Linear infinite-particle operators
Linear infinite-particle operators

Author: V. A. Malyshev Robert Adol_fovich Minlos

Publisher: American Mathematical Soc.

Published: 1995-02-13

Total Pages: 314

ISBN-13: 9780821897607

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The main subject of this book can be viewed in various ways. From the standpoint of functional analysis, it studies spectral properties of a certain class of linear operators; from the viewpoint of probability theory, it is concerned with the analysis of singular Markov processes; and, from the vantage point of mathematical physics, it analyzes the dynamics of equilibrium systems in quantum statistical physics and quantum field theory. Malyshev and Minlos describe two new approaches to the subject which have not been previously treated in monograph form. They also present background material which makes the book accessible and useful to researchers and graduate students working in functional analysis, probability theory, and mathematical physics.

Second Order Elliptic Equations and Elliptic Systems
Second Order Elliptic Equations and Elliptic Systems

Author: Ya-Zhe Chen

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 266

ISBN-13: 0821819240

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There are two parts to the book. In the first part, a complete introduction of various kinds of a priori estimate methods for the Dirichlet problem of second order elliptic partial differential equations is presented. In the second part, the existence and regularity theories of the Dirichlet problem for linear and nonlinear second order elliptic partial differential systems are introduced. The book features appropriate materials and is an excellent textbook for graduate students. The volume is also useful as a reference source for undergraduate mathematics majors, graduate students, professors, and scientists.

Modular forms and Hecke operators
Modular forms and Hecke operators

Author: A. N. Andrianov V. G. Zhuravlev

Publisher: American Mathematical Soc.

Published: 1995-08-28

Total Pages: 350

ISBN-13: 9780821897621

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The concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms. As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning. Specifically, the theory provided a framework for discovering certain multiplicative properties of the number of integer representations of quadratic forms by quadratic forms. Now that the theory has matured, the time is right for this detailed and systematic exposition of its fundamental methods and results. Features: The book starts with the basics and ends with the latest results, explaining the current status of the theory of Hecke operators on spaces of holomorphic modular forms of integer and half-integer weight congruence-subgroups of integral symplectic groups. Hecke operators are considered principally as an instrument for studying the multiplicative properties of the Fourier coefficients of modular forms. It is the authors' intent that Modular Forms and Hecke Operators help attract young researchers to this beautiful and mysterious realm of number theory.

Modular Forms and Hecke Operators
Modular Forms and Hecke Operators

Author: A. N. Andrianov

Publisher: American Mathematical Soc.

Published: 2016-01-29

Total Pages: 346

ISBN-13: 1470418681

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he concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms. As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning. Specifically, the theory provided a framework for discovering certain multiplicative properties of the number of integer representations of quadratic forms by quadratic forms. Now that the theory has matured, the time is right for this detailed and systematic exposition of its fundamental methods and results. Features: The book starts with the basics and ends with the latest results, explaining the current status of the theory of Hecke operators on spaces of holomorphic modular forms of integer and half-integer weight congruence-subgroups of integral symplectic groups.Hecke operators are considered principally as an instrument for studying the multiplicative properties of the Fourier coefficients of modular forms. It is the authors' intent that Modular Forms and Hecke Operators help attract young researchers to this beautiful and mysterious realm of number theory.

Number Theory 1
Number Theory 1

Author: Kazuya Kato

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 180

ISBN-13: 9780821808634

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This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series.

Algebraic Geometry 1
Algebraic Geometry 1

Author: Kenji Ueno

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 180

ISBN-13: 9780821808627

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By studying algebraic varieties over a field, this book demonstrates how the notion of schemes is necessary in algebraic geometry. It gives a definition of schemes and describes some of their elementary properties.

Best Approximation by Linear Superpositions (approximate Nomography)
Best Approximation by Linear Superpositions (approximate Nomography)

Author: S. I͡A. Khavinson

Publisher: American Mathematical Soc.

Published: 1997-01-01

Total Pages: 188

ISBN-13: 9780821897737

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This book deals with problems of approximation of continuous or bounded functions of several variables by linear superposition of functions that are from the same class and have fewer variables. The main topic is the space of linear superpositions D considered as a sub-space of the space of continous functions C(X) on a compact space X. Such properties as density of D in C(X), its closedness, proximality, etc. are studied in great detail. The approach to these and other problems based on duality and the Hahn-Banach theorem is emphasized. Also, considerable attention is given to the discussion of the Diliberto-Straus algorithm for finding the best approximation of a given function by linear superpositions.

Elliptic Functions and Elliptic Integrals
Elliptic Functions and Elliptic Integrals

Author: Viktor Vasil_evich Prasolov

Publisher: American Mathematical Soc.

Published: 1997-09-16

Total Pages: 202

ISBN-13: 9780821897805

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This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.

Local Properties of Distributions of Stochastic Functionals
Local Properties of Distributions of Stochastic Functionals

Author: Yu. A. Davydov, M. A. Lifshits, andN. V. Smorodina

Publisher: American Mathematical Soc.

Published: 1998-02-10

Total Pages: 208

ISBN-13: 9780821897836

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This book investigates the distributions of functionals defined on the sample paths of stochastic processes. It contains systematic exposition and applications of three general research methods developed by the authors. (i) The method of stratifications is used to study the problem of absolute continuity of distribution for different classes of functionals under very mild smoothness assumptions. It can be used also for evaluation of the distribution density of the functional. (ii) The method of differential operators is based on the abstract formalism of differential calculus and proves to be a powerful tool for the investigation of the smoothness properties of the distributions. (iii) The superstructure method, which is a later modification of the method of stratifications, is used to derive strong limit theorems (in the variation metric) for the distributions of stochastic functionals under weak convergence of the processes. Various application examples concern the functionals of Gaussian, Poisson and diffusion processes as well as partial sum processes from the Donsker-Prokhorov scheme. The research methods and basic results in this book are presented here in monograph form for the first time. The text would be suitable for a graduate course in the theory of stochastic processes and related topics.

Discreteness and Continuity in Problems of Chaotic Dynamics
Discreteness and Continuity in Problems of Chaotic Dynamics

Author: Michael L. Blank

Publisher: American Mathematical Soc.

Published: 1997-01-01

Total Pages: 184

ISBN-13: 9780821897751

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This book presents the study of ergodic properties of so-called chaotic dynamical systems. One of the central topics is the interplay between deterministic and quasi-stochastic behaviour in chaotic dynamics and between properties of continuous dynamical systems and those of their discrete approximations. Using simple examples, the author describes the main phenomena known in chaotic dynamical systems, studying topics such as the operator approach in chaotic dynamics, stochastic stability, and the so-called coupled systems. The last two chapters are devoted to problems of numerical modeling of chaotic dynamics.