Geer, Gerard van der; Moonen, Ben J.J.; Schoof, Rene (Eds.)

Number Fields and Function Fields - Two Parallel Worlds

Series: Progress in Mathematics, Vol. 239
2005, Approx. 300 p. 10 illus., Hardcover
ISBN: 0-8176-4397-4

About this book

Ever since the analogy between number fields and function fields was discovered in the 19th century it has been and remains a source of inspiration for new ideas. A deeper understanding of this correlation could have tremendous consequences for the search for a unified approach that has become a sort of Holy Grail. The arrival of Arakelov's new geometry that tries to put the archimedean places on a par with the finite ones gave a new impetus and led to spectacular success in Faltings' hands. There are numerous further examples where ideas or techniques from the more geometrically-oriented world of function fields have led to new insights in the more arithmetically-oriented world of number fields, or vice versa.

These invited articles by leading researchers in the field explore various aspects of the parallel worlds of function fields and number fields. Topics range from Arakelov geometry, the search for a theory of varieties over the field with one element, via Eisenstein series to Drinfeld modules, and t-motives.

This volume is aimed at a wide audience of graduate students, mathematicians, and researchers interested in geometry and arithmetic and their connections.

Contributors include: G. Boeckle; T. van den Boogaart; H. Brenner; F. Breuer; K. Conrad; A. Deitmar; C. Deninger; B. Edixhoven; G. Faltings; G. Harder; U. Hartl; R. de Jong; K. Koehler; U. Kuehn; J. Lagarias; T. Oda; R. Pink; D. Roessler; U. Stuhler; and A. Werner.

Table of contents

Preface.- G. Boeckle.- H. Brenner.- F. Breuer and R. Pink.- K. Conrad.- A. Deitmar.- C. Deninger and A. Werner.- B. Edixhoven and T. van den Boogaart.- G. Faltings.- G. Harder and U. Stuhler.- U. Hartl.- R. de Jong.- K. Koehler.- U. Kuehn.- J. Lagarias.- T. Oda.- D. Roessler.

Knapp, Anthony W.

Basic Real Analysis and Advanced Real Analysis

Series: Cornerstones
2005, Approx. 1150 p.
ISBN: 0-8176-4407-5

About this textbook

Basic Real Analysis and Advanced Real Analysis systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics.

Key topics and features:

* The development proceeds from the particular to the general, often introducing examples well before a theory that incorporates them

* Incorporates, in the text and especially in the problems, material in which real analysis is used in algebra, in topology, in complex analysis, in probability, in differential geometry, and in applied mathematics of various kinds

* The texts include many examples and hundreds of problems, and each provides a lengthy separate section giving hints or complete solutions for most of the problems

Because they focus on what every young mathematician needs to know about real analysis, the books are ideal both as course texts and for self-study, especially for graduate students preparing for qualifying examinations. Their scope and approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, mathematical physics, and differential equations. Indeed, their clarity and breadth make them a welcome addition to the personal library of every mathematician.

Written for:

Advanced undergraduates, graduate students, pure and applied mathematicians working in analytic areas such as statisitics, math physics, electrical engineering, and differential equations

Bellouquid, A., Delitala, Marcello Edoardo

Mathematical Modeling of Complex Biological Systems
A Kinetic Theory Approach

Series: Modeling and Simulation in Science, Engineering and Technology
2006, Approx. 200 p. 30 illus., Hardcover
ISBN: 0-8176-4395-8

About this book

This book develops methods using mathematical kinetic theory to describe the evolution of several socio-biological systems. Specifically, it deals with modeling and simulations of biological systems constituted by large populations of interacting cells, whose dynamics follow both the rules of mechanics as well as their own ability to organize movement and biological functions. Proposed is a new biological model focused on the analysis of competition between cells of an aggressive host and cells of the immune system.

Modeling in kinetic theory may represent a way to understand phenomena of nonequilibrium statistical mechanics that is not described by the traditional macroscopic approach. A kinetic theory approach involves deriving suitable evolution equations for the statistical distribution function of the microscopic state of each element in a given system. Macroscopic information is obtained by suitable moments of the distribution function. The authors focus on models that refer to the Boltzmann equation (generalized Boltzmann models) with the dynamics of populations of several interacting individuals (kinetic population models). The book follows the classical research line applied to modeling real systems, linking the phenomenological observation of systems to modeling and simulations. Computational algorithms are applied and qualitative analysis techniques are used to identify the prediction ability of specific models.

