogeneous boundary conditions.

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Book SynopsisStein's method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007, the authors discovered that one can combine Stein's method with the powerful Malliavin calculus of variations, in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail, for instance: fourth moment theorems on the Wiener chaos, density estimates, BreuerâMajor theorems for fractional processes, recursive cumulant computations, optimal rates and universality results for homogeneous sums. Largely self-contained, the book is perfect for self-study. It will appeal to researchers and graduate students in probability and statistics, especially those who wiTrade Review'This monograph is a nice and excellent introduction to Malliavin calculus and its application to deducing quantitative central limit theorems in combination with Stein's method for normal approximation. It provides a self-contained and appealing presentation of the recent work developed by the authors, and it is well tailored for graduate students and researchers.' David Nualart, Mathematical Reviews'The book contains many examples and exercises which help the reader understand and assimilate the material. Also bibliographical comments at the end of each chapter provide useful references for further reading.' Bulletin of the American Mathematical SocietyTable of ContentsPreface; Introduction; 1. Malliavin operators in the one-dimensional case; 2. Malliavin operators and isonormal Gaussian processes; 3. Stein's method for one-dimensional normal approximations; 4. Multidimensional Stein's method; 5. Stein meets Malliavin: univariate normal approximations; 6. Multivariate normal approximations; 7. Exploring the Breuer–Major Theorem; 8. Computation of cumulants; 9. Exact asymptotics and optimal rates; 10. Density estimates; 11. Homogeneous sums and universality; Appendix 1. Gaussian elements, cumulants and Edgeworth expansions; Appendix 2. Hilbert space notation; Appendix 3. Distances between probability measures; Appendix 4. Fractional Brownian motion; Appendix 5. Some results from functional analysis; References; Index.

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Book SynopsisThe unique feature of this compact student''s introduction to Mathematica and the Wolfram Language is that the order of the material closely follows a standard mathematics curriculum. As a result, it provides a brief introduction to those aspects of the Mathematica software program most useful to students. Used as a supplementary text, it will help bridge the gap between Mathematica and the mathematics in the course, and will serve as an excellent tutorial for former students. There have been significant changes to Mathematica since the second edition, and all chapters have now been updated to account for new features in the software, including natural language queries and the vast stores of real-world data that are now integrated through the cloud. This third edition also includes many new exercises and a chapter on 3D printing that showcases the new computational geometry capabilities that will equip readers to print in 3D.Trade Review'This book is an easy-to-read introduction to Mathematica. It is interspersed with helpful hints that make interacting with Mathematica more efficient and examples to test the reader's comprehension. This book is good for learning how to use Mathematica to graph functions, perform algebraic manipulation, and approach topics from calculus and linear algebra. This new version shines some light on entity objects and accessing Wolfram's curated data which is needed because their structure is unintuitive and because of their growing prominence in the Wolfram ecosystem. The new final chapter on 3D printing gives readers the tools to quickly design and 3D print physical objects that embody mathematical surfaces. These two additions showcase recent advances in the Wolfram Language and ensure that the whole book remains relevant and up to date.' Christopher Hanusa, Queens College, City University of New York'Mathematica has the power to unravel some of the current mysteries of mathematics – but only if you know how to ask it the right questions. The 3rd edition of The Student's Introduction to Mathematica and the Wolfram Language can be your well-used guide for such exploration. Beginning and experienced Mathematica users will easily learn from the pages of this book especially given the recent changes to Mathematica. Even more, the 3rd edition moves into a new dimension, giving details on 3D printing! Grab one for yourself and another for a student you know.' Tim Chartier, Davidson College, North Carolina'This text, including the exercises and solutions, is written in a student-friendly style … Unlike most tutorial introductions to Mathematica, the authors go to significant lengths to provide explanations and rationales underlying what a newcomer would likely find confusing … I believe that this book would be a useful addition to any student's library in a college or university that uses Mathematica.' Marvin Schaefer, MAA ReviewsTable of ContentsPreface; 1. Getting started; 2. Working with Mathematica®; 3. Functions and their graphs; 4. Algebra; 5. Calculus; 6. Multivariable calculus; 7. Linear algebra; 8. Programming; 9. 3D printing; Index.

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Book SynopsisThe language of ends and (co)ends provides a natural and general way of expressing many phenomena in category theory, in the abstract and in applications. Yet although category-theoretic methods are now widely used by mathematicians, since (co)ends lie just beyond a first course in category theory, they are typically only used by category theorists, for whom they are something of a secret weapon. This book is the first systematic treatment of the theory of (co)ends. Aimed at a wide audience, it presents the (co)end calculus as a powerful tool to clarify and simplify definitions and results in category theory and export them for use in diverse areas of mathematics and computer science. It is organised as an easy-to-cite reference manual, and will be of interest to category theorists and users of category theory alike.Table of ContentsPreface; 1. Dinaturality and (co)ends; 2. Yoneda and Kan; 3. Nerves and realisations; 4. Weighted (co)limits; 5. Profunctors; 6. Operads; 7. Higher dimensional (co)ends; Appendix A. Review of category theory; Appendix B; References; Index.

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Book SynopsisIn recent years the development of new classification and regression algorithms based on deep learning has led to a revolution in the fields of artificial intelligence, machine learning, and data analysis. The development of a theoretical foundation to guarantee the success of these algorithms constitutes one of the most active and exciting research topics in applied mathematics. This book presents the current mathematical understanding of deep learning methods from the point of view of the leading experts in the field. It serves both as a starting point for researchers and graduate students in computer science, mathematics, and statistics trying to get into the field and as an invaluable reference for future research.Table of Contents1. The modern mathematics of deep learning Julius Berner, Philipp Grohs, Gitta Kutyniok and Philipp Petersen; 2. Generalization in deep learning Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio; 3. Expressivity of deep neural networks Ingo Gühring, Mones Raslan and Gitta Kutyniok; 4. Optimization landscape of neural networks René Vidal, Zhihui Zhu and Benjamin D. Haeffele; 5. Explaining the decisions of convolutional and recurrent neural networks Wojciech Samek, Leila Arras, Ahmed Osman, Grégoire Montavon and Klaus-Robert Müller; 6. Stochastic feedforward neural networks: universal approximation Thomas Merkh and Guido Montúfar; 7. Deep learning as sparsity enforcing algorithms A. Aberdam and J. Sulam; 8. The scattering transform Joan Bruna; 9. Deep generative models and inverse problems Alexandros G. Dimakis; 10. A dynamical systems and optimal control approach to deep learning Weinan E, Jiequn Han and Qianxiao Li; 11. Bridging many-body quantum physics and deep learning via tensor networks Yoav Levine, Or Sharir, Nadav Cohen and Amnon Shashua.

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Book SynopsisI Fundamentals.- 1 Ordinary Differential Equations.- 2 Difference Equations.- II Local Analysis.- 3 Approximate Solution of Linear Differential Equations.- 4 Approximate Solution of Nonlinear Differential Equations.- 5 Approximate Solution of Difference Equations.- 6 Asymptotic Expansion of Integrals.- III Perturbation Methods.- 7 Perturbation Series.- 8 Summation of Series.- IV Global Analysis.- 9 Boundary Layer Theory.- 10 WKB Theory.- 11 Multiple-Scale Analysis.Trade Review"This book is a reprint of the original published by McGraw-Hill \ref [MR0538168 (80d:00030)]. The only changes are the addition of the Roman numeral I to the title and the provision of a subtitle, "Asymptotic methods and perturbation theory". This latter improvement is much needed, as the original title suggested that this was a teaching book for undergraduate scientists and engineers. It is not, but is an excellent introduction to asymptotic and perturbation methods for master's degree students or beginning research students. Certain parts of it could be used for a course in asymptotics for final year undergraduates in applied mathematics or mathematical physics. This is a book that has stood the test of time and I cannot but endorse the remarks of the original reviewer. It is written in a fresh and lively style and the many graphs and tables, comparing the results of exact and approximate methods, were in advance of its time. I have owned a copy of the original for over twenty years, using it on a regular basis, and, after the original had gone out of print, lending it to my research students. Springer-Verlag has done a great service to users of, and researchers in, asymptotics and perturbation theory by reprinting this classic." (A.D. Wood, Mathematical Reviews) Table of ContentsI Preface. 1 Ordinary Differential Equations. 2 Difference Equations. 3 Approximate Solution of Linear Differential Equations. 4 Approximate Solution of Nonlinear Equations. 5 Approximate Solution of Difference Equations. 6 Asymptotic Expansion of Integrals. 7 Perturbation Series. 8 Summation of Series. 9 Boundary Layer Theory. 10 WKB Theory. 11 Multiple Scales Analysis. Appendix, References, Index

