Introduction to Algebraic Independence Theory
Introduction to Algebraic Independence Theory

Author: Yuri V. Nesterenko

Publisher: Springer

Published: 2003-07-01

Total Pages: 257

ISBN-13: 3540445501

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In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.

Lectures on the Theory of Algebraic Numbers
Lectures on the Theory of Algebraic Numbers

Author: E. T. Hecke

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 251

ISBN-13: 1475740921

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. . . if one wants to make progress in mathematics one should study the masters not the pupils. N. H. Abel Heeke was certainly one of the masters, and in fact, the study of Heeke L series and Heeke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book, and Heeke's Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master, Andre Weil: "To improve upon Heeke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task. " We have tried to remain as close as possible to the original text in pre serving Heeke's rich, informal style of exposition. In a very few instances we have substituted modern terminology for Heeke's, e. g. , "torsion free group" for "pure group. " One problem for a student is the lack of exercises in the book. However, given the large number of texts available in algebraic number theory, this is not a serious drawback. In particular we recommend Number Fields by D. A. Marcus (Springer-Verlag) as a particularly rich source. We would like to thank James M. Vaughn Jr. and the Vaughn Foundation Fund for their encouragement and generous support of Jay R. Goldman without which this translation would never have appeared. Minneapolis George U. Brauer July 1981 Jay R.

Contributions to the Theory of Transcendental Numbers
Contributions to the Theory of Transcendental Numbers

Author: Gregory Chudnovsky

Publisher: American Mathematical Soc.

Published: 1984

Total Pages: 464

ISBN-13: 0821815008

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Contains a collection of papers devoted primarily to transcendental number theory and diophantine approximations. This title includes a text of the author's invited address on his work on the theory of transcendental numbers to the 1978 International Congress of Mathematicians in Helsinki.

Simplicial Complexes of Graphs
Simplicial Complexes of Graphs

Author: Jakob Jonsson

Publisher: Springer Science & Business Media

Published: 2007-11-15

Total Pages: 376

ISBN-13: 3540758585

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A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory.

Zeta Functions of Groups and Rings
Zeta Functions of Groups and Rings

Author: Marcus du Sautoy

Publisher: Springer Science & Business Media

Published: 2008

Total Pages: 217

ISBN-13: 354074701X

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Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic behaviour of these functions together with an investigation of functional equations. Many important examples of zeta functions are calculated and recorded providing an important data base of explicit examples and methods for calculation.

Stochastic Calculus for Fractional Brownian Motion and Related Processes
Stochastic Calculus for Fractional Brownian Motion and Related Processes

Author: Yuliya Mishura

Publisher: Springer Science & Business Media

Published: 2008-01-02

Total Pages: 411

ISBN-13: 3540758720

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This volume examines the theory of fractional Brownian motion and other long-memory processes. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. It proves that the market with stock guided by the mixed model is arbitrage-free without any restriction on the dependence of the components and deduces different forms of the Black-Scholes equation for fractional market.

Introduction to Modern Number Theory
Introduction to Modern Number Theory

Author: Yu. I. Manin

Publisher: Springer Science & Business Media

Published: 2006-03-30

Total Pages: 519

ISBN-13: 3540276920

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This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.

Axiom of Choice
Axiom of Choice

Author: Horst Herrlich

Publisher: Springer

Published: 2006-07-21

Total Pages: 207

ISBN-13: 3540342680

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AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom. It is shunned by some, used indiscriminately by others. This treatise shows paradigmatically that disasters happen without AC and they happen with AC. Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

Random Times and Enlargements of Filtrations in a Brownian Setting
Random Times and Enlargements of Filtrations in a Brownian Setting

Author: Roger Mansuy

Publisher: Springer Science & Business Media

Published: 2006-02-10

Total Pages: 167

ISBN-13: 3540294074

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In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azéma-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration. The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the progressive kind. It is accessible to researchers and graduate students working in stochastic calculus and excursion theory, and more broadly to mathematicians acquainted with the basics of Brownian motion.

Combinatorial Stochastic Processes
Combinatorial Stochastic Processes

Author: Jim Pitman

Publisher: Springer

Published: 2006-07-21

Total Pages: 257

ISBN-13: 3540342664

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The purpose of this text is to bring graduate students specializing in probability theory to current research topics at the interface of combinatorics and stochastic processes. There is particular focus on the theory of random combinatorial structures such as partitions, permutations, trees, forests, and mappings, and connections between the asymptotic theory of enumeration of such structures and the theory of stochastic processes like Brownian motion and Poisson processes.