Integral calculus and equations Books

Book SynopsisThis textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.Table of ContentsChapter 01- Real Numbers.- Chapter 02- Metric Spaces.- Chapter 03- Real Sequences and Series.- Chapter 04- Real Function Limits.- Chapter 05- Continuous Functions.- Chapter 06- Derivatives.- Chapter 07- The Riemann Integral.- Chapter 08- Differential Analysis in Rn.- Chapter 09- Integration in Rn.- Chapter 10- Topics on Vector Calculus and Vector Analysis.

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Book SynopsisThe theory of homogenization replaces a real composite material with an imaginary homogeneous one, to describe the macroscopic properties of the composite using the properties of the microscopic structure. This work illustrates the relevant mathematics, logic and methodology with examples.Trade Review'serve as good textbook for a post-graduate course' ZAMMTable of Contents1. Weak and weak - convergence in Banach spaces ; 2. Rapidly oscillating periodic functions ; 3. Some classes of Sobolev spaces ; 4. Some variational elliptic problems ; 5. Examples of periodic composite materials ; 6. Homogenization of elliptic equations: the convergence result ; 7. The multiple-scale method ; 8. Tartar's method of oscillating test functions ; 9. The two-scale convergence method ; 10. Homogenization in linearized elasticity ; 11. Homogenization of the heat equation ; 12. Homogenization of the wave equation ; 13. General Approaches to the non-periodic case ; References

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Book SynopsisHeun''s equation is a second-order differential equation which crops up in a variety of forms in a wide range of problems in applied mathematics. These include integral equations of potential theory, wave propogation, electrostatic oscillation, and Schrodinger''s equation. This volume brings together important research work for the first time, providing an important resource for all those interested in this mathematical topic. Both the current theory and the main areas of application are surveyed, and includes contributions from authoritative researchers such as Felix Arscott (Canada), P. Maroni (France), and Gerhard Wolf (Germany).Trade ReviewThere is a wealth of important results and open problems and the book is a welcome addition to the literature on these important special functions and their applications. * B D Sleeman, Zbl. Math. 847/96. *Table of ContentsA. HEUN'S EQUATION ; I: GENERAL AND POWER SERIES ; II: HYPERGEOMETRIC FUNCTION SERIES ; B. CONFLUENT HEUN EQUATION ; C. DOUBLE CONFLUENT HEUN EQUATION ; D. BICONFLUENT HEUN EQUATION ; E. TRICONFLUENT HEUN EQUATION

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Book SynopsisA breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more.Trade Review"...an important tool...gives the newest results in this field...shows an important application of vector integration..." (Bulletin of the Belgian Mathematical Society, Vol 11(1), 2004) "...it can be expected that...just like the author's 1967 volume, this book will stimulate further research on vector stochastic integration and can serve as a graduate-level reference work." (Mathematical Reviews Issue 2001h) "Dense, detailed, comprehensive introduction. Contains...material only found before in journals..." (American Mathematical Monthly, March 2002) "...a highly technical book." (The Mathematical Gazette, March 2002) "The author of this important and interesting book is a well-known specialist on vector measures." (Zentralblatt Math, Vol.974, No. 24 2001)Table of ContentsVector Integration. The Stochastic Integral. Martingales. Processes with Finite Variation. Processes with Finite Semivariation. The Itô Formula. Stochastic Integration in the Plane. Two-Parameter Martingales. Two-Parameter Processes with Finite Variation. Two-Parameter Processes with Finite Semivariation. References.

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

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Book SynopsisThis two-volume text in harmonic analysis is appropriate for advanced undergraduate students with a strong background in mathematical analysis and for beginning graduate students wishing to specialize in analysis. With numerous exercises and problems it is suitable for independent study as well as for use as a course text.Trade ReviewReview of the set: 'The two-volume set under review is a worthy addition to this tradition from two of the younger generation of researchers. It is remarkable that the authors have managed to fit all of this into [this number of] smaller-than-average pages without omitting to provide motivation and helpful intuitive remarks. Altogether, these books are a most welcome addition to the literature of harmonic analysis.' Gerald B. Folland, Mathematical ReviewsTable of ContentsPreface; Acknowledgements; 1. Fourier series: convergence and summability; 2. Harmonic functions, Poisson kernel; 3. Conjugate harmonic functions, Hilbert transform; 4. The Fourier Transform on Rd and on LCA groups; 5. Introduction to probability theory; 6. Fourier series and randomness; 7. Calderón–Zygmund theory of singular integrals; 8. Littlewood–Paley theory; 9. Almost orthogonality; 10. The uncertainty principle; 11. Fourier restriction and applications; 12. Introduction to the Weyl calculus; References; Index.

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Book SynopsisThis book is devoted to the mathematical theory of regularization methods and gives an account of the currently available results about regularization methods for linear and nonlinear ill-posed problems.Trade Review`It is written in a very clear style, the material is well organized, and there is an extensive bibliography with 290 items. There is no doubt that this book belongs to the modern standard references on ill-posed and inverse problems. It can be recommended not only to mathematicians interested in this, but to students with a basic knowledge of functional analysis, and to scientists and engineers working in this field.' Mathematical Reviews Clippings, 97k `... it will be an extremely valuable tool for researchers in the field, who will find under the same cover and with unified notation material that is otherwise scattered in extremely diverse publications.' SIAM Review, 41:2 (1999) Table of ContentsPreface. 1. Introduction: Examples of Inverse Problems. 2. Ill-Posed Linear Operator Equations. 3. Regularization Operators. 4. Continuous Regularization Methods. 5. Tikhonov Regularization. 6. Iterative Regularization Methods. 7. The Conjugate Gradient Method. 8. Regularization with Differential Operators. 9. Numerical Realization. 10. Tikhonov Regularization of Nonlinear Problems. 11. Iterative Methods for Nonlinear Problems. A. Appendix: A.1. Weighted Polynomial Minimization Problems. A.2. Orthogonal Polynomials. A.3. Christoffel Functions. Bibliography. Index.

