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The seminar on Stochastic Analysis and Mathematical Physics of the Ca­ tholic University of Chile, started in Santiago in 1984, has being followed and enlarged since 1995 by a series of international workshops aimed at pro­ moting a wide-spectrum dialogue between experts on the fields of classical and quantum stochastic analysis, mathematical physics, and physics. This volume collects most of the contributions to the Fourth Interna­ tional Workshop on Stochastic Analysis and Mathematical Physics (whose Spanish abbreviation is "ANESTOC"; in English, "STAMP"), held in San­ tiago, Chile, from January 5 to 11, 2000. The workshop style stimulated a vivid exchange of ideas which finally led to a number of written con­ tributions which I am glad to introduce here. However, we are currently submitted to a sort of invasion of proceedings books, and we do not want to increase our own shelves with a new one of the like. On the other hand, the editors of conference proceedings have to use different exhausting and com­ pulsive strategies to persuade authors to write and provide texts in time, a task which terrifies us. As a result, this volume is aimed at smoothly start­ ing a new kind of publication. What we would like to have is a collection of books organized like our seminar.

Over the last years, stochastic analysis has had an enormous progress with the impetus originating from different branches of mathematics: PDE's and the Malliavin calculus, quantum physics, path space analysis on curved manifolds via probabilistic methods, and more. This volume contains selected contributions which were presented at the 8th Silivri Workshop on Stochastic Analysis and Related Topics, held in September 2000 in Gazimagusa, North Cyprus. The topics include stochastic control theory, generalized functions in a nonlinear setting, tangent spaces of manifold-valued paths with quasi-invariant measures, and applications in game theory, theoretical biology and theoretical physics. Contributors: A.E. Bashirov, A. Bensoussan and J. Frehse, U. Capar and H. Aktuglul, A.B. Cruzeiro and Kai-Nan Xiang, E. Hausenblas, Y. Ishikawa, N. Mahmudov, P. Malliavin and U. Taneri, N. Privault, A.S. Üstünel

This volume contains the proceedings of the International Workshop on Operator Theory and Applications held at the University of Algarve in Faro, Portugal, September 12-15, in the year 2000. The main topics of the conference were !> Factorization Theory; !> Operator Theoretical Methods in Diffraction Theory;

This book contains the proceedings of the conference "Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics" that was held in the second week of June 2001 at Graz University of Technology, in the capital of Styria, southeastern province of Austria.

This volume comprises the specially prepared lecture notes of a a Summer School on "Factorization and Integrable Systems" held in September 2000 at the University of Algarve in Portugal. The main aim of the school was to review the modern factorization theory and its application to classical and quantum integrable systems.

Consisting of 16 refereed original contributions, this volume presents a diversified collection of recent results in control of distributed parameter systems. Topics addressed include - optimal control in fluid mechanics - numerical methods for optimal control of partial differential equations - modeling and control of shells - level set methods - mesh adaptation for parameter estimation problems - shape optimization Advanced graduate students and researchers will find the book an excellent guide to the forefront of control and estimation of distributed parameter systems.

From the reviews: "This is an excellent exposition about abelian Reidemeister torsions for three-manifolds." -Zentralblatt Math"This monograph contains a wealth of information many topologists will find very handy.

Euklid, der Geometer aus Alexandria, und sein uberwiegend geometrisches literarisches Werk bilden den Mittelpunkt dieses Buches zur Geschichte der Mathematik. Es ist bis heute nicht mit letzter Sicherheit geklart, ob es einen Mathematiker mit dem Namen Euklid wirklich gegeben, wann er gelebt hat und ob die ihm zugeschriebenen Lehrbucher zur Mathematik und zur mathematischen Physik tatsachlich nur von ihm verfasst wurden. Trotz dieser Einschrankungen breitet der Autor das derzeit verfugbare Wissen uber die voreuklidische griechische Mathematik, die Traditionslinien der euklidischen Geometrie und die uber Jahrhunderte andauernde Wirkungsgeschichte der euklidischen Werke umfassend aus. Dabei wird an vielen Stellen deutlich, wie unzureichend unsere Kenntnisse uber die Rezeptionsgeschichte und wie luckenhaft das antike Quellenmaterial auch nach zweitausend Jahren immer noch sind. Das Buch spurt den vielfaltigen historischen und interkulturellen Aspekten und Facetten der Mathematik nach.

