Seminaires et Congres 25 (2012), xv+383 pages
Familles orthogonales et semigroupes en analyse et probabilites - Ecole CIMPA Merida, Venezuela, 2006
L'Ecole CIMPA-UNESCO Familles orthogonales et semigroupes en analyse et probabilites qui s'est tenue en 2006 a Merida, Venezuela, a ete organisee avec la collaboration de trois universites venezueliennes (UCV, USB et ULA). Cette Ecole avait pour objectif de presenter la theorie moderne des semigroupes des operateurs, lies aux developpements polynomiaux orthogonaux. Cette theorie est aujourd'hui tres riche et est reliee a des domaines mathematiques varies, tels que l'analyse harmonique, la theorie des probabilites, la theorie des matrices aleatoires, le calcul stochastique et la theorie du controle. Les chapitres de ce volume sont issus des cours de cette Ecole et mettent l'accent sur les interactions de tous ces domaines.
The Workshop CIMPA-UNESCO ``Orthogonal families and semigroups in analysis and probability'' was held in 2006 in Merida, Venezuela and was organized with the collaboration of three Venezuelian universities(UCV, USB and ULA). The objective of the Workshop was to present the modern theory of operator semigroups, related to polynomial orthogonal expansions. This theory comprises nowadays a vast body of knowledge and has interconnections with several other areas, including harmonic analysis, probability, random matrices, stochastic calculus and control theory. The chapters of this volume originate from the lectures of this Workshop and they stress the interplay of all these domains.
ISBN : 978-2-85629-340-9
ISBN: 978-1-1180-7205-9
Hardcover
602 pages
March 2012
Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel’s theory of Abelian equations, casus irreducibili, and the Galois theory of origami.
In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including:
・ The contributions of Lagrange, Galois, and Kronecker
・ How to compute Galois groups
・ Galois’s results about irreducible polynomials of prime or prime-squared degree
・ Abel’s theorem about geometric constructions on the lemniscates
・ Galois groups of quartic polynomials in all characteristics
Throughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple? and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study.
Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.
ISBN: 978-1-1180-9139-5
Hardcover
296 pages
July 2012
This book provides an introduction to Galois theory and focuses on one central theme - the solvability of polynomials by radicals. Both classical and modern approaches to the subject are described in turn in order to have the former (which is relatively concrete and computational) provide motivation for the latter (which can be quite abstract). The theme of the book is historically the reason that Galois theory was created, and it continues to provide a platform for exploring both classical and modern concepts. This book examines a number of problems arising in the area of classical mathematics, and a fundamental question to be considered is: For a given polynomial equation (over a given field), does a solution in terms of radicals exist? That the need to investigate the very existence of a solution is perhaps surprising and invites an overview of the history of mathematics. The classical material within the book includes theorems on polynomials, fields, and groups due to such luminaries as Gauss, Kronecker, Lagrange, Ruffini and, of course, Galois. These results figured prominently in earlier expositions of Galois theory, but seem to have gone out of fashion. This is unfortunate since, aside from being of intrinsic mathematical interest, such material provides powerful motivation for the more modern treatment of Galois theory presented later in the book. Over the course of the book, three versions of the Impossibility Theorem are presented: the first relies entirely on polynomials and fields, the second incorporates a limited amount of group theory, and the third takes full advantage of modern Galois theory. This progression through methods that involve more and more group theory characterizes the first part of the book. The latter part of the book is devoted to topics that illustrate the power of Galois theory as a computational tool, but once again in the context of solvability of polynomial equations by radicals.
Series: Physics Research and Technology
Pub. Date: 2012 3rd Quarter
7 x 10 (NBC - C)
Binding: Hardcover
ISBN: 978-1-62081-063-7
The purpose of this book is to provide a reference for researchers and postgraduates. The problems are taken from quantum physics, but the method has important applications in any field of science involving the natural modes. The reader is assumed to be familiar with the elements of the quantum theory of scattering and the elements of the analytic functions theory. The described method can be applied in electromagnetism, atomic and molecular physics, nuclear physics, particle physics, solid state physics with application in nanoscience and electronic devices, chemical physics, and optics.
Preface
1. Introduction
2. Natural modes and quantum scattering by a potential
3. Riemann surface approach to bound and resonant states. Global method for all S-matrix poles analysis
4. Riemann surface approach to bound and resonant states for central rectangular potential
5. Riemann surface approach to bound and resonant states for central rectangular potential followed by a rectangular barrier
6. Riemann surface approach to bound and resonant states for central rectangular potential with Coulomb barrier
7. A particular case of exotic resonant states for a central potential with Coulomb barrier: the di-nuclear parent quasimolecular states
8. Riemann surface approach to bound and resonant states for the two-channel model with square potentials
9. Riemann surface approach to bound and resonant states for the 1D twin rectangular complex potentials. Subluminal and superluminal traversal times
10. Riemann surface approach to bound and resonant states for a nonlocal potential
11. Jump phenomenon induced by potential strength variation and the influence of exotic resonant states.
12. Local degeneracy for the exotic resonant states
13. A Analytic manifold Rg on which the function k(g) de_ned by the entire relation
Index
Cloth | March 2012 | ISBN: 9780691151199
280 pp. | 6 x 9 | 52 line illus. 16 tables
Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.
The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and in the process venture to the very frontiers of modern mathematics. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.
Avner Ash is professor of mathematics at Boston College. Robert Gross is associate professor of mathematics at Boston College. They are the coauthors of Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Princeton).
"Assuming only what every mathematically inclined freshman should know, this book leads the reader to an understanding of one of the most important conjectures in current number theory--whose proof is one of the Clay Mathematics Institute's million-dollar prize problems. The book is carefully and clearly written, and can be recommended without hesitation."--Peter Swinnerton-Dyer, University of Cambridge
"The Birch and Swinnerton-Dyer Conjecture is one of the great insights in number theory from the twentieth century, and Ash and Gross write with care and a clear love of the subject. Elliptic Tales will have wide appeal."--Peter Sarnak, Princeton University