There is a recent and increasing interest in harmonic analysis of non-smooth geometries. Real-world examples where these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. Notions of harmonic analysis focus on transforms and expansions and involve dual variables. In this book on smooth and non-smooth harmonic analysis, the notion of dual variables will be adapted to fractals. In addition to harmonic analysis via Fourier duality, the author also covers multiresolution wavelet approaches as well as a third tool, namely, $L^2$ spaces derived from appropriate Gaussian processes. The book is based on a series of ten lectures delivered in June 2018 at a CBMS conference held at Iowa State University.

A functional calculus is a construction which associates with an operator or a family of operators a homomorphism from a function space into a subspace of continuous linear operators, i.e. a method for defining "functions of an operator". Perhaps the most familiar example is based on the spectral theorem for bounded self-adjoint operators on a complex Hilbert space.This book contains an exposition of several such functional calculi. In particular, there is an exposition based on the spectral theorem for bounded, self-adjoint operators, an extension to the case of several commuting self-adjoint operators and an extension to normal operators. The Riesz operational calculus based on the Cauchy integral theorem from complex analysis is also described. Finally, an exposition of a functional calculus due to H. Weyl is given.

This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of n-dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors. Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds.

This book gives a rigorous treatment of selected topics in classical analysis, with many applications and examples. The exposition is at the undergraduate level, building on basic principles of advanced calculus without appeal to more sophisticated techniques of complex analysis and Lebesgue integration. Among the topics covered are Fourier series and integrals, approximation theory, Stirling's formula, the gamma function, Bernoulli numbers and polynomials, the Riemann zeta function, Tauberian theorems, elliptic integrals, ramifications of the Cantor set, and a theoretical discussion of differential equations including power series solutions at regular singular points, Bessel functions, hypergeometric functions, and Sturm comparison theory. Preliminary chapters offer rapid reviews of basic principles and further background material such as infinite products and commonly applied inequalities. This book is designed for individual study but can also serve as a text for second-semester courses in advanced calculus. Each chapter concludes with an abundance of exercises. Historical notes discuss the evolution of mathematical ideas and their relevance to physical applications. Special features are capsule scientific biographies of the major players and a gallery of portraits. Although this book is designed for undergraduate students, others may find it an accessible source of information on classical topics that underlie modern developments in pure and applied mathematics.

This pioneering book presents a study of the interrelationships among operator calculus, graph theory, and quantum probability in a unified manner, with significant emphasis on symbolic computations and an eye toward applications in computer science.Presented in this book are new methods, built on the algebraic framework of Clifford algebras, for tackling important real world problems related, but not limited to, wireless communications, neural networks, electrical circuits, transportation, and the world wide web. Examples are put forward in Mathematica throughout the book, together with packages for performing symbolic computations.

An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.

This is a book on many variable calculus. It is the second volume of a set of two. It includes proofs of all theorems presented, either in the text itself, or in an appendix. It also includes a sufficient introduction to linear algebra to allow the accurate presentation of many variable calculus.

The use of elementary linear algebra in presenting the topics of multi- variable calculus is more extensive than usual in this book. It makes many of these topics easier to understand and remember. The book will prepare readers for more advanced math courses and also for courses in physical science.

This is a book on many variable calculus. It is the second volume of a set of two. It includes proofs of all theorems presented, either in the text itself, or in an appendix. It also includes a sufficient introduction to linear algebra to allow the accurate presentation of many variable calculus.

The use of elementary linear algebra in presenting the topics of multi- variable calculus is more extensive than usual in this book. It makes many of these topics easier to understand and remember. The book will prepare readers for more advanced math courses and also for courses in physical science.

This is a book on single variable calculus including most of the important applications of calculus. It also includes proofs of all theorems presented, either in the text itself, or in an appendix. It also contains an introduction to vectors and vector products which is developed further in Volume 2.

While the book does include all the proofs of the theorems, many of the applications are presented more simply and less formally than is often the case in similar titles.

This is a book on single variable calculus including most of the important applications of calculus. It also includes proofs of all theorems presented, either in the text itself, or in an appendix. It also contains an introduction to vectors and vector products which is developed further in Volume 2.

While the book does include all the proofs of the theorems, many of the applications are presented more simply and less formally than is often the case in similar titles.

