Classroom-tested and lucidly written, Multivariable Calculus gives a thorough and rigoroustreatment of differential and integral calculus of functions of several variables. Designed as ajunior-level textbook for an advanced calculus course, this book covers a variety of notions,including continuity , differentiation, multiple integrals, line and surface integrals, differentialforms, and infinite series. Numerous exercises and examples throughout the book facilitatethe student's understanding of important concepts.The level of rigor in this textbook is high; virtually every result is accompanied by a proof. Toaccommodate teachers' individual needs, the material is organized so that proofs can be deemphasizedor even omitted. Linear algebra for n-dimensional Euclidean space is developedwhen required for the calculus; for example, linear transformations are discussed for the treatmentof derivatives.Featuring a detailed discussion of differential forms and Stokes' theorem, Multivariable Calculusis an excellent textbook for junior-level advanced calculus courses and it is also usefulfor sophomores who have a strong background in single-variable calculus. A two-year calculussequence or a one-year honor calculus course is required for the most successful use of thistextbook. Students will benefit enormously from this book's systematic approach to mathematicalanalysis, which will ultimately prepare them for more advanced topics in the field.

Complex Dynamics: Families and Friends features contributions by many of the leading mathematicians in the field, such as Mikhail Lyubich, John Milnor, Mitsuhiro Shishikura, and William Thurston. Some of the chapters, including an introduction by Thurston to the general subject of complex dynamics, are classic manuscripts that were never published before but have influenced the field for more than two decades. Other chapters contain fresh, original work and bring readers to the current frontier of research. The title reflects the fruitful interplay between diverse mathematical fields bound together by the common theme of complex dynamics, including hyperbolic geometry, number theory, group theory, combinatorics, general dynamics, and many more. At the same time, the title alludes to the spirit of mathematical friendship among the researchers in this area. This book is a tribute to John Hubbard, one of the most inspiring pioneers in the field of complex dynamics.

The literature on the spectral analysis of second order elliptic differential operators contains a great deal of information on the spectral functions for explicitly known spectra. The same is not true, however, for situations where the spectra are not explicitly known. Over the last several years, the author and his colleagues have developed new, innovative methods for the exact analysis of a variety of spectral functions occurring in spectral geometry and under external conditions in statistical mechanics and quantum field theory. Spectral Functions in Mathematics and Physics presents a detailed overview of these advances. The author develops and applies methods for analyzing determinants arising when the external conditions originate from the Casimir effect, dielectric media, scalar backgrounds, and magnetic backgrounds. The zeta function underlies all of these techniques, and the book begins by deriving its basic properties and relations to the spectral functions. The author then uses those relations to develop and apply methods for calculating heat kernel coefficients, functional determinants, and Casimir energies. He also explores applications in the non-relativistic context, in particular applying the techniques to the Bose-Einstein condensation of an ideal Bose gas. Self-contained and clearly written, Spectral Functions in Mathematics and Physics offers a unique opportunity to acquire valuable new techniques, use them in a variety of applications, and be inspired to make further advances.

This book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs. The authors concentrate on three types of absorbing sets: attractors, exponential attractors, and inertial manifolds. They present the fundamental properties of these sets, and then proceed to show the existence of some of these sets for a number of dynamical systems generated by well-known physical models. In particular, they consider in full detail two particular PDEEs: a semilinear version of the heat equation and a corresponding version of the dissipative wave equation. These examples illustrate the most important features of the theory of semiflows and provide a sort of template that can be applied to the analysis of other models. The material builds in a careful, gradual progression, developing the background needed by newcomers to the field, and culminating in a more detailed presentation of the main topics than found in most sources. The authors' approach to and treatment of the subject builds the foundation for more advanced references and research on global attractors, exponential attractors, and inertial manifolds.

As a relatively new area in mathematics, stochastic partial differential equations (PDEs) are still at a tender age and have not yet received much attention in the mathematical community. Filling the void of an introductory text in the field, Stochastic Partial Differential Equations introduces PDEs to students familiar with basic probability theory and Ito's equations, highlighting several computational and analytical techniques. Without assuming specific knowledge of PDEs, the text includes many challenging problems in stochastic analysis and treats stochastic PDEs in a practical way. The author first brings the subject back to its root in classical concrete problems. He then discusses a unified theory of stochastic evolution equations and describes a few applied problems, including the random vibration of a nonlinear elastic beam and invariant measures for stochastic Navier-Stokes equations. The book concludes by pointing out the connection of stochastic PDEs to infinite-dimensional stochastic analysis. By thoroughly covering the concepts and applications of stochastic PDEs at an introductory level, this text provides a guide to current research topics and lays the groundwork for further study.

