Here is a modern introduction to the theory of tensor algebra and tensor analysis. It discusses tensor algebra and introduces differential manifold. Coverage also details tensor analysis, differential forms, connection forms, and curvature tensor. In addition, the book investigates Riemannian and pseudo-Riemannian manifolds in great detail. Throughout, examples and problems are furnished from the theory of relativity and continuum mechanics.

This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists, at levels ranging from advanced undergraduate students to professional scientists. The book bridges the acknowledged gap between the different languages used by mathematicians and physicists. For students of mathematics the author shows that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which goes beyond the usual curriculum in physics.

Our book is devoted to the topological fixed point theory both for single-valued and multivalued mappings in locally convex spaces, including its application to boundary value problems for ordinary differential equations (inclusions) and to (multivalued) dynamical systems. It is the first monograph dealing with the topo- logical fixed point theory in non-metric spaces. Although the theoretical material was tendentially selected with respect to ap- plications, we wished to have a self-consistent text (see the scheme below). There- fore, we supplied three appendices concerning almost-periodic and derivo-periodic single-valued {multivalued) functions and (multivalued) fractals. The last topic which is quite new can be also regarded as a contribution to the fixed point theory in hyperspaces. Nevertheless, the reader is assumed to be at least partly famil- iar in some related sections with the notions like the Bochner integral, the Au- mann multivalued integral, the Arzela-Ascoli lemma, the Gronwall inequality, the Brouwer degree, the Leray-Schauder degree, the topological (covering) dimension, the elemens of homological algebra, ...Otherwise, one can use the recommended literature. Hence, in Chapter I, the topological and analytical background is built. Then, in Chapter II (and partly already in Chapter I), topological principles necessary for applications are developed, namely: the fixed point index theory (resp. the topological degree theory), the Lefschetz and the Nielsen theories both in absolute and relative cases, periodic point theorems, topological essentiality, continuation-type theorems.

The theory of functional equations has been developed in a rapid and productive way in the second half of the Twentieth Century. First of all, this is due to the fact that the mathematical applications raised the investigations of newer and newer types of functional equations. At the same time, the self development of this theory was also very fruitful. This can be followed in many monographs that treat and discuss the various methods and approaches. These developments were also essentially influenced by a number jour nals, for instance, by the Publicationes Mathematicae Debrecen (founded in 1953) and by the Aequationes Mathematicae (founded in 1968), be cause these journals published papers from the field of functional equa tions readily and frequently. The latter journal also publishes the yearly report of the International Symposia on Functional Equations and a comprehensive bibliography of the most recent papers. At the same time, there are periodically and traditionally organized conferences in Poland and in Hungary devoted to functional equations and inequali ties. In 2000, the 38th International Symposium on Functional Equations was organized by the Institute of Mathematics and Informatics of the University of Debrecen in Noszvaj, Hungary. The report about this meeting can be found in Aequationes Math. 61 (2001), 281-320."

This monograph is devoted to a rapidly developing area of research of the qualitative theory of difference and functional differential equations. In fact, in the last 25 years Oscillation Theory of difference and functional differential equations has attracted many researchers. This has resulted in hundreds of research papers in every major mathematical journal, and several books. In the first chapter of this monograph, we address oscillation of solutions to difference equations of various types. Here we also offer several new fundamental concepts such as oscillation around a point, oscillation around a sequence, regular oscillation, periodic oscillation, point-wise oscillation of several orthogonal polynomials, global oscillation of sequences of real valued functions, oscillation in ordered sets, ( , R, )-oscillate, oscillation in linear spaces, oscillation in Archimedean spaces, and oscillation across a family. These concepts are explained through examples and supported by interesting results. In the second chapter we present recent results pertaining to the oscil lation of n-th order functional differential equations with deviating argu ments, and functional differential equations of neutral type. We mainly deal with integral criteria for oscillation. While several results of this chapter were originally formulated for more complicated and/or more general differ ential equations, we discuss here a simplified version to elucidate the main ideas of the oscillation theory of functional differential equations. Further, from a large number of theorems presented in this chapter we have selected the proofs of only those results which we thought would best illustrate the various strategies and ideas involved."

