Series: Australian Mathematical Society Lecture Series (No. 20)
Paperback (ISBN-13: 9780521749893)
A complex reflection is a linear transformation which fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or arrangement of mirrors. This book gives a complete classification of all groups of transformations of n-dimensional complex space which are generated by complex reflections, using the method of line systems. In particular: irreducible groups are studied in detail, and are identified with finite linear groups; reflection subgroups of reflection groups are completely classified; the theory of eigenspaces of elements of reflection groups is discussed fully; an appendix outlines links to representation theory, topology and mathematical physics. Containing over 100 exercises ranging in difficulty from elementary to research level, this book is ideal for honours and graduate students, or for researchers in algebra, topology and mathematical physics.
* Contains the only complete account of the modern theory of eigenspaces
of elements of reflection groups * A comprehensive appendix contains suggestions
for research projects in a variety of fields * Exercises range in difficulty
to suit both graduate students and researchers
Introduction; 1. Preliminaries; 2. The groups G(m, p, n); 3. Polynomial invariants; 4. Poincare series and characterisations of reflection groups; 5. Quaternions and the finite subgroups of SU2(C); 6. Finite unitary reflection groups of rank two; 7. Line systems; 8. The Shepherd and Todd classification; 9. The orbit map, harmonic polynomials and semi-invariants; 10. Covariants and related polynomial identities; 11. Eigenspace theory and reflection subquotients; 12. Reflection cosets and twisted invariant theory; A. Some background in commutative algebra; B. Forms over finite fields; C. Applications and further reading; D. Tables; Bibliography; Index of notation; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 117)
Hardback (ISBN-13: 9780521888509)
Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory. Assuming as little technical background as possible, the book starts with basic algebraic and topological concepts, but already presented from the modern viewpoint advocated by Grothendieck. This enables a systematic yet accessible development of the theories of fundamental groups of algebraic curves, fundamental groups of schemes, and Tannakian fundamental groups. The connection between fundamental groups and linear differential equations is also developed at increasing levels of generality. Key applications and recent results, for example on the inverse Galois problem, are given throughout.
* Provides key applications and recent results throughout * A friendly
introduction to material that isnft usually taught until advanced graduate
level * Class tested by the author with graduate and advanced undergraduate
students
Foreword; 1. Galois theory of fields; 2. Fundamental groups in topology; 3. Riemann surfaces; 4. Fundamental groups of algebraic curves; 5. Fundamental groups of schemes; 6. Tannakian fundamental groups; Bibliography; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 127)
Hardback (ISBN-13: 9780521519267)
Aggregation is the process of combining several numerical values into a single representative value, and an aggregation function performs this operation. These functions arise wherever aggregating information is important: applied and pure mathematics (probability, statistics, decision theory, functional equations), operations research, computer science, and many applied fields (economics and finance, pattern recognition and image processing, data fusion, etc.). This is a comprehensive, rigorous and self-contained exposition of aggregation functions. Classes of aggregation functions covered include triangular norms and conorms, copulas, means and averages, and those based on nonadditive integrals. The properties of each method, as well as their interpretation and analysis, are studied in depth, together with construction methods and practical identification methods. Special attention is given to the nature of scales on which values to be aggregated are defined (ordinal, interval, ratio, bipolar). It is an ideal introduction for graduate students and a unique resource for researchers.
* Up-to-date and complete survey on aggregation functions * Presents a
unified view of aggregation functions with unified notations and terminology
in accordance with mathematical standards * Essential reference for researchers
in a broad range of fields
Preface; 1. Introduction; 2. Properties for aggregation; 3. Conjunctive and disjunctive aggregation functions; 4. Means and averages; 5. Aggregation functions based on nonadditive integrals; 6. Construction methods; 7. Aggregation on specific scale types; 8. Aggregation on ordinal scales; 9. Aggregation on bipolar scales; 10. Behavioral analysis of aggregation functions; 11. Identification of aggregation functions; A. Aggregation of infinitely many arguments; B. Examples and applications; List of symbols; Bibliography; Index.
Nonclassical logics play an ever-increasing role in various disciplines from mathematics, informatics and computer science to artificial intelligence, cognitive science, linguistics and philosophy.
The authors develop a uniform framework of relational semantics to mediate the connection between logical calculi and their semantics through algebra, resulting in a lucid and conceptually clear presentation. Among the familiar logics covered are normal modal logics such as K and S5 as well as substructural logics such as relevance logics, linear logic and Lambek calculi. Less-familiar and new logical systems are treated with equal deftness.
Suitable for use as a graduate textbook in nonclassical logic, this book will also please experts with gems such as the chapter on topological duality theory. Even novices can find their way eased into the field by an appendix that provides a concise introduction into the relevant parts of universal algebra.
Katalin Bimbo is Research Associate at Indiana University.
J. Michael Dunn is Oscar R. Ewing Professor Emeritus of Philosophy and Professor Emeritus of Computer Science, of Cognitive Science and of Informatics at Indiana University.
8/1/2008
ISBN (Paperback): 1575865742
ESI Lectures in Mathematics and Physics
ISBN 978-3-03719-057-9
October 2008, 284 pages, softcover, 17 x 24 cm.
Ludwig Eduard Boltzmann (1844*1906) was an Austrian physicist famous for
his founding contributions in the fields of statistical mechanics and statistical
thermodynamics. He was one of the most important advocates for atomic theory
when that scientific model was still highly controversial. To commemorate
the 100th anniversary of his death in Duino, the International Symposium
gBoltzmann's Legacyh was held at the Erwin Schrodinger International
Institute for Mathematical Physics in June 2006.
This text covers a wide spectrum of topics ranging from equilibrium statistical and nonequilibrium statistical physics, ergodic theory and chaos to basic questions of biology and historical accounts of Boltzmann's work. Besides the lectures presented at the symposium the volume also contains contributions specially written for this occasion. The articles give a broad overview of Boltzmann's legacy to the sciences from the standpoint of some of present day's leading scholars in the field.
The book addresses students and researchers in mathematics, physics and the history of science.
ISBN : 9782705666996
Descriptif
Depuis son emergence dans la seconde moitie du XIXe siecle, les theories homologiques et cohomologiques ont dynamise et considerablement enrichi de nombreux domaines des mathematiques. Elles sont parvenues a fertiliser et a eclairer certains travaux en physique theorique (cohomologie BRST des theories de jauge, theorie des champs, theories conformes, homologie de Morse et supersymetrie, systemes dynamiques/dots) et a inspirer de nombreux developpement mathematiques recents.
Thierry Masson est ancien eleve de lfENS de Cachan, agrege de Mathematiques et docteur es sciences. Il est depuis 1997 chercheur en physique- mathematiques au CNRS au Laboratoie de Physique Theorique dfOrsay. Ses domaines de recherche sont principalement orientes sur la geometrie non commutative et ses applications a la physique theorique.