This 1996 book unites the study of dynamical systems and numerical solution of differential equations. The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulated as dynamical systems and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strange attractors under numerical approximation. This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.

"Provides a thorough introduction to the algebraic theory of systems of differential equations, as developed by the Japanese school of M. Sato and his colleagues. Features a complete review of hyperfunction-microfunction theory and the theory of D-modules. Strikes the perfect balance between analytic and algebraic aspects."

Complete coverage of Differential Equations for one or two semester course Includes coverage of PDEs and linear algebra Chapter on Power Series and series solutions added to this edition Focus on modeling and applications Emphasis on numerical solutions

Building on the author's previous book in the series, Complex Analysis with Applications to Flows and Fields (CRC Press, 2010), Transcendental Representations with Applications to Solids and Fluids focuses on four infinite representations: series expansions, series of fractions for meromorphic functions, infinite products for functions with infinitely many zeros, and continued fractions as alternative representations. This book also continues the application of complex functions to more classes of fields, including incompressible rotational flows, compressible irrotational flows, unsteady flows, rotating flows, surface tension and capillarity, deflection of membranes under load, torsion of rods by torques, plane elasticity, and plane viscous flows. The two books together offer a complete treatment of complex analysis, showing how the elementary transcendental functions and other complex functions are applied to fluid and solid media and force fields mainly in two dimensions. The mathematical developments appear in odd-numbered chapters while the physical and engineering applications can be found in even-numbered chapters. The last chapter presents a set of detailed examples. Each chapter begins with an introduction and concludes with related topics. Written by one of the foremost authorities in aeronautical/aerospace engineering, this self-contained book gives the necessary mathematical background and physical principles to build models for technological and scientific purposes. It shows how to formulate problems, justify the solutions, and interpret the results.

A Complete Treatment of Current Research Topics in Fourier Transforms and Sinusoids Sinusoids: Theory and Technological Applications explains how sinusoids and Fourier transforms are used in a variety of application areas, including signal processing, GPS, optics, x-ray crystallography, radioastronomy, poetry and music as sound waves, and the medical sciences. With more than 200 illustrations, the book discusses electromagnetic force and sychrotron radiation comprising all kinds of waves, including gamma rays, x-rays, UV rays, visible light rays, infrared, microwaves, and radio waves. It also covers topics of common interest, such as quasars, pulsars, the Big Bang theory, Olbers' paradox, black holes, Mars mission, and SETI. The book begins by describing sinusoids-which are periodic sine or cosine functions-using well-known examples from wave theory, including traveling and standing waves, continuous musical rhythms, and the human liver. It next discusses the Fourier series and transform in both continuous and discrete cases and analyzes the Dirichlet kernel and Gibbs phenomenon. The author shows how invertibility and periodicity of Fourier transforms are used in the development of signals and filters, addresses the general concept of communication systems, and explains the functioning of a GPS receiver. The author then covers the theory of Fourier optics, synchrotron light and x-ray diffraction, the mathematics of radioastronomy, and mathematical structures in poetry and music. The book concludes with a focus on tomography, exploring different types of procedures and modern advances. The appendices make the book as self-contained as possible.

Summarizes information scattered in the technical literature on a subject too new to be included in most textbooks, but which is of interest to statisticians, and those who use statistics in science and education, at an advanced undergraduate or higher level. Overviews recent research on constructin

This volume presents the general theory of generalized functions, including the Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and Poisson integral transforms and operational calculus, with the traditional material augmented by the theory of Fourier series, abelian theorems, and boundary values of helomorphic functions for one and several variables. The author addresses several facets in depth, including convolution theory, convolution algebras and convolution equations in them, homogenous generalized functions, and multiplication of generalized functions. This book will meet the needs of researchers, engineers, and students of applied mathematics, control theory, and the engineering sciences.

Combining mathematical theory, physical principles, and engineering problems, Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables. As part of a series of books for engineers and scientists exploring advanced mathematics, Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as: Gauss and Stokes' theorems in the differential geometry, tensor calculus, and theory of potential fields Self-adjoint and non-self-adjoint problems for linear differential equations and nonlinear problems with large deformations Multipolar expansions and Green's functions for elastic strings and bars, potential and rotational flow, electro- and magnetostatics, and more This third volume in the series Mathematics and Physics for Science and Technology is designed to complete the theory of functions and its application to potential fields, relating generalized functions to broader follow-on topics like differential equations. Featuring step-by-step examples with interpretations of results and discussions of assumptions and their consequences, Generalized Calculus with Applications to Matter and Forces enables readers to construct mathematical-physical models suited to new observations or novel engineering devices.