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Book SynopsisA comprehensive exposition of the properties of semiconcave functions and their various applications, particularly to optimal control problems. It is suitable for graduate students and researchers in optimal control, the calculus of variations, and PDEs.Trade Review"The main purpose of this book is to provide a systematic study of the notion of semiconcave functions, as well as a presentation of mathematical fields in which this notion plays a fundamental role. Many results are extracted from articles by the authors and their collaborators, with simplified—and often new—presentation and proofs.... One of the most attractive features of this book is the interplay between several fields of mathematical analysis.... Despite the many topics addressed in the book, the required mathematical background for reading it is limited because all the necessary notions are not only recalled, but also carefully explained, and the main results proved. The book will be found very useful by experts in nonsmooth analysis, nonlinear control theory and PDEs, in particular, as well as by advanced graduate students in this field. They will appreciate the many detailed examples, the clear proofs and the elegant style of presentation, the fairly comprehensive and up-to-date bibliography and the very pertinent historical and bibliographical comments at the end of each chapter." —Mathematical ReviewsTable of ContentsA Model Problem.- Semiconcave Functions.- Generalized Gradients and Semiconcavity.- Singularities of Semiconcave Functions.- Hamilton-Jacobi Equations.- Calculus of Variations.- Optimal Control Problems.- Control Problems with Exit Time.

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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 SynopsisMotivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg's Proof of Szemeredi's Theorem.- Actions of Locally Compact Groups.- Geodesic Flow on Quotients of the Hyperbolic Plane.- Nilrotation.- More Dynamics on Quotients of the Hyperbolic Plane.- Appendix A: Measure Theory.- Appendix B: Functional Analysis.- Appendix C: Topological GroupsTrade ReviewFrom the reviews:“The book is an introduction to ergodic theory and dynamical systems. … The book is intended for graduate students and researchers with some background in measure theory and functional analysis. Definitely, it is a book of great interest for researchers in ergodic theory, homogeneous dynamics or number theory.” (Antonio Díaz-Cano Ocaña, The European Mathematical Society, January, 2014)“A book with a wider perspective on ergodic theory, and yet with a focus on the interaction with number theory, remained a glaring need in the overall context of the development of the subject. … The book under review goes a long way in fulfilling this need. … it covers a good deal of conventional ground in ergodic theory … . a very welcome addition and would no doubt inspire interest in the area among researchers as well as students, and cater to it successfully.” (S. G. Dani, Ergodic Theory and Dynamical Systems, Vol. 32 (3), June, 2012)“The book under review is an introductory textbook on ergodic theory, written with applications to number theory in mind. … it aims both to provide the reader with a solid comprehensive background in the main results of ergodic theory, and of reaching nontrivial applications to number theory. … The book should also be very appealing to more advanced readers already conducting research in representation theory or number theory, who are interested in understanding the basis of the recent interaction with ergodic theory.” (Barak Weiss, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 114, 2012)“This introductory book, which goes beyond the standard texts and allows the reader to get a glimpse of modern developments, is a timely and welcome addition to the existing and ever-growing ergodic literature. … This book is highly recommended to graduate students and indeed to anyone who is interested in acquiring a better understanding of contemporary developments in mathematics.” (Vitaly Bergelson, Mathematical Reviews, Issue 2012 d)“The book contains a presentation of the ergodic theory field, focusing mainly on results applicable to number theory. … of interest for researchers, specialists, professors and students that work within some other areas than precisely the ergodic theory. … ‘Ergodic Theory. With a view toward number theory’ is now an indispensable reference in the domain and offers important instruments of research for other theoretical fields.” (Adrian Atanasiu, Zentralblatt MATH, Vol. 1206, 2011)Table of ContentsMotivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg’s Proof of Szemeredi’s Theorem.- Actions of Locally Compact Groups.- Geodesic Flow on Quotients of the Hyperbolic Plane.- Nilrotation.- More Dynamics on Quotients of the Hyperbolic Plane.- Appendix A: Measure Theory.- Appendix B: Functional Analysis.- Appendix C: Topological Groups

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Book SynopsisThis two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and useful to graduates and researchers in pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. The first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional CalderÃnâZygmund and LittlewoodâPaley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; CoifmanâMeyer theory; Carleson's resolution of the Lusin conjecture; CalderÃn's commutators and the Cauchy integral on Lipschitz curves. TTrade ReviewReview of the set: 'The two-volume set under review is a worthy addition to this tradition from two of the younger generation of researchers. It is remarkable that the authors have managed to fit all of this into [this number of] smaller-than-average pages without omitting to provide motivation and helpful intuitive remarks. Altogether, these books are a most welcome addition to the literature of harmonic analysis.' Gerald B. Folland, Mathematical ReviewsTable of ContentsPreface; Acknowledgements; 1. Leibniz rules and gKdV equations; 2. Classical paraproducts; 3. Paraproducts on polydiscs; 4. Calderón commutators and the Cauchy integral; 5. Iterated Fourier series and physical reality; 6. The bilinear Hilbert transform; 7. Almost everywhere convergence of Fourier series; 8. Flag paraproducts; 9. Appendix: multilinear interpolation; Bibliography; Index.