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Book SynopsisWith a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus.  Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse''s lemma and the Poincaré lemma.  The ideas behind most topics can be understood with just two or three variables.  The book incorporates modern computational tools to give visualization real power.  Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps.  The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books.  This is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics.  Prerequisites are an introduction to linear algebra and multivariable calculus.  There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry.  The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.Trade ReviewFrom the reviews:“Many concepts in calculus and linear algebra have obvious geometric interpretations. … This book differs from other advanced calculus works … it can serve as a useful reference for professors. … it is the adopted course resource, its inclusion in a college library’s collection should be determined by the size and interests of the mathematics faculty. Summing Up … . Upper-division undergraduate through professional collections.” (C. Bauer, Choice, Vol. 48 (8), April, 2011)“The author of this book sees an opportunity to bring back a more geometric, visual and physically-motivated approach to the subject. … The author makes exceptionally good use of two and three-dimensional graphics. Drawings and figures are abundant and strongly support his exposition. Exercises are plentiful and they cover a range from routine computational work to proofs and extensions of results from the text. … Strong students … are likely to be attracted by the approach and the serious meaty content.” (William J. Satzer, The Mathematical Association of America, January, 2011)“A new geometric and visual approach to advanced calculus is presented. … The book can be useful a textbook for beginners as well as a source of supplementary material for university teachers in calculus and analysis. … the book meets a wide auditorium among undergraduate and graduate students in mathematics, physics, economics and in other fields which essentially use mathematical models. It is also very interesting for teachers and instructors in Calculus and Mathematical Analysis.” (Sergei V. Rogosin, Zentralblatt MATH, Vol. 1205, 2011)Table of Contents1 Starting Points.-1.1 Substitution.- Exercises.- 1.2 Work and path integrals.- Exercises.- 1.3 Polar coordinates.- Exercises.- 2 Geometry of Linear Maps.- 2.1 Maps from R2 to R2.- Exercises.- 2.2 Maps from Rn to Rn.- Exercises.- 2.3 Maps from Rn to Rp, n 6= p.- Exercises.- 3 Approximations.- 3.1 Mean-value theorems.- Exercises.- 3.2 Taylor polynomials in one variable.- Exercises.- 3.3 Taylor polynomials in several variables.- Exercises.- 4 The Derivative.- 4.1 Differentiability.- Exercises.- 4.2 Maps of the plane.- Exercises.- 4.3 Parametrized surfaces.- Exercises.- 4.4 The chain rule.- Exercises.- 5 Inverses.- 5.1 Solving equations.- Exercises.- 5.2 Coordinate Changes.- Exercises.- 5.3 The Inverse Function Theorem.- Exercises.- 6 Implicit Functions.- 6.1 A single equation.- Exercises.- 6.2 A pair of equations.- Exercises.- 6.3 The general case.- Exercises.- 7 Critical Points.- 7.1 Functions of one variable.- Exercises.- 7.2 Functions of two variables.- Exercises.- 7.3 Morse’s lemma.- Exercises.- 8 Double Integrals.- 8.1 Example: gravitational attraction.- Exercises.- 8.2 Area and Jordan content.- Exercises.- 8.3 Riemann and Darboux integrals.- Exercises.- 9 Evaluating Double Integrals.- 9.1 Iterated integrals.- Exercises.- 9.2 Improper integrals.- Exercises.- 9.3 The change of variables formula.- 9.4 Orientation.- Exercises.- 9.5 Green’s Theorem.- Exercises.- 10 Surface Integrals.- 10.1 Measuring flux.- Exercises.- 10.2 Surface area and scalar integrals.- Exercises.- 10.3 Differential forms.- Exercises.- 11 Stokes’ Theorem.- 11.1 Divergence.- Exercises.- 11.2 Circulation and Vorticity.- Exercises.- 11.3 Stokes’ Theorem.- 11.4 Closed and Exact Forms.- Exercises

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Book SynopsisThis book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions.The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics.The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis. Trade Review“The book is carefully prepared and well presented, and I recommend the book … for students who have just mastered vector calculus and Maxwellian electromagnetism.” (Hirokazu Nishimura, zbMATH 1433.58001, 2020)Table of ContentsPrelude: Linear algebra.- Exterior algebra.- Clifford algebra.- Mappings of inner product spaces.- Spinors in inner product spaces.- Interlude: Analysis.- Exterior calculus.- Hodge decompositions.- Hypercomplex analysis.- Dirac equations.- Multivector calculus on manifolds.- Two index theorems.

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Book SynopsisThis book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided. We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications. After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc. The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics. Table of ContentsHolomorphic functions.- Meromorphic functions.- Riemann's theorem.- Harmonic functions.- Riemann surfaces and their modules.- Compact Riemann surfaces and algebraic curves.- Riemann-Roch theorem and theta functions.- Integrable Systems.- The formula for the conformal mapping of an arbitrary domain into the unit disk.

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Book SynopsisConservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks. Table of ContentsIntroduction.- Boundary Control.- Decentralized Control.- Distributed Control.- Lagrangian Control.- Hamilton-Jacobi Equations.- Appendix A: Balance Laws with Boundary.- Conservation Laws on Networks.

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Book SynopsisThis volume covers some of the most seminal research in the areas of mathematical analysis and numerical computation for nonlinear phenomena. Collected from the international conference held in honor of Professor Yoshikazu Giga’s 60th birthday, the featured research papers and survey articles discuss partial differential equations related to fluid mechanics,　electromagnetism, surface diffusion, and evolving interfaces. Specific focus is placed on topics such as the solvability of the Navier-Stokes equations and the regularity, stability, and symmetry of their solutions, analysis of a living fluid, stochastic effects and numerics for Maxwell’s equations, nonlinear heat equations in critical spaces, viscosity solutions describing various kinds of interfaces, numerics for evolving interfaces, and a hyperbolic obstacle problem. Also included in this volume are an introduction of Yoshikazu Giga’s extensive academic career and a long list of his published work. Students and researchers in mathematical analysis and computation will find interest in this volume on theoretical study for nonlinear phenomena. Table of ContentsPartial differential equations and mathematical fluid mechanics, Matthis Hieber (TU Darmstadt).- Applied mathematics and mathematical biology, Ryo Kobayashi (Hiroshima University).- Nonlinear partial differential equations, calculus of variations, phase transformations, and composite materials, Robert V. Kohn (Courant Institute, NYU).- Nonlinear partial differential equations, calculus of variations, and computations for complex fluids, Chun Liu (Penn State University).- Partial differential equations and mathematical fluid mechanics, Yasunori Maekawa (Tohoku University).- Mathematics and computations in meterology, and fluid mechanics, Alex Mahalov (Arizona State University).- Nonlinear partial differential equations, mathematics and computations for crystal growth, Takeshi Ohtsuka (Gunma University).- Calculus of variations and mathematical analysis of phase transitions, Piotr Rybka (University of Warsaw).- Partial differential equations and mathematical fluid mechanics, Jurgen Saal (Dusseldorf University).- Multi-scale modeling and computations, and computational interface problems, Richard Tsai (University of Texas).- Plasma physics and fluid mechanics, Zensho Yoshida (University of Tokyo).

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Book SynopsisThis book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.Trade Review “The reviewer recommends young mathematics and physics majors to open the book and to keep it on their bookshelves. Indeed, the reviewer even envies young students who can study differential forms with such a fascinating book.” (Hirokazu Nishimura, zbMath 1419.58001, 2019)Table of Contents

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Book SynopsisThis edition contains detailed solutions of selected exercises. Many readers have requested this, because it makes the book more suitable for self-study. At the same time new exercises (without solutions) have beed added. They have all been placed in the end of each chapter, in order to facilitate the use of this edition together with previous ones. Several errors have been corrected and formulations have been improved. This has been made possible by the valuable comments from (in alphabetical order) Jon Bohlin, Mark Davis, Helge Holden, Patrick Jaillet, Chen Jing, Natalia Koroleva,MarioLefebvre,Alexander Matasov,Thilo Meyer-Brandis, Keigo Osawa, Bjorn Thunestvedt, Jan Uboe and Yngve Williassen. I thank them all for helping to improve the book. My thanks also go to Dina Haraldsson, who once again has performed the typing and drawn the ?gures with great skill. Blindern, September 2002 Bernt Oksendal xv Preface to Corrected Printing, Fifth Edition The main corrections and improvements in this corrected printing are from Chapter 12. I have bene?tted from useful comments from a number of p- ple, including (in alphabetical order) Fredrik Dahl, Simone Deparis, Ulrich Haussmann, Yaozhong Hu, Marianne Huebner, Carl Peter Kirkebo, Ni- lay Kolev, Takashi Kumagai, Shlomo Levental, Geir Magnussen, Anders Oksendal, Jur . . gen Pottho?, Colin Rowat, Stig Sandnes, Lones Smith, S- suo Taniguchi and Bjorn Thunestvedt. I want to thank them all for helping me making the book better. I also want to thank Dina Haraldsson for pro?cient typing.Trade ReviewFrom the reviews of the fifth edition: "This is a highly readable and refreshingly rigorous introduction to stochastic calculus. … This is not a watered-down treatment. It is a serious introduction that starts with fundamental measure-theoretic concepts and ends, coincidentally, with the Black-Scholes formula as one of several examples of applications. This is the best single resource for learning the stochastic calculus … ." (riskbook.com, 2002) From the reviews of the sixth edition: "The book … has evolved from a 200-page typewritten booklet to a modern classic. Part of its charm and success is the fact that the author does not bother too much with the (for the novice) cumbersome rigorous theory … . This does not mean that the book is not rigorous, it is just the timing and dosage of mathematical rigour … that is palatable for undergraduates … . a highly readable account, suitable for self-study and for use in the classroom." (René L. Schilling, The Mathematical Gazette, March, 2005) "This is the sixth edition of the classical and excellent book on stochastic differential equations. The main difference with the next to last edition is the addition of detailed solutions of selected exercises … . This is certainly an excellent idea in view to test its ability of applications of the concepts … . certainly one of the best books on the subject, it will be very helpful to any graduate students and also very valuable for any analysts of financial market." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004) "This is now the sixth edition of the excellent book on stochastic differential equations and related topics. … the presentation is successfully balanced between being easily accessible for a broad audience and being mathematically rigorous. The book is a first choice for courses at graduate level in applied stochastic differential equations. The inclusion of detailed solutions to many of the exercises in this edition also makes it very useful for self-study." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1025, 2003)Table of ContentsSome Mathematical Preliminaries.- Itô Integrals.- The Itô Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