Ever since the groundbreaking work of J.J. Kohn in the early 1960s, there has been a significant interaction between the theory of partial differential equations and the function theory of several complex variables. Partial Differential Equations and Complex Analysis explores the background and plumbs the depths of this symbiosis. The book is an excellent introduction to a variety of topics and presents many of the basic elements of linear partial differential equations in the context of how they are applied to the study of complex analysis. The author treats the Dirichlet and Neumann problems for elliptic equations and the related Schauder regularity theory, and examines how those results apply to the boundary regularity of biholomorphic mappings. He studies the ?-Neumann problem, then considers applications to the complex function theory of several variables and to the Bergman projection.

The analysis and topology of elliptic operators on manifolds with singularities are much more complicated than in the smooth case and require completely new mathematical notions and theories. While there has recently been much progress in the field, many of these results have remained scattered in journals and preprints. Starting from an elementary level and finishing with the most recent results, this book gives a systematic exposition of both analytical and topological aspects of elliptic theory on manifolds with singularities. The presentation includes a review of the main techniques of the theory of elliptic equations, offers a comparative analysis of various approaches to differential equations on manifolds with singularities, and devotes considerable attention to applications of the theory. These include Sobolev problems, theorems of Atiyah-Bott-Lefschetz type, and proofs of index formulas for elliptic operators and problems on manifolds with singularities, including the authors' new solution to the index problem for manifolds with nonisolated singularities. A glossary, numerous illustrations, and many examples help readers master the subject. Clear exposition, up-to-date coverage, and accessibility-even at the advanced undergraduate level-lay the groundwork for continuing studies and further advances in the field.

Classification of Lipschitz Mappings presents a systematic, self-contained treatment of a new classification of Lipschitz mappings and its application in many topics of metric fixed point theory. Suitable for readers interested in metric fixed point theory, differential equations, and dynamical systems, the book only requires a basic background in functional analysis and topology. The author focuses on a more precise classification of Lipschitzian mappings. The mean Lipschitz condition introduced by Goebel, Japon Pineda, and Sims is relatively easy to check and turns out to satisfy several principles: Regulating the possible growth of the sequence of Lipschitz constants k(Tn) Ensuring good estimates for k0(T) and k (T) Providing some new results in metric fixed point theory

Among the theoretical methods for solving many problems of applied mathematics, physics, and technology, asymptotic methods often provide results that lead to obtaining more effective algorithms of numerical evaluation. Presenting the mathematical methods of perturbation theory, Introduction to Asymptotic Methods reviews the most important methods of singular perturbations within the scope of application of differential equations. The authors take a challenging and original approach based on the integrated mathematical-analytical treatment of various objects taken from interdisciplinary fields of mechanics, physics, and applied mathematics. This new hybrid approach will lead to results that cannot be obtained by standard theories in the field. Emphasizing fundamental elements of the mathematical modeling process, the book provides comprehensive coverage of asymptotic approaches, regular and singular perturbations, one-dimensional non-stationary non-linear waves, Pade approximations, oscillators with negative Duffing type stiffness, and differential equations with discontinuous nonlinearities. The book also offers a method of construction for canonical variables transformation in parametric form along with a number of examples and applications. The book is applications oriented and features results and literature citations that have not been seen in the Western Scientific Community. The authors emphasize the dynamics of the development of perturbation methods and present the development of ideas associated with this wide field of research.

For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science. Covering both theoretical and practical aspects, the book presents: Key mathematical and physical ideas in chaos theory Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance Until the authors' recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.

Are some areas of fast Fourier transforms still unclear to you? Do the notation and vocabulary seem inconsistent? Does your knowledge of their algorithmic aspects feel incomplete? The fast Fourier transform represents one of the most important advancements in scientific and engineering computing. Until now, however, treatments have been either brief, cryptic, intimidating, or not published in the open literature. Inside the FFT Black Box brings the numerous and varied ideas together in a common notational framework, clarifying vague FFT concepts. Examples and diagrams explain algorithms completely, with consistent notation. This approach connects the algorithms explicitly to the underlying mathematics. Reviews and explanations of FFT ideas taken from engineering, mathematics, and computer science journals teach the computational techniques relevant to FFT. Two appendices familiarize readers with the design and analysis of computer algorithms, as well. This volume employs a unified and systematic approach to FFT. It closes the gap between brief textbook introductions and intimidating treatments in the FFT literature. Inside the FFT Black Box provides an up-to-date, self-contained guide for learning the FFT and the multitude of ideas and computing techniques it employs.

