Series: Numerical Methods and Algorithms, Vol. 4
2005, XVI, 296 p., Hardcover
ISBN: 0-387-24271-6
About this book
The Schur complement plays an important role in matrix analysis,
statistics, numerical analysis, and many other areas of
mathematics and its applications. This book describes the Schur
complement as a rich and basic tool in mathematical research and
applications and discusses many significant results that
illustrate its power and fertility. The eight chapters of the
book cover themes and variations on the Schur complement,
including its historical development, basic properties,
eigenvalue and singular value inequalities, matrix inequalities
in both finite and infinite dimensional settings, closure
properties, and applications in statistics, probability, and
numerical analysis.
Although the book is primarily intended to serve as a research
reference, it will also be useful for graduate and advanced
undergraduate courses in mathematics, applied mathematics, and
statistics. The contributing authorsf exposition makes most of
the material accessible to readers with a sound foundation in
linear algebra.
Table of contents
Preface.- Historical Introduction: Issai Schur and the Early
Development of the Schur Complement.- Basic Properties of the
Schur Complement.- Eigenvalue and Singular Value Inequalities of
Schur Complements.- Block Matrix Techniques.- Closure Properties.-
Schur Complements and Matrix Inequalities: Operator-Theoretic
Approach.- Schur Complements in Statistics and Probability.-
Schur Complements and Applications in Numerical Analysis.-
Bibliography.- Notation.- Index.
Series: Texts in Theoretical Computer Science. An EATCS Series,
2005, Approx. 286 p., Hardcover
ISBN: 3-540-23949-9
About this textbook
Randomness is a powerful phenomenon that can be harnessed to
solve various problems in all areas of computer science.
Randomized algorithms are often more efficient, simpler and,
surprisingly, also more reliable than their deterministic
counterparts. Computing tasks exist that require billions of
years of computer work when solved using the fastest known
deterministic algorithms, but they can be solved using randomized
algorithms in a few minutes with negligible error probabilities.
Introducing the fascinating world of randomness, this book
systematically teaches the main algorithm design paradigms ?
foiling an adversary, abundance of witnesses, fingerprinting,
amplification, and random sampling, etc. ? while also providing a
deep insight into the nature of success in randomization. Taking
sufficient time to present motivations and to develop the
reader's intuition, while being rigorous throughout, this text is
a very effective and efficient introduction to this exciting
field.
Table of contents
Introduction.- Fundamentals.- Foiling the Adversary.-
Fingerprinting.- Success Amplification and Random Sampling.-
Abundance of Witnesses.- Optimization and Random Rounding.-
Appendix: Fundamentals of Mathematics.- References.- Index.
Series: Texts in Theoretical Computer Science. An EATCS
Series,
2005, Approx. 480 p., Hardcover
About this textbook
Modern cryptology increasingly employs mathematically rigorous
concepts and methods from complexity theory. Conversely, current
research topics in complexity theory are often motivated by
questions and problems from cryptology. This book takes account
of this situation, and therefore its subject is what may be
dubbed "cryptocomplexity'', a kind of symbiosis of these two
areas.
This book is written for undergraduate and graduate students of
computer science, mathematics, and engineering, and can be used
for courses on complexity theory and cryptology, preferably by
stressing their interrelation. Moreover, it may serve as a
valuable source for researchers, teachers, and practitioners
working in these fields. Starting from scratch, it works its way
to the frontiers of current research in these fields and provides
a detailed overview of their history and their current research
topics and challenges.
Table of contents
Introduction to Cryprocomplexity.- Foundations of Computer
Science and Mathematics.- Foundations of Complexity Theory.-
Foundations of Cryptology.- Hierarchies Based on NP.- Randomized
Algorithms and Complexity Classes.- RSA Cryptosystem, Primality,
and Factoring.- Other Public-Key Cryptosystems and Protocols.-
References.- List of Figures.- List of Tables.- Index.
Series: Springer Monographs in Mathematics,
2005, XVI, 653 p., Hardcover
ISBN: 3-540-24170-1
About this book
The theory of Lie algebras and algebraic groups has been an area
of active research for the last 50 years. It intervenes in many
different areas of mathematics: for example invariant theory,
Poisson geometry, harmonic analysis, mathematical physics. The
aim of this book is to assemble in a single volume the algebraic
aspects of the theory, so as to present the foundations of the
theory in characteristic zero. Detailed proofs are included and
some recent results are discussed in the final chapters. All the
prerequisites on commutative algebra and algebraic geometry are
included.
Table of contents
Preface
1. Results on topological spaces,2. Rings and modules, 3.
Integral extensions , 4. Factorial rings, 5. Field extensions
6. Finitely generated algebras, 7. Gradings and filtrations, 8.
Inductive limits, 9. Sheaves of functions , 10. Jordan
decomposition and some basic results on groups, 11. Algebraic
sets, 12. Prevarieties and varieties, 13. Projective varieties
14. Dimension, 15. Morphisms and dimension, 16. Tangent spaces,
17. Normal varieties, 18. Root systems, 19. Lie algebras
20. Semisimple and reductive Lie algebras, 21. Algebraic groups,
22. Affine algebraic groups, 23. Lie algebra of an algebraic
group, 24. Correspondence between groups and Lie algebras, 25.
Homogeneous spaces and quotients, 26. Solvable groups
27. Reductive groups, 28. Borel subgroups, parabolic subgroups
and Cartan subgroups, 29. Cartan subalgebras, Borel subalgebras
and parabolic subalgebras, 30. Representations of semisimple Lie
algebras, 32. S-triples, 33. Polarizations
34. Results on orbits, 35. Centralizers, 36. s -root systems, 37.
Symmetric Lie algebras, 38. Semisimple symmetric Lie algebras, 39.
Sheets of Lie algebras, 40. Index and linear forms, References,
Index of notations, Index of terminologies
Series: Algorithms and Computation in Mathematics, Vol. 14
2005, XIII, 425 p., 44 illus., 11 in color., Hardcover
ISBN: 3-540-24326-7
About this textbook
This book provides a general introduction to modern mathematical
aspects in computing with multivariate polynomials and in solving
algebraic systems. It presents the state of the art in several
symbolic, numeric, and symbolic-numeric techniques, including
effective and algorithmic methods in algebraic geometry and
computational algebra, complexity issues, and applications
ranging from statistics and geometric modelling to robotics and
vision.
Graduate students, as well as researchers in related areas, will
find an excellent introduction to currently interesting topics.
These cover Groebner and border bases, multivariate resultants,
residues, primary decomposition, multivariate polynomial
factorization, homotopy continuation, complexity issues, and
their applications.
Table of contents
A.Dickenstein, I.Z.Emiris: Preface.- 1 E.Cattani, A.Dickenstein:
Introduction to Residues and Resultants.- 2 D.A.Cox: Solving
Equations via Algebras.- 3 M.Elkadi, B.Mourrain: Symbolic-numeric
Methods for Solving Polynomial Equations and Applications.- 4 A.Kehrein,
M.Kreuzer, L.Robbiano: An Algebraist's View on Border Bases.- 5 M.Stillman:
Tools for Computing primary Decompositions and Applications to
Ideals Associated to Bayesian Networks.- 6 J.Sabia: Algorithms
and Their Complexities.- 7 I.Z.Emiris: Toric Resultants and
Applications to Geometric Modelling.- 8 A.J.Sommese, J.Verschelde,
Ch.W.Wampler: Introduction to Numerical Algebraic Geometry.- 9 G.Cheze,
A.Galligo: Four Lectures on Polynomial Absolute Factorization.-
References.- Index.