Bøger i Lectures in Mathematics. Eth Zurich serien

Eigenschaften von Singularitäten . . . . . . . . . . . . . . . . . . 49 4. 1 Der Umgebungsrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4. 2 Gute Repräsentanten von Abbildungskeimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4. 3 Monodromie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4. 4 Die Monodromie einer quadratischen Singularität (lokaler Fall) . . . . . . . . . . 65 5 Die U ntersuchung von Milnorfasern . . . . . . . . . . . . . . . . . ¿. . . ¿. . ¿¿. ¿ ¿. . . . . . . 73 5. 1 Milnorfasem von ebenen Kurvensingularitäten . . . . . . ¿. . ¿. . . ¿. . ¿. . . . . . . . . 73 5. 2 Milnorfasem von Hyperfiä. chensingularitäten . . . . . . . . . . . . . . ¿. . ¿. . . . . . . . . . 81 6 Die Beec r hnung d er M o n oro d mie . . . . . . . . . . . . . . . . . . . ¿. . . . . . ¿. . ¿ . . . . . . . . . 87 6. 1 Die Morsifikation . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¿. . . . . . . 87 6. 2 Die Monodromie der ebenen Kurvensingularitäten in Cl. . . . . . . . . . 88 6. 3 Dynkin-Dia. gramm und Monodromiegruppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6. 4 Die Monodromie beim Addieren von FUnktionskeimen . . . . . . . . . . . . .

The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion.

This book introduces several remarkable new probabilistic objects that combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial differential equations (PDE).

The basic idea is to isolate a property that on one hand can be formulated solely in terms of the distance function and on the other hand is characteristic of nonpositive sectional curvature on a Riemannian manifold, and then to take this property as an axiom for defining a metric space of nonposi tive curvature.

Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.

The Baum-Connes conjecture is part of A Connes' non-commutative geometry programme. This book presents an introduction to the Baum-Connes conjecture. It starts by defining the objects in both sides of the conjecture, then the assembly map which connects them. It illustrates the main tool to attack the conjecture (Kasparov's theory).

This book studies time-dependent partial differential equations and their numerical solution, developing the analytic and the numerical theory in parallel, and placing special emphasis on the discretization of boundary conditions.

one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field".

Der Begriff der Splinefunktionen wurde von I. J. Schoenberg 1946 eingefUhrt. "Spline" ist der Name eines Zeichengerates, welches auf mechanischem Weg Interpolatio- aufgaben lost. Dieses Gerat besteht aus einer flexiblen, oft mehrere Meter langen Latte, die auf dem Zeichenbrett aufliegt und dort an bestimmten Stellen durch Gewichte festgehalten wird. Die Form, die die Latte annimmt, hangt von den Elastizitatseigenschaften der Latte abo -, " , , , , , , \ , \ \ I , , , ," -"', , , , , J::>----" , I I , I I I , , , , , , , , , , , " ) Fig. 1: Latteninterpo1ation Po1ynominterpo1ation _ - - - - - - - --0 Wir konnen natUrlich versuchen, ein mathematisches Modell fUr dieses mechanische Zeichengerat zu machen, d. h. die Gestalt solcher Kurven mathematisch zu erfassen. - 2 - Die Theorie der Balkenbiegung verlangt, dass die mittlere 2 K quadratische KrUmmung, ("strain energy", Spannungs- J energie) minimiert wird. Lasst sich die Kurve als Graph einer Funktion f auf dem Intervall [a,b] schreiben, so erhalt man mit dem bekannten Ausdruck fUr die Krlimmung K [f" (t) P --------------dt ~ min (1) t [1 +f' (t)2J5/2 a Statt dieses schwierige Extremalproblem zu losen, begnUgt man sich damit, (2) zu minimieren. Die Extremalfunktion fUr das Funktional (2) ist stUckweise ein kubisches Polynom; die Polyn- stUcke gehen an den Bruchstellen so glatt ineinander Uber, dass die Funktion zweimal stetig differenzierbar ist.