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Book SynopsisConcrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions.Trade ReviewFrom the reviews:“This monograph is a thorough and masterful work on non-linear analysis designed to be read and studied by graduate students and professional mathematical researchers. The overall perspective and choice of material is highly novel and original. … It is a unique account of some key areas of modern analysis which will surely turn out to be invaluable for many researchers in this and related areas.” (David Applebaum, The Mathematical Gazette, Vol. 98 (541), March, 2014)“The present monograph is quite extensive and interesting. It is divided into twelve chapters on different topics on Functional calculus and an appendix on non-atomic measure spaces. … The book has many historical comments and remarks which clarify the developments of the theory. It has also an extensive bibliography with 258 references. … will be very useful for all interested readers in Real-Functional Analysis and Probability.” (Francisco L. Hernandez, The European Mathematical Society, January, 2012)“The monograph under review aims at analyzing properties such as Hölder continuity, differentiability and analyticity of various types of nonlinear operators which arises in the study of differential and integral equations and in applications to problems of statistics and probability. … this is an interesting book which contains a lot of material.” (Massimo Lanza de Cristoforis, Mathematical Reviews, Issue 2012 e)Table of ContentsPreface.- 1 Introduction and Overview.- 2 Definitions and Basic Properties of Extended Riemann-Stieltjes integrals.- 3 Phi-variation and p-variation; Inequalities for Integrals.- 4 Banach Algebras.- 5 Derivatives and Analyticity in Normed Spaces.- 6 Nemytskii Operators on Function Spaces.- 7 Nemytskii Oerators on Lp Spaces.- 8 Two-Function Composition.- 9 Product Integration.- 10 Nonlinear Differential and Integral Equations.- 11 Fourier Series.- 12 Stochastic Processes and Phi-Variation.- Appendix Nonatomic Measure Spaces.- References.- Subject Index.- Author Index.- Index of Notation.

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Book SynopsisIn its Second Edition, this in-depth study of the vibrations of fractal strings interlinks number theory, spectral geometry and fractal geometry. Includes a geometric reformulation of the Riemann hypothesis and a new final chapter on recent topics and results.Trade Review“This interesting volume gives a thorough introduction to an active field of research and will be very valuable to graduate students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)“In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a self-contained manner the results … are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)“The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a self-contained manner, the results (including some fundamental ones) are completely proved. … the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory.” (Nicolae-Adrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH"This highly original self-contained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." -- Mathematical Reviews (Review of previous book by authors)"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is self-contained, intelligent and well paced." -- Bulletin of the London Mathematical Society (Review of previous book by authors)"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics." -- Simulation News Europe (Review of previous book by authors)Table of ContentsPreface.- Overview.- Introduction.- 1. Complex Dimensions of Ordinary Fractal Strings.- 2. Complex Dimensions of Self-Similar Fractal Strings.- 3. Complex Dimensions of Nonlattice Self-Similar Strings.- 4. Generalized Fractal Strings Viewed as Measures.- 5. Explicit Formulas for Generalized Fractal Strings.- 6. The Geometry and the Spectrum of Fractal Strings.- 7. Periodic Orbits of Self-Similar Flows.- 8. Fractal Tube Formulas.- 9. Riemann Hypothesis and Inverse Spectral Problems.- 10. Generalized Cantor Strings and their Oscillations.- 11. Critical Zero of Zeta Functions.- 12 Fractality and Complex Dimensions.- 13. Recent Results and Perspectives.- Appendix A. Zeta Functions in Number Theory.- Appendix B. Zeta Functions of Laplacians and Spectral Asymptotics.- Appendix C. An Application of Nevanlinna Theory.- Bibliography.- Author Index.- Subject Index.- Index of Symbols.- Conventions.- Acknowledgements.

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Book SynopsisBurstein, and Lax's Calculus with Applications and Computing offers meaningful explanations of the important theorems of single variable calculus.Trade ReviewFrom the book reviews:“The book under review, a little over 500 pages co-authored with Maria Terrell, is a first-approximation to Lax’s dream come true: a ‘thorough revision’ of the 1976 Lax-Burstein-Lax. … This reviewer will attempt to use them as a pedagogical tool when teaching single-variable calculus or introductory analysis in the future. … It is filled with beautiful ideas that are elegantly explained and chock-full with problems that will enchant both the experienced teacher and the curious novice.” (Tushar Das, MAA Reviews, December, 2014)“The text starts with introductory facts on real numbers, sequences and limits, followed by chapters aimed at differential and integral calculus. … The text is accompanied by a lot of worked examples, figures and applications. Together with detailed proofs of theorems this makes the text suitable also for self-study.” (Vladimír Janiš, zbMATH, 2014)Table of Contents1 Numbers and Limits.- 2 Functions and Continuity.- 3 The Derivative and Differentiation.- 4 The Theory of Differentiable Functions.- 5 Applications of the Derivative.- 6 Integration.- 7 Methods for Integration.- 8 Approximation of Integrals.- 9 Complex Numbers.- 10 Differential Equations.- 11 Probability.- Answers to Selected Probems.- Index.

