This book shows the importance of studying semilocal convergence in iterative methods through Newton's method and addresses the most important aspects of the Kantorovich's theory including implicated studies. Kantorovich's theory for Newton's method used techniques of functional analysis to prove the semilocal convergence of the method by means of the well-known majorant principle. To gain a deeper understanding of these techniques the authors return to the beginning and present a deep-detailed approach of Kantorovich's theory for Newton's method, where they include old results, for a historical perspective and for comparisons with new results, refine old results, and prove their most relevant results, where alternative approaches leading to new sufficient semilocal convergence criteria for Newton's method are given. The book contains many numerical examples involving nonlinear integral equations, two boundary value problems and systems of nonlinear equations related to numerous physical phenomena. The book is addressed to researchers in computational sciences, in general, and in approximation of solutions of nonlinear problems, in particular.

This book gives a rigorous and practical treatment of integral equations and aims to tackle the solution of integral equations using a blend of abstract structural results and more direct, down-to-earth mathematics. The interplay between these two approaches is a central feature of the text, and it allows a thorough account to be given of many of the types of integral equation that arise, particularly in numerical analysis and fluid mechanics. Because it is not always possible to find explicit solutions to the problems posed, much attention is devoted to obtaining qualitative information and approximations and the associated error estimates.

Ordered vector spaces and cones made their debut in mathematics at the beginning of the twentieth century. They were developed in parallel (but from a different perspective) with functional analysis and operator theory. Before the 1950s, ordered vector spaces appeared in the literature in a fragmented way. Their systematic study began around the world after 1950 mainly through the efforts of the Russian, Japanese, German, and Dutch schools. Since cones are being employed to solve optimization problems, the theory of ordered vector spaces is an indispensable tool for solving a variety of applied problems appearing in several diverse areas, such as engineering, econometrics, and the social sciences. For this reason this theory plays a prominent role not only in functional analysis but also in a wide range of applications. This is a book about a modern perspective on cones and ordered vector spaces. It includes material that has not been presented earlier in a monograph or a textbook. With many exercises of varying degrees of difficulty, the book is suitable for graduate courses. Most of the new topics currently discussed in the book have their origins in problems from economics and finance. Therefore, the book will be valuable to any researcher and graduate student who works in mathematics, engineering, economics, finance, and any other field that uses optimization techniques.

A rigorous introduction to the abstract theory of partial differential equations progresses from the theory of distribution and Sobolev spaces to Fredholm operations, the Schauder fixed point theorem and Bochner integrals.

Professor Arnold is a prolific and versatile mathematician who has done striking work in differential equations and geometrical aspects of analysis. In this volume are collected seven of his survey articles from Russian Mathematical Surveys on singularity theory, the area to which he has made most contribution. These surveys contain Arnold's own analysis and synthesis of a decade's work. All those interested in singularity theory will find this an invaluable compilation. Professor C. T. C. Wall has written an introduction outlining the significance and content of the articles.

This book develops a new theory of multi-parameter singular integrals associated with Carnot-Caratheodory balls. Brian Street first details the classical theory of Calderon-Zygmund singular integrals and applications to linear partial differential equations. He then outlines the theory of multi-parameter Carnot-Caratheodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. Street then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. "Multi-parameter Singular Integrals" will interest graduate students and researchers working in singular integrals and related fields."

1. Historical Remarks Convex Integration theory, ?rst introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods.

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.

Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.

"Ergodic Theory with a view towards Number Theory" will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results, we study the associated Kolmogorov equations, the large-time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton- Jacobi-Bellman equations. In the literature there exists a large number of works (mostly in finite dimen sion) dealing with these arguments in the case of bounded Lipschitz-continuous coefficients and some of them concern the case of coefficients having linear growth. Few papers concern the case of non-Lipschitz coefficients, but they are mainly re lated to the study of the existence and the uniqueness of solutions for the stochastic system. Actually, the study of any further properties of those systems, such as their regularizing properties or their ergodicity, seems not to be developed widely enough. With these notes we try to cover this gap."

This textbook provides a mathematical introduction to linear systems, with a focus on the continuous-time models that arise in engineering applications such as electrical circuits and signal processing. The book introduces linear systems via block diagrams and the theory of the Laplace transform, using basic complex analysis. The book mainly covers linear systems with finite-dimensional state spaces. Graphical methods such as Nyquist plots and Bode plots are presented alongside computational tools such as MATLAB. Multiple-input multiple-output (MIMO) systems, which arise in modern telecommunication devices, are discussed in detail. The book also introduces orthogonal polynomials with important examples in signal processing and wireless communication, such as Telatar's model for multiple antenna transmission. One of the later chapters introduces infinite-dimensional Hilbert space as a state space, with the canonical model of a linear system. The final chapter covers modern applications to signal processing, Whittaker's sampling theorem for band-limited functions, and Shannon's wavelet. Based on courses given for many years to upper undergraduate mathematics students, the book provides a systematic, mathematical account of linear systems theory, and as such will also be useful for students and researchers in engineering. The prerequisites are basic linear algebra and complex analysis.

Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.

The book uses classical problems to motivate a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil and McShane, showing how new theories of integration were developed to solve problems that earlier integration theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and could be used separately in teaching a portion of an introductory real analysis course. There is a sufficient supply of exercises to make this book useful as a textbook.

In diesem Lehrbuch wird die klassische Lebesguesche Mass- und Integrationstheorie stringent entwickelt und dargestellt - trotz grossem Tiefgang ist das Buch dadurch gut lesbar. Die einzelnen Abschnitte werden ausserdem durch zahlreiche Beispiele und Aufgaben illustriert und erganzt. Das Buch ist somit sowohl zum Selbststudium als auch als Nachschlagewerk sehr gut geeignet. Grundkenntnisse aus Mengenlehre und Analysis sowie gelegentlich auch Linearer Algebra und Topologie werden vorausgesetzt. Bei Bedarf koennen diese in den beiden Buchern Grundkonzepte der Mathematik und Analysis einer Veranderlichen der Autoren U. Storch und H. Wiebe nachgelesen werden.

Now in its second edition, this textbook serves as an introduction to probability and statistics for non-mathematics majors who do not need the exhaustive detail and mathematical depth provided in more comprehensive treatments of the subject. The presentation covers the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. More classical examples such as Montmort's problem, the ballot problem, and Bertrand's paradox are now included, along with applications such as the Maxwell-Boltzmann and Bose-Einstein distributions in physics. Key features in new edition: * 35 new exercises * Expanded section on the algebra of sets * Expanded chapters on probabilities to include more classical examples * New section on regression * Online instructors' manual containing solutions to all exercises< Advanced undergraduate and graduate students in computer science, engineering, and other natural and social sciences with only a basic background in calculus will benefit from this introductory text balancing theory with applications. Review of the first edition: This textbook is a classical and well-written introduction to probability theory and statistics. ... the book is written 'for an audience such as computer science students, whose mathematical background is not very strong and who do not need the detail and mathematical depth of similar books written for mathematics or statistics majors.' ... Each new concept is clearly explained and is followed by many detailed examples. ... numerous examples of calculations are given and proofs are well-detailed." (Sophie Lemaire, Mathematical Reviews, Issue 2008 m)

This classroom-tested text is intended for a one-semester course in Lebesgue's theory. With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students. The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis. In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text. The presentation is elementary, where -algebras are not used in the text on measure theory and Dini's derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue's theory are found in the book. http://online.sfsu.edu/sergei/MID.htm

Dieses Buch vermittelt ein solides Grundwissen uber Masstheorie, indem es die wichtigsten Teile derselben in detaillierten, gut nachvollziehbaren Schritten darlegt sowie mit zahlreichen Beispielen verbindet. Viele UEbungsaufgaben unterschiedlicher Schwierigkeitsgrade unterstutzen dabei das Verstandnis des Stoffes. Zur Selbstkontrolle werden im Anhang Loesungen zu samtlichen UEbungsaufgaben angegeben. Anwendungen der Masstheorie in der Stochastik werden in Kapiteln uber bedingte Erwartungen und Likelihood-Funktionen aufgezeigt. Die benoetigten Vorkenntnisse sind auf ein Minimum beschrankt, da zu Beginn in ubersichtlicher Form notwendige Grundlagen aus Mengenlehre und Theorie der reellen Zahlen wiederholt und vertieft werden.

Book 7 in the Princeton Mathematical Series. Originally published in 1961. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Das Buch ist eine kompakte, leicht lesbare Einfuhrung in die Mass- und Integrationstheorie samt Wahrscheinlichkeitstheorie, in der auch auf den fur das Verstandnis wichtigen Bezug zur klassischen Analysis, etwa in Abschnitten uber Funktionen von beschrankter Variation oder dem Hauptsatz der Differential- und Integralrechnung eingegangen wird. Trotz seines verhaltnismassig geringen Umfangs behandelt es alle wesentlichen Themen dieser Fachgebiete, wie Mengensysteme, Mengenfunktionen Massfortsetzung, Unabhangigkeit, Lebesgue-Stieltjes-Masse, Verteilungsfunktionen, messbare Funktionen, Zufallsvariable, Integral, Erwartungswert, Konvergenzsatze, Transformationssatze, Produktraume, Satz von Fubini, Zerlegungssatze, Funktionen von beschrankter Variation, Hauptsatz der Differential- und Integralrechnung, Lp-Raume, Bedingte Erwartungen, Gesetze der grossen Zahlen, Ergodensatze, Martingale, Verteilungskonvergenz, charakteristische Funktionen und die Grenzverteilungssatze von Lindeberg und Feller."