How do we measure happiness? Focusing on subjective measures as a proxy for welfare and well-being, this book finds ways to do that. Subjective measures have been used by psychologists, sociologists, political scientists, and, more recently, economists to answer a variety of scientifically and politically relevant questions. Van Praag, a pioneer in this field since 1971, and Ferrer-i-Carbonell present in this book a generally applicable methodology for the analysis of subjective satisfaction. Drawing on a range of surveys on people's satisfaction with their jobs, income, housing, marriages, and government policy, among other areas of life, this book shows how satisfaction with life "as a whole" is an aggregate of these domain satisfactions. Using German, British, Dutch, and Russian data, the authors cover a wide range of topics, even some not usually considered part of economic study. The book makes a distinction between actual satisfaction levels and individual norms, and in this way complements Van Praag's earlier work within the Leyden School with his later work in "happiness research". Among the many topics covered, the authors discuss: individuals' memory and anticipation processes and the estimation of adaptation phenomena (how individuals adapt to changing circumstances); the effect of reference groups on income norms and satisfaction with income; the importance of climate for well-being, including the development of a climate-equivalence index; the trade-offs between chronic diseases and income when well-being is kept constant; the damage of aircraft noise on well-being; the construction of a new talent tax tariff; and inequality from a satisfaction perspective, including the definition of "satisfaction inequalities", a natural extension of income inequality and poverty. This groundbreaking book presents new and fruitful methodology that consitutes a welcome addition to the social sciences.

Curious Curves is self-contained and unified in presentation. This book is suitable for a topics course, capstone course, or senior seminar; it is also intended for independent study by students and others interested in mathematics.Curves can often provide a better representation of natural phenomena than do the figures of classical geometry. Thus the content - presented with an emphasis on the geometric intuition characteristic of the study of curves - is highly relevant not only for people working in mathematics, but also those in other sciences. The explanations are detailed and illustrative to capture the interest of the reader, as well as complete to provide the necessary background information needed to go further into the subject.

An unusual supplement to every calculus textbook, Misteaks and How to Find Them before the Teacher Does is popular with students and teachers alike. Teachers love the way it encourages students to truly think about mathematics rather than simply plugging numbers into equations to crank out answers, and students love the author's straightforward, tongue-in-cheek style. The title of this light-hearted and amusing book might well have been "Going Gray in Elementary Calculus and How to Avoid it." Changing the metaphor, Barry has hit the nail on the finger in hundreds of fine examples. --Philip J. Davis, coauthor of The Mathematical Experience. "How I wish that something like this had been available when I was a student!" --Ralph P. Boas, former editor of The American Mathematical Monthly. Bonus: Solution to LeWitt Puzzle

This book makes accessible to calculus students in high school, college and university a range of counter-examples to "conjectures" that many students erroneously make. In addition, it urges readers to construct their own examples by tinkering with the ones shown here in order to enrich the example spaces to which they have access, and to deepen their appreciation of conspectus and conditions applying to theorems.

This book makes accessible to calculus students in high school, college and university a range of counter-examples to "conjectures" that many students erroneously make. In addition, it urges readers to construct their own examples by tinkering with the ones shown here in order to enrich the example spaces to which they have access, and to deepen their appreciation of conspectus and conditions applying to theorems.

Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and $n$-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series. The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed.

The positive response to the publication of Blanton's English translations of Euler's "Introduction to Analysis of the Infinite" confirmed the relevance of this 240 year old work and encouraged Blanton to translate Euler's "Foundations of Differential Calculus" as well. The current book constitutes just the first 9 out of 27 chapters. The remaining chapters will be published at a later time. With this new translation, Euler's thoughts will not only be more accessible but more widely enjoyed by the mathematical community.

This accessible introduction to Calculus is designed to demonstrate how calculus applies to various fields of study. The text is packed with real data and real-life applications to business, economics, social and life sciences. Applications using real data enhances student motivation. Many of these applications include source lines, to show how mathematics is used in the real world. * NEW! Conceptual problems ask students to put the concepts and results into their own words. These problems are marked with an icon to make them easier to assign. * More opportunities for the use of graphing calculator, including screen shots and instructions, and the use of icons that clearly identify each opportunity for the use of spreadsheets or graphing calculator. * Work problems appear throughout the text, giving the student the chance to immediately reinforce the concept or skill they have just learned. * Chapter Reviews contain a variety of features to help synthesize the ideas of the chapter, including: Objectives Check, Important Terms and Concepts, True-False Items, Fill in the Blanks and Review Exercises. * Includes Mathematical Questions from Professional Exams (CPA)

Since the publication of the first edition of this book, the area of mathematical finance has grown rapidly, with financial analysts using more sophisticated mathematical concepts, such as stochastic integration, to describe the behavior of markets and to derive computing methods. Maintaining the lucid style of its popular predecessor, Introduction to Stochastic Calculus Applied to Finance, Second Edition incorporates some of these new techniques and concepts to provide an accessible, up-to-date initiation to the field. New to the Second Edition Complements on discrete models, including Rogers' approach to the fundamental theorem of asset pricing and super-replication in incomplete markets Discussions on local volatility, Dupire's formula, the change of numeraire techniques, forward measures, and the forward Libor model A new chapter on credit risk modeling An extension of the chapter on simulation with numerical experiments that illustrate variance reduction techniques and hedging strategies Additional exercises and problems Providing all of the necessary stochastic calculus theory, the authors cover many key finance topics, including martingales, arbitrage, option pricing, American and European options, the Black-Scholes model, optimal hedging, and the computer simulation of financial models. They succeed in producing a solid introduction to stochastic approaches used in the financial world.

