A collection of articles summarizing the state of knowledge in a large portion of modern homotopy theory. This welcome reference for many new results and recent methods is addressed to all mathematicians interested in homotopy theory and in geometric aspects of group theory.

When the DFG (Deutsche Forschungsgemeinschaft) launched its collabora tive research centre or SFB (Sonderforschungsbereich) 438 "Mathematical Modelling, Simulation, and Verification in Material-Oriented Processes and Intelligent Systems" in July 1997 at the Technische Vniversitat Munchen and at the Vniversitat Augsburg, southern Bavaria got its second nucleus of the still young discipline scientific computing. Whereas the first and older one, FORTWIHR, the Bavarian Consortium for High Performance Scientific Com puting, had put its main emphasis on the supercomputing aspect, this new initiative was now expected to focus on the mathematical part. Consequently, throughout all of the five main research topics (A) adaptive materials and thin layers, (B) adaptive materials in medicine, (C) robotics, aeronautics, and automobile technology, (D) microstructured devices and systems, and (E) transport processes in flows, mathematical aspects play a predominant role. The formation of the SFB 438 and its scientific program are inextricably linked with the name of Karl-Heinz Hoffmann. As full professor for applied mathematics in Augsburg (1981-1991) and in Munchen (since 1992) and as dean of the faculty of mathematics at the TV Munchen, he was the driv ing force of this fascinating, but not always easy-to-realize idea of bringing together scientists from mathematics, physics, engineering, informatics, and medicine for joint efforts in modern applied mathematics. However, scarcely work had begun when the successful captain was called to take command on a bigger boat."

Towards the end of the 1960s, a number of quite different circumstances combined to launch a period of intense activity in the digital processing of electron micro graphs. First, many years of work on correcting the resolution-limiting aberrations of electron microscope objectives had shown that these optical impediments to very high resolution could indeed be overcome, but only at the cost of immense exper imental difficulty; thanks largely to the theoretical work of K. -J. Hanszen and his colleagues and to the experimental work of F. Thon, the notions of transfer func tions were beginning to supplant or complement the concepts of geometrical optics in electron optical thinking; and finally, large fast computers, capable of manipu lating big image matrices in a reasonable time, were widely accessible. Thus the idea that recorded electron microscope images could be improved in some way or rendered more informative by subsequent computer processing gradually gained ground. At first, most effort was concentrated on three-dimensional reconstruction, particu larly of specimens with natural symmetry that could be exploited, and on linear operations on weakly scattering specimens (Chap. l). In 1973, however, R. W. Gerchberg and W. O. Saxton described an iterative algorithm that in principle yielded the phase and amplitude of the electron wave emerging from a strongly scattering speci men."

Domain decomposition is an active, interdisciplinary research area that is devoted to the development, analysis and implementation of coupling and decoupling strategies in mathematics, computational science, engineering and industry. A series of international conferences starting in 1987 set the stage for the presentation of many meanwhile classical results on substructuring, block iterative methods, parallel and distributed high performance computing etc. This volume contains a selection from the papers presented at the 15th International Domain Decomposition Conference held in Berlin, Germany, July 17-25, 2003 by the world's leading experts in the field. Its special focus has been on numerical analysis, computational issues, complex heterogeneous problems, industrial problems, and software development.

The contributions in this book by leading international experts in the field of electromagnetic field computation cover a wide area of contemporary research activities. They clearly underline the important role of modeling, analysis and numerical methods to provide powerful tools for the simulation of electromagnetic phenomena. The main topics range from the mathematical analysis of Maxwell's equations including its proper spatial discretizations (edge elements, boundary element methods, finite integration), and efficient iterative solution techniques (multigrid, domain decomposition) to multiscale aspects in micromagnetics. The reader will get acquainted with many facets of modern computational techniques and its applications to relevant problems in electromagnetism.

A collection of articles summarizing the state of knowledge in a large portion of modern homotopy theory. This welcome reference for many new results and recent methods is addressed to all mathematicians interested in homotopy theory and in geometric aspects of group theory.