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Book SynopsisThese notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.Table of ContentsFormally self-adjoint differential expressions.- Appendix to section 1: The separation of the Dirac operator.- Fundamental properties and general assumptions.- Appendix to section 2: Proof of the Lagrange identity for n>2.- The minimal operator and the maximal operator.- Deficiency indices and self-adjoint extensions of T0.- The solutions of the inhomogeneous differential equation (?-?)u=f; Weyl's alternative.- Limit point-limit circle criteria.- Appendix to section 6: Semi-boundedness of Sturm-Liouville type operators.- The resolvents of self-adjoint extensions of T0.- The spectral representation of self-adjoint extensions of T0.- Computation of the spectral matrix ?.- Special properties of the spectral representation, spectral multiplicities.- L2-solutions and essential spectrum.- Differential operators with periodic coefficients.- Appendix to section 12: Operators with periodic coefficients on the half-line.- Oscillation theory for regular Sturm-Liouville operators.- Oscillation theory for singular Sturm-Liouville operators.- Essential spectrum and absolutely continuous spectrum of Sturm-Liouville operators.- Oscillation theory for Dirac systems, essential spectrum and absolutely continuous spectrum.- Some explicitly solvable problems.

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Book SynopsisThe main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.Trade ReviewFrom the reviews:"The book under review is an introduction to the field of linear partial differential equations and to standard methods for their numerical solution. … The balanced combination of mathematical theory with numerical analysis is an essential feature of the book. … The book is easily accessible and concentrates on the main ideas while avoiding unnecessary technicalities. It is therefore well suited as a textbook for a beginning graduate course in applied mathematics." (A. Ostermann, IMN - Internationale Mathematische Nachrichten, Vol. 59 (198), 2005)"This book, which is aimed at beginning graduate students of applied mathematics and engineering, provides an up to date synthesis of mathematical analysis, and the corresponding numerical analysis, for elliptic, parabolic and hyperbolic partial differential equations. … This widely applicable material is attractively presented in this impeccably well-organised text. … Partial differential equations with numerical methods covers a lot of ground authoritatively and without ostentation and with a constant focus on the needs of practitioners." (Nick Lord, The Mathematical Gazette, March, 2005)"Larsson and Thomée … discuss numerical solution methods of linear partial differential equations. They explain finite difference and finite element methods and apply these concepts to elliptic, parabolic, and hyperbolic partial differential equations. … The text is enhanced by 13 figures and 150 problems. Also included are appendixes on mathematical analysis preliminaries and a connection to numerical linear algebra. Summing Up: Recommended. Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, March, 2004)"This book presents a very well written and systematic introduction to the finite difference and finite element methods for the numerical solution of the basic types of linear partial differential equations (PDE). … the book is very well written, the exposition is clear, readable and very systematic." (Emil Minchev, Zentralblatt MATH, Vol. 1025, 2003)"The author’s purpose is to give an elementary, relatively short, and readable account of the basic types of linear partial differential equations, their properties, and the most commonly used methods for their numerical solution. … We warmly recommend it to advanced undergraduate and beginning graduate students of applied mathematics and/or engineering at every university of the world." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 71, 2005)"The presentation of the book is smart and very classical; it is more a reference book for applied mathematicians … . The convergence results, error estimates, variation formulations, all the theorems proofs, are very clear and well presented, the annexes A and B summary the necessary background for the understanding, without redundant generalisation or forgotten matter. The bibliography is presented by theme, well targeted on the topic of the book." (Anne Lemaitre, Physicalia Magazine, Vol. 28 (1), 2006)“Offers basic theory of linear partial differential equations and discusses the most commonly used numerical methods to solve these equations. … There are two appendices providing some extra basic material, useful to help understanding some of the theoretical principles that might be unfamiliar to unexperienced readers and students. The text is elementary and meant for students in mathematics, physics, engineering. … The bibliography is well arranged according to the important issues, which makes it easy to get informed about possible references for further study.” (Paula Bruggen, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsA Two-Point Boundary Value Problem.- Elliptic Equations.- Finite Difference Methods for Elliptic Equations.- Finite Element Methods for Elliptic Equations.- The Elliptic Eigenvalue Problem.- Initial-Value Problems for Ordinary Differential Equations.- Parabolic Equations.- Finite Difference Methods for Parabolic Problems.- The Finite Element Method for a Parabolic Problem.- Hyperbolic Equations.- Finite Difference Methods for Hyperbolic Equations.- The Finite Element Method for Hyperbolic Equations.- Some Other Classes of Numerical Methods.

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Book SynopsisM. Brelot: Historical introduction.- H. Bauer: Harmonic spaces and associated Markov processes.- J.M. Bony: Opérateurs elliptiques dégénérés associés aux axiomatiques de la theorie du potentiel.- J. Deny: Méthodes hilbertiennes en theory du potentiel.- J.L. Doob: Martingale theory – Potential theory.- G. Mokobodzki: Cônes de potentiels et noyaux subordonnés.Table of ContentsM. Brelot: Historical introduction.- H. Bauer: Harmonic spaces and associated Markov processes.- J.M. Bony: Opérateurs elliptiques dégénérés associés aux axiomatiques de la theorie du potentiel.- J. Deny: Méthodes hilbertiennes en theory du potentiel.- J.L. Doob: Martingale theory – Potential theory.- G. Mokobodzki: Cônes de potentiels et noyaux subordonnés.

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Book SynopsisThis book provides a concise and meticulous introduction to functional analysis. Since the topic draws heavily on the interplay between the algebraic structure of a linear space and the distance structure of a metric space, functional analysis is increasingly gaining the attention of not only mathematicians but also scientists and engineers. The purpose of the text is to present the basic aspects of functional analysis to this varied audience, keeping in mind the considerations of applicability. A novelty of this book is the inclusion of a result by Zabreiko, which states that every countably subadditive seminorm on a Banach space is continuous. Several major theorems in functional analysis are easy consequences of this result.The entire book can be used as a textbook for an introductory course in functional analysis without having to make any specific selection from the topics presented here. Basic notions in the setting of a metric space are defined in terms of sequences. These include total boundedness, compactness, continuity and uniform continuity. Offering concise and to-the-point treatment of each topic in the framework of a normed space and of an inner product space, the book represents a valuable resource for advanced undergraduate students in mathematics, and will also appeal to graduate students and faculty in the natural sciences and engineering. The book is accessible to anyone who is familiar with linear algebra and real analysis.Trade Review“The title of this book indicates that it is mainly devoted to linear maps on linear spaces. … All chapters are accompanied by useful exercises of varying levels of difficulty, which help the readers to develop their knowledge on the topics. The solutions of the exercises are given at the end of the book. … This textbook is essentially addressed to people working in engineering and sciences branches.” (Mohammad Sal Moslehian, zbMATH 1352.46001, 2017)Table of ContentsChapter 1. Preliminaries.- Chapter 2. Basic Framework.- Chapter 3. Bounded Linear Maps.- Chapter 4. Dual Spaces, Transposes and Adjoints.- Chapter 5. Spectral Theory.

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Book SynopsisThis book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 ≤ k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke’s mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke’s existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard’s regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

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Book SynopsisThe Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kähler cases. In this book, the unsolved cases of the conjecture will be discussed.It will be shown that the problem is closely related to the geometry of moduli spaces of test configurations for polarized algebraic manifolds. Another important tool in our approach is the Chow norm introduced by Zhang. This is closely related to Ding’s functional, and plays a crucial role in our differential geometric study of stability. By discussing the Chow norm from various points of view, we shall make a systematic study of the existence problem of extremal Kähler metrics.Trade Review“The concise style of exposition likely means that this monograph is best suited for experts with background knowledge in canonical Kähler metrics. … It can be recommended also to those who would like a review of important results concerning the generalised Kähler-Einstein metrics, with various examples, and the moduli space of Lp-spaces.” (Yoshinori Hashimoto, Mathematical Reviews, May, 2023)Table of ContentsIntroduction.- The Donaldson-Futaki invariant.- Canonical Kähler metrics.- Norms for test configurations.- Stabilities for polarized algebraic manifolds.- The Yau-Tian-Donaldson conjecture.- Stability theorem.- Existence problem.- Canonical Kähler metrics on Fano manifolds.- Geometry of pseudo-normed graded algebras.- Solutions.

