Felix Klein
Lectures on Mathematics
Description
In the late summer of 1893, following the Congress of Mathematicians held in Chicago, Felix Klein gave two weeks of lectures on the current state of mathematics. Rather than offering a universal perspective, Klein presented his personal view of the most important topics of the time. It is remarkable how most of the topics continue to be important today. Originally published in 1893 and republished by the AMS in 1911, we are
pleased to bring this work into print once more with this new edition.
Klein begins by highlighting the works of Clebsch and of Lie. In particular, he discusses Clebsch's work on Abelian functions and compares his approach to the theory with Riemann's more geometrical point of view. Klein devotes two lectures to Sophus Lie, focussing on his contributions to geometry, including sphere geometry and contact geometry.
Klein's ability to connect different mathematical disciplines clearly comes through in his lectures on mathematical developments. For instance, he discusses recent progress in non-Euclidean geometry by emphasizing the connections to projective geometry and the role of transformation groups. In his descriptions of analytic function theory and of recent work in hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric point of view. He discusses Galois theory and solutions of algebraic equations of degree five or higher by reducing them to normal forms that might be solved by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved "by elliptic functions". This also leads to Klein's well-known work connecting the quintic to the group of the icosahedron.
Klein expounds on the roles of intuition and logical thinking in mathematics. He reflects on the influence of physics and the physical world on mathematics and, conversely, on the influence of mathematics on physics and the other natural sciences. The discussion is strikingly similar to today's discussions about "physical mathematics".
There are a few other topics covered in the lectures which are somewhat removed from Klein's own work. For example, he discusses Hilbert's proof of the transcendence of certain types of numbers (including $\pi$ and $e$), which Klein finds much simpler than the methods used by Lindemann to show the transcendence of $\pi$. Also, Klein uses the example of quadratic forms (and forms of higher degree) to explain the need
for a theory of ideals as developed by Kummer.
Klein's look at mathematics at the end of the 19th Century remains compelling today, both as history and as mathematics. It is delightful and fascinating to observe from a one-hundred year retrospect, the musings of one of the masters of an earlier era.
Contents
Lecture I.: Clebsch
Lecture II.: Sophus Lie
Lecture III.: Sophus Lie
Lecture IV.: On the real shape of algebraic curves and surfaces
Lecture V.: Theory of functions and geometry
Lecture VI.: On the mathematical character of space-intuition and the relation of pure mathematics to the applied sciences
Lecture VII.: The transcendency of the numbers $e$ and $\pi$
Lecture VIII.: Ideal numbers
Lecture IX.: The solution of higher algebraic equations
Lecture X.: On some recent advances in hyperelliptic and Abelian functions
Lecture XI.: The most recent researches in non-Euclidean geometry
Lecture XII.: The study of mathematics at G?ttingen
The development of mathematics at the German Universities
Details:
Series: AMS Chelsea Publishing
ISBN: 0-8218-2733-2
Paging: 109 pp.
Binding: Hardcover
Edited by: David R. McDonald, University of Ottawa, ON, Canada,
and Stephen R. E. Turner, University of Cambridge, England
Analysis of Communication Networks: Call Centres, Traffic and Performance
Description
This volume consists of the proceedings of the Workshop on Analysis and Simulation of Communication Networks held at The Fields Institute (Toronto). The workshop was divided into two main themes, entitled "Stability and Load Balancing of a Network of Call Centres" and "Traffic and Performance".
The call center industry is large and fast-growing. In order to provide top-notch customer service, it needs good mathematical models. The first part of the volume focuses on probabilistic issues involved in optimizing the performance of a call center. While this was the motivating application, many of the papers are also applicable to more general distributed queueing networks.
The second part of the volume discusses the characterization of traffic streams and how to estimate their impact on the performance of a queueing system. The performance of queues under worst-case traffic flows or flows with long bursts is treated. These studies are motivated by questions about buffer dimensioning and call admission control in ATM or IP networks.
This volume will serve researchers as a comprehensive, state-of-the-art reference source on developments in this rapidly expanding field.
