1926; 535 pp; hardcover
ISBN-13: 978-0-8218-3976-8
This edition consists largely of a reproduction of the first edition (which was based on lectures on Elementary Analysis given at Queen's College, Galway, from 1902-1907), with additional theorems and examples. Additional material includes a discussion of the solution of linear differential equations of the second order; a discussion of elliptic function formulae; expanded treatment of asymptomatic series; a discussion of trigonometrical series, including Stokes's transformation and Gibbs's phenomenon; and an expanded Appendix II that includes an account of Napier's invention of logarithms.
Sequences and limits
Series of positive terms
Series in general
Absolute convergence
Double series
Infinite products
Series of variable terms
Power series
Special power series
Trigonometrical formulae
Complex series and products
Special complex series and functions
Non-convergent series
Asymptotic series
Trigonometrical series
Appendix I. Arithmetic theory of irrational numbers and limits
Appendix II. Definitions of the logarithmic and exponential functions
Appendix III. Some theorems on infinite integrals and gamma-functions
Miscellaneous examples
Index of special integrals, products, and series
General index
University Lecture Series, Volume: 58
2012; 90 pp; softcover
ISBN-13: 978-0-8218-7559-9
Expected publication date is January 15, 2012.
Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, "The shortest and best way between two truths of the real domain often passes through the imaginary one." Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics.
Topics discussed include weighted approximation on the line, Muntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction theory, the Riesz convexity theorem, the Paley-Wiener theorem, the Titchmarsh convolution theorem, the Gleason-Kahane-?elazko theorem, and the Fatou-Julia-Baker theorem. The discussion begins with the world's shortest proof of the fundamental theorem of algebra and concludes with Newman's almost effortless proof of the prime number theorem. Four brief appendices provide all necessary background in complex analysis beyond the standard first year graduate course. Lovers of analysis and beautiful proofs will read and reread this slim volume with pleasure and profit.
Graduate students and research mathematicians interested in analysis.
Mathematical Surveys and Monographs, Volume: 180
2012; 380 pp; hardcover
ISBN-13: 978-0-8218-6920-8
Expected publication date is February 11, 2012.
This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups.
The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.
Research mathematicians interested in algebraic groups.
CRM Monograph Series, Volume: 30
2012; 140 pp; hardcover
ISBN-13: 978-0-8218-7582-7
Expected publication date is March 3, 2012.
This monograph studies moduli problems associated to algebraic dynamical systems. It is an expanded version of the notes for a series of lectures delivered at a workshop on Moduli Spaces and the Arithmetic of Dynamical Systems at the Bellairs Research Institute, Barbados, in 2010.
The author's g oal is to provide an overview, with enough details and pointers to the existing literature, to give the reader an entry into this exciting area of current research. Topics covered include:
(1) Construction and properties of dynamical moduli spaces for self-maps of projective space.
(2) Dynatomic modular curves for the space of quadratic polynomials.
(3) The theory of canonical heights associated to dynamical systems.
(4) Special loci in dynamical moduli spaces, in particular the locus of post-critically finite maps.
(5) Field of moduli and fields of definition for dynamical systems.
Titles in this series are co-published with the Centre de Recherches Mathematiques.
Graduate students and research mathematicians interested in dynamical systems, number theory, and algebraic geometry.
Moduli spaces associated to dynamical systems
The geometry of dynamical moduli spaces
Dynamical moduli spaces-Further topics
Dynatomic polynomials and dynamical modular curves
Canonical heights
Postcritically finite maps
Field of moduli and field of definition
Schedule of talks at the Bellairs workshop
Glossary
Bibliography
Index
Graduate Studies in Mathematics, Volume: 131
2012; approx. 500 pp; hardcover
ISBN-13: 978-0-8218-6867-6
Expected publication date is April 25, 2012.
Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. This book develops the theory of Lie superalgebras, their enveloping algebras, and their representations.
The book begins with five chapters on the basic properties of Lie superalgebras, including explicit constructions for all the classical simple Lie superalgebras. Borel subalgebras, which are more subtle in this setting, are studied and described. Contragredient Lie superalgebras are introduced, allowing a unified approach to several results, in particular to the existence of an invariant bilinear form on $\mathfrak{g}$.
The enveloping algebra of a finite dimensional Lie superalgebra is studied as an extension of the enveloping algebra of the even part of the superalgebra. By developing general methods for studying such extensions, important information on the algebraic structure is obtained, particularly with regard to primitive ideals. Fundamental results, such as the Poincare-Birkhoff-Witt Theorem, are established.
Representations of Lie superalgebras provide valuable tools for understanding the algebras themselves, as well as being of primary interest in applications to other fields. Two important classes of representations are the Verma modules and the finite dimensional representations. The fundamental results here include the Jantzen filtration, the Harish-Chandra homomorphism, the ?apovalov determinant, supersymmetric polynomials, and Schur-Weyl duality. Using these tools, the center can be explicitly described in the general linear and orthosymplectic cases.
In an effort to make the presentation as self-contained as possible, some background material is included on Lie theory, ring theory, Hopf algebras, and combinatorics.
Graduate students interested in Lie algebras, Lie superalgebras, quantum groups, string theory, and mathematical physics.
Introduction
The classical simple Lie superalgebras. I
Borel subalgebras and Dynkin-Kac diagrams
The classical simple Lie superalgebras. II
Contragredient Lie superalgebras
The PBW Theorem and filtrations on enveloping algebras
Methods from ring theory
Enveloping algebras of classical simple Lie superalgebras
Verma modules. I
Verma modules. II
Schur-Weyl duality
Supersymmetric polynomials
The center and related topics
Finite dimensional representations of classical Lie superalgebras
Prime and primitive ideals in enveloping algebras
Cohomology of Lie superalgebras
Zero divisors in enveloping algebras
Affine Lie superalgebras and number theory
Appendix A
Appendix B
Bibliography
Index
Graduate Studies in Mathematics, Volume: 132
2012; approx. 291 pp; hardcover
ISBN-13: 978-0-8218-7430-1
Expected publication date is April 13, 2012.
The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.
Graduate students and research mathematicians interested in random matrix theory.
Preparatory material
Random matrices
Related articles
Bibliography
Index