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Book SynopsisThis third of the three-volume book is targeted as a basic course in algebraic topology and topology for fiber bundles for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, and algebra. Topics covered in this volume include homotopy theory, homology and cohomology theories, homotopy theory of fiber bundles, Euler characteristic, and the Betti number. It also includes certain classic problems such as the Jordan curve theorem along with the discussions on higher homotopy groups and establishes links between homotopy and homology theories, axiomatic approach to homology and cohomology as inaugurated by Eilenberg and Steenrod. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power and active learning of the subject, all the while covering a wide range of theory and applications in a balanced unified way.Table of Contents1. Prerequisite Concepts of Topology, Algebra and Category Theory.- 2. Homotopy Theory: Fundamental and Higher Homotopy Groups.- 3. Homology and Cohomology Theories: An Axiomatic Approach with Consequences.- 4. Topology of Fiber Bundles.- 5. Homotopy Theory of Bundles.- 6. Some Applications of Algebraic Topology.- 7. Brief History on Algebraic Topology and Fiber Bundles.

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Book SynopsisThis unique book provides a new and well-motivated introduction to calculus and analysis, historically significant fundamental areas of mathematics that are widely used in many disciplines. It begins with familiar elementary high school geometry and algebra, and develops important concepts such as tangents and derivatives without using any advanced tools based on limits and infinite processes that dominate the traditional introductions to the subject. This simple algebraic method is a modern version of an idea that goes back to René Descartes and that has been largely forgotten. Moving beyond algebra, the need for new analytic concepts based on completeness, continuity, and limits becomes clearly visible to the reader while investigating exponential functions.The author carefully develops the necessary foundations while minimizing the use of technical language. He expertly guides the reader to deep fundamental analysis results, including completeness, key differential equations, definite integrals, Taylor series for standard functions, and the Euler identity. This pioneering book takes the sophisticated reader from simple familiar algebra to the heart of analysis. Furthermore, it should be of interest as a source of new ideas and as supplementary reading for high school teachers, and for students and instructors of calculus and analysis.Table of ContentsTangents and Double Points; Derivatives by Algebra; Exponential Functions; Completeness of Real Numbers; The Base of the Natural Exponential and Logarithm Functions; Continuity of Functions; Differentiability; Chain Rule and Other Rules for Derivatives; Derivatives of Trigonometric Functions; Mean Value Inequality and Theorem; Basic Differential Equations; Motion with Constant Acceleration; Linear and Higher Order Approximations; The Antiderivative Problem; Definite Integrals; Fundamental Theorem of Calculus; Integrability of Monotonic Functions; Integrability of Functions with Bounded Derivative; Substitution; Integration by Parts; Taylor's Theorem; Analytic Functions; The Euler Identity;

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Book SynopsisI Fundamentals.- 1 Ordinary Differential Equations.- 2 Difference Equations.- II Local Analysis.- 3 Approximate Solution of Linear Differential Equations.- 4 Approximate Solution of Nonlinear Differential Equations.- 5 Approximate Solution of Difference Equations.- 6 Asymptotic Expansion of Integrals.- III Perturbation Methods.- 7 Perturbation Series.- 8 Summation of Series.- IV Global Analysis.- 9 Boundary Layer Theory.- 10 WKB Theory.- 11 Multiple-Scale Analysis.Trade Review"This book is a reprint of the original published by McGraw-Hill \ref [MR0538168 (80d:00030)]. The only changes are the addition of the Roman numeral I to the title and the provision of a subtitle, "Asymptotic methods and perturbation theory". This latter improvement is much needed, as the original title suggested that this was a teaching book for undergraduate scientists and engineers. It is not, but is an excellent introduction to asymptotic and perturbation methods for master's degree students or beginning research students. Certain parts of it could be used for a course in asymptotics for final year undergraduates in applied mathematics or mathematical physics. This is a book that has stood the test of time and I cannot but endorse the remarks of the original reviewer. It is written in a fresh and lively style and the many graphs and tables, comparing the results of exact and approximate methods, were in advance of its time. I have owned a copy of the original for over twenty years, using it on a regular basis, and, after the original had gone out of print, lending it to my research students. Springer-Verlag has done a great service to users of, and researchers in, asymptotics and perturbation theory by reprinting this classic." (A.D. Wood, Mathematical Reviews) Table of ContentsI Preface. 1 Ordinary Differential Equations. 2 Difference Equations. 3 Approximate Solution of Linear Differential Equations. 4 Approximate Solution of Nonlinear Equations. 5 Approximate Solution of Difference Equations. 6 Asymptotic Expansion of Integrals. 7 Perturbation Series. 8 Summation of Series. 9 Boundary Layer Theory. 10 WKB Theory. 11 Multiple Scales Analysis. Appendix, References, Index

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Book SynopsisThis text in multivariable calculus fosters comprehension through meaningful explanations. Written with students in mathematics, the physical sciences, and engineering in mind, it extends concepts from single variable calculus such as derivative, integral, and important theorems to partial derivatives, multiple integrals, Stokes’ and divergence theorems. Students with a background in single variable calculus are guided through a variety of problem solving techniques and practice problems. Examples from the physical sciences are utilized to highlight the essential relationship between calculus and modern science. The symbiotic relationship between science and mathematics is shown by deriving and discussing several conservation laws, and vector calculus is utilized to describe a number of physical theories via partial differential equations. Students will learn that mathematics is the language that enables scientific ideas to be precisely formulated and that science is a source for the development of mathematics. Trade Review“The presentation of the material is guided by applications so that physics and engineering students will find the text engaging and see the relevance of multivariable calculus to their work. The text contains over 500 exercises with answers and/or solutions to half provided at the back of the book, enabling students to gauge their understanding of the content as they proceed. A well-written, engaging text. Summing Up: Highly recommended. Upper-division undergraduates and professionals.” (J. T. Zerger, Choice, Vol. 56 (03), November, 2018)“This book belongs to a collection aimed at third- and fourth-year undergraduate mathematics students at North American universities. … There are more than 200 figures to help the reader to understand the explanations and about 500 problems. … I think this book can be recommended since, moreover, it is very pedagogical.” (Richard Becker, Mathematical Reviews, October, 2018)“Lax and Terrell’s sequel to their Calculus With Applications presents a first course in multivariable calculus that fits in just over 400 pages. Even instructors who use standard texts will find much of value in this refreshing first edition. The book is written with a wide range of STEM students in mind, and its exposition remains remarkably fluid without scarificing precision. Every section of each chapter ends with an excellent collection of exercises, which should be graciously welcomed by independent learners and instructors alike.” (Tushar Das, MAA Reviews, September, 2018)“The main achievement of the authors is that they essentially have simplified the teaching of the old topics to make a place for new ones. The proofs are exposited to encourage understanding, not meaningless rigor. … the presented book is a useful tool for all mathematicians (not only for students) and I find it regrettable that this book was not written when I was a student.” (Andrey Zahariev, zbMATH 1396.26002, 2018)Table of Contents1. Vectors and matrices.- 2. Functions.- 3. Differentiation.- 4. More about differentiation.- 5. Applications to motion.- 6. Integration.- 7. Line and surface integrals.- 8. Divergence and Stokes’ Theorems and conservation laws.- 9. Partial differential equations.- Answers to selected problems.- Index.

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Book Synopsis

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Book Synopsis

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Book SynopsisPerfected over three editions and more than forty years, this field- and classroom-tested reference: * Uses the method of maximum likelihood to a large extent to ensure reasonable, and in some cases optimal procedures. * Treats all the basic and important topics in multivariate statistics. * Adds two new chapters, along with a number of new sections. * Provides the most methodical, up-to-date information on MV statistics available.Trade Review"…suitable for a graduate-level course on multivariate analysis…an important reference on the bookshelves of many scientific researchers and most practicing statisticians." (Journal of the American Statistical Association, September 2004) “…really well written. The edition will be certainly welcomed…” (Zentralblatt Math, Vo.1039, No.08, 2004) "…a wonderful textbook…that covers the mathematical theory of multivariate statistical analysis…" (Clinical Chemistry, Vol. 50, No. 2, May 2004) "...remains an authoritative work that can still be highly recommended..." (Short Book Reviews, 2004) "...still a very serious and comprehensive book on the statistical theory of multivariate analysis." (Technometrics, Vol. 46, No. 1, February 2004) “...remains a mathematically rigorous development of statistical methods for observations consisting of several measurements or characteristics of each subject and a study of their properties.” (Quarterly of Applied Mathematics, Vol. LXI, No. 4, December 2003)Table of ContentsPreface to the Third Edition. Preface to the Second Edition. Preface to the First Edition. 1. Introduction. 2. The Multivariate Normal Distribution. 3. Estimation of the Mean Vector and the Covariance Matrix. 4. The Distributions and Uses of Sample Correlation Coefficients. 5. The Generalized T2-Statistic. 6. Classification of Observations. 7. The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance. 8. Testing the General Linear Hypothesis: Multivariate Analysis of Variance 9. Testing Independence of Sets of Variates. 10. Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices. 11. Principal Components. 12. Cononical Correlations and Cononical Variables. 13. The Distributions of Characteristic Roots and Vectors. 14. Factor Analysis. 15. Pattern of Dependence; Graphical Models. Appendix A: Matrix Theory. Appendix B: Tables. References. Index.

