Bücher der Reihe Advanced Lectures in Mathematics

"A very simple but instructive problem was treated by Jacob Steiner, the famous representative of geometry at the University of Berlin in the early nineteenth century. Three villages A,B ,C are to be joined by a system of roads of minimum length. " Due to this remark of Courant and Robbins (1941), a problem received its name that actually reaches two hundred years further back and should more appropriately be attributed to the French mathematician Pierre Fermat. At the end of his famous treatise "Minima and Maxima" he raised the question to find for three given points in the plane a fourth one in such a way that the sum of its distances to the given points is minimized - that is, to solve the problem mentioned above in its mathematical abstraction. It is known that Evangelista Torricelli had found a geometrical solution for this problem already before 1640. During the last centuries this problem was rediscovered and generalized by many mathematicians, including Jacob Steiner. Nowadays the term "Steiner prob­ lem" refers to a problem where a set of given points PI, . . . ,Pn have to be connected in such a way that (i) any two of the given points are joined and (ii) the total length (measured with respect to some predefined cost function) is minimized.

The aim of these notes is to cover the basic algebraic tools and results behind the scenes in the foundations of Real and Complex Analytic Geometry. The author has learned the subject through the works of many mathematicians, to all of whom he is indebted. However, as the reader will immediately realize, he was specially influenced by the writings of S.S. Abhyankar and J .-C. Tougeron. In any case, the presentation of all topics is always as elementary as it can possibly be, even at the cost of making some arguments longer. The background formally assumed consists of: 1) Polynomials: roots, factorization, discriminant; real roots, Sturm's Theorem, formally real fields; finite field extensions, Primitive Element Theorem. 2) Ideals and modules: prime and maximal ideals; Nakayama's Lemma; localiza­ tion. 3) Integral dependence: finite ring extensions and going-up. 4) Noetherian rings: primary decomposition, associated primes, Krull's Theorem. 5) Krull dimension: chains of prime ideals, systems of parameters; regular systems of parameters, regular rings. These topics are covered in most texts on Algebra and/or Commutative Algebra. Among them we choose here as general reference the following two: ¿ M. Atiyah, I.G. Macdonald: Introduction to Commutative Algebra, 1969, Addison-Wesley: Massachusetts; quoted [A-McD] . ¿ S. Lang: Algebra, 1965, Addison-Wesley: Massachusetts; quoted [L].

This book is a slightly augmented version of a set of lec­ tures on optimization which I held at the University of Got­ tingen in the winter semester 1983/84. The lectures were in­ tended to give an introduction to the foundations and an im­ pression of the applications of optimization theory. Since in­ finite dimensional problems were also to be treated and one could only assume a minimal knowledge of functional analysis, the necessary tools from functional analysis were almost com­ pletely developed during the course of the semester. The most important aspects of the course are the duality theory for convex programming and necessary optimality conditions for nonlinear optimization problems; here we strive to make the geometric background particularly clear. For lack of time and space we were not able to go into several important problems in optimization - e. g. vector optimization, geometric program­ ming and stability theory. I am very grateful to various people for their help in pro­ ducing this text. R. Schaback encouraged me to publish my lec­ tures and put me in touch with the Vieweg-Verlag. W. BrUbach and O. Herbst proofread the manuscript; the latter also pro­ duced the drawings and assembled the index. I am indebted to W. LUck for valuable suggestions for improvement. I am also particularly grateful to R. Switzer, who translated the German text into English. Finally I wish to thank Frau P. Trapp for her Gare and patience in typing the final version.

This text introduces at a moderate speed and in a thorough way the basic concepts of the theory of stochastic integrals and Ito calculus for sem i­ martingales. There are many reasons to study this subject. We are fascinated by the contrast between general measure theoretic arguments and concrete probabilistic problems, and by the own flavour of a new differential calculus. For the beginner, a lot of work is necessary to go through this text in detail. As areward it should enable her or hirn to study more advanced literature and to become at ease with a couple of seemingly frightening concepts. Already in this introduction, many enjoyable and useful facets of stochastic analysis show up. We start out having a glance at several elementary predecessors of the stochastic integral and sketching some ideas behind the abstract theory of semimartingale integration. Having introduced martingales and local martingales in chapters 2 - 4, the stochastic integral is defined for locally uniform limits of elementary processes in chapter S. This corresponds to the Riemann integral in one-dimensional analysis and it suffices for the study of Brownian motion and diffusion processes in the later chapters 9 and 12.

This work illustrates two basic principles in the calculus of variations - the questions of existence of solutions and, closely related, the problem of regularity of minimizers. It concentrates on techniques, and presents methods useful for applications in regularity theorems.

The body of mathematics developed in the last forty years or so which can be put under the heading Singularity Theory is quite large. And the excellent introductions to this vast sub­ ject which are already available (for instance [AGVJ, [BGJ, [GiJ, [GGJ, [LmJ, [Mr], [WsJ or the more advanced [Ln]) cover necessarily only apart of even the most basic topics. The aim of the present book is to introduce the reader to a few important topics from ZoaaZ Singularity Theory. Some of these topics have already been treated in other introductory books (e.g. right and contact finite determinacy of function germs) while others have been considered only in papers (e.g. Mather's Lemma, classification of simple O-dimensional complete intersection singularities, singularities of hyperplane sections and of dual mappings of projective hypersurfaces). Even in the first case, we feel that our treatment is different from the introductions mentioned above - the general reason being that we give special attention to the aompZex anaZytia situation and to the connections with AZgebraia Geometry. We offer now a detailed description of the contents, pOint­ ing out special aspects and new material (i.e. previously un­ published, though for the most part surely known to the~ts!). Chapter 1 is a short introduction for the beginner. We recall here two basic results (the Submersion Theorem and Morse Lemma) and make a few comments on what is meant by the local behaviour of a function or of a plane algebraic curve.

The purpose of coding theory is the design of efficient systems for the transmission of information. The mathematical treatment leads to certain finite structures: the error-correcting codes. Surprisingly problems which are interesting for the design of codes turn out to be closely related to problems studied partly earlier and independently in pure mathematics. In this book, examples of such connections are presented. The relation between lattices studied in number theory and geometry and error-correcting codes is discussed. The book provides at the same time an introduction to the theory of integral lattices and modular forms and to coding theory.In the 3rd edition, again numerous corrections and improvements have been made and the text has been updated.

Diese Einführung in die Welt der Wavelets ist gedacht für Studierende der Mathematik in oberen Semestern, aber auch für mathematisch interessierte Ingenieure. Sie hat zum Ziel, die notwendigen mathematischen Grundlagen und die eigentlichen Wavelet-Konstruktionen sowie die zugehörigen Algorithmen im Zusammenhang darzustellen. Die (für Studierende) abstrakten Inhalte der "höheren Analysis" werden konkret an Beispielen mathematisch durchsichtig gemacht, z.B. an signaltechnische Erfahrungen von Anwendern. Zahlreiche Figuren und durchgerechnete Beispiele bereichern den Band.