Developed in this book are several deep connections between time-frequency (Fourier/Gabor) analysis and time-scale (wavelet) analysis, emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. While researchers at the forefront of these areas are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger community of practitioners about the precise strengths and limitations of Fourier/Gabor analysis versus wavelets. This book fills that gap by presenting the interface of time-frequency and time-scale methods as a rich area of work. "Foundations of Time-Frequency and Time-Scale Methods" will be suitable for applied mathematicians and engineers in signal/image processing and communication theory, as well as researchers and students in mathematical analysis, signal analysis, and mathematical physics.

This work, consisting of expository articles as well as research papers, highlights recent developments in nonlinear analysis and differential equations. The material is largely an outgrowth of autumn school courses and seminars held at the University of Lisbon and has been thoroughly refereed.

Several topics in ordinary differential equations and partial differential equations are the focus of key articles, including:

* periodic solutions of systems with p-Laplacian type operators (J. Mawhin)

* bifurcation in variational inequalities (K. Schmitt)

* a geometric approach to dynamical systems in the plane via twist theorems (R. Ortega)

* asymptotic behavior and periodic solutions for Navier--Stokes equations (E. Feireisl)

* mechanics on Riemannian manifolds (W. Oliva)

* techniques of lower and upper solutions for ODEs (C. De Coster and P. Habets)

A number of related subjects dealing with properties of solutions, e.g., bifurcations, symmetries, nonlinear oscillations, are treated in other articles.

This volume reflects rich and varied fields of research and will be a useful resource for mathematicians and graduate students in the ODE and PDE community.

This book offers an essential introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for providing an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, lies in the strenuous mathematics demands that even the simplest physical cases entail. Graduate courses in physics rarely offer enough time to cover the theory of Hilbert space and operators, as well as distribution theory, with sufficient mathematical rigor. Accordingly, compromises must be found between full rigor and the practical use of the instruments. Based on one of the authors's lectures on functional analysis for graduate students in physics, the book will equip readers to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. It also includes a brief introduction to topological groups, and to other mathematical structures akin to Hilbert space. Exercises and solved problems accompany the main text, offering readers opportunities to deepen their understanding. The topics and their presentation have been chosen with the goal of quickly, yet rigorously and effectively, preparing readers for the intricacies of Hilbert space. Consequently, some topics, e.g., the Lebesgue integral, are treated in a somewhat unorthodox manner. The book is ideally suited for use in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.

This book provides an overview of some of the most active topics in the theory of transformation groups over the past decades and stresses advances obtained in the last dozen years. The emphasis is on actions of Lie groups on manifolds and CW complexes. Manifolds and actions of Lie groups on them are studied in the linear, semialgebraic, definable, analytic, smooth, and topological categories. Equivalent vector bundles play an important role.

The work is divided into fifteen articles and will be of interest to anyone researching or studying transformations groups. The references make it easy to find details and original accounts of the topics surveyed, including tools and theories used in these accounts.

This volume contains contributions written by participants of the 4th Workshop on Operator Theory in Krein Spaces and Applications, held at the TU Berlin, Germany, December 17 to 19, 2004. The workshop covered topics from spectral, perturbation, and extension theory of linear operators and relations in inner product spaces.

This research-level book presents up-to-date information concerning recent developments in convex functions and partial orderings and some applications in mathematics, statistics, and reliability theory. The book will serve researchers in mathematical and statistical theory and theoretical and applied reliabilists.
Key Features
* Presents classical and newly published results on convex functions and related inequalities
* Explains partial ordering based on arrangement and their applications in mathematics, probability, statsitics, and reliability
* Demonstrates the connection of partial ordering with other well-known orderings such as majorization and Schur functions
* Will generate further research and applications

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 1997.

