Alberich-Carraminana, M., University of Barcelona, Spain

Geometry of the Plane Cremona Maps

2002. XVI, 257 pp. Softcover
3-540-42816-X

This book provides a self-contained exposition of the theory of plane Cremona maps, reviewing the classical theory. The book updates, correctly proves and generalises a number of classical results by allowing any configuration of singularities for the base points of the plane Cremona maps. It also presents some material which has only appeared in research papers and includes new, previously unpublished results. This book will be useful as a reference text for any researcher who is interested in the topic of plane birational maps.

Contents:
1. Preliminaries 1.1 Blowing-ups 1.2 Weighted clusters 1.3 Birational maps of surfaces 2. Plane Cremona maps 2.1 Base points 2.2 Principal curves 2.3 Contractile curves 2.4 Characteristic matrix 2.5 Equations of condition 2.6 Noether's inequality 2.7 Further relations 2.8 Quadratic plane Cremona maps 2.9 Transforming curves 3. Clebsch's theorems and jacobian 3.1 A Clebsch's theorem 3.2 The entries of the characteristic matrix 3.3 On symmetry of chararcteristics 3.4 Further properties 3.5 Jacobian of the homaloidal net 4. Composition 4.1 Composition of two plane Cremona maps 4.2 Consequences 5. Characteristic matrices 5.1 Homaloidal nets 5.2 Homaloidal types 5.3 On proper homaloidal types 5.4 Characteristic matrices 5.5 Exceptional types 5.6 On proper exceptional types 5.7 Weyl groups 6. Total principal and special homaloidal curves 6.1 Virtual versus effective behaviour 6.2 Non-expansive corresponding base points 6.3 Generic versus effective behaviour 6.4 Irreducible homaloidal curves 6.5 Special homaloidal curves 7 Inverse Cremona map 7.1 Non-expected contractile curves 7.2 Proximity among base points of the inverse 7.3 Inverse map and total principle curves 7.4 Consequences 8. Noether's factorization theorem 8.1 Criterion for homaloidal nets 8.2 Complexity and major base points 8.3 Resolution into the Jonquieres maps 8.4 Resolution into quadratic maps 8.5 Resolution into ordinary quadratic maps

Series: Lecture Notes in Mathematics. VOL. 1769

Gluesing-Luerssen, H., University of Oldenburg, Germany

Linear Delay-Differential Systems with Commensurate Delays:
An Algebraic Approach

2002. VII, 176 pp. Softcover
3-540-42821-6

The book deals with linear time-invariant delay-differential equations with commensurated point delays in a control-theoretic context. The aim is to show that with a suitable algebraic setting a behavioral theory for dynamical systems described by such equations can be developed. The central object is an operator algebra which turns out to be an elementary divisor domain and thus provides the main tool for investigating the corresponding matrix equations. The book also reports the results obtained so far for delay-differential systems with noncommensurate delays. Moreover, whenever possible it points out similarities and differences to the behavioral theory of multidimensional systems, which is based on a great deal of algebraic structure itself. The presentation is introductory and self-contained. It should also be accessible to readers with no background in delay-differential equations or behavioral systems theory. The text should interest researchers and graduate students.

Contents:
Introduction.- The Algebraic Framework.- The Algebraic Structure of H_0. Divisibility Properties. Matrices over H_0. Systems over Rings: A Brief Survery. The Nonfinitely Generated Ideals of H_0. The Ring H as a Convolution Algebra. Computing the Bezout Identity.- Behaviors of Delay-Differential Systems. The Lattice of Behaviors. Input/Output Systems. Transfer Classes and Controllable Systems. Subbehaviors and Interconnections. Assigning the Characteristic Function. Biduals of Nonfinitely Generated Ideals.- First-Order Representations. Multi-Operator Systems. The Realization Procedure of Fuhrmann. First-Order Realizations. Some Minimality Issues.

Series: Lecture Notes in Mathematics. VOL. 1770

Bourbaki, N.