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

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Book SynopsisThe inverse scattering problem is central to many areas of science and technology such as radar, sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. In this fourth edition, a number of significant additions have been made including a new chapter on transmission eigenvalues and a new section on the impedance boundary condition where particular attention has been made to the generalized impedance boundary condition and to nonlocal impedance boundary conditions. Brief discussions on the generalized linear sampling method, the method of recursive linearization, anisotropic media and the use of target signatures in inverse scattering theory have also been added.Table of ContentsIntroduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- Ill-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle Scattering.- AcousticWaves in an Inhomogeneous Medium.- ElectromagneticWaves in an Inhomogeneous Medium.- Transmission Eigenvalues.- The Inverse Medium Problem.

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Book SynopsisThis open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics.Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn.Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability.Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online. For errata and updates, visit https://measure.axler.net/Trade Review“This textbook is addressed to students with a good background in undergraduate real analysis. Students are encouraged to actively study the theory by working on the exercises that are found at the end of each section. Definitions and theorems are printed in yellow and blue boxes, respectively, giving a clear visual aid of the content.” (Marta Tyran-Kamińska, Mathematical Reviews, May, 2021)“The book will become an invaluable reference for graduate students and instructors. Those interested in measure theory and real analysis will find the monograph very useful since the book emphasizes getting the students to work with the main ideas rather than on proving all possible results and it contains a rather interesting selection of topics which makes the book a nice presentation for students and instructors as well.” (Oscar Blasco, zbMATH 1435.28001, 2020)Table of ContentsAbout the Author.- Preface for Students.- Preface for Instructors.- Acknowledgments.- 1. Riemann Integration.- 2. Measures.- 3. Integration.- 4. Differentiation.- 5. Product Measures.- 6. Banach Spaces.- 7. L^p Spaces.- 8. Hilbert Spaces.- 9. Real and Complex Measures.- 10. Linear Maps on Hilbert Spaces.- 11. Fourier Analysis.- 12. Probability Measures.- Photo Credits.- Bibliography.- Notation Index.- Index.- Colophon: Notes on Typesetting.

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Book SynopsisThis textbook focuses on one of the most valuable skills in multivariable and vector calculus: visualization. With over one hundred carefully drawn color images, students who have long struggled picturing, for example, level sets or vector fields will find these abstract concepts rendered with clarity and ingenuity. This illustrative approach to the material covered in standard multivariable and vector calculus textbooks will serve as a much-needed and highly useful companion. Emphasizing portability, this book is an ideal complement to other references in the area. It begins by exploring preliminary ideas such as vector algebra, sets, and coordinate systems, before moving into the core areas of multivariable differentiation and integration, and vector calculus. Sections on the chain rule for second derivatives, implicit functions, PDEs, and the method of least squares offer additional depth; ample illustrations are woven throughout. Mastery Checks engage students in material on the spot, while longer exercise sets at the end of each chapter reinforce techniques. An Illustrative Guide to Multivariable and Vector Calculus will appeal to multivariable and vector calculus students and instructors around the world who seek an accessible, visual approach to this subject. Higher-level students, called upon to apply these concepts across science and engineering, will also find this a valuable and concise resource.Trade Review“The book is self-contained. … It is suitable as a textbook for students having completed courses in single variable calculus and linear algebra. Alternatively, the book can be used as a reference text to complement the textbooks in advanced calculus, giving the students a different visual perspective.” (Mihail Voicu, zbMATH 1441.26002, 2020)Table of Contents1. Preliminary Ideas.- 2. Introduction to Differentiation.- 3. Applications of the Differential Calculus.- 4. Introduction to Integration.- 5. Vector Calculus.- Glossary of Symbols.- Bibliography.- Index.

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Book SynopsisThis book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.Trade Review“‘Lectures on convex geometry’ is an excellent graduate book about convex geometry. … The book is very well-organized. … The presentation is clear, well-composed and illustrated. The problems at the end of each subchapter are carefully selected and revised. The whole text is readable, interesting and easy to learn from. … the book is excellent and it can serve the studies of the future generation students in convex geometry.” (Gergely Kiss, zbMATH 1487.52001, 2022)“The book is informative, very interesting, and mathematically accessible, and the authors have achieved the purpose that they outline above. As the title suggests, Lectures on convex geometry is well suited to be used as the prescribed textbook for graduate courses in convex geometry; this is because of its pedagogical style and the quality of the exercises. It will also be useful to students intending to pursue a research career in the area … .” (Daniel John Fresen, Mathematical Reviews, June, 2022)Table of ContentsPreface.- Preliminaries and Notation.- 1. Convex Sets.- 2. Convex Functions.- 3. Brunn-Minkowski Theory.- 4. From Area Measures to Valuations.- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises.- References.- Index.

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Book SynopsisThis book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.Trade Review“‘Lectures on convex geometry’ is an excellent graduate book about convex geometry. … The book is very well-organized. … The presentation is clear, well-composed and illustrated. The problems at the end of each subchapter are carefully selected and revised. The whole text is readable, interesting and easy to learn from. … the book is excellent and it can serve the studies of the future generation students in convex geometry.” (Gergely Kiss, zbMATH 1487.52001, 2022)“The book is informative, very interesting, and mathematically accessible, and the authors have achieved the purpose that they outline above. As the title suggests, Lectures on convex geometry is well suited to be used as the prescribed textbook for graduate courses in convex geometry; this is because of its pedagogical style and the quality of the exercises. It will also be useful to students intending to pursue a research career in the area … .” (Daniel John Fresen, Mathematical Reviews, June, 2022)Table of ContentsPreface.- Preliminaries and Notation.- 1. Convex Sets.- 2. Convex Functions.- 3. Brunn-Minkowski Theory.- 4. From Area Measures to Valuations.- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises.- References.- Index.