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Book SynopsisPresents a novel approach to conducting meta-analysis using structural equation modeling. Structural equation modeling (SEM) and meta-analysis are two powerful statistical methods in the educational, social, behavioral, and medical sciences. They are often treated as two unrelated topics in the literature. This book presents a unified framework on analyzing meta-analytic data within the SEM framework, and illustrates how to conduct meta-analysis using the metaSEM package in the R statistical environment. Meta-Analysis: A Structural Equation Modeling Approach begins by introducing the importance of SEM and meta-analysis in answering research questions. Key ideas in meta-analysis and SEM are briefly reviewed, and various meta-analytic models are then introduced and linked to the SEM framework. Fixed-, random-, and mixed-effects models in univariate and multivariate meta-analyses, three-level meta-analysis, and meta-analytic structural equation modeling, areTrade Review"This book will be a valuable resource for statistical and academic researchers and graduate students carrying out meta-analyses, and will also be useful to researchers and statisticians using SEM in biostatistics. cover, would sit well on the bookshelves of those interested in this increasingly important field of scientific endeavour." (Zentralblatt MATH, 1 June 2015)Table of ContentsPreface xiii Acknowledgments xv List of abbreviations xvii List of figures xix List of tables xxi 1 Introduction 1 1.1 What is meta-analysis? 1 1.2 What is structural equation modeling? 2 1.3 Reasons for writing a book on meta-analysis and structural equation modeling 3 1.4 Outline of the following chapters 6 1.5 Concluding remarks and further readings 8 2 Brief review of structural equation modeling 13 2.1 Introduction 13 2.2 Model specification 14 2.3 Common structural equation models 18 2.4 Estimation methods, test statistics, and goodness-of-fit indices 25 2.5 Extensions on structural equation modeling 38 2.6 Concluding remarks and further readings 42 3 Computing effect sizes for meta-analysis 48 3.1 Introduction 48 3.2 Effect sizes for univariate meta-analysis 50 3.3 Effect sizes for multivariate meta-analysis 57 3.4 General approach to estimating the sampling variances and covariances 60 3.5 Illustrations Using R 68 3.6 Concluding remarks and further readings 78 4 Univariate meta-analysis 81 4.1 Introduction 81 4.2 Fixed-effects model 83 4.3 Random-effects model 87 4.4 Comparisons between the fixed- and the random-effects models 93 4.5 Mixed-effects model 96 4.6 Structural equation modeling approach 100 4.7 Illustrations using R 105 4.8 Concluding remarks and further readings 116 5 Multivariate meta-analysis 121 5.1 Introduction 121 5.2 Fixed-effects model 124 5.3 Random-effects model 127 5.4 Mixed-effects model 134 5.5 Structural equation modeling approach 136 5.6 Extensions: mediation and moderation models on the effect sizes 140 5.7 Illustrations using R 145 5.8 Concluding remarks and further readings 174 6 Three-level meta-analysis 179 6.1 Introduction 179 6.2 Three-level model 183 6.3 Structural equation modeling approach 188 6.4 Relationship between the multivariate and the three-level meta-analyses 195 6.5 Illustrations using R 200 6.6 Concluding remarks and further readings 210 7 Meta-analytic structural equation modeling 214 7.1 Introduction 214 7.2 Conventional approaches 218 7.3 Two-stage structural equation modeling: fixed-effects models 223 7.4 Two-stage structural equation modeling: random-effects models 233 7.5 Related issues 235 7.6 Illustrations using R 244 7.7 Concluding remarks and further readings 273 8 Advanced topics in SEM-based meta-analysis 279 8.1 Restricted (or residual) maximum likelihood estimation 279 8.2 Missing values in the moderators 289 8.3 Illustrations using R 294 8.4 Concluding remarks and further readings 309 9 Conducting meta-analysis with Mplus 313 9.1 Introduction 313 9.2 Univariate meta-analysis 314 9.3 Multivariate meta-analysis 327 9.4 Three-level meta-analysis 346 9.5 Concluding remarks and further readings 353 A A brief introduction to R, OpenMx, and metaSEM packages 356 A.1 R 357 A.2 OpenMx 362 A.3 metaSEM 364 References 368 Index 369

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Book SynopsisAn accessible guide to the multivariate time series tools used in numerous real-world applications Multivariate Time Series Analysis: With R and Financial Applications is the much anticipated sequel coming from one of the most influential and prominent experts on the topic of time series.Table of ContentsPreface xv Acknowledgements xvii 1 Multivariate Linear Time Series 1 1.1 Introduction, 1 1.2 Some Basic Concepts, 5 1.3 Cross-Covariance and Correlation Matrices, 8 1.4 Sample CCM, 9 1.5 Testing Zero Cross-Correlations, 12 1.6 Forecasting, 16 1.7 Model Representations, 18 1.8 Outline of the Book, 22 1.9 Software, 23 Exercises, 23 2 Stationary Vector Autoregressive Time Series 27 2.1 Introduction, 27 2.2 VAR(1) Models, 28 2.3 VAR(2) Models, 37 2.4 VAR(p) Models, 41 2.5 Estimation, 44 2.6 Order Selection, 61 2.7 Model Checking, 66 2.8 Linear Constraints, 80 2.9 Forecasting, 82 2.10 Impulse Response Functions, 89 2.11 Forecast Error Variance Decomposition, 96 2.12 Proofs, 98 Exercises, 100 3 Vector Autoregressive Moving-Average Time Series 105 3.1 Vector MA Models, 106 3.2 Specifying VMA Order, 112 3.3 Estimation of VMA Models, 113 3.4 Forecasting of VMA Models, 126 3.5 VARMA Models, 127 3.6 Implications of VARMA Models, 139 3.7 Linear Transforms of VARMA Processes, 141 3.8 Temporal Aggregation of VARMA Processes, 144 3.9 Likelihood Function of a VARMA Model, 146 3.10 Innovations Approach to Exact Likelihood Function, 155 3.11 Asymptotic Distribution of Maximum Likelihood Estimates, 160 3.12 Model Checking of Fitted VARMA Models, 163 3.13 Forecasting of VARMA Models, 164 3.14 Tentative Order Identification, 166 3.15 Empirical Analysis of VARMA Models, 176 3.16 Appendix, 192 Exercises, 194 4 Structural Specification of VARMA Models 199 4.1 The Kronecker Index Approach, 200 4.2 The Scalar Component Approach, 212 4.3 Statistics for Order Specification, 220 4.4 Finding Kronecker Indices, 222 4.5 Finding Scalar Component Models, 226 4.6 Estimation, 237 4.7 An Example, 245 4.8 Appendix: Canonical Correlation Analysis, 259 Exercises, 262 5 Unit-Root Nonstationary Processes 265 5.1 Univariate Unit-Root Processes, 266 5.2 Multivariate Unit-Root Processes, 279 5.3 Spurious Regressions, 290 5.4 Multivariate Exponential Smoothing, 291 5.5 Cointegration, 294 5.6 An Error-Correction Form, 297 5.7 Implications of Cointegrating Vectors, 300 5.8 Parameterization of Cointegrating Vectors, 302 5.9 Cointegration Tests, 303 5.10 Estimation of Error-Correction Models, 313 5.11 Applications, 319 5.12 Discussion, 326 5.13 Appendix, 327 Exercises, 328 6 Factor Models and Selected Topics 333 6.1 Seasonal Models, 333 6.2 Principal Component Analysis, 341 6.3 Use of Exogenous Variables, 345 6.4 Missing Values, 357 6.5 Factor Models, 364 6.6 Classification and Clustering Analysis, 386 Exercises, 394 7 Multivariate Volatility Models 399 7.1 Testing Conditional Heteroscedasticity, 401 7.2 Estimation of Multivariate Volatility Models, 407 7.3 Diagnostic Checks of Volatility Models, 409 7.4 Exponentially Weighted Moving Average, 414 7.5 BEKK Models, 417 7.6 Cholesky Decomposition and Volatility Modeling, 420 7.7 Dynamic Conditional Correlation Models, 428 7.8 Orthogonal Transformation, 434 7.9 Copula-Based Models, 443 7.10 Principal Volatility Components, 454 Exercises, 461 Appendix A Review of Mathematics and Statistics 465 A.1 Review of Vectors and Matrices, 465 A.2 Least-Squares Estimation, 477 A.3 Multivariate Normal Distributions, 478 A.4 Multivariate Student-t Distribution, 479 A.5 Wishart and Inverted Wishart Distributions, 480 A.6 Vector and Matrix Differentials, 481 Index 489

