This 2003 book presents min-max methods through a study of the different faces of the celebrated Mountain Pass Theorem (MPT) of Ambrosetti and Rabinowitz. The reader is led from the most accessible results to the forefront of the theory, and at each step in this walk between the hills, the author presents the extensions and variants of the MPT in a complete and unified way. Coverage includes standard topics, but it also covers other topics covered nowhere else in book form: the non-smooth MPT; the geometrically constrained MPT; numerical approaches to the MPT; and even more exotic variants. Each chapter has a section with supplementary comments and bibliographical notes, and there is a rich bibliography and a detailed index to aid the reader. The book is suitable for researchers and graduate students. Nevertheless, the style and the choice of the material make it accessible to all newcomers to the field.

Fourier analysis has many scientific applications - in physics, number theory, combinatorics, signal processing, probability theory, statistics, option pricing, cryptography, acoustics, oceanography, optics and diffraction, geometry, and other areas. In signal processing and related fields, Fourier analysis is typically thought of as decomposing a signal into its component frequencies and their amplitudes.
This practical, applications-based professional handbook comprehensively covers the theory and applications of Fourier Analysis, spanning topics from engineering mathematics, signal processing and related multidimensional transform theory, and quantum physics to elementary deterministic finance and even the foundations of western music theory.
As a definitive text on Fourier Analysis, Handbook of Fourier Analysis and Its Applications is meant to replace several less comprehensive volumes on the subject, such as Processing of Multifimensional Signals by Alexandre Smirnov, Modern Sampling Theory by John J. Benedetto and Paulo J.S.G. Ferreira, Vector Space Projections by Henry Stark and Yongyi Yang and Fourier Analysis and Imaging by Ronald N. Bracewell. In addition to being primarily used as a professional handbook, it includes sample problems and their solutions at the end of each section and thus serves as a textbook for advanced undergraduate students and beginning graduate students in courses such as: Multidimensional Signals and Systems, Signal Analysis, Introduction to Shannon Sampling and Interpolation Theory, Random Variables and Stochastic Processes, and Signals and Linear Systems.

The concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, engineering, transportation, sociology, chemistry, biology and other fields. If one can formulate the equilibrium problem in the form of a mathematical model, solutions of the corresponding problem can be used for forecasting the future behavior of very complex systems and, also, for correcting the the current state of the system under control.
This book presents a unifying look on different equilibrium concepts in economics, including several models from related sciences.
- Presents a unifying look on different equilibrium concepts and also the present state of investigations in this field
- Describes static and dynamic input-output models, Walras, Cassel-Wald, spatial price, auction market, oligopolistic equilibrium models, transportation and migration equilibrium models
- Covers the basics of theory and solution methods both for the complementarity and variational inequality problems
- The methods are illustrated by applications and exercises to economic equilibrium models

Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introduction to Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristics and Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute.Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

In 1781, Gaspard Monge defined the problem of ""optimal transportation"", or the transferring of mass with the least possible amount of work, with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is at once an introduction to the field of optimal transportation and a survey of the research on the topic over the last 15 years. The book is intended for graduate students and researchers, and it covers both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis.

This book succinctly covers the key topics of numerical methods. While it is basically a survey of the subject, it has enough depth for the student to walk away with the ability to implement the methods by writing computer programs or by applying them to problems in physics or engineering. The author manages to cover the essentials while avoiding redundancies and using well-chosen examples and exercises. The exposition is supplemented by numerous figures. Work estimates and pseudo codes are provided for many algorithms, which can be easily converted to computer programs. Topics covered include interpolation, the fast Fourier transform, iterative methods for solving systems of linear and nonlinear equations, numerical methods for solving ODEs, numerical methods for matrix eigenvalue problems, approximation theory, and computer arithmetic.In general, the author assumes only a knowledge of calculus and linear algebra. The book is suitable as a text for a first course in numerical methods for mathematics students or students in neighboring fields, such as engineering, physics, and computer science.

Written by a well-known expert in the field, the focus of this book is on an innovative mathematical theory which applies to classical models of physics such as shock waves and balance laws. The text is based on early works in common with P.L. Lions (field medalist).

This is a handbook of Gamma-convergence, which is a theoretical tool used to study problems in Applied Mathematics where varying parameters are present, with many applications that range from Mechanics to Computer Vision. The book is directed to Applied Mathematicians in all fields and to Engineers with a theoretical background.