A remarkable interplay exists between the fields of elliptic functions and orthogonal polynomials. In the first monograph to explore their connections, Elliptic Polynomials combines these two areas of study, leading to an interesting development of some basic aspects of each. It presents new material about various classes of polynomials and about the odd Jacobi elliptic functions and their inverses. The term elliptic polynomials refers to the polynomials generated by odd elliptic integrals and elliptic functions. In studying these, the authors consider such things as orthogonality and the construction of weight functions and measures, finding structure constants and interesting inequalities, and deriving useful formulas and evaluations. Although some of the material may be familiar, it establishes a new mathematical field that intersects with classical subjects at many points. Its wealth of information on important properties of polynomials and clear, accessible presentation make Elliptic Polynomials valuable to those in real and complex analysis, number theory, and combinatorics, and will undoubtedly generate further research.

This important new book sets forth a comprehensive description of various mathematical aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important mechanical applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics (MHD), shallow water, and solid dynamics equations. This treatment provides-for the first time in book form-a collection of recipes for applying higher-order non-oscillatory shock-capturing schemes to MHD modelling of physical phenomena. The authors also address a number of original "nonclassical" problems, such as shock wave propagation in rods and composite materials, ionization fronts in plasma, and electromagnetic shock waves in magnets. They show that if a small-scale, higher-order mathematical model results in oscillations of the discontinuity structure, the variety of admissible discontinuities can exhibit disperse behavior, including some with additional boundary conditions that do not follow from the hyperbolic conservation laws. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. The authors formulate several selection rules, which in some cases easily allow a correct, physically realizable choice. This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes Mathematical Aspects of Numerical Solution of Hyperbolic Systems particularly valuable not only to those interested the development of numerical methods, but to physicists and engineers who strive to solve increasingly complicated nonlinear equations.

Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems. Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: "Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?" Along with this problem, many inverse problems for heat and wave equations are solved. The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas. Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.

The theory of distributions is most often presented as L. Schwartz originally presented it: as a theory of the duality of topological vector spaces. Although this is a sound approach, it can be difficult, demanding deep prior knowledge of functional analysis. The more elementary treatments that are available often consider distributions as limits of sequences of functions, but these usually present the theoretical foundations in a form too simplified for practical applications. Distributions, Integral Transforms and Applications offers an approachable introduction to the theory of distributions and integral transforms that uses Schwartz's description of distributions as linear continous forms on topological vector spaces. The authors use the theory of the Lebesgue integral as a fundamental tool in the proofs of many theorems and develop the theory from its beginnings to the point of proving many of the deep, important theorems, such as the Schwartz kernel theorem and the Malgrange-Ehrenpreis theorem. They clearly demonstrate how the theory of distributions can be used in cases such as Fourier analysis, when the methods of classical analysis are insufficient. Accessible to anyone who has completed a course in advanced calculus, this treatment emphasizes the remarkable connections between distributional theory, classical analysis, and the theory of differential equations and leads directly to applications in various branches of mathematics.

The mathematical analysis of contact problems, with or without friction, is an area where progress depends heavily on the integration of pure and applied mathematics. This book presents the state of the art in the mathematical analysis of unilateral contact problems with friction, along with a major part of the analysis of dynamic contact problems without friction. Much of this monograph emerged from the authors' research activities over the past 10 years and deals with an approach proven fruitful in many situations. Starting from thin estimates of possible solutions, this approach is based on an approximation of the problem and the proof of a moderate partial regularity of the solution to the approximate problem. This in turn makes use of the shift (or translation) technique - an important yet often overlooked tool for contact problems and other nonlinear problems with limited regularity. The authors pay careful attention to quantification and precise results to get optimal bounds in sufficient conditions for existence theorems. Unilateral Contact Problems: Variational Methods and Existence Theorems a valuable resource for scientists involved in the analysis of contact problems and for engineers working on the numerical approximation of contact problems. Self-contained and thoroughly up to date, it presents a complete collection of the available results and techniques for the analysis of unilateral contact problems and builds the background required for further research on more complex problems in this area.

Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators. Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the book focuses on the existence of new exact solutions on linear invariant subspaces for nonlinear operators and their crucial new properties. This practical reference deals with various partial differential equations (PDEs) and models that exhibit some common nonlinear invariant features. It begins with classical as well as more recent examples of solutions on invariant subspaces. In the remainder of the book, the authors develop several techniques for constructing exact solutions of various nonlinear PDEs, including reaction-diffusion and gas dynamics models, thin-film and Kuramoto-Sivashinsky equations, nonlinear dispersion (compacton) equations, KdV-type and Harry Dym models, quasilinear magma equations, and Green-Naghdi equations. Using exact solutions, they describe the evolution properties of blow-up or extinction phenomena, finite interface propagation, and the oscillatory, changing sign behavior of weak solutions near interfaces for nonlinear PDEs of various types and orders. The techniques surveyed in Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics serve as a preliminary introduction to the general theory of nonlinear evolution PDEs of different orders and types.