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet.

Once again, the proceedings of the Badgastein Symposium, this one being the 22nd in an uninterrupted series, offer a comprehensive record of the scientific events. Highlights at this meeting were certainly reports ofthe exciting new developments ofpositron imaging with Anger cameras, both with conventional high-energy collimators and with coincidence techniques. Preliminary c1inical results are promising and seem to offer a true alternative to sophisticated PET -technology with its for many laboratories prohibitive costs. The general nature ofthe symposium is documented by the large range oftopics covered so that all major topics of interest in both the basic sciences of nuc1ear medicine and c1inical ap­ plication are found. In accordance with the general trends in nuc1ear medicine, there has been a substantial increase in oncological work compared to the more traditional topics. This is also reflected in the Badgastein lecture, given by Professor P. Rigo, who presented an excellent survey ofnuc1ear oncology.

The International Meetings on Approximation Theory attempt to keep track in particular of fun damental advances in the theory of function approximation, for example by (or thogonal) polynomials, (weighted) interpolation, multivariate quasi-interpolation, splines, radial basis functions and several others.

The seminar Symplectic Geometry at the University of Berne in summer 1992 showed that the topic of this book is a very active field, where many different branches of mathematics come tog9ther: differential geometry, topology, partial differential equations, variational calculus, and complex analysis.

The present lectures are based on a course deli vered by the authors at the Uni versi ty of Bucharest, in the winter semester 1985-1986. Without aiming at completeness, the topics selected cover all the major questions concerning hyponormal operators. Our main purpose is to provide the reader with a straightforward access to an active field of research which is strongly related to the spectral and perturbation theories of Hilbert space operators, singular integral equations and scattering theory. We have in view an audience composed especially of experts in operator theory or integral equations, mathematical physicists and graduate students. The book is intended as a reference for the basic results on hyponormal operators, but has the structure of a textbook. Parts of it can also be used as a second year graduate course. As prerequisites the reader is supposed to be acquainted with the basic principles of functional analysis and operator theory as covered for instance by Reed and Simon [1]. A t several stages of preparation of the manuscript we were pleased to benefit from proper comments made by our cOlleagues: Grigore Arsene, Tiberiu Constantinescu, Raul Curto, Jan Janas, Bebe Prunaru, Florin Radulescu, Khrysztof Rudol, Konrad Schmudgen, Florian-Horia Vasilescu. We warmly thank them all. We are indebted to Professor Israel Gohberg, the editor of this series, for his constant encouragement and his valuable mathematical advice. We wish to thank Mr. Benno Zimmermann, the Mathematics Editor at Birkhauser Verlag, for cooperation and assistance during the preparation of the manuscript.