Composite materials are widely used in industry and include such well known examples as superconductors and optical fibers. However, modeling these materials is difficult, since they often has different properties at different points. The mathematical theory of homogenization is designed to handle this problem. The theory uses an idealized homogenous material to model a real composite while taking into account the microscopic structure. This introduction to homogenization theory develops the natural framework of the theory with four chapters on variational methods for partial differential equations. It then discusses the homogenization of several kinds of second-order boundary value problems. It devotes separate chapters to the classical examples of stead and non-steady heat equations, the wave equation, and the linearized system of elasticity. It includes numerous illustrations and examples.

Mathematical theory of discrete time decision processes, also known as stochastic control, is based on two major ideas: backward induction and conditioning. It has a large number of applications in almost all branches of the natural sciences. The aim of these notes is to give a self-contained introduction to this theory and its applications. Our intention was to give a global and mathematically precise picture of the subject and present well motivated examples. We cover systems with complete or partial information as well as with complete or partial observation. We have tried to present in a unified way several topics such as dynamic programming equations, stopping problems, stabilization, Kalman-Bucy filter, linear regulator, adaptive control and option pricing. The notes discuss a large variety of models rather than concentrate on general existence theorems.

In this book we study the degree theory and some of its applications in analysis. It focuses on the recent developments of this theory for Sobolev functions, which distinguishes this book from the currently available literature. We begin with a thorough study of topological degree for continuous functions. The contents of the book include: degree theory for continuous functions, the multiplication theorem, Hopf`s theorem, Brower`s fixed point theorem, odd mappings, Jordan`s separation theorem. Following a brief review of measure theory and Sobolev functions and study local invertibility of Sobolev functions. These results are put to use in the study variational principles in nonlinear elasticity. The Leray-Schauder degree in infinite dimensional spaces is exploited to obtain fixed point theorems. We end the book by illustrating several applications of the degree in the theories of ordinary differential equations and partial differential equations.

Noted mathematician offers basic treatment of theory of analytic functions of a complex variable, touching on analytic functions of several real or complex variables as well as the existence theorem for solutions of differential systems where data is analytic. Also included is a systematic, though elementary, exposition of theory of abstract complex manifolds of one complex dimension. Topics include power series in one variable, holomorphic functions, Cauchy's integral, more. Exercises. 1973 edition.

This is the first book devoted to lattice methods, a recently developed way of calculating multiple integrals in many variables. Multiple integrals of this kind arise in fields such as quantum physics and chemistry, statistical mechanics, Bayesian statistics and many others. Lattice methods are an effective tool when the number of integrals are large. The book begins with a review of existing methods before presenting lattice theory in a thorough, self-contained manner, with numerous illustrations and examples. Group and number theory are included, but the treatment is such that no prior knowledge is needed. Not only the theory but the practical implementation of lattice methods is covered. An algorithm is presented alongside tables not available elsewhere, which together allow the practical evaluation of multiple integrals in many variables. Most importantly, the algorithm produces an error estimate in a very efficient manner. The book also provides a fast track for readers wanting to move rapidly to using lattice methods in practical calculations. It concludes with extensive numerical tests which compare lattice methods with other methods, such as the Monte Carlo.

Guicciardini presents a comprehensive survey of both the research and teaching of Newtonian calculus, the calculus of "fluxions," over the period between 1700 and 1810. Although Newton was one of the inventors of calculus, the developments in Britain remained separate from the rest of Europe for over a century. While it is usually maintained that after Newton there was a period of decline in British mathematics, the author's research demonstrates that the methods used by researchers of the period yielded considerable success in laying the foundations and investigating the applications of the calculus. Even when "decline" set in, in mid century, the foundations of the reform were being laid, which were to change the direction and nature of the mathematics community. The book considers the importance of Isaac Newton, Roger Cotes, Brook Taylor, James Stirling, Abraham de Moivre, Colin Maclaurin, Thomas Bayes, John Landen and Edward Waring. This will be a useful book for students and researchers in the history of science, philosophers of science and undergraduates studying the history of mathematics.

Collected together in this book are ten state-of-the-art expository articles on the most important topics in optimization, written by leading experts in the field. The book therefore provides a primary reference for those performing research in some area of optimization or for those who have some basic knowledge of optimization techniques but wish to learn the most up-to-date and efficient algorithms for particular classes of problems. The first sections of each chapter are expository and therefore accessible to master's level graduate students. However, the chapters also contain advanced material on current topics of interest to researchers. For instance there are chapters which describe the polynomial-time linear programming algorithms of Khachian and Karmarkar and the techniques used to solve combinatorial and integer programming problems, an order of magnitude larger than was possible just a few years ago. Overall a comprehensive yet lively and up-to-date discussion of the state-of-the-art in optimization is presented in this book.