Die Anwendung der Laplace-Transformation in den Naturwissenschaften und der Technik gewinnt standig an Bedeutung. Dies fuhrt zwangslaufig dazu, dass diese Methode in die Stoffplane fur Mathematik der meisten Fachrichtungen an Technischen Hochschulen und Fachhochschulen aufgenommen werden wird. Im Hinblick auf ihre Verwendung in anderen Fachern, erscheint es sinnvoll, mit dem Studium moglichst fruh zu beginnen, spatestens jedoch im dritten Semester. Dies wiederum bedingt, dass nur Kenntnisse vorausgesetzt werden konnen, die im ersten und zweiten Semester vermittelt wurden. Unter diesem Gesichtspunkt ist dieses Arbeits- und ubungsbuch entstanden. Es soll dem Studenten vom dritten Semester aufwarts ermoglichen, so weit in die Theorie und Praxis der Laplace-Transformation vorzudringen, dass er gewohnliche Differentialgleichungen mit konstanten Koeffizienten und Differentialgleichungssysteme, wie sie bei der Behandlung von Schwingungsproblemen auftreten, selbstandig losen kann. Daruberhinaus soll der Stu dent in die Lage versetzt werden, mit fortschreitender Kenntnis in der Mathematik, weiter fuhrende Werke uber die Theorie der Laplace-Transformation zu lesen. Das Buch ist folgendermassen aufgebaut: Im ersten Kapitel werden in zahlreichen Beispielen Funktionen in den Bildraum transfor miert, um den Leser mit dem Umgang mit Laplace -Transformierten vertraut zu machen. Im zweiten Kapitel werden die Eigenschaften der Laplace-Transformation untersucht. Im dritten Kapitel wird die Laplace-Transformation zur Losung von Differentialgleichun gen benutzt. Im vierten Kapitel steht die Anwendung auf technische Probleme im Vordergrund. Alle Beispiele im Text sind ausflihrlich durchgerechnet. Am Schluss jeden Kapitels sind Aufgaben gestellt, deren Losungen im Anhang angegeben werden, so dass der Leser uber prufen kann, ob er den Inhalt des Kapitels verstanden hat."

Angesichts der derzeitigen Situation an der Universitaten, den vielfaltigen Belastungen durch Selbstverwaltungsaufgaben und Lehrveranstaltungen, stellt die Anfertigung einer grosseren Monographie ein Unterfangen dar, das sich kaum noch realisieren lasst. Das gilt um so mehr, wenn es sich wie im vorliegenden Fall um eine sehr komplexe, gleichzeitig eng mit zwei Teildis ziplinen verbundene Thematik handelt und versucht werden soll, neue Per spektiven aufzuzeigen und neue Anstosse zu geben. Mein besonderer Dank gilt deswegen dem Leiter der Abteilung I - In nen- und EG-Politik, Politische Theorie - des Instituts fur Politikwissen schaft der Universitat Tubingen, Herrn Prof. Dr. Rudolf Hrbek, der mich zu dem Vorhaben ermuntert und mir im universitaren Alltagsbetrieb die not wendigen Freiraume fur seine Verwirklichung verschafft hat. Der Arbeitszu sammenhang der Abteilung I hat daruber hinaus aber auch insofern zu der vorliegenden Studie beigetragen, als eine ganze Reihe von in diesem Rahmen entstandenen Arbeiten die empirische Basis fur die nachfolgend prasentierten Uberlegungen wesentlich verbreitern helfen haben. Dies gilt namentlich fur die Magisterarbeiten von Frank und Peter Berg zur Umweltpolitik, Karin Heiniein zur Wahrungspolitik, Christian Roth zur Sozial- und Jurgen Wagner zur Medienpolitik der EU."

Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have for many years proved to be indispensible in the study of partial differential equations and variational problems. Many developments of the basic theory since its inception arise in response to concrete problems, for example, with the (ubiquitous) sets with fractal boundaries.

The theory will probably enjoy substantial further growth, but even now a connected account of the mature parts of it makes a useful addition to the literature. Accordingly, the main themes of this book are Banach spaces and spaces of Sobolev type based on them; integral operators of Hardy type on intervals and on trees; and the distribution of the approximation numbers (singular numbers in the Hilbert space case) of embeddings of Sobolev spaces based on generalised ridged domains.

This timely book will be of interest to all those concerned with the partial differential equations and their ramifications. A prerequisite for reading it is a good graduate course in real analysis.

The first systematic presentation of the theory of dynamical systems under the influence of randomness, this book includes products of random mappings as well as random and stochastic differential equations. The basic multiplicative ergodic theorem is presented, providing a random substitute for linear algebra. On its basis, many applications are detailed. Numerous instructive examples are treated analytically or numerically.

L.Cesari: Appunti sulla teoria delle superficie continue.- C.Y. Pauc: D riv?'s et int grants. Fonctions de cellule.