Henning Krause, University of Bielefeld, Germany
The Spectrum of a Module Category
Description
These notes present an introduction into the spectrum of the category of modules over a ring. We discuss the general theory of pure-injective modules and concentrate on the isomorphism classes of indecomposable pure-injective modules which form the underlying set of this spectrum. The interplay between the
spectrum and the category of finitely presented modules provides new insight into the geometrical and homological properties of the category of finitely presented modules. Various applications from representation theory of finite dimensional algebras are included.
Contents
Introduction
The functor category
Definable subcategories
Left approximations
duality
Ideals in the category of finitely presented modules
Endofinite modules
Krull-Gabriel dimension
The infinite radical
Functors between module categories
Tame algebras
Rings of definable scalars
Reflective definable subcategories
Sheaves
Tame hereditary algebras
Coherent rings
Appendix A. Locally coherent Grothendieck categories
Appendix B. Dimensions
Appendix C. Finitely presented functors and ideals
Bibliography
Details:
Publisher: American Mathematical Society
Series: Memoirs of the American Mathematical Society Volume: 149
Publication Year: 2001
ISBN: 0-8218-2618-2
Paging: 125 pp.
Binding: Softcover
William M. Kantor, University of Oregon, Eugene, OR,
and Akos Seress, Ohio State University, Columbus, OHBlack Box Classical Groups
Description
If a black box simple group is known to be isomorphic to a classical group over a field of known characteristic, a Las Vegas algorithm is used to produce an explicit isomorphism. The proof relies on the geometry of the classical groups rather than on difficult group-theoretic background. This algorithm has applications to matrix group questions and to nearly linear time algorithms for permutation groups. In particular, we upgrade all known nearly linear time Monte Carlo permutation group algorithms to nearly linear Las Vegas algorithms when the input group has no composition factor isomorphic to an exceptional group of Lie type or a 3-dimensional unitary group.
Contents
Introduction
Preliminaries
Special linear groups: mathrm{PSL}(d,q)
Orthogonal groups: mathrm{P}Omega^varepsilon(d,q)
Symplectic groups: mathrm{PSp}(2m,q)
Unitary groups: mathrm{PSU}(d,q)
Proofs of Theorems 1.1 and 1.1', and of corollaries 1.2-1.4
Permutation group algorithms
Concluding remarks
References
Details:
Publisher: American Mathematical Society
Series: Memoirs of the American Mathematical Society, Volume: 149
Publication Year: 2001
ISBN: 0-8218-2619-0
Paging: 168 pp.
Binding: Softcover
Macques Francheteau and Guy Metivier
Existence de Chocs Faibles pour des Systemes
Quasi-Lineaires Hyperboliques Multidimensionnels
Description
In this work, the authors consider weak shocks for systems of conservation laws in any space dimension. The main result is the construction on a space-time domain,
independent of the parameter varepsilon, of families of weak solutions u^varepsilon, discontinuous along a smooth hypersurface Sigma^varepsilon, with jumps of order varepsilon. For a fixed varepsilon, the problem can be recast as a nonlinear mixed hyperbolic problem with a free noncharacteristic boundary.
It has been solved by A. Majda. When varepsilon tends to zero, the front tends to be characteristic. This induces a loss of stability and regularity. As a consequence, the classical nonlinear methods based on Picard's iterations and differentiation of the equations do not apply. In this work, to prove the suitable a priori estimates and to construct the solutions, the authors use more sophisticated methods, such as the paradifferential calculus and Nash-Moser-type iteration schemes. An important application of the results concern Euler's equations of gas dynamics. They apply to the full system and to the isentropic system. The authors construct and compare weak shock solutions of these two systems.
Contents
Introduction
Resultats principaux
Les etapes des preuves
Estimations preliminaires
Operateurs de traces et de relevement de traces
Compatibilites, constructions de solutions approchees
Paralinearisation
Estimations d'energie conormales
Estimations a priori pour le probleme non-lineaire
Le theoreme de prolongement ・varepsilon fix・
Prolongement de la resularit・
Application au systeme d'Euler. Comparaison des solutions
Bibliographie
Series: Asterisque, Number: 268
Publication Year: 2000
ISBN: 2-85629-092-2
Paging: 198 pp.