These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the following spring.

The subject of these notes is counting and related topics, viewed from a computational perspective. These notes will be of value not only to teachers of postgraduate courses on these topics, but also to established researchers.

This volume presents new results in probability theory and partial differential equations related to asymptotic

On the other hand, Mather started with a specific class of area-preserving annulus mappings, the so-called monotone twist maps. These maps appear in mechanics as Poincare maps. In 1982, Mather succeeded to make essential progress in this field and to prove the existence of a class of closed invariant subsets which are now called Mather sets.

Assume that after preconditioning we are given a fixed point problem x = Lx + f (*) where L is a bounded linear operator which is not assumed to be symmetric and f is a given vector.

The group of Hamiltonian diffeomorphisms Ham(M, 0) of a symplectic mani fold (M, 0) plays a fundamental role both in geometry and classical mechanics. They turn out to be very different from the usual circle of problems considered in symplectic topology and thus extend significantly our vision of the symplectic world.

The aim of these lecture notes is to give an essentially self-contained introduction to the basic regularity theory for energy minimizing maps, including recent developments concerning the structure of the singular set and asymptotics on approach to the singular set.

The notes in this volume were written as a part of a Nachdiplom course that I gave at the ETH in the summer semester of 1995. Such a course began with the fundamentals of group cohomology, and then investigated the structure of cohomology rings, and their maximal ideal spectra.

Combinatorial group theory is a loosely defined subject, with close connections to topology and logic.

This book discloses a fascinating connection between optimal stopping problems in probability and free-boundary problems. Areas of application include financial mathematics, financial engineering, and mathematical statistics.

One question treated at length concerns the low temperature behavior of short-range spin glasses: whether and in what sense Parisi's analysis of the meanfield (or "infinite-range") model is relevant.

Offers an introduction to combinatorial torsions of cellular spaces and manifolds with emphasis on torsions of 3-dimensional manifolds. This book describes the results of G Meng, C H Taubes and the author on the connections between the refined torsions and the Seiberg-Witten invariant of 3-manifolds.

Although one might argue whether this golden age is really foregone, and discuss the "height" of the technology involved, this quotation is closely related to the main motivations of Part II: this technology, which includes stochastic calculus for general discontinuous semi-martingales, enlargement of filtrations, .

For one, The uniformization theory of canonical Kahler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical Kahler metrics on complex manifolds.

This title provides and introduction to harmonic maps between a surface and a symmetric manifold and constant mean curvature surfaces as completely integrable systems. It should help the reader to access the ideas of the theory and to aquire a unified perspective of the subject.

First, it was clear that the classical approach, using the theory of extremal quasi-conformal mappings (in this approach we completely avoid the use of quasi-conformal maps) was not easily applicable to the theory of minimal surfaces, a field of interest of the author over many years.

These notes form the contents of a Nachdiplomvorlesung given at the Forschungs­ institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. K. Chandrasekharan and Prof. Jurgen Moser have encouraged me to write them up for inclusion in the series, published by Birkhiiuser, of notes of these courses at the ETH. Dr. Albert Stadler produced detailed notes of the first part of this course, and very intelligible class-room notes of the rest. Without this work of Dr. Stadler, these notes would not have been written. While I have changed some things (such as the proof of the Serre duality theorem, here done entirely in the spirit of Serre's original paper), the present notes follow Dr. Stadler's fairly closely. My original aim in giving the course was twofold. I wanted to present the basic theorems about the Jacobian from Riemann's own point of view. Given the Riemann-Roch theorem, if Riemann's methods are expressed in modern language, they differ very little (if at all) from the work of modern authors.

This book examines the well-posedness theory for nonlinear hyperbolic systems of conservation laws, recently completed by the author together with his collaborators. It also introduces the reader to the developing theory of nonclassical (undercompressive) entropy solutions.