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Book SynopsisThis popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory.Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as: limit theorems for sums of random variables martingales percolation Markov chains and electrical networks construction of stochastic processes Poisson point process and infinite divisibility large deviation principles and statistical physics Brownian motion stochastic integrals and stochastic differential equations. The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts.This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology.Table of Contents1 Basic Measure Theory.- 2 Independence.- 3 Generating Functions.- 4 The Integral.- 5 Moments and Laws of Large Numbers.- 6 Convergence Theorems.- 7 Lp-Spaces and the Radon–Nikodym Theorem.- 8 Conditional Expectations.- 9 Martingales.- 10 Optional Sampling Theorems.- 11 Martingale Convergence Theorems and Their Applications.- 12 Backwards Martingales and Exchangeability.- 13 Convergence of Measures.- 14 Probability Measures on Product Spaces.- 15 Characteristic Functions and the Central Limit Theorem.- 16 Infinitely Divisible Distributions.- 17 Markov Chains.- 18 Convergence of Markov Chains.- 19 Markov Chains and Electrical Networks.- 20 Ergodic Theory.- 21 Brownian Motion.- 22 Law of the Iterated Logarithm.- 23 Large Deviations.- 24 The Poisson Point Process.- 25 The Itô Integral.- 26 Stochastic Differential Equations.- References.- Notation Index.- Name Index.- Subject Index.

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Book SynopsisThis multidisciplinary volume is the second in the STEAM-H series to feature invited contributions on mathematical applications in naval engineering. Seeking a more holistic approach that transcends current scientific boundaries, leading experts present interdisciplinary instruments and models on a broad range of topics. Each chapter places special emphasis on important methods, research directions, and applications of analysis within the field. Fundamental scientific and mathematical concepts are applied to topics such as microlattice materials in structural dynamics, acoustic transmission in low Mach number liquid flow, differential cavity ventilation on a symmetric airfoil, Kalman smoother, metallic foam metamaterials for vibration damping and isolation, seal whiskers as a bio-inspired model for the reduction of vortex-induced vibrations, multidimensional integral for multivariate weighted generalized Gaussian distributions, minimum uniform search track placement for rectangular regions, antennas in the maritime environment, the destabilizing impact of non-performers in multi-agent groups, inertial navigation accuracy with bias modeling.Carefully peer-reviewed and pedagogically presented for a broad readership, this volume is perfect to graduate and postdoctoral students interested in interdisciplinary research. Researchers in applied mathematics and sciences will find this book an important resource on the latest developments in naval engineering. In keeping with the ideals of the STEAM-H series, this volume will certainly inspire interdisciplinary understanding and collaboration.Table of ContentsIntroductory Chapter: Mathematical Sciences and Naval Engineering Research.- Microlattice Materials and their potential applications in structural dynamics and acoustics.- Alternative Approach to Cagniard Method for Transient Ocean Acoustic Modeling.- Acoustic transmission in a low Mach number liquid flow.- Lift Production using Differential Cavity Ventilation.- An Exact Solution for a Kalman Smoother.- Metallic Foam Metamaterials for Vibration Damping and Isolation.- The other Navy Seals: Seal Whiskers as a Bio-Inspired Model for the Resolution of Vortex Induced Vibrations.- A series of multidimensional integral identities with applications to multivariate weighted generalized Gaussian Distributions.- Minimum Uniform Track Placement for Rectangular Regions.- Antenna Behavior in the Maritime Environment.- The Destabilizing impact of Non-Performers in Multi-Agent Groups.- Improving Inertial Navigation Accuracy with Bias Modeling.