Even though the theories of operational calculus and integral transforms are centuries old, these topics are constantly developing, due to their use in the fields of mathematics, physics, and electrical and radio engineering. Operational Calculus and Related Topics highlights the classical methods and applications as well as the recent advances in the field.

Combining the best features of a textbook and a monograph, this volume presents an introduction to operational calculus, integral transforms, and generalized functions, the backbones of pure and applied mathematics. The text examines both the analytical and algebraic aspects of operational calculus and includes a comprehensive survey of classical results while stressing new developments in the field. Among the historical methods considered are Oliver Heaviside's algebraic operational calculus and Paul Dirac's delta function. Other discussions deal with the conditions for the existence of integral transforms, Jan Mikusiński's theory of convolution quotients, operator functions, and the sequential approach to the theory of generalized functions.

Benefits...

- Discusses theory "and" applications of integral transforms

- Gives inversion, complex-inversion, and Dirac's delta distribution formulas, among others

- Offers a short survey of actual results of finite integral transforms, in particular convolution theorems

Because Operational Calculus and Related Topics provides examples and illustrates the applications to various disciplines, it is an ideal reference for mathematicians, physicists, scientists, engineers, and students.

This textbook provides a calculus-based introduction to economics. Students blessed with a working knowledge of the calculus would find that this text facilitates their study of the basic analytical framework of economics. The textbook examines a wide range of micro and macro topics, including prices and markets, equity versus efficiency, Rawls versus Bentham, accounting and the theory of the firm, optimal lot size and just in time, monopoly and competition, exchange rates and the balance of payments, inflation and unemployment, fiscal and monetary policy, IS-LM analysis, aggregate demand and supply, speculation and rational expectations, growth and development, exhaustiable resources and over-fishing. While the content is similar to that of conventional introductory economics textbook, the assumption that the reader knows and enjoys the calculus distinguishes this book from the traditional text.

This textbook provides a calculus-based introduction to economics. Students blessed with a working knowledge of the calculus would find that this text facilitates their study of the basic analytical framework of economics. The textbook examines a wide range of micro and macro topics, including prices and markets, equity versus efficiency, Rawls versus Bentham, accounting and the theory of the firm, optimal lot size and just in time, monopoly and competition, exchange rates and the balance of payments, inflation and unemployment, fiscal and monetary policy, IS-LM analysis, aggregate demand and supply, speculation and rational expectations, growth and development, exhaustiable resources and over-fishing. While the content is similar to that of conventional introductory economics textbook, the assumption that the reader knows and enjoys the calculus distinguishes this book from the traditional text.

This important book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudodifferential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book.A Guide to Distribution Theory and Fourier Transforms can serve as a textbook for parts of a course on Applied Analysis or Methods of Mathematical Physics, and in fact it is used that way at Cornell.

This text is a practical course in complex calculus that covers the applications, but does not assume the full rigour of a real analysis background. Topics covered include algebraic and geometric aspects of complex numbers, differentiation, contour integration, evaluation of finite and infinite real integrals, summation of series and the fundamental theorm of algebra. The Residue Theorem for evaluatting complex integrals is presented in such a way that those wishing to study the subject at a deeper level should not need to unlearn anything presented here. A working knowledge of real calculus is assumed as is an acquaintance with complex numbers. This book is accessible to a any student who has studied calculus at upper school level and will be of interest to undergraduate students of applied mathematics, physical sciences and engineering.

Comprehensive Reference Work on Multivariate Analysis and Its Applications The first edition of this book, by Mardia, Kent and Bibby, has been widely used globally for over 40 years. This second edition brings many topics up to date, with a special emphasis on recent developments. A wide range of material in multivariate analysis is covered, including the classical themes of multivariate normal theory, multivariate regression, inference, multidimensional scaling, factor analysis, cluster analysis and principal component analysis. The book also now covers modern developments such as graphical models, robust estimation, statistical learning, and high-dimensional methods. The book expertly blends theory and application, providing numerous worked examples and exercises at the end of each chapter. The reader is assumed to have a basic knowledge of mathematical statistics at an undergraduate level together with an elementary understanding of linear algebra. There are appendices which provide a background in matrix algebra, a summary of univariate statistics, a collection of statistical tables and a discussion of computational aspects. The work includes coverage of: Basic properties of random vectors, normal distribution theory, and estimation Hypothesis testing, multivariate regression, and analysis of variance Principal component analysis, factor analysis, and canonical correlation analysis Cluster analysis and multidimensional scaling New advances and techniques, including statistical learning, graphical models and regularization methods for high-dimensional data Although primarily designed as a textbook for final year undergraduates and postgraduate students in mathematics and statistics, the book will also be of interest to research workers and applied scientists.