Contents
A. R. Ward and W. Whitt -- Predicting response times in processor-sharing queues
D. A. Stanford and W. K. Grassmann -- Bilingual server call centres
R. J. Williams -- On dynamic scheduling of a parallel server system with complete resource pooling
Y. C. Teh -- Dynamic scheduling for queueing networks derived from discrete-review policies
S. R. E. Turner -- Large derviations for join the shorter queue
D. R. McDonald and S. R. E. Turner -- Comparing load balancing algorithms for distributed queueing networks
P. W. Glynn and A. J. Zeevi -- Estimating tail probabilities in queues via extremal statistics
G. Kesidis and T. Konstantopoulos -- Extremal traffic and worst-case performance for queues with shaped arrivals
D. J. Daley and R. A. Vesilo -- Long range dependence of inputs and outputs of some classical queues
S. Grishechkin, M. Devetsikiotis, I. Lambadaris, and C. Hobbs -- On `catastrophic' behavior of queueing networks
Details:
Series: Fields Institute Communications, Volume: 28
Publication Year: 2000
ISBN: 0-8218-1991-7
Paging: 200 pp.
Binding: Hardcover
Edited by:
Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu,
and Jie-Tai Yu, University of Hong Kong, China,
and Efim I. Zelmanov, Yale University, New Haven, CT
Combinatorial and Computational Algebra
Description
This volume presents articles based on the talks at the International Conference on Combinatorial and Computational Algebra held at the University of Hong Kong (China). The conference was part of the Algebra Program at the Institute of Mathematical Research and the Mathematics Department at the University of Hong Kong. Topics include recent developments in the following areas: combinatorial and computational aspects of group theory, combinatorial and computational aspects of associative and nonassociative algebras, automorphisms of polynomial algebras and the Jacobian conjecture, and combinatorics and coding theory.
This volume can serve as a solid introductory guide for advanced graduate students, as well as a rich and up-to-date reference source for contemporary researchers in the field.
Contents
Combinatorial and computational aspects of group theory
E. Aljadeff and A. R. Magid -- Deformations and liftings of representations
A. Yu. Ol'shanskii and M. V. Sapir -- Embeddings of relatively free groups into finitely presented groups
A. Shalev -- Fixed point ratios, character ratios, and Cayley graphs
Combinatorial and computational aspects of associative and nonassociative algebras
L. A. Bokut, Y. Fong, and W.-F. Ke -- Gr?bner-Shirshov bases and composition lemma for associative conformal algebras: An example
Y. Fong -- Derivations in near-ring theory
A. A. Mikhalev and J.-T. Yu -- Automorphic orbits of elements of free algebras with the Nielsen-Schreier property
A. V. Mikhalev and I. A. Pinchuk -- Universal central extensions of the matrix Lie superalgebras sl$(m,n,A)$
G. F. Pilz -- The useful world of one-sided distributive systems
E. Zelmanov -- On the structure of conformal algebras
Automorphisms of polynomial algebras and the Jacobian conjecture
L. A. Campbell -- Unipotent Jacobian matrices and univalent maps
V. Drensky and J.-T. Yu -- Automorphisms and coordinates of polynomial algebras
A. van den Essen -- On Bass' inverse degree approach to the Jacobian conjecture and exponential automorphisms
A. van den Essen and P. van Rossum -- A note on possible counterexamples to the Abhyankar-Sathaye conjecture constructed by Shpilrain and Yu
W. D. Neumann and P. G. Wightwick -- Algorithms for polynomials in two variables
V. Shpilrain and J.-T. Yu -- Peak reduction technique in commutative algebra: A survey
D. Wright -- Reversion, trees, and the Jacobian conjecture
Combinatorics and coding theory
W.-C. W. Li -- Various constructions of good codes
M.-K. Siu -- Combinatorics and algebra: A medley of problems? A medley of techniques?
Details:
Series: Contemporary Mathematics, Volume: 264
Publication Year: 2000
ISBN: 0-8218-1984-4
Paging: approximately 304 pp.
Binding: Softcover
Edited by: Michael Semenov-Tian-Shansky,
Steklov Mathematical Institute, St. Petersburg, Russia
L. D. Faddeev's Seminar on Mathematical Physics
Description
Professor L. D. Faddeev's seminar at Steklov Mathematical Institute (St. Petersburg, Russia) has a record of more than 30 years of intensive work which has helped to shape modern mathematical physics. This collection, honoring Professor Faddeev's 65th anniversary, has been prepared by his students and colleagues.