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Book SynopsisThis volume is part of collection of contributions devoted to analytical and experimental techniques of dynamical systems, presented at the 15th International Conference “Dynamical Systems: Theory and Applications”, held in Łódź, Poland on December 2-5, 2019. The wide selection of material has been divided into three volumes, each focusing on a different field of applications of dynamical systems.The broadly outlined focus of both the conference and these books includes bifurcations and chaos in dynamical systems, asymptotic methods in nonlinear dynamics, dynamics in life sciences and bioengineering, original numerical methods of vibration analysis, control in dynamical systems, optimization problems in applied sciences, stability of dynamical systems, experimental and industrial studies, vibrations of lumped and continuous systems, non-smooth systems, engineering systems and differential equations, mathematical approaches to dynamical systems, and mechatronics.Table of ContentsOn the spinning motion of a disc under the influence a gyrostatic moment (Gamiel).- Suppression of impact oscillations in a railway current collection system with an additional oscillatory system (Nishiyama).- On Qualitative Analysis of Lattice Dynamical System of Two- and Three-Dimensional Biopixels Array: Bifurcations and Transition to Chaos (Martsenyuk).- Response sensitivity of damper-connected adjacent structural systems subjected to fully non-stationary random excitations (Genovese).- Analysis of switching strategies for the optimization of periodic chemical reactions with controlled flow-rate (Zuyev).- Quaternion based free-floating space manipulator dynamics modeling using the dynamically equivalent manipulator approach (Jarzębowska).- Slosh analysis on a full car model with SDRE control and hydraulic damper (Balthazar).- A comparison of the common types of nonlinear energy sinks (Saeed).- Stability of three wheeled narrow vehicle (Weigel-Milleret).- Testing and analysis of vibrations of a tension transmission with a thermally sealed belt (Szymański).- Modeling and experimental tests on motion resistance of double-flange rollers of rubber track systems due to sliding friction between the rollers and guide lugs of rubber tracks (Chołodowski).- Structural dynamic response of the coupling between transmission lines and tower under random excitation (Machado).- Experimental assessment of the test station support structure rigidity by the vibration diagnostics method (Šmeringaiová).- Experimental dynamical analysis of a mechatronic analogy of the human circulatory system (Olejnik).- Robust design of inhibitory neuronal networks displaying rhythmic activity (Taylor).- Nonlinear dynamics of the industrial city's atmospheric ventilation: New differential equations model and chaos (Buyadzhi).- Biomechanical analysis of different foot morphology during standing on a dynamic support surface (Shu).- Comparison of various fractional order controllers on a poorly damped system (Birs).- Asymptotic analysis of submerged spring pendulum motion in liquid (Amer).- Parametric identification of non linear structures using Particle Swarm Optimization based on power flow balance criteria (Rajan).- Vibration and buckling of laminated plates of complex form under in-plane uniform and non-uniform loading (Linnik).- Dynamical systems and stability in fractional solid mechanics (Beda).- Stability of coupled systems of stochastic Cohen-Grossberg neural networks with time delays, impulses and Markovian switching (Tojtovska).- Stability of steady states with regular or chaotic behaviour in time (Mikhlin).- Modelling of torsional vibrations in a motorcycle steering system (Dębowski).- Free vibration frequencies of simply supported bars with variable cross section (Szlachetka).- On dynamics of a rigid block on visco-elastic foundation (Garziera).

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Book SynopsisThis book is divided into two parts, the first one to study the theory of differentiable functions between Banach spaces and the second to study the differential form formalism and to address the Stokes' Theorem and its applications. Related to the first part, there is an introduction to the content of Linear Bounded Operators in Banach Spaces with classic examples of compact and Fredholm operators, this aiming to define the derivative of Fréchet and to give examples in Variational Calculus and to extend the results to Fredholm maps. The Inverse Function Theorem is explained in full details to help the reader to understand the proof details and its motivations. The inverse function theorem and applications make up this first part. The text contains an elementary approach to Vector Fields and Flows, including the Frobenius Theorem. The Differential Forms are introduced and applied to obtain the Stokes Theorem and to define De Rham cohomology groups. As an application, the final chapter contains an introduction to the Harmonic Functions and a geometric approach to Maxwell's equations of electromagnetism.Trade Review“A specific feature of the book is the abundance of examples from mechanics, physics, calculus of variations, illustrating the abstract concepts introduced in the main text. … There are a lot of exercises spread through the book, some elementary, while others are more advanced. The book can be used as supplementary material for undergraduate or graduate level courses, as well as by the students in physics interested in a mathematical treatment of some important problems in their domain.” (Stefan Cobzaş, zbMATH 1479.46001, 2022)Table of ContentsIntroduction.- Chapter 1. Differentiation in R^n.- Chapter 2. Linear Operators in Banach Spaces.- Chapter 3. Differentiation in Banach Spaces.- Chapter 4. Vector Fields.- Chapter 5. Vectors Integration, Potential Theory.- Chapter 6. Differential Forms, Stoke’s Theorem.- Chapter 7. Applications to the Stoke’s Theorem.- Appendix A. Basics of Analysis.- Appendix B. Differentiable Manifolds, Lie Groups.- Appendix C. Tensor Algebra.- Bibliography.- Index.

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Book SynopsisThis brief explores the Krasnosel'skiĭ-Man (KM) iterative method, which has been extensively employed to find fixed points of nonlinear methods. Table of Contents1. Introduction.- 2. Notation and Mathematical Foundations.-3. The Krasnoselskii-Mann Iteration.- 4. Relations of the Krasnosel'skii-Mann Iteration and the Operator Splitting Methods.- 5. The Inertial Krasnoselskii-Mann Iteration.- 6. The Multi-step Inertial Krasnoselskii-Mann Iteration.- 7. Relaxation Parameters of the Krasnoselskii-Mann Iteration.- 8. Two Applications.

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Book SynopsisThis monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems – as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis – will find this text to be a valuable addition to the mathematical literature.Table of ContentsIntroduction.- Geometric Measure Theory.- Calderon-Zygmund Theory for Boundary Layers in UR Domains.- Boundedness and Invertibility of Layer Potential Operators.- Controlling the BMO Semi-Norm of the Unit Normal.- Boundary Value Problems in Muckenhoupt Weighted Spaces.- Singular Integrals and Boundary Problems in Morrey and Block Spaces.- Singular Integrals and Boundary Problems in Weighted Banach Function Spaces.