The book will be a valuable resource for applied mathematicians as well as researchers in the field of biological sciences. It may also be used for advanced graduate courses in biological systems modeling with applications to collective social behavior, immunology, and epidemiology.

Table of contents

Introduction to the Aims and Contents of the Book.- Mathematical Frameworks of the Generalized Kinetic (Boltzmann) Theory.- Modeling the Immune Competition.- Qualitative Analysis of the Chauchy Problem.- Simulations and Biological Interpretations.- Models with Space Structure and Macroscopic Equations.- Conclusions and Perspectives.- Appendix.- References

Boutet de Monvel, Louis

Toeplitz Operators and Deformation-Algebras

Series: Progress in Mathematics, Preliminary entry 400
2007, Approx. 200 p., Hardcover
ISBN: 0-8176-3226-3

About this book

This work provides a comprehensive treatment of Toeplitz operators and closely related mathematical theories, in particular, star-products or deformation algebras. Toeplitz operators are closely related to pseudodifferential operators. They give rise to an essentially isomorphic symbolic calculus and to much of pseudodifferential analysis, as applied to results on propagation of singularities. Of significance is the fact that Toeplitz operators are useful for problems in deformation quantization. The symbolic calculus of pseudodifferential and Toeplitz operators is also a special case of star-product. The book will be suitable for graduate students and mathematicians in the areas of differential geometry and operator theory.

Table of contents

Preface * Review of Microanalysis * The Bergman and Szego Kernels * Toeplitz Operators * Symbolic Calculus and Deformation Algebras * Bibliography * Index

Aftalion, Amandine

Vortices in Bose-Einstein Condensates

Series: Progress in Nonlinear Differential Equations and Their Applications, Preliminary entry 1101
2007, Approx. 160 p. 25 illus., Hardcover
ISBN: 0-8176-4392-3

About this book

One of the key issues related to superfluidity is the existence of vortices. In very recent experiments on Bose?Einstein condensates, vortices have been observed by rotating the trap holding the atoms. In contrast to a classical fluid for which the equilibrium velocity corresponds to solid body rotation, a quantum fluid such as a Bose?Einstein condensate can rotate only through the nucleation of quantized vortices. This monograph is dedicated to the mathematical modeling of these phenomena.

One of the experiments studied focuses on rotating the trap holding the atoms. At low velocity, no modification of the condensate is observed, while beyond some critical value, vortices appear in the system. There are two interesting regimes: one close to the critical velocity where there is only one vortex; and another one at high rotation values, for which a dense lattice is observed. Another experiment that is studied consists of superfluid flow around an obstacle. At low velocity, the flow is stationary; while at larger velocity, vortices are nucleated from the boundary of the obstacle.

The mathematical tools employed are energy estimates, Gamma convergence, and homogenization techniques. The mathematical analysis is made in the framework of the Gross?Pitaevskii energy. Results are presented and open problems related to recent experiments are explained.

Kai Liu The University of Liverpool, UK

Stability of Infinite Dimensional Stochastic Differential
and Equations Applications

Series: Monographs & Surveys in Pure & Applied Math Volume: 135
ISBN: 158488598X
Publication Date: 8/19/2005
Number of Pages: 312

Presents up-to-date material in an accessible way
Features an in-depth discussion of infinite dimensions, focusing on Hilbert spaces
Investigates stability for linear and nonlinear equations
Provides stability models for both finite and infinite dimensions
Includes concrete applications and reference material that is ideal for students, researchers, engineers, and scientists

Stability of Infinite Dimensional Stochastic Differential Equations with Applications presents up-to-date, complex material in an accessible way. Focusing mainly on Hilbert spaces, this book features an in-depth discussion of infinite dimensions, including the notion of L2-stability in mean. It investigates stability for the essential classes of linear stochastic evolution equations. Additional material explores topics related to the stability of nonlinear systems and equations. With various stability models and applications for both finite and infinite dimensions, this text is an ideal reference for graduate students, researchers, engineers, and scientists interested in this area.

The work can serve as a reference for mathematical researchers and theoretical physicists interested in superfluidity and quantum condensates, and can also complement a graduate seminar in elliptic PDEs or modeling of physical experiments.

Table of Contents

Stochastic Differential Equations in Infinite Dimensions. Stability of Linear Stochastic Differential Equations. Stability of Non Linear Stochastic Differential Equations. Stability of Stochastic Functional Differential Equations. Some Related Topics of Stability and Applications.