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.

From Measures to Ito Integrals gives a clear account of measure theory, leading via L2-theory to Brownian motion, Ito integrals and a brief look at martingale calculus. Modern probability theory and the applications of stochastic processes rely heavily on an understanding of basic measure theory. This text is ideal preparation for graduate-level courses in mathematical finance and perfect for any reader seeking a basic understanding of the mathematics underpinning the various applications of Ito calculus.

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 2nd edition of LNM 523 is based on the two first authors' mathematical approach of this theory presented in its 1st edition in 1976. An entire new chapter on the current forefront of research has been added. Except for this new chapter and the correction of a few misprints, the basic material and presentation of the first edition has been maintained. At the end of each chapter the reader will also find notes with further bibliographical information.

This book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods. The problems addressed are challenging and complex, and often require the use of several refined techniques to overcome the major difficulties encountered. The lectures, given during the course "Partial Differential Equations and Geometric Measure Theory'' in Cetraro, June 2-7, 2014, should help to encourage further research in the area. The enthusiasm of the speakers and the participants of this CIME course is reflected in the text.

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 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."

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.

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.

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.

Jurgen Beetz fuhrt zuerst in den Ursprung der erdachten Geschichten der Mathematik aus der Steinzeit ein. Im Anschluss daran stellt er die zentrale Fragestellung der "Infinitesimalrechnung" anhand eines einfachen Beispiels dar. Dann erlautert der Autor die Grundproblematik des Integrierens: die Flache zwischen einer beliebigen Stelle einer Funktion y=f(x) und der x-Achse festzustellen. Es gibt einige bekannte Satze, z. B. den Mittelwertsatz und den Hauptsatz der Integralrechnung. Hauptanwendungsgebiet der Integralrechnung ist das Loesen von Differentialgleichungen: Gleichungen, die Funktionen und ihre Ableitungen nebeneinander enthalten.

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.

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.

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

W.K. ALLARD: On the first variation of area and generalized mean curvature.- F.J. ALMGREN Jr.: Geometric measure theory and elliptic variational problems.- E. GIUSTI: Minimal surfaces with obstacles.- J. GUCKENHEIMER: Singularities in soap-bubble-like and soap-film-like surfaces.- D. KINDERLEHRER: The analyticity of the coincidence set in variational inequalities.- M. MIRANDA: Boundaries of Caciopoli sets in the calculus of variations.- L. PICCININI: De Giorgi 's measure and thin obstacles.

In May 2006, The University of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The goal of this minicourse was to introduce graduate students and recent Ph.D.s to various modern topics in stochastic PDEs, and to bring together several experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic partial differential equations. This monograph contains an up-to-date compilation of many of those lectures. Particular emphasis is paid to showcasing central ideas and displaying some of the many deep connections between the mentioned disciplines, all the time keeping a realistic pace for the student of the subject.

This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Ito's formula, the optional stopping theorem and Girsanov's theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Ito, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

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.

Les deux premiers volumes sont consacrA(c)s aux fonctions dans R ou C, y compris la thA(c)orie A(c)lA(c)mentaire des sA(c)ries et intA(c)grales de Fourier et une partie de celle des fonctions holomorphes. L'exposA(c) non strictement linA(c)aire, combine indications historiques et raisonnements rigoureux. Il montre la diversitA(c) des voies d'accA]s aux principaux rA(c)sultats afin de familiariser le lecteur avec les mA(c)thodes de raisonnement et idA(c)es fondamentales plutAt qu'avec les techniques de calcul, point de vue utile aussi aux personnes travaillant seules.
Les volumes 3 et 4 traitent principalement des fonctions analytiques (thA(c)orie de Cauchy, thA(c)orie analytique des nombres et fonctions modulaires), ainsi que du calcul diffA(c)rentiel sur les variA(c)tA(c)s, avec un exposA(c) de l'intA(c)grale de Lebesgue, en suivant d'assez prA]s le cA(c)lA]bre cours donnA(c) longtemps par l'auteur A l'UniversitA(c) Paris 7.
On reconnaA(R)tra dans ce nouvel ouvrage le style inimitable de l'auteur, et pas seulement par son refus de l'A(c)criture condensA(c)e en usage dans ce nombreux manuels.