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Book SynopsisThis textbook describes selected topics in functional analysis as powerful tools of immediate use in many fields within applied mathematics, physics and engineering. It follows a very reader-friendly structure, with the presentation and the level of exposition especially tailored to those who need functional analysis but don’t have a strong background in this branch of mathematics. For every tool, this work emphasizes the motivation, the justification for the choices made, and the right way to employ the techniques. Proofs appear only when necessary for the safe use of the results. The book gently starts with a road map to guide reading. A subsequent chapter recalls definitions and notation for abstract spaces and some function spaces, while Chapter 3 enters dual spaces. Tools from Chapters 2 and 3 find use in Chapter 4, which introduces distributions. The Linear Functional Analysis basic triplet makes up Chapter 5, followed by Chapter 6, which introduces the concept of compactness. Chapter 7 brings a generalization of the concept of derivative for functions defined in normed spaces, while Chapter 8 discusses basic results about Hilbert spaces that are paramount to numerical approximations. The last chapter brings remarks to recent bibliographical items. Elementary examples included throughout the chapters foster understanding and self-study. By making key, complex topics more accessible, this book serves as a valuable resource for researchers, students, and practitioners alike that need to rely on solid functional analysis but don’t need to delve deep into the underlying theory.Table of ContentsRoad Map.- Basic Concepts.- Dual of a Normed Space.- Sobolev Spaces, Distributions.- The Three Basic Principles.- Compactness.- Function Derivatives in Normed Spaces.- Hilbert Bases and Approximations.

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Book SynopsisThis textbook provides a mathematical introduction to linear systems, with a focus on the continuous-time models that arise in engineering applications such as electrical circuits and signal processing. The book introduces linear systems via block diagrams and the theory of the Laplace transform, using basic complex analysis.The book mainly covers linear systems with finite-dimensional state spaces. Graphical methods such as Nyquist plots and Bode plots are presented alongside computational tools such as MATLAB. Multiple-input multiple-output (MIMO) systems, which arise in modern telecommunication devices, are discussed in detail. The book also introduces orthogonal polynomials with important examples in signal processing and wireless communication, such as Telatar’s model for multiple antenna transmission. One of the later chapters introduces infinite-dimensional Hilbert space as a state space, with the canonical model of a linear system. The final chapter covers modern applications to signal processing, Whittaker’s sampling theorem for band-limited functions, and Shannon’s wavelet.Based on courses given for many years to upper undergraduate mathematics students, the book provides a systematic, mathematical account of linear systems theory, and as such will also be useful for students and researchers in engineering. The prerequisites are basic linear algebra and complex analysis.Table of Contents- 1. Linear Systems and Their Description. - 2. Solving Linear Systems by Matrix Theory. - 3. Eigenvalues and Block Decompositions of Matrices. - 4. Laplace Transforms. - 5. Transfer Functions, Frequency Response, Realization and Stability. - 6. Algebraic Characterizations of Stability. - 7. Stability and Transfer Functions via Linear Algebra. - 8. Discrete Time Systems. - 9. Random Linear Systems and Green’s Functions. - 10. Hilbert Spaces. - 11. Wireless Transmission and Wavelets. - 12. Solutions to Selected Exercises.

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Book SynopsisThis book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the study of measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.Table of ContentsPreface.- 1. Logic and Sets.- 2. Integers.- 3. Construction of the Real Numbers.- 4. The Real Numbers.- 5. Real Sequences.- 6. Some Applications of Real Sequences.- 7. Real Series.- 8. Additional Topics in Real Series.- 9. Functions and Limits.- 10. Continuity.- 11. Function Sequences and Series.- 12. Power Series.- 13. Differentiation.- 14. Some Applications of Differentiation.- 15. Riemann and Darboux Integration.- 16. The Fundamental Theorem of Calculus.- 17. Taylor and MacLaurin Series.- 18. Introduction to Measure Theory.- 19. Lebesgue Integration.- 20. Double Integrals.- Solutions to the Exercises.- Bibliography.- Index.

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Book SynopsisThis book covers problems involving a variety of fractional differential equations, as well as some involving the generalized Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. The authors highlight the existence, uniqueness, and stability results for various classes of fractional differential equations based on the most recent research in the area. The book discusses the classic and novel fixed point theorems related to the measure of noncompactness in Banach spaces and explains how to utilize them as tools. The authors build each chapter upon the previous one, helping readers to develop their understanding of the topic. The book includes illustrated results, analysis, and suggestions for further study.Table of ContentsIntroduction.- Preliminary Background.- Hybrid Fractional Differential Equations.- Fractional Differential Equations with Retardation and Anticipation.- Impulsive Fractional Differential Equations with Retardation and Anticipation.- Coupled Systems for Fractional Differential Equations.

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Book SynopsisThis textbook offers a complete one-semester course in probability, covering the essential topics necessary for further study in the areas of probability and statistics. The book begins with a review of the fundamentals of measure theory and integration. Probability measures, random variables, and their laws are introduced next, along with the main analytic tools for their investigation, accompanied by some applications to statistics. Questions of convergence lead to classical results such as the law of large numbers and the central limit theorem with their applications also to statistical analysis and more. Conditioning is the next main topic, followed by a thorough introduction to discrete time martingales. Some attention is given to computer simulation. Through the text, over 150 exercises with full solutions not only reinforce the concepts presented, but also provide students with opportunities to develop their problem-solving skills, and make this textbook suitable for guided self-study. Based on years of teaching experience, the author's expertise will be evident in the clear presentation of material and the carefully chosen exercises. Assuming familiarity with measure and integration theory as well as elementary notions of probability, the book is specifically designed for teaching in parallel with a first course in measure theory. An invaluable resource for both instructors and students alike, it offers ideal preparation for further courses in statistics or probability, such as stochastic calculus, as covered in the author's book on the topic.Table of Contents1 Elements of Measure Theory.- 2 Probability.- 3 Convergence.- 4 Conditioning.- 5 Martingales.- 6 Complements.- 7 Solutions.