Topics covered in the volume include classical and quantum integrable systems (both analytic and algebraic aspects), quantum groups and generalizations, quantum field theory, and deformation quantization. Included is a history of the seminar highlighting important developments, such as the invention of the quantum inverse
scattering method and of quantum groups. The book will serve nicely as a comprehensive, up-to-date resource on the topic.
Contents
M. Semenov-Tian-Shansky -- Some personal historic notes on our seminar
E. Meinrenken and A. Alekseev -- An elementary derivation of certain classical dynamical $r$-matrices
I. Ya. Aref'eva and O. A. Rytchkov -- Incidence matrix description of intersection $p$-brane solutions
A. I. Bobenko and Yu. B. Suris -- A discrete time Lagrange top and discrete elastic curves
A. M. Budylin and V. S. Buslaev -- The Gelfand-Levitan-Marchenko equation and the long-time asymptotics of the solutions of the nonlinear Schrodinger equation
R. M. Kashaev and A. Yu. Volkov -- From the tetrahedron equation to universal $R$-matrices
A. N. Kirillov -- On some quadratic algebras
V. Korepin and N. Slavnov -- Quantum inverse scattering method and correlation functions
A. Losev, N. Nekrasov, and S. Shatashvili -- Testing Seiberg-Witten solution
J. M. Maillet and J. S. de Santos -- Drinfeld twists and algebraic Bethe Ansatz
V. B. Matveev -- Darboux transformations, covariance theorems and integrable systems
A. L. Pirozerski and M. A. Semenov-Tian-Shansky -- Generalized $q$-deformed Gelfand-Dickey structures on the group of $q$-pseudodifference operators
A. K. Pogrebkov -- On time evolutions associated with the nonstationary Schrodinger equation
N. Reshetikhin and L. A. Takhtajan -- Deformation quantization of Kohler manifolds
E. K. Sklyanin -- Canonicity of BucFunction Theory in Several Complex Variables
F. A. Smirnov -- Quasi-classical study of form factors in finite volume
V. Tarasov -- Completeness of the hypergeometric solutions of the $qKZ$ equation at level zero
Details:
Series: American Mathematical Society Translations--Series 2, Volume: 201
Subseries: Advances in the Mathematical Sciences
Publication Year: 2000
ISBN: 0-8218-2133-4
Paging: 321 pp.
Binding: Hardcover
Toshio Nishino, Kyushu University, Fukuoka, Japan
Function Theory in Several Complex Variables
Description
Kiyoshi Oka, at the beginning of his research, regarded the collection of problems which he encountered in the study of domains of holomorphy as large mountains which separate today and tomorrow. Thus, he believed that there could be no essential progress in analysis without climbing over these mountains ... this book is a
worthwhile initial step for the reader in order to understand the mathematical world which was created by Kiyoshi Oka.
--From the Preface
This book explains results in the theory of functions of several complex variables which were mostly established from the late nineteenth century through the middle of
the twentieth century. In the work, the author introduces the mathematical world created by his advisor, Kiyoshi Oka.
In this volume, Oka's work is divided into two parts. The first is the study of analytic functions in univalent domains in ${\mathbf C}^n$. Here Oka proved that three
concepts are equivalent: domains of holomorphy, holomorphically convex domains, and pseudoconvex domains; and moreover that the PoincarEproblem, the
Cousin problems, and the Runge problem, when stated properly, can be solved in domains of holomorphy satisfying the appropriate conditions. The second part of
Oka's work established a method for the study of analytic functions defined in a ramified domain over ${\mathbf C}^n$ in which the branch points are considered as
interior points of the domain. Here analytic functions in an analytic space are treated, which is a slight generalization of a ramified domain over ${\mathbf C}^n$.
In writing the book, the author's goal was to bring to readers a real understanding of Oka's original papers. This volume is an English translation of the original Japanese edition, published by the University of Tokyo Press (Japan). It would make a suitable course text for advanced graduate level introductions to several
complex variables.
Contents
Fundamental theory
Holomorphic functions and domains of holomorphy
Implicit functions and analytic sets
The PoincarE Cousin, and Runge problems
Pseudoconvex domains and pseudoconcave sets
Holomorphic mappings
Theory of analytic spaces
Ramified domains
Analytic sets and holomorphic functions
Analytic spaces
Normal pseudoconvex spaces
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs, Publication Year: 2001
ISBN: 0-8218-0816-8
Paging: approximately 450 pp.
Binding: Hardcover