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Book SynopsisProvides a sound, intuitive understanding of the basic concepts students need as they pursue careers in business, economics, and the life and social sciences. In this book, the author applies real-world orientation to concepts, problem-solving approach, straight forward and concise writing style, and comprehensive exercise sets.Table of ContentsChapter 1: Functions, Graphs, and Limits1.1Functions1.2The Graph of a Function1.3Lines and Linear Functions1.4Functional Models1.5Limits1.6One-Sided Limits and ContinuityChapter 2: Differentiation: Basic Concepts2.1The Derivative2.2Techniques of Differentiation2.3Product and Quotient Rules; Higher-Order Derivatives2.4The Chain Rule2.5Marginal Analysis and Approximations Using Increments2.6Implicit Differentiation and Related RatesChapter 3: Additional Applications of the Derivative3.1 Increasing and Decreasing Functions; Relative Extrema3.2 Concavity and Points of Inflection3.3 Curve Sketching3.4 Optimization; Elasticity of Demand3.5 Additional Applied OptimizationChapter 4: Exponential and Logarithmic Functions4.1 Exponential Functions; Continuous Compounding4.2 Logarithmic Functions4.3 Differentiation of Exponential and Logarithmic Functions4.4 Additional Applications; Exponential ModelsChapter 5: Integration5.1 Indefinite Integration and Differential Equations5.2 Integration by Substitution5.3 The Definite Integral and the Fundamental Theorem of Calculus5.4 Applying Definite Integration: Distribution of Wealth and Average Value5.5 Additional Applications to Business and Economics5.6 Additional Applications to the Life and Social SciencesChapter 6: Additional Topics in Integration6.1 Integration by Parts; Integral Tables6.2 Numerical Integration6.3 Improper Integrals6.4 Introduction to Continuous ProbabilityChapter 7: Calculus of Several Variables7.1 Functions of Several Variables7.2 Partial Derivatives7.3 Optimizing Functions of Two Variables7.4 The Method of Least-Squares7.5 Constrained Optimization: The Method of Lagrange Multipliers7.6 Double IntegralsAppendix A: Algebra ReviewA.1 A Brief Review of AlgebraA.2 Factoring Polynomials and Solving Systems of EquationsA.3 Evaluating Limits with L’Hopital’s RuleA.4 The Summation Notation

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Book SynopsisStatistical science s first coordinated manual of methods for analyzing ordered categorical data, now fully revised and updated, continues to present applications and case studies in fields as diverse as sociology, public health, ecology, marketing, and pharmacy.Table of ContentsPreface. 1. Introduction. 1.1. Ordinal Categorical Scales. 1.2. Advantages of Using Ordinal Methods. 1.3. Ordinal Modeling Versus Ordinary Regession Analysis. 1.4. Organization of This Book. 2. Ordinal Probabilities, Scores, and Odds Ratios. 2.1. Probabilities and Scores for an Ordered Categorical Scale. 2.2. Ordinal Odds Ratios for Contingency Tables. 2.3. Confidence Intervals for Ordinal Association Measures. 2.4. Conditional Association in Three-Way Tables. 2.5. Category Choice for Ordinal Variables. Chapter Notes. Exercises. 3. Logistic Regression Models Using Cumulative Logits. 3.1. Types of Logits for An Ordinal Response. 3.2. Cumulative Logit Models. 3.3. Proportional Odds Models: Properties and Interpretations. 3.4. Fitting and Inference for Cumulative Logit Models. 3.5. Checking Cumulative Logit Models. 3.6. Cumulative Logit Models Without Proportional Odds. 3.7. Connections with Nonparametric Rank Methods. Chapter Notes. Exercises. 4. Other Ordinal Logistic Regression Models. 4.1. Adjacent-Categories Logit Models. 4.2. Continuation-Ratio Logit Models. 4.3. Stereotype Model: Multiplicative Paired-Category Logits. Chapter Notes. Exercises. 5. Other Ordinal Multinomial Response Models. 5.1. Cumulative Link Models. 5.2. Cumulative Probit Models. 5.3. Cumulative Log-Log Links: Proportional Hazards Modeling. 5.4. Modeling Location and Dispersion Effects. 5.5. Ordinal ROC Curve Estimation. 5.6. Mean Response Models. Chapter Notes. Exercises. 6. Modeling Ordinal Association Structure. 6.1. Ordinary Loglinear Modeling. 6.2. Loglinear Model of Linear-by-Linear Association. 6.3. Row or Column Effects Association Models. 6.4. Association Models for Multiway Tables. 6.5. Multiplicative Association and Correlation Models. 6.6. Modeling Global Odds Ratios and Other Associations. Chapter Notes. Exercises. 7. Non-Model-Based Analysis of Ordinal Association. 7.1. Concordance and Discordance Measures of Association. 7.2. Correlation Measures for Contingency Tables. 7.3. Non-Model-Based Inference for Ordinal Association Measures. 7.4. Comparing Singly Ordered Multinomials. 7.5. Order-Restricted Inference with Inequality Constraints. 7.6. Small-Sample Ordinal Tests of Independence. 7.7. Other Rank-Based Statistical Methods for Ordered Categories. Appendix: Standard Errors for Ordinal Measures. Chapter Notes. Exercises. 8. Matched-Pairs Data with Ordered Categories. 8.1. Comparing Marginal Distributions for Matched Pairs. 8.2. Models Comparing Matched Marginal Distributions. 8.3. Models for The Joint Distribution in A Square Table. 8.4. Comparing Marginal Distributions for Matched Sets. 8.5. Analyzing Rater Agreement on an Ordinal Scale. 8.6. Modeling Ordinal Paired Preferences. Chapter Notes. Exercises. 9. Clustered Ordinal Responses: Marginal Models. 9.1. Marginal Ordinal Modeling with Explanatory Variables. 9.2. Marginal Ordinal Modeling: GEE Methods. 9.3. Transitional Ordinal Modeling, Given the Past. Chapter Notes. Exercises. 10. Clustered Ordinal Responses: Random Effects Models. 10.1. Ordinal Generalized Linear Mixed Models. 10.2. Examples of Ordinal Random Intercept Models. 10.3. Models with Multiple Random Effects. 10.4. Multilevel (Hierarchical) Ordinal Models. 10.5. Comparing Random Effects Models and Marginal Models. Chapter Notes. Exercises. 11. Bayesian Inference for Ordinal Response Data. 11.1. Bayesian Approach to Statistical Inference. 11.2. Estimating Multinomial Parameters. 11.3. Bayesian Ordinal Regression Modeling. 11.4. Bayesian Ordinal Association Modeling. 11.5. Bayesian Ordinal Multivariate Regression Modeling. 11.6. Bayesian Versus Frequentist Approaches to Analyzing Ordinal Data. Chapter Notes. Exercises. Appendix Software for Analyzing Ordinal Categorical Data. Bibliography. Example Index. Subject Index.

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Book SynopsisThis book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.Table of ContentsPreface to the Second Edition.- Preface to the First Edition.- A Study Guide.- 1. Methods of Proof.- 2. Algebra.- 3. Real Analysis.- 4. Geometry and Trigonometry.- 5. Number Theory.- 6. Combinatorics and Probability.- Solutions.- Index of Notation.- Index.