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Book SynopsisThis is the revised and enlarged 2nd edition of the authors’ original text, which was intended to be a modest complement to Grenander's fundamental memoir on stochastic processes and related inference theory. The present volume gives a substantial account of regression analysis, both for stochastic processes and measures, and includes recent material on Ridge regression with some unexpected applications, for example in econometrics.The first three chapters can be used for a quarter or semester graduate course on inference on stochastic processes. The remaining chapters provide more advanced material on stochastic analysis suitable for graduate seminars and discussions, leading to dissertation or research work. In general, the book will be of interest to researchers in probability theory, mathematical statistics and electrical and information theory.Trade Review“A wonderful text with a very high pedagogical and scientific quality, on inference theory in stochastic processes, important for researchers in probability theory, mathematical statistics and electrical and information theory.” (Prof. Dr. Manuel Alberto M. Ferreira, Acta Scientiae et Intellectus, Vol. 2 (1), 2016)“This book is the revised and enlarged edition of the author's original text … . The book is well written and will be of interest for researchers in probability theory and mathematical statistics.” (N. G. Gamkrelidze, zbMATH 1341.62036, 2016)Table of Contents1.Introduction and Preliminaries.- 2.Some Principles of Hypothesis Testing.- 3.Parameter Estimation and Asymptotics.- 4.Inferences for Classes of Processes.- 5.Likelihood Ratios for Processes.- 6.Sampling Methods for Processes.- 7.More on Stochastic Inference.- 8.Prediction and Filtering of Processes.- 9.Nonparametric Estimation for Processes.- Bibliography.- Index.

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Book SynopsisBased on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points — which are rarely treated in books at this level — and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.Trade Review"This book, in its second edition, provides the basic concepts of real analysis. ... I strongly recommend it to everyone who wishes to study real mathematical analysis." (Catalin Barbu, zbMATH 1329.26003, 2016)Table of ContentsReal Numbers.- A Taste of Topology.- Functions of a Real Variable.- Function Spaces.- Multivariable Calculus.- Lebesgue Theory.

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Book SynopsisThis comprehensive treatment of multivariable calculus focuses on the numerous tools that MATLAB® brings to the subject, as it presents introductions to geometry, mathematical physics, and kinematics. Covering simple calculations with MATLAB®, relevant plots, integration, and optimization, the numerous problem sets encourage practice with newly learned skills that cultivate the reader’s understanding of the material. Significant examples illustrate each topic, and fundamental physical applications such as Kepler’s Law, electromagnetism, fluid flow, and energy estimation are brought to prominent position. Perfect for use as a supplement to any standard multivariable calculus text, a “mathematical methods in physics or engineering” class, for independent study, or even as the class text in an “honors” multivariable calculus course, this textbook will appeal to mathematics, engineering, and physical science students. MATLAB® is tightly integrated into every portion of this book, and its graphical capabilities are used to present vibrant pictures of curves and surfaces. Readers benefit from the deep connections made between mathematics and science while learning more about the intrinsic geometry of curves and surfaces. With serious yet elementary explanation of various numerical algorithms, this textbook enlivens the teaching of multivariable calculus and mathematical methods courses for scientists and engineers.Trade Review“The book is addressed to students as well as to instructors of calculus. It helps to understand multivariable analysis utilysing visualization of such geometric structures like domains, curves and surfaces. It also develops the skill of students to use a powerful software for solving modern problems.” (Ivan Podvigin, zbMATH 1400.26001, 2019)Table of Contents1. Introduction.- 2. Vectors and Graphics.- 3. Geometry of Curves.- 4. Kinematics.- 5. Directional Derivatives.- 6. Geometry of Surfaces.- 7. Optimization in Several Variables.- 8. Multiple Integrals.- 9. Multidimensional Calculus.- 10. Physical Applications of Vector Calculus.- 11. MATLAB Tips.- Sample Solutions.- Index.

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Book SynopsisThis book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods. The problems addressed are challenging and complex, and often require the use of several refined techniques to overcome the major difficulties encountered. The lectures, given during the course "Partial Differential Equations and Geometric Measure Theory'' in Cetraro, June 2–7, 2014, should help to encourage further research in the area. The enthusiasm of the speakers and the participants of this CIME course is reflected in the text.Table of ContentsAlberto Farina and Enrico Valdinoci:Introduction.-Alessio Figalli:Global Existence for the Semi-Geostrophic Equations via Sobolev Estimates for Monge-Ampère.-Ireneo Peral Alonso: On Some Elliptic and Parabolic Equations Related to Growth Models.- Enrico Valdinoci: All Functions are (locally) S-harmonic (up to a small error) – and Applications

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Book SynopsisFunctions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.Trade ReviewFrom the reviews of the original French edition: "... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.J. Mawhin in Zentralblatt Mathematik (1999) Table of ContentsDifferential and Integral Calculus.- The Riemann Integral.- Integrability Conditions.- The “Fundamental Theorem” (FT).- Integration by parts.- Taylor’s Formula.- The change of variable formula.- Generalised Riemann integrals.- Approximation Theorems.- Radon measures in ? or ?.- Schwartz distributions.- Asymptotic Analysis.- Truncated expansions.- Summation formulae.- Harmonic Analysis and Holomorphic Functions.- Analysis on the unit circle.- Elementary theorems on Fourier series.- Dirichlet’s method.- Analytic and holomorphic functions.- Harmonic functions and Fourier series.- From Fourier series to integrals.

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