This book is a self-contained exposition of the spectral theory of Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients. It includes an introduction to Carleson curves, Muckenhoupt weights, weighted norm inequalities, local principles, Wiener-Hopf factorization, and Banach algebras generated by idempotents. Some basic phenomena in the field and the techniques for treating them came to be understood only in recent years and are comprehensively presented here for the first time.
The material has been polished in an effort to make advanced topics accessible to a broad readership. The book is addressed to a wide audience of students and mathematicians interested in real and complex analysis, functional analysis and operator theory.

This book gives an account of an ellipsoidal calculus and ellipsoidal techniques developed by the authors. The text ranges from a specially developed theory of exact set-valued solutions to the description of ellipsoidal calculus, related ellipsoidal-based methods and examples worked out with computer graphics.

This volume presents the lectures given during the second French-Uzbek Colloquium on Algebra and Operator Theory which took place in Tashkent in 1997, at the Mathematical Institute of the Uzbekistan Academy of Sciences. Among the algebraic topics discussed here are deformation of Lie algebras, cohomology theory, the algebraic variety of the laws of Lie algebras, Euler equations on Lie algebras, Leibniz algebras, and real K-theory. Some contributions have a geometrical aspect, such as supermanifolds. The papers on operator theory deal with the study of certain types of operator algebras. This volume also contains a detailed introduction to the theory of quantum groups. Audience: This book is intended for graduate students specialising in algebra, differential geometry, operator theory, and theoretical physics, and for researchers in mathematics and theoretical physics.

The stability problem for approximate homomorphisms, or the Ulam stability problem, was posed by S. M. Ulam in the year 1941. The solution of this problem for various classes of equations is an expanding area of research. In particular, the pursuit of solutions to the Hyers-Ulam and Hyers-Ulam-Rassias stability problems for sets of functional equations and ineqalities has led to an outpouring of recent research.

This volume, dedicated to S. M. Ulam, presents the most recent results on the solution to Ulam stability problems for various classes of functional equations and inequalities. Comprised of invited contributions from notable researchers and experts, this volume presents several important types of functional equations and inequalities and their applications to problems in mathematical analysis, geometry, physics and applied mathematics.

"Functional Equations in Mathematical Analysis" is intended for researchers and students in mathematics, physics, and other computational and applied sciences.

Weak convergence is a basic tool of modern nonlinear analysis because it enjoys the same compactness properties that finite dimensional spaces do: basically, bounded sequences are weak relatively compact sets. Nonetheless, weak conver gence does not behave as one would desire with respect to nonlinear functionals and operations. This difficulty is what makes nonlinear analysis much harder than would normally be expected. Parametrized measures is a device to under stand weak convergence and its behavior with respect to nonlinear functionals. Under suitable hypotheses, it yields a way of representing through integrals weak limits of compositions with nonlinear functions. It is particularly helpful in comprehending oscillatory phenomena and in keeping track of how oscilla tions change when a nonlinear functional is applied. Weak convergence also plays a fundamental role in the modern treatment of the calculus of variations, again because uniform bounds in norm for se quences allow to have weak convergent subsequences. In order to achieve the existence of minimizers for a particular functional, the property of weak lower semicontinuity should be established first. This is the crucial and most delicate step in the so-called direct method of the calculus of variations. A fairly large amount of work has been devoted to determine under what assumptions we can have this lower semicontinuity with respect to weak topologies for nonlin ear functionals in the form of integrals. The conclusion of all this work is that some type of convexity, understood in a broader sense, is usually involved."