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Book SynopsisDieses vierfarbige Lehrbuch wendet sich an Studierende der Mathematik in Bachelor- und Lehramts-Studiengängen. Es bietet in einem Band ein lebendiges Bild der mathematischen Inhalte, die üblicherweise im ersten Studienjahr behandelt werden (und etliches mehr). Mathematik-Studierende finden wichtige Begriffe, Sätze und Beweise ausführlich und mit vielen Beispielen erklärt und werden an grundlegende Konzepte und Methoden herangeführt.Im Mittelpunkt stehen das Verständnis der mathematischen Zusammenhänge und des Aufbaus der Theorie sowie die Strukturen und Ideen wichtiger Sätze und Beweise. Es wird nicht nur ein in sich geschlossenes Theoriengebäude dargestellt, sondern auch verdeutlicht, wie es entsteht und wozu die Inhalte später benötigt werden.Herausragende Merkmale sind:- durchgängig vierfarbiges Layout mit mehr als 600 Abbildungen- prägnant formulierte Kerngedanken bilden die Abschnittsüberschriften- Selbsttests in kurzen Abständen ermöglichen Lernkontrollen während des Lesens- farbige Merkkästen heben das Wichtigste hervor- „Unter-der-Lupe“-Boxen zoomen in Beweise hinein, motivieren und erklären Details- „Hintergrund-und-Ausblick“-Boxen stellen Zusammenhänge zu anderen Gebieten und weiterführenden Themen her- Zusammenfassungen zu jedem Kapitel sowie Übersichtsboxen- mehr als 400 Verständnisfragen, Rechenaufgaben und Aufgaben zu Beweisen- deutsch-englisches Symbol- und Begriffsglossar Der inhaltliche Schwerpunkt liegt auf den Themen der Vorlesungen Analysis 1 und 2 sowie Linearer Algebra 1 und 2. Behandelt werden darüber hinaus Inhalte und Methodenkompetenzen, die vielerorts im ersten Studienjahr der Mathematikausbildung vermittelt werden.Hinweise, Lösungswege und Ergebnisse zu allen Aufgaben des Buchs stehen als PDF-Dateien auf http://sn.pub/extras in dem Ordner für das Werk Arens et al, „Mathematik“, Copyrightjahr 2018 zur Verfügung. Das Buch wird allen Studierenden der Mathematik vom Beginn des Studiums bis in höhere Semester hinein ein verlässlicher Begleiter sein.Für die 2. Auflage ist es vollständig durchgesehen, an zahlreichen Stellen didaktisch weiter verbessert und um einige Themen ergänzt worden.Stimme zur ersten Auflage:„Besonders gut gefallen mir die Übersichtlichkeit und die Verständlichkeit, besonders aber die Sichtbarmachung der Verbindung von Analysis und linearer Algebra, die in den Erstsemestervorlesungen oft zu kurz kommt.” Sylvia Prinz, Institut für Mathematikdidaktik, Universität zu KölnTable of ContentsVorwort.- 1 Was ist Mathematik und was tun Mathematiker?- 2 Logik, Mengen, Abbildungen − die Sprache der Mathematik.- 2.1 Junktoren und Quantoren.- 2.2 Grundbegriffe aus der Mengenlehre.- 2.3 Abbildungen.- 2.4 Relationen.- Zusammenfassung.- Aufgaben.- 3 Algebraische Strukturen − ein Blick hinter die Rechenregeln.- 3.1 Gruppen.- 3.2 Homomorphismen.- 3.3 Körper.- 3.4 Ringe.- Zusammenfassung.- Aufgaben.- 4 Zahlbereiche − Basis nicht nur der Analysis.- 4.1 Reelle Zahlen.- 4.2 Körperaxiome für die reellen Zahlen.- 4.3 Anordnungsaxiome für die reellen Zahlen.- 4.4 Ein Vollständigkeitsaxiom für die reellen Zahlen.- 4.5 Natürliche Zahlen und vollständige Induktion.- 4.6 Ganze Zahlen und rationale Zahlen.- 4.7 Komplexe Zahlen: Ihre Arithmetik und Geometrie.- Zusammenfassung.- Aufgaben.- 5 Lineare Gleichungssysteme − ein Tor zur linearen Algebra.- 5.1 Erste Lösungsversuche.- 5.2 Das Lösungsverfahren von Gauß und Jordan.- 5.3 Das Lösungskriterium und die Struktur der Lösung.- Zusammenfassung.- Aufgaben.- 6 Vektorräume − von Basen und Dimensionen.- 6.1 Der Vektorraumbegriff.- 6.2 Beispiele von Vektorräumen.- 6.3 Untervektorräume.- 6.4 Basis und Dimension.- 6.5 Summe und Durchschnitt von Untervektorräumen.- Zusammenfassung.- Aufgaben.- 7 Analytische Geometrie − Rechnen statt Zeichnen.- 7.1 Punkte und Vektoren im Anschauungsraum.- 7.2 Das Skalarprodukt im Anschauungsraum.- 7.3 Weitere Produkte von Vektoren im Anschauungsraum.- 7.4 Abstände zwischen Punkten, Geraden und Ebenen.- 7.5 Wechsel zwischen kartesischen Koordinatensystemen.- Zusammenfassung.- Aufgaben.- 8 Folgen − der Weg ins Unendliche.- 8.1 Der Begriff einer Folge.- 8.2 Konvergenz.- 8.3 Häufungspunkte und Cauchy-Folgen.- Zusammenfassung.- Aufgaben.- 9 Funktionen und Stetigkeit − ε trifft auf δ.- 9.1 Grundlegendes zu Funktionen.- 9.2 Beschränkte und monotone Funktionen.- 9.3 Grenzwerte für Funktionen und die Stetigkeit.- 9.4 Abgeschlossene, offene, kompakte Mengen.- 9.5 Stetige Funktionen mit kompaktem Definitionsbereich, Zwischenwertsatz.- Zusammenfassung.- Aufgaben.- 10 Reihen − Summieren bis zum Letzten.- 10.1 Motivation und Definition.- 10.2 Kriterien für Konvergenz.- 10.3 Absolute Konvergenz.- 10.4 Kriterien für absolute Konvergenz.- Zusammenfassung.- Aufgaben.- 11 Potenzreihen − Alleskönner unter den Funktionen.- 11.1 Definition und Grundlagen.- 11.2 Die Darstellung von Funktionen durch Potenzreihen.- 11.3 Die Exponentialfunktion.- 11.4 Trigonometrische Funktionen.- 11.5 Der Logarithmus.- Zusammenfassung.- Aufgaben.- 12 Lineare Abbildungen und Matrizen − Brücken zwischen Vektorräumen.- 12.1 Definition und Beispiele.- 12.2 Verknüpfungen von linearen Abbildungen.- 12.3 Kern, Bild und die Dimensionsformel.- 12.4 Darstellungsmatrizen.- 12.5 Das Produkt von Matrizen.- 12.6 Das Invertieren von Matrizen.- 12.7 Elementarmatrizen.- 12.8 Basistransformation.- 12.9 Der Dualraum.- Zusammenfassung.- Aufgaben.- <13 Determinanten − Kenngrößen von Matrizen.- 13.1 Die Definition der Determinante.- 13.2 Determinanten von Endomorphismen.- 13.3 Berechnung der Determinante.- 13.4 Anwendungen der Determinante.- Zusammenfassung.- Aufgaben.- 14 Normalformen − Diagonalisieren und Triangulieren.- 14.1 Diagonalisierbarkeit.- 14.2 Eigenwerte und Eigenvektoren.- 14.3 Berechnung der Eigenwerte und Eigenvektoren.- 14.4 Algebraische und geometrische Vielfachheit.- 14.5 Die Exponentialfunktion für Matrizen.- 14.6 Das Triangulieren von Endomorphismen.- 14.7 Die Jordan-Normalform.- 14.8 Die Berechnung einer Jordan-Normalform und Jordan-Basis.- Zusammenfassung.- Aufgaben.- 15 Differenzialrechnung − die Linearisierung von Funktionen.- 15.1 Die Ableitung.- 15.2 Differenziationsregeln.- 15.3 Der Mittelwertsatz.- 15.4 Verhalten differenzierbarer Funktionen.- 15.5 Taylorreihen.- Zusammenfassung.- Aufgaben.- 16 Integrale − von lokal zu global.- 16.1 Integration von Treppenfunktionen.- 16.2 Das Lebesgue-Integral.- 16.3 Stammfunktionen.- 16.4 Integrationstechniken.- 16.5 Integration über unbeschränkte Intervalle oder Funktionen.- 16.6 Parameterabhängige Integrale.- 16.7 Weitere Integrationsbegriffe.- Zusammenfassung.- Aufgaben.- 17 Euklidische und unitäre Vektorräume − orthogonales Diagonalisieren.- 17.1 Euklidische Vektorräume.- 17.2 Norm, Abstand, Winkel, Orthogonalität.- 17.3 Orthonormalbasen und orthogonale Komplemente.- 17.4 Unitäre Vektorräume.- 17.5 Orthogonale und unitäre Endomorphismen.- 17.6 Selbstadjungierte Endomorphismen.- 17.7 Normale Endomorphismen.- Zusammenfassung.- Aufgaben.- 18 Quadriken − vielseitig nutzbare Punktmengen.- 18.1 Symmetrische Bilinearformen.- 18.2 Hermitesche Sesquilinearformen.- 18.3 Quadriken und ihre Hauptachsentransformation.- 18.4 Die Singulärwertzerlegung.- 18.5 Die Pseudoinverse einer linearen Abbildung.- Zusammenfassung.- Aufgaben.- 19 Funktionenräume − Analysis und lineare Algebra Hand in Hand.- 19.1 Metrische Räume und ihre Topologie, normierte Räume.- 19.2 Konvergenz und Stetigkeit in metrischen Räumen.- 19.3 Kompaktheit.- 19.4 Zusammenhangsbegriffe.- 19.5 Vollständigkeit.- 19.6 Banach- und Hilberträume.- Zusammenfassung.- Aufgaben.- 20 Differenzialgleichungen − Funktionen sind gesucht.- 20.1 Begriffsbildungen.- 20.2 Elementare analytische Techniken.- 20.3 Existenz und Eindeutigkeit.- 20.4 Grundlegende numerische Verfahren.- Zusammenfassung.- Aufgaben .- 21 Funktionen mehrerer Variablen − Differenzieren im Raum.- 21.1 Einführung.- 21.2 Differenzierbarkeitsbegriffe: Totale und partielle Differenzierbarkeit.- 21.3 Differenziationsregeln.- 21.4 Mittelwertsätze und Schranksätze.- 21.5 Höhere partielle Ableitungen und der der Vertauschungssatz von H. A. Schwarz.- 21.6 Taylor-Formel und lokale Extrema.- 21.7 Der Lokale Umkehrsatz.- 21.8 Der Satz über implizite Funktionen.- Zusammenfassung.- Aufgaben.- 22 Gebietsintegrale − das Ausmessen von Mengen.- 22.1 Definition und Eigenschaften.- 22.2 Die Berechnung von Integralen.- 22.3 Die Transformationsformel.- 22.4 Wichtige Koordinatensysteme.- Zusammenfassung.- Aufgaben.- 23 Vektoranalysis − im Zentrum steht der Gauß'sche Satz.- 23.1 Kurven und Kurvenintegrale.- 23.2 Flächen und Flächenintegrale.- 23.3 Der Gauß’sche Satz.- Zusammenfassung.- Aufgaben.- 24 Optimierung − ein sehr generelles Problem.- 24.1 Lineare Optimierung.- 24.2 Das Simplex-Verfahren.- 24.3 Dualitätstheorie.- Zusammenfassung.- Aufgaben.- 25 Elementare Zahlentheorie − Teiler und Vielfache.- 25.1 Teilbarkeit.- 25.2 Der euklidische Algorithmus.- 25.3 Der Fundamentalsatz der Arithmetik.- 25.4 ggT und kgV.- 25.5 Zahlentheoretische Funktionen.- 25.6 Rechnen mit Kongruenzen.- Zusammenfassung.- Aufgaben.- 26 Elemente der diskreten Mathematik − die Kunst des Zählens.- 26.1 Einführung in die Graphentheorie.- 26.2 Einführung in die Kombinatorik.- 26.3 Erzeugende Funktionen.- Zusammenfassung.- Aufgaben.- Hinweise zu den Aufgaben.- Lösungen zu den Aufgaben.- Symbolglossar.- Index.