Generally it is not sufficiently appreciated that electron microscopy is in fact a diffraction method. In essential aspects electron microscopes are more closely related to X-ray diffracto- meters than to light microscopes. In electron microscopes monochromatized radiation and coherent illumination (never used in light microscopy) correspond in X-ray diffractometers to the primary beam with a small divergence. Imaging ina general sense can take place in interference experiments between a primary beam and a scattered beam, or between diffe- rent deflected scattered beams. This leads to the realization of an old dream in diffracto- metry, namely to a general experimental solution of the "phase problem". The most im- pressive analogy, however, concerns the potential of the electron microscope as a tool for structure determination (where the radiation wavelenght is smaller than the atomic distan- ces). It was therefore considered timely to treat this topic in this series. It was a fortunate cioncidence that in 1976 a Workshop on "Unconventional Electron Microscope Methods for the Investigation of Molecular Structures" (sponsored by the European Molecular Biology Organisation, the Deutsche Forschungsgemeinschaft and the Max-Planck-Gesell- schaft) took place, and that most speakers presenting introductory lectures agreed to publish their contributions in an expanded version in this volume. This volume is thus not a symposium report in the usual sense since it contains the majority of these introductory lectures only.

Es ist meistens ein zufalliger AnlaB, der zu dem Entstehen eines Werkes dieser Art ftihrt: Vor mehreren lahren hatten einige Kollegen an der Technischen Uni- versiHit Mtinchen eine Ringvorlesung tiber Biophysik durchgeftihrt, in welcher nicht nur Physiker sondern auch Chemiker, Biochemiker und Biologen zu W orte kamen, mit dem Wunsch, die physikalisch orientierten Prinzipien ihrer Disziplinen darzustellen. Aus dieser V orlesung ist dieses Buch hervorgegangen - allerdings in nicht unbetrachtlich erweiterter Form und mit z. T. neuen Autoren. Was ist Biophysik? Wie immer bei Grenzgebieten Wit es schwer, prazise Defini- tionen zu finden. Es ist ferner unmoglich, Biophysik zu betreiben, wenn man nicht gewisse Grundkenntnisse der Biologie, der Physik, der physikalischen Chemie, der Chemie und der Biochemie besitzt. Ftir den Entwurf eines biophysikalischen Lehr- buches ergibt sich damit sofort die Frage, ob man den Studenten auf die Literatur dieser N achbargebiete verweisen solI, wobei ihm dann die Auswahl des notwendigen Wissensstoffes tiberlassen ware. Wir waren der Meinung, daB es ntitzlicher und zeitsparender ist, wenn er den ausgewahlten "Zusatzwissensstoff' in konzentrierter Form im Rahmen der Biophysik-Vorlesung geboten bekommt. Auch in diesem Buch wird man daher Beitrage tiber die Struktur und Funktion der Zelle, tiber den chemischen Bau von biogenen Makromolekiilen, aber auch tiber theoretische Chemie usw. finden. Wiederholen wir die Frage, was ist Biophysik? MtiB man Physiologie, Elektro- medizin, Strahlenmedizin usw. auch hinzurechnen? Das Bild der Biophysik hat sich in den letzten lahren sehr gewandelt.

Mit Beitragen zahlreicher Fachwissenschaftler

Dieses Numerik-Lehrbuch hat sich seit seinem Erscheinen zu einem Standardwerk der Numerischen Mathematik entwickelt und wird in zahlreichen Lehrveranstaltungen zur Einfuhrung in die Numerik als Begleittext verwendet. Der Erfolg dieses Lehrbuchs liegt in der Verbindung analytischer Strenge in der Prasentation der grundlegenden Prinzipien der Numerischen Mathematik und praktischer Anwendung durch Bereitstellung und Diskussion fundamentaler algorithmischer Werkzeuge.

Die in den vergangenen Jahren in den Bereichen der Numerik und des Wissenschaftlichen Rechnens erfolgte Entwicklung neuer Methodologien und daraus resultierender numerischer Verfahren erfordert eine adaquate Anpassung der Darstellung der Grundlagen, die Aufnahme neuer algorithmischer Techniken sowie eine kritische Beurteilung existenter Methoden. Dies ist durch die vorliegende Neubearbeitung dieses Lehrbuchs geschehen."