Binding: Softcover
Sigurdur Helgason, Massachusetts Institute of Technology, Cambridge, MA
Differential Geometry and Symmetric Spaces
"Remarkably well written ... might be used as a textbook for how to write mathematics."
-- Bulletin of the AMS
Description
Sigurdur Helgason's Differential Geometry and Symmetric Spaces was quickly recognized as a remarkable and important book. For many years, it was the standard text both for Riemannian geometry and for the analysis and geometry of symmetric spaces. Several generations of mathematicians relied on it for its clarity and careful attention to detail.
Although much has happened in the field since the publication of this book, as demonstrated by Helgason's own three-volume expansion of the original work, this single volume is still an excellent overview of the subjects. For instance, even though there are now many competing texts, the chapters on differential geometry and Lie groups continue to be among the best treatments of the subjects available. There is also a well-developed treatment of Cartan's classification and structure theory of symmetric spaces. The last chapter, on functions on symmetric spaces, remains an excellent introduction to the study of spherical functions, the theory of invariant
differential operators, and other topics in harmonic analysis. This text is rightly called a classic.
Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces.
Contents
Elementary differential geometry
Lie groups and Lie algebras
Structure of semisimple Lie algebras
Symmetric spaces
Decomposition of symmetric spaces
Symmetric spaces of the noncompact type
Symmetric spaces of the compact type
Hermitian symmetric spaces
On the classification of symmetric spaces
Functions on symmetric spaces
Bibliography
List of notational conventions
Symbols frequently used
Author index
Subject index
Reviews for the first edition
Details:
Publisher: American Mathematical Society
Series: AMS Chelsea Publishing
Publication Year: 1962
ISBN: 0-8218-2735-9
Paging: 486 pp.
Binding: Hardcover
Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst
Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide
Software, Environments, and Tools 11
Large-scale problems of engineering and scientific computing often require solutions of eigenvalue and related problems. This book gives a unified overview of theory, algorithms, and practical software for eigenvalue problems. It organizes this large body of material to make it accessible for the first time to the many nonexpert users who need to choose the best state-of-the-art algorithms and software for their problems. Using an informal decision tree, just enough theory is introduced to identify the relevant mathematical structure that determines the best algorithm for each problem.
The algorithms and software at the "leaves" of the decision tree range from the classical QR algorithm, which is most suitable for small dense matrices, to iterative algorithms for very large generalized eigenvalue problems. Algorithms are presented in a unified style as templates, with
different levels of detail suitable for readers ranging from beginning students to experts. The authors' comprehensive treatment includes a treasure of further bibliographic information.
Editors' Affiliations
Zhaojun Bai, University of California, Davis, Davis, California; James Demmel, University of California, Berkeley, Berkeley, California; Jack Dongarra, University of Tennessee, Knoxville, Tennessee, and Oak Ridge National Laboratory, Oak Ridge, Tennessee; Axel Ruhe, Chalmers
Tekniska Hskola, Geborg, Sweden; Henk van der Vorst, Utrecht University, Utrecht, The Netherlands.
Contents
List of Symbols and Acronyms; List of Iterative Algorithm Templates; List of Direct Algorithms; List of Figures; List of Tables; Chapter 1: Introduction; Chapter 2: A Brief Tour of Eigenproblems; Chapter 3: An Introduction to Iterative Projection Methods; Chapter 4: Hermitian
Eigenvalue Problems; Chapter 5: Generalized Hermitian Eigenvalue Problems; Chapter 6: Singular Value Decomposition; Chapter 7: Non-Hermitian Eigenvalue Problems; Chapter 8: Generalized Non-Hermitian Eigenvalue Problems; Chapter 9: Nonlinear Eigenvalue Problems; Chapter 10: Common Issues; Chapter 11: Preconditioning Techniques; Appendix: Of Things Not Treated; Bibliography; Index
2000 / xxx + 410 pages / Softcover / ISBN 0-89871-471-0