This book is an introduction to level set methods and dynamic implicit surfaces. These are powerful techniques for analyzing and computing moving fronts in a variety of different settings. While it gives many examples of the utility of the methods to a diverse set of applications, it also gives complete numerical analysis and recipes, which will enable users to quickly apply the techniques to real problems. The book begins with a description of implicit surfaces and their basic properties, then devises the level set geometry and calculus toolbox, including the construction of signed distance functions. Part II adds dynamics to this static calculus. Topics include the level set equation itself, Hamilton-Jacobi equations, motion of a surface normal to itself, re-initialization to a signed distance function, extrapolation in the normal direction, the particle level set method and the motion of co-dimension two (and higher) objects. Part III is concerned with topics taken from the fields of Image Processing and Computer Vision. These include the restoration of images degraded by noise and blur, image segmentation with active contours (snakes), and reconstruction of surfaces from unorganized data points. Part IV is dedicated to Computational Physics. It begins with one phase compressible fluid dynamics, then two-phase compressible flow involving possibly different equations of state, detonation and deflagration waves, and solid/fluid structure interaction. Next it discusses incompressible fluid dynamics, including a computer graphics simulation of smoke, free surface flows, including a computer graphics simulation of water, and fully two-phase incompressible flow. Additional related topics include incompressible flames with applications to computer graphics and coupling a compressible and incompressible fluid. Finally, heat flow and Stefan problems are discussed. A student or researcher working in mathematics, computer graphics, science, or engineering interested in any dynamic moving front, which might change its topology or develop singularities, will find this book interesting and useful.

This book is intended as a self-contained exposition of hyperbolic functional dif ferential inequalities and their applications. Its aim is to give a systematic and unified presentation of recent developments of the following problems: (i) functional differential inequalities generated by initial and mixed problems, (ii) existence theory of local and global solutions, (iii) functional integral equations generated by hyperbolic equations, (iv) numerical method of lines for hyperbolic problems, (v) difference methods for initial and initial-boundary value problems. Beside classical solutions, the following classes of weak solutions are treated: Ca ratheodory solutions for quasilinear equations, entropy solutions and viscosity so lutions for nonlinear problems and solutions in the Friedrichs sense for almost linear equations. The theory of difference and differential difference equations ge nerated by original problems is discussed and its applications to the constructions of numerical methods for functional differential problems are presented. The monograph is intended for different groups of scientists. Pure mathemati cians and graduate students will find an advanced theory of functional differential problems. Applied mathematicians and research engineers will find numerical al gorithms for many hyperbolic problems. The classical theory of partial differential inequalities has been described exten sively in the monographs 138, 140, 195, 225). As is well known, they found applica tions in differential problems. The basic examples of such questions are: estimates of solutions of partial equations, estimates of the domain of the existence of solu tions, criteria of uniqueness and estimates of the error of approximate solutions."

This text serves as an introduction to the use of nonlinear symmetries in studying, simplifying and solving nonlinear equations. Part One provides a self-contained introduction to the theory. This emphasizes an intuitive understanding of jet spaces and the geometry of differential equations, and a special treatment of evolution problems and dynamical systems, including original results. In Part Two the theory is applied to equivariant dynamics, to bifurcation theory and to gauge symmetries, reporting recent results by the author. In particular, the fundamental results of equivariant bifurcation theory are extended to the case of nonlinear symmetries. The final part of the book gives an overview of new developments, including a number of applications, mainly in the physical sciences. A list of references dealing with nonlinear symmetries completes the volume. This volume should be of interest to researchers in mathematics and mathematical physics.

This edited book aims at presenting current research activities in the field of robust variable-structure systems. The scope equally comprises highlighting novel methodological aspects as well as presenting the use of variable-structure techniques in industrial applications including their efficient implementation on hardware for real-time control. The target audience primarily comprises research experts in the field of control theory and nonlinear dynamics but the book may also be beneficial for graduate students.

This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.

The book addresses mathematicians and physicists, including graduate students, who are interested in quantum dynamical systems and applications of operator algebras and ergodic theory. It is the only monograph on this topic. Although the authors assume a basic knowledge of operator algebras, they give precise definitions of the notions and in most cases complete proofs of the results which are used.

Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. In fact, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power series diverging everywhere. In this book the author presents the classical theory of meromorphic systems of ODE in the new light shed upon it by the recent achievements in the theory of summability of formal power series.

This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations. The key topics include operators as "sums of squares" of real and complex vector fields, nonlinear evolution equations, local solvability, and hyperbolic questions.