Series: Springer Monographs in Mathematics
1st Edition., 2011, XVI, 476 p. 15 illus., Hardcover
ISBN: 978-0-85729-159-2
Due: November 2010
The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations.
Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. Fundamental results are organized in a unified way and existing proofs refined. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock
modules in chapter eight.
This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.
Preliminary.- Classification of Harish-Chandra Modules.- The Jantzen Filtration.- Determinant Formulae.- Verma Modules I: Preliminaries.- Verma Modules II: Structure Theorem.- A Duality among Verma Modules.- Fock Modules.- Rational Vertex Operator Algebras.- Coset Constructions for sl2.- Unitarisable Harish-Chandra Modules.- Homological Algebras.- Lie p-algebras.- Vertex Operator Algebras.
Series: Universitext
1st Edition., 2011, VI, 392 p. 14 illus., 2 in color., Softcover
ISBN: 978-0-85729-182-0
Due: November 2010
Written as a hybrid between a research monograph and a textbook the first half of this book is concerned with basic concepts for the study of Banach algebras that, in a sense, are not too far from being commutative. Essentially, the algebra under consideration either has a sufficiently large center or is subject to a higher order commutator property (an algebra with a so-called polynomial identity or in short: Pl-algebra). In the second half of the book, a number of selected examples are used to demonstrate how this theory can be successfully applied to problems in operator theory and numerical analysis.</p>
Distinguished by the consequent use of local principles (non-commutative Gelfand theories), PI-algebras, Mellin techniques and limit operator techniques, each one of the applications presented in chapters 4, 5 and 6 forms a theory that is up to modern standards and interesting in its own right.
Written in a way that can be worked through by the reader with fundamental knowledge of analysis, functional analysis and algebra, this book will be accessible to 4th year students of mathematics or physics whilst also being of interest to researchers in the areas of operator theory, numerical analysis, and the general theory of Banach algebras.
Banach algebras.- Local principles.- Banach algebras generated by idempotents.- Singular integral operators.- Convolution operators.- Algebras of operator sequences.
Series: Algebra and Applications, Vol. 13
1st Edition., 2011, XIX, 197 p. 1 illus., Hardcover
ISBN: 978-0-85729-156-1
Due: October 2010
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provide the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects.
The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups.
At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.
Structure of SL2(Fq).- The Geometry of the Drinfeld Curve.- Harish-Chandra Induction.- Deligne-Lusztig Induction.- The Character Table.- More about Characters of G and of its Sylow Subgroups.- Unequal Characteristic: Generalities.- Unequal Characteristic: Equivalences of Categories.- Unequal Characteristic: Simple Modules, Decomposition Matrices.- Equal Characteristic.- Special Cases.- Deligne-Lusztig Theory: an Overview*.- Appendices: l-Adic Cohomology.- Block Theory.- Review of Reflection Groups
Series: Lecture Notes in Mathematics, Vol. 2010
Subseries: Fondazione C.I.M.E., Firenze
1st Edition., 2011, X, 234 p., Softcover
ISBN: 978-3-642-15966-4
Due: December 2010
This volume presents a review of advanced technological problems in the glass industry and of the mathematics involved. It is amazing that such a seemingly small research area is extremely rich and calls for an impressively large variety of mathematical methods, including numerical simulations of considerable complexity. The problems treated here are very typical of the field of glass manufacturing and cover a large spectrum of complementary subjects: injection molding by various techniques, radiative heat transfer in glass, nonisothermal flows and fibre spinning. The book can certainly be useful not only to applied mathematicians, but also to physicists and engineers, who can find in it an overview of the most advanced models and methods.
Mathematical modelling of glass forming processes.- Radiative heat transfer and applications for glass production processes.- I. Radiative heat transfer and applications for glass production processes.- II. Nonisothermal flows and fibres drawing.
Series: Springer Undergraduate Mathematics Series
1st Edition., 2011, XIV, 268 p. 69 illus., Softcover
ISBN: 978-0-85729-147-9
Due: December 17, 2010
Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject.
It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into `lecture' sized pieces, motivated and illustrated by numerous theoretical and computational examples.
Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors.
The book covers key foundation topics:
o Taylor series methods
o Runge--Kutta methods
o Linear multistep methods
o Convergence
o Stability
and a range of modern themes:
o Adaptive stepsize selection
o Long term dynamics
o Modified equations
o Geometric integration
o Stochastic differential equations
The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via www.springer.com
An Introduction.- Euler’s Method.- The Taylor Series Method.- Linear Multistep Methods-I.- Linear Multistep Methods-II.- Linear Multistep Methods -III.- Linear Multistep Methods-IV.- Linear Multistep Methods-V.- Runge-Kutta Method-I: Order Conditions.- Runge-Kutta Methods-II Absolute Stability.- Adaptive Step Size Selection.- Long-Term Dynamics.- Modified Equations.- Geometric Integration Part I-Invariants.- Geometric Integration Part II-Hamiltonian Dynamics.- Stochastic Differential Equations.- A Glossary and Notation.- B Taylor Series.- C Jacobians and Variational Equations.- D Constant-Coefficient Difference Equations.