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Book SynopsisFirst Part.- I The Complex Plane and Elementary Functions.- II Analytic Functions.- III Line Integrals and Harmonic Functions.- IV Complex Integration and Analyticity.- V Power Series.- VI Laurent Series and Isolated Singularities.- VII The Residue Calculus.- Second Part.- VIII The Logarithmic Integral.- IX The Schwarz Lemma and Hyperbolic Geometry.- X Harmonic Functions and the Reflection Principle.- XI Conformal Mapping.- Third Part.- XII Compact Families of Meromorphic Functions.- XIII Approximation Theorems.- XIV Some Special Functions.- XV The Dirichlet Problem.- XVI Riemann Surfaces.- Hints and Solutions for Selected Exercises.- References.- List of Symbols.Table of Contents* The Complex Plane and Elementary Functions * Analytic Functions * Line Integrals and Harmonic Functions * Complex Integration and Analyticity * Power Series * Laurent Series and Isolated Singularities * The Residue Calculus * The Logarithmic Integral * The Schwarz Lemma and Hyperbolic Geometry * Harmonic Functions and the Reflection Principle * Conformal Mapping * Compact Families of Meromorphic Functions * Approximation Theorems * Some Special Functions * The Dirichlet Problem * Riemann Surfaces

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Book SynopsisApplied Calculus for Business, Economics, and the Social and Life Sciences, Expanded Edition provides a sound, intuitive understanding of the basic concepts students need as they pursue careers in business, economics, and the life and social sciences. Students achieve success using this text as a result of the author''s applied and real-world orientation to concepts, problem-solving approach, straight forward and concise writing style, and comprehensive exercise sets. More than 100,000 students worldwide have studied from this text!Table of ContentsChapter 1: Functions, Graphs, and Limits1.1 Functions1.2 The Graph of a Function1.3 Linear Functions1.4 Functional Models1.5 Limits1.6 One-Sided Limits and ContinuityChapter 2: Differentiation: Basic Concepts2.1 The Derivative2.2 Techniques of Differentiation2.3 Product and Quotient Rules; Higher-Order Derivatives2.4 The Chain Rule2.5 Marginal Analysis and Approximations Using Increments2.6 Implicit Differentiation and Related RatesChapter 3: Additional Applications of the Derivative3.1 Increasing and Decreasing Functions; Relative Extrema3.2 Concavity and Points of Inflection3.3 Curve Sketching3.4 Optimization; Elasticity of Demand3.5 Additional Applied OptimizationChapter 4: Exponential and Logarithmic Functions4.1 Exponential Functions; Continuous Compounding4.2 Logarithmic Functions4.3 Differentiation of Exponential and Logarithmic Functions4.4 Applications; Exponential ModelsChapter 5: Integration5.1 Indefinite Integration with Applications5.2 Integration by Substitution5.3 The Definite Integral and the Fundamental Theorem of Calculus5.4 Applying Definite Integration: Area Between Curves and Average Value5.5 Additional Applications to Business and Economics5.6 Additional Applications to the Life and Social SciencesChapter 6: Additional Topics in Integration6.1 Integration by Parts; Integral Tables6.2 Numerical Integration6.3 Improper IntegralsChapter 7: Calculus of Several Variables7.1 Functions of Several Variables7.2 Partial Derivatives7.3 Optimizing Functions of Two Variables7.4 The Method of Least-Squares7.5 Constrained Optimization: The Method of Lagrange Multipliers7.6 Double IntegralsChapter 8: Trigonometric Functions8.1 Angle Measurement; Trigonometric Functions8.2 Derivatives of Trigonometric Functions8.3 Integrals of Trigonometric FunctionsChapter 9: Differential Equations9.1 Introduction to Differential Equations9.2 First-Order Linear Differential Equations9.3 Additional Applications of Differential Equations9.4 Approximate Solutions of Differential Equations9.5 Difference Equations; The Cobweb ModelChapter 10: Probability and Calculus10.1 Continuous Probability Distributions10.2 Expected Value and Variance10.3 Normal DistributionsChapter 11: Infinite Series and Taylor Series Approximations11.1 Infinite Series; Geometric Series11.2 Tests for Convergence11.3 Functions as Power Series; Taylor SeriesAppendix A: Algebra ReviewA.1 A Brief Review of AlgebraA.2 Factoring Polynomials and Solving Systems of EquationsA.3 Evaluating Limits with L’Hopital’s RuleA.4 The Summation Notation

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Book SynopsisThis textbook is designed for a year-long introductory course in Functional Analysis and the theory of Operator Algebras. It guides graduate students and researchers through a wide range of topics including Hilbert spaces, Weak Topologies and C*-algebras. With numerous problems and examples, it is suitable for classroom teaching and self-learning.Table of ContentsPreface; Notation; 1. Preliminaries; 2. Normed Linear Spaces; 3. Hilbert Spaces; 4. Dual Spaces; 5. Operators on Banach Spaces; 6. Weak Topologies; 7. Spectral Theory; 8. C*-Algebras; 9. Measure and Integration; 10. Normal Operators on Hilbert Spaces; Appendices; A.1 The Stone–Weierstrass Theorem; A.2 The Radon–Nikodym Theorem; Bibliography; Index.

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Book SynopsisThis concise introduction covers all of the measure theory and probability most useful for statisticians. Originating from the authors' own graduate course, it is perfect for a two-term course or for self-study. It is especially useful to graduate students in related fields who want to shore up their mathematical foundation.Table of ContentsPreface; Acknowledgements; 1. Point sets and certain classes of sets; 2. Measures: general properties and extension; 3. Measurable functions and transformations; 4. The integral; 5. Absolute continuity and related topics; 6. Convergence of measurable functions, Lp-spaces; 7. Product spaces; 8. Integrating complex functions, Fourier theory and related topics; 9. Foundations of probability; 10. Independence; 11. Convergence and related topics; 12. Characteristic functions and central limit theorems; 13. Conditioning; 14. Martingales; 15. Basic structure of stochastic processes; References; Index.

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Book SynopsisThis modern introduction to operator theory on spaces with indefinite inner product discusses the geometry and the spectral theory of linear operators on these spaces, the deep interplay with complex analysis, and applications to interpolation problems. The text covers the key results from the last four decades in a readable way with full proofs provided throughout. Step by step, the reader is guided through the intricate geometry and topology of spaces with indefinite inner product, before progressing to a presentation of the geometry and spectral theory on these spaces. The author carefully highlights where difficulties arise and what tools are available to overcome them. With generous background material included in the appendices, this text is an excellent resource for researchers in operator theory, functional analysis, and related areas as well as for graduate students.Table of Contents1. Inner product spaces; 2. Angular operators; 3. Subspaces of Kreĭn spaces; 4. Linear operators on Kreĭn spaces; 5. Selfadjoint projections and unitary operators; 6. Techniques of induced Kreĭn spaces; 7. Plus/minus-operators; 8. Geometry of contractive operators; 9. Invariant maximal semidefinite subspaces; 10. Hankel operators and interpolation problems; 11. Spectral theory for selfadjoint operators; 12. Quasi-contractions; 13. More on definitisable operators; Appendix; References; Symbol index; Subject index.

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