Elements of Mathematics. Lie Groups and Lie Algebras
Chapters 4-6

2002. XII, 300 pp. 43 figs. Hardcover
3-540-42650-7
Book category: Monograph
Publication language: English

>From the reviews of the French edition: "This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a (B,N)-pair structure, or "Tits systems". A historical note provides a survey of the contexts in which groups generated by reflections have arisen. A brief introduction includes almost the only other mention of Lie groups and algebras to be found in the volume. Thus the presentation here is really quite independent of Lie theory. The choice of such an approach makes for an elegant, self-contained treatment of some highly interesting mathematics, which can be read with profit and with relative ease by a very wide circle of readers (and with delight by many, if the reviewer is at all representative)."
(G.B. Seligman in MathReviews)

Contents: Ch. IV. Coxeter Groups and Tits Systems: Coxeter Groups. Tits Systems.- Ch. V. Groups Generated by Reflections: Hyperplanes, Chambers and Facets. Reflections. Groups of Displacements Generated by Reflections. The Geometric Representation of a Coxeter Group. Invariants in the Symmetric Algebra. The Coxeter Transformation.- Ch. VI. Root Systems: Root Systems. Affine Weyl Group. Exponential Invariants. Classification of Root Systems.- Historical Note.

Kac, V., MIT, Cambridge, MA, USA Cheung, P., MIT, Cambridge, MA, USA

Quantum Calculus

2002. Approx. 130 pp. 2 figs. Softcover
0-387-95341-8

Simply put, quantum calculus is ordinary calculus without taking limits. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. As this book develops quantum calculus along the lines of traditional calculus, the reader discovers, with a remarkable inevitability, many important notions and results of classical mathematics.
This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. It is based on lectures and seminars given by Professor Kac over the last few years at MIT.
Victor Kac is Professor of Mathematics at MIT. He is an author of 4 books and over a hundred research papers. He was awarded the Wigner Medal for his work on Kac-Moody algebras that has numerous applications to mathematics and theoretical physics. He is a honorary member of the Moscow Mathematical Society.
Pokman Cheung graduated from MIT in 2001 after three years of undergraduate studies. He is presently a graduate student at Stanford University.

Keywords: Quantum Calculus, Quantum Groups

Contents: From the contents: q-derivative and h-derivative.- Generalized Taylor's formula for polynomials.- q-analogue of (x-a)n, n an integer, and q-derivatives of binomials.- q-Taylor's formula for polynomials.- Gauss' binomial formula and a non-commutative binomial formula.- Properties of q-binomial coefficients.- q-binomial coefficients and linear algebra over finite fields.- q-Taylor's formula for formal power series and Heine's binomial formula.- Two Euler identities and two q-exponential functions.- q-trigonometric functions.- Jacobi's triple product identity.- Classical partition function and Euler's product formula.- q-hypergeometric functions and Heine's formula.

Series: Universitext.

Martinez, A., Universita di Bologna, Italy

An Introduction to Semiclassical and Microlocal Analysis

2002. Approx. 200 pp. Hardcover
0-387-95344-2

This book presents most of the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics. Both the standard C? pseudodifferential calculus and the analytic microlocal analysis are developed, in a context which remains intentionally global so that only the relevant difficulties of the theory are encountered. The originality lies in the fact that the main features of analytic microlocal analysis are derived from a single and elementary a priori estimate.
Various exercises illustrate the chief results of each chapter while introducing the reader to further developments of the theory. Applications to the study of the Schrodinger operator are also discussed, to further the understanding of new notions or general results by replacing them in the context of quantum mechanics. This book is aimed at non-specialists of the subject and the only required prerequisite is a basic knowledge of the theory of distributions.
Andre Martinez is currently Professor of Mathematics at the University of Bologna, Italy, after having moved from France where he was Professor at Paris-Nord University. He has published many research articles in semiclassical quantum mechanics, in particular related to the Born-Oppenheimer approximation, phase-space tunneling, scattering theory and resonances.

Contents: Introduction.- Semiclassical Pseudodifferential Calculus.- Microlocalization.- Applications to the Solutions of Analytic Linear PDEs.- Complements: Symplectic Aspects.- Appendix: List of Formulae.- Bibliography.- Index.- List of Notations.

Series: Universitext.