Kiechle, H., University of Hamburg, Germany

Theory of K-Loops

2002. X, 186 pp. Softcover
3-540-43262-0

The book contains the first systematic exposition of the current known theory of K-loops, as well as some new material. In particular, big classes of examples are constructed. The theory for sharply 2-transitive groups is generalized to the theory of Frobenius groups with many involutions. A detailed discussion of the relativistic velocity addition based on the author's construction of K-loops from classical groups is also included. The first chapters of the book can be used as a text, the later chapters are research notes, and only partially suitable for the classroom. The style is concise, but complete proofs are given. The prerequisites are a basic knowledge of algebra such as groups, fields, and vector spaces with forms.

Keywords: K-loop, Bol loops, Kikkawa loops, Frobenius groups with many Involutions Mathematics Subject Classification ( 2000 ): 20N05

Contents: Introduction.- Preliminaries.- Left Loops and Transversals.- The Left Inverse Property and Kikkawa Loops.- Isotopy Theory.- Nuclei and the Autotopism Group.- Bol Loops and K-Loops.- Frobenius Ggroups with Mmany Involutions.- Loops with Fibrations.- K-Loops from Classical Groups over Ordered Fields.- Relativistic Velocity Addition.- K-Loops from the General Linear Groups over Rings.- Derivations.

Series: Lecture Notes in Mathematics. VOL. 1778

Osaki, S., Nanzan University, Aichi, Japan (Ed.)

Stochastic Models in Reliability and Maintenance

2002. XII, 338 pp. 46 figs., 11 tabs. Hardcover
3-540-43133-0

This book contains 12 contributions on stochastic models in reliability and maintenance. Written by the leading researchers on each topic, each contribution surveys the current status on stochastic models emphasizing mathematical formulation and optimization applications. Each contribution is self-contained and has a thorough bibliography. The topics include renewal processes, semi-Markov processes, Markovian deterioration models, maintenance and replacement models, software reliability models and Monte-Carlo simulation. This book provides researchers, reliability engineers and graduate students with the current status of the field and future developments of the subject.

Keywords: Stochastic models, Reliability, Maintenance, Replacement models

Bruinier, J.H., University of Heidelberg, Germany

Borcherds Products on O(2,l)
and Chern Classes of Heegner Divisors

2002. VIII, 152 pp. Softcover
3-540-43320-1

Around 1994 R. Borcherds discovered a new type of meromorphic modular form on the orthogonal group $O(2,n)$. These "Borcherds products" have infinite product expansions analogous to the Dedekind eta-function. They arise as multiplicative liftings of elliptic modular forms on $(SL)_2(R)$. The fact that the zeros and poles of Borcherds products are explicitly given in terms of Heegner divisors makes them interesting for geometric and arithmetic applications. In the present text the Borcherds' construction is extended to Maass wave forms and is used to study the Chern classes of Heegner divisors. A converse theorem for the lifting is proved.

Keywords: Automorphic form, orthogonal group, Heegner divisor, Chern class, Weil representation

Contents: Introduction.- Vector valued modular forms for the metaplectic group. The Weil representation. Poincare series and Einstein series. Non-holomorphic Poincare series of negative weight.- The regularized theta lift. Siegel theta functions. The theta integral. Unfolding against F. Unfolding against theta.- The Fourier theta lift. Lorentzian lattices. Lattices of signature (2,l). Modular forms on orthogonal groups. Borcherds products.- Some Riemann geometry on O(2,l). The invariant Laplacian. Reduction theory and L^p-estimates. Modular forms with zeros and poles on Heegner divisors.- Chern classes of Heegner divisors. A lifting into cohomology. Modular forms with zeros and poles on Heegner divisors II.

Series: Lecture Notes in Mathematics. VOL. 1780

Grune, L., Johann-Wolfgang-Goethe Universitat, Frankfurt, Germany

Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization

2002. X, 231 pp. Softcover
3-540-43391-0

This book provides an approach to the study of perturbation and discretization effects on the long-time behavior of dynamical and control systems. It analyzes the impact of time and space discretizations on asymptotically stable attracting sets, attractors, asumptotically controllable sets and their respective domains of attractions and reachable sets. Combining robust stability concepts from nonlinear control theory, techniques from optimal control and differential games and methods from nonsmooth analysis, both qualitative and quantitative results are obtained and new algorithms are developed, analyzed and illustrated by examples.

Keywords: Dynamical Systems, Numerical Systems, Control Systems, Long-Time Behavior, Perturbation and Discretization Mathematics Subject Classification ( 2000 ): 37B25, 93D30, 65L40, 93B05, 93B35, 49L25

Contents: Dynamics, Perturbation and Discretization.- Setup and Preliminaries.- Strongly Attracting Sets.- Weakly Attracting Sets.- Relation between Discretization and Perturbation.- Discretizations of Attractive Sets.- Domains of Attraction.- Appendices. Viscosity Solutions. Comparison Functions. Numerical Examples.

Series: Lecture Notes in Mathematics. VOL. 1783

Arias de Reyna, J., Universidad de Sevilla, Spain

Pointwise convergence of Fourier series

2002. XVIII, 175 pp. Softcover
3-540-43270-1

This book contains a detailed exposition of Carleson-Hunt theorem following the proof of Carleson: to this day this is the only one giving better bounds. It points out the motivation of every step in the proof. Thus the Carleson-Hunt theorem becomes accessible to any analyst.The book also contains the first detailed exposition of the fine results of Hunt, Sjolin, Soria, etc on the convergence of Fourier Series. Its final chapters present original material. With both Fefferman's proof and the recent one of Lacey and Thiele in print, it becomes more important than ever to understand and compare these two related proofs with that of Carleson and Hunt. These alternative proofs do not yield all the results of the Carleson-Hunt proof. The intention of this monograph is to make Carleson's proof accessible to a wider audience, and to explain its consequences for the pointwise convergence of Fourier series for functions in spaces near ${cal L}^1$, filling a well-known gap in the literature.

Keywords: Fourier series, convergence, maximal operator, Carleson theorem Mathematics Subject Classification ( 2000 ): 42A20

Contents: Part I. Fourier series and Hilbert Transform.- Hardy-Littlewood maximal function.- Fourier Series.- Hilbert Transform.- Part II. The Carleson-Hunt Theorem.- The Basic Step.- Maximal inequalities.- Growth of Partial Sums.- Carleson Analysis of the Function.- Allowed pairs.- Pair Interchange Theorems.- All together.- Part III. Consequences.- Some spaces of functions.- The Maximal Operator of Fourier series.

Series: Lecture Notes in Mathematics. VOL. 1785