den; daB er nieht allzu unglueklieh gewahlt ist, solI der Anhang darlegen. Dort wird gezeigt, daB dieser Begriff sehr veralIgemeinerungsfahig ist. DaB aueh heute noeh beaehtenswerte Weehselbeziehungen zwischen der darstellenden Geometrie und neueren Zweigen der geometrisehen Wissen­ sehaften bestehen, solI ebenfalIs der Anhang auseinandersetzen. Mit Hilfe von Begriffsbildungen der modernen Theorie der Gewebe werden namlich die ge­ laufigen Methoden der darstellenden Geometrie in neuem Licht gezeigt und stark veralIgemeinert. Von den Anwendungen habe ieh diejenigen Teile etwas ausfUhrlieher be­ handeIt, welche ungefahr den Stoff einer Spezialvorlesung ausmaehen, die ieh jeweils im Sommer an der ETH. haIten muB, namlieh die Grundlagen der Photogrammetrie und die sphiirische Geometrie. Umfangreiehere und selbstan­ dige Anwendungsgebiete wie die N omographie oder kinematische Geometrie sol­ len vielIeieht spater einmal dargestellt werden. Der Faehmann wird vielleieht einige ihm liebgewordene Einzelheiten ver­ missen. Er moge als Entsehuldigung annehmen, daB ieh einerseits naeh reif­ lieher Oberlegung nur die Teile weggelassen habe, die mir dem heutigen Stand der mathematisehen und teehnisehen Wissensehaften nieht mehr angemessen sehienen, und daB ieh andererseits den Preis des Buehes mogliehst niedrig haIten wollte. Was endlieh die Figuren anbetrifft, so habe ieh mieh bemuht, einfaeh zu bleiben und mit wenigen Linien auszukommen. Es ist namlieh fUr den Leser ein reeht unangenehmes Gesehaft, eine komplizierte Figur ent­ wirren zu mussen, bei deren Entstehung er nieht zugegen war und die nur den Zeichner freut. Den Herren Prof. Dr. C. BURRI von der ETH. und O. SCHLAPFER von der Oberrealsehule Zurich verdanke ieh einige Anregungen und Verbesserungen.

Chapter IV contains papers on harmonic analysis in connection with approximation and, finally, Chapter V is devoted to approximation by splines, algebraic polynomials, rational functions, and to Pade approximation.

The Indian National Science Academy on the occasion ofthe Golden Jubilee Celebration (Fifty years of India's Independence) decided to publish a number of monographs on the selected fields. In reponse to this assignment, we invited several eminent Number Theorists to contribute expository/research articles for this monograph on Number Theory.

The present volume contains manuscripts of lectures or topics related to the lectures which were given at the conference on "Inverse Problems" at the mathematical Research Institute at Oberwolfach.

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets.The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations.- - -The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)

Emil Artin and Helmut Hasse

Contains refereed research or review articles presented at the 7th Seminar on Stochastic Analysis, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verita) in Ascona , Switzerland, in May 2011.

Harmonic maps between Riemannian manifolds were first established by James Eells and Joseph H. Sampson in 1964. Wave maps are harmonic maps on Minkowski spaces and have been studied since the 1990s. Yang-Mills fields, the critical points of Yang-Mills functionals of connections whose curvature tensors are harmonic, were explored by a few physicists in the 1950s, and biharmonic maps (generalizing harmonic maps) were introduced by Guoying Jiang in 1986. The book presents an overview of the important developments made in these fields since they first came up. Furthermore, it introduces biwave maps (generalizing wave maps) which were first studied by the author in 2009, and bi-Yang-Mills fields (generalizing Yang-Mills fields) first investigated by Toshiyuki Ichiyama, Jun-Ichi Inoguchi and Hajime Urakawa in 2008. Other topics discussed are exponential harmonic maps, exponential wave maps and exponential Yang-Mills fields.

George Britton, Synnl/Jve Liaaen-Jensen and Hanspeter Pfander This book, Volume 2 in the series Carotenoids, is the first book to be published that is devoted entirely to the total synthesis of carotenoids, but it is timely in view of the rapid development and the growing diversification of the carotenoid field.

This book deals with the modeling, analysis and simulation of problems arising in the life sciences, and especially in biological processes. The models and findings presented result from intensive discussions with microbiologists, doctors and medical staff, physicists, chemists and industrial engineers and are based on experimental data. They lead to a new class of degenerate density-dependent nonlinear reaction-diffusion convective equations that simultaneously comprise two kinds of degeneracy: porous-medium and fast-diffusion type degeneracy. To date, this class is still not clearly understood in the mathematical literature and thus especially interesting. The author both derives realistic life science models and their above-mentioned governing equations of the degenerate types and systematically studies these classes of equations. In each concrete case well-posedness, the dependence of solutions on boundary conditions reflecting some properties of the environment, and the large-time behavior of solutions are investigated and in some instances also studied numerically.

Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering.

We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.

We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.