Series: London Mathematical Society Lecture Note Series (No. 367)
Paperback (ISBN-13: 9780521133128)
This book focuses on the behaviour of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium.
* A modern theoretical treatment that includes new results and proofs *
Contains introductory material and summaries of key points to make the
book easily accessible to non-specialists * Its rigorous presentation means
the book is still suitably comprehensive for mathematicians
Introduction; 1. Metric Measure spaces; 2. Lie groups and matrix ensembles;
3. Entropy and concentration of measure; 4. Free entropy and equilibrium;
5. Convergence to equilibrium; 6. Gradient ows and functional inequalities;
7. Young tableaux; 8. Random point fields and random matrices; 9. Integrable
operators and differential equations; 10. Fluctuations and the Tracy*Widom
distribution; 11. Limit groups and Gaussian measures; 12. Hermite polynomials;
13. From the Ornstein*Uhlenbeck process to Burger's equation; 14. Noncommutative
probability spaces; References; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 130)
Hardback (ISBN-13: 9780521199605)
This is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization. The main idea behind the book is to study modules G over a ring R via their endomorphism ring EndR(G). The author discusses a wealth of results that classify G and EndR(G) via numerous properties, and in particular results from point set topology are used to provide a complete characterization of the direct sum decomposition properties of G. For graduate students this is a useful introduction, while the more experienced mathematician will discover that the book contains results that are not otherwise available. Each chapter contains a list of exercises and problems for future research, which provide a springboard for students entering modern professional mathematics.
* Contains over 200 exercises as well as problems for further research
* Serves as an introduction for graduate students wishing to study modules
over endomorphism rings and related problems * Presents results and perspectives
that are not available elsewhere
Preface; 1. Preliminary results; 2. Class number of an Abelian group; 3. Mayer-Vietoris sequences; 4. Lifting units; 5. The conductor; 6. Conductors and groups; 7. Invertible fractional ideals; 8. L-Groups; 9. Modules and homotopy classes; 10; Tensor functor equivalences; 11. Characterizing endomorphisms; 12. Projective modules; 13. Finitely generated modules; 14. Rtffr E-projective groups; 15. Injective endomorphism modules; 16. A diagram of categories; 17. Diagrams of Abelian groups; 18. Marginal isomorphisms; References; Index.
Series: Cambridge Series in Statistical and Probabilistic Mathematics
Hardback (ISBN-13: 9780521877220)
Exact statistical inference may be employed in diverse fields of science and technology. As problems become more complex and sample sizes become larger, mathematical and computational difficulties can arise that require the use of approximate statistical methods. Such methods are justified by asymptotic arguments but are still based on the concepts and principles that underlie exact statistical inference. With this in perspective, this book presents a broad view of exact statistical inference and the development of asymptotic statistical inference, providing a justification for the use of asymptotic methods for large samples. Methodological results are developed on a concrete and yet rigorous mathematical level and are applied to a variety of problems that include categorical data, regression, and survival analyses. This book is designed as a textbook for advanced undergraduate or beginning graduate students in statistics, biostatistics, or applied statistics but may also be used as a reference for academic researchers.
* Offers a lucid treatise of basic statistical inference * Provides an
appraisal of their limitations in real applications * Explains the role
of asymptotic methods in statistical inference
1. Motivation and basic tools; 2. Estimation theory; 3. Hypothesis testing; 4. Elements of statistical decision theory; 5. Stochastic processes: an overview; 6. Stochastic convergence and probability inequalities; 7. Asymptotic distributions; 8. Asymptotic behavior of estimators and tests; 9. Categorical data models; 10. Regression models; 11. Weak convergence and Gaussian processes.
Series: New Mathematical Monographs (No. 14)
Hardback (ISBN-13: 9780521764919)
This is a coherent explanation for the existence of the 26 known sporadic simple groups originally arising from many unrelated contexts. The given proofs build on the close relations between general group theory, ordinary character theory, modular representation theory and algorithmic algebra described in the first volume. The author presents a new algorithm by which 25 sporadic simple groups can be constructed (the smallest Mathieu group M11 can be omitted for theoretical reasons), and demonstrates that it is not restricted to sporadic simple groups. He also describes the constructions of various groups and proves their uniqueness whenever possible. The computational existence proofs are documented in the accompanying DVD. The author also states several open problems related to the theorem asserting that there are exactly 26 groups, and R. Brauer's warning that there may be infinitely many. Some of these problems require new experiments with the author's algorithm.