Provides a survey of the theoretical results on systems of nonlinear equations in finite dimension and the major iterative methods for their computational solution. Originally published in 1970, it offers a research-level presentation of the principal results known at that time. Although the field has developed since the book originally appeared, it remains a major background reference for the literature before 1970. In particular, Part II contains the only relatively complete introduction to the existence theory for finite-dimensional nonlinear equations from the viewpoint of computational mathematics. Over the years semilocal convergence results have been obtained for various methods, especially with an emphasis on error bounds for the iterates. The results and proof techniques introduced here still represent a solid basis for this topic.

No one working in duality should be without a copy of Convex Analysis and Variational Problems. This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references.

This unabridged republication of the 1980 text, an established classic in the field, is a resource for many important topics in elliptic equations and systems and is the first modern treatment of free boundary problems. Variational inequalities (equilibrium or evolution problems typically with convex constraints) are carefully explained in An Introduction to Variational Inequalities and Their Applications. They are shown to be extremely useful across a wide variety of subjects, ranging from linear programming to free boundary problems in partial differential equations. Exciting new areas like finance and phase transformations along with more historical ones like contact problems have begun to rely on variational inequalities, making this book a necessity once again.

This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues such as Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are knowledge of the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations.

Guicciardini presents a comprehensive survey of both the research and teaching of Newtonian calculus, the calculus of "fluxions", over the period between 1700 and 1810. Although Newton was one of the inventors of calculus, the developments in Britain remained separate from the rest of Europe for over a century. While it is usually maintained that after Newton there was a period of decline in British mathematics, the author's research demonstrates that the methods used by researchers of the period yielded considerable success in laying the foundations and investigating the applications of the calculus. Even when "decline" set in, in mid century, the foundations of the reform were being laid, which were to change the direction and nature of the mathematics community. The book considers the importance of Isaac Newton, Roger Cotes, Brook Taylor, James Stirling, Abraham de Moivre, Colin Maclaurin, Thomas Bayes, John Landen and Edward Waring. This will be a useful book for students and researchers in the history of science, philosophers of science and undergraduates studying the history of mathematics.

The calculus of variations is a classical area of mathematical analysi s-300 years old-yet its myriad applications in science and technology continue to hold great interest and keep it an active area of research . These two volumes contain the refereed proceedings of the internatio nal conference on Calculus of Variations and Related Topics held at th e Technion-Israel Institute of Technology in March 1998. The conferenc e commemorated 300 years of work in the field and brought together man y of its leading experts. The papers in the first volume focus on cri tical point theory and differential equations. The other volume deals with variational aspects of optimal control. Together they provide a u nique opportunity to review the state-of-the-art of the calculus of va riations, as presented by an international panel of masters in the fie ld.

This book deals with a class of mathematical problems which involve the minimization of the sum of a volume and a surface energy and have lately been refered to as 'free discontinuity problems'. Examples of such problems come from fracture mechanics, image analysis, or the theory of phase transitions. A systematic introduction to this field, this book is highly suitable for graduate students, bridging the gap between research level texts and elementary textbooks on measure theory and calculus of variation. The first half of the book contains a comprehensive and updated treatment of the theory of Functions of Bounded Variation and of the mathematical prerequisites of that theory, that is Abstract Measure Theory and Geometric Measure Theory.

This text discusses existence and necessary conditions, such as Potryagin's maximum principle, for optimal control problems described by ordinary and partial differential equations. These necessary conditions are obtained from KuhnTucker theorems for nonlinear programming problems in infinite dimensional spaces. The optimal control problems include control constraints, state constraints and target conditions. Evolution partial differential equations are studied using semigroup theory, abstract differential equations in linear spaces, integral equations and interpolation theory. Existence of optimal controls is established for arbitrary control sets by means of a general theory of relaxed controls. Applications include nonlinear systems described by partial differential equations of hyperbolic and parabolic type and results on convergence of suboptimal controls.

This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues such as Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are knowledge of the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations.

This book combines the efforts of a distinguished team of authors, who are all renowned mathematicians and expositors, and provides a modern introduction to the calculus of variations. By focusing on the one-dimensional case it remains relatively free of technicalities, and therefore provides a useful overview of the theory at a level suitable for graduate students.