The Arithmetic of Dynamical Systems

Author: J.H. Silverman

Publisher: Springer Science & Business Media

ISBN: 038769904X

Category: Mathematics

Page: 511

View: 2506

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

Advanced Topics in the Arithmetic of Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business Media

ISBN: 1461208513

Category: Mathematics

Page: 528

View: 1382

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.

The Arithmetic of Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business Media

ISBN: 9780387094946

Category: Mathematics

Page: 513

View: 7962

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.

Introduction to Dynamics

Author: I. C. Percival,D. Richards

Publisher: Cambridge University Press

ISBN: 9780521281492

Category: Mathematics

Page: 228

View: 3529

A new approach to dynamics that takes account of recent advances that have wide applications in the sciences and engineering. It introduces the subject at an undergraduate level by means of elementary qualitative theory of differential equations, the geometry of phase curves, and the theory of stability.

Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby

Author: Joseph Auslander,Aimee Johnson,Cesar E. Silva

Publisher: American Mathematical Soc.

ISBN: 1470422999

Category: Dynamical systems and ergodic theory -- Arithmetic and non-Archimedean dynamical systems -- Non-Archimedean Fatou and Julia sets

Page: 316

View: 5781

This volume contains the proceedings of three conferences in Ergodic Theory and Symbolic Dynamics: the Oxtoby Centennial Conference, held from October 30–31, 2010, at Bryn Mawr College; the Williams Ergodic Theory Conference, held from July 27–29, 2012, at Williams College; and the AMS Special Session on Ergodic Theory and Symbolic Dynamics, held from January 17–18, 2014, in Baltimore, MD. This volume contains articles covering a variety of topics in measurable, symbolic and complex dynamics. It also includes a survey article on the life and work of John Oxtoby, providing a source of information about the many ways Oxtoby's work influenced mathematical thought in this and other fields.

An Experimental Introduction to Number Theory

Author: Benjamin Hutz

Publisher: American Mathematical Soc.

ISBN: 1470430975

Category: Number theory

Page: 376

View: 8894

This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems. The mathematical prerequisites for this book are few. Early chapters contain topics such as integer divisibility, modular arithmetic, and applications to cryptography, while later chapters contain more specialized topics, such as Diophantine approximation, number theory of dynamical systems, and number theory with polynomials. Students of all levels will be drawn in by the patterns and relationships of number theory uncovered through data driven exploration.

Dynamical Systems, Number Theory and Applications

'A Festschrift in Honor of Armin Leutbecher''s 80th Birthday'

Author: Thomas Hagen,Florian Rupp,Jürgen Scheurle

Publisher: World Scientific

ISBN: 9814699888

Category: Mathematics

Page: 280

View: 6117

This volume consists of a selection of research-type articles on dynamical systems, evolution equations, analytic number theory and closely related topics. A strong emphasis is on a fair balance between theoretical and more applied work, thus spanning the chasm between abstract insight and actual application. Several of the articles are expected to be in the intersection of dynamical systems theory and number theory. One article will likely relate the topics presented to the academic achievements and interests of Prof. Leutbecher and shed light on common threads among all the contributions. Contents:PrefaceBiographical Note on Armin Leutbecher (S Walcher)Das Jahr 1934 ... (J Fischer)Explicit Expressions for Equivariant Minimal Lagrangian Surfaces (J F Dorfmeister & H Ma)Rational Parameter Rays of the Multibrot Sets (D Eberlein, S Mukherjee & D Schleicher)The Matovich-Pearson Equations Revisited (T Hagen)Diffeomorphisms with Stable Manifolds as Basin Boundaries (S Hayes & Ch Wolf)A New Type of Functional Equations of Euler Products (B Heim)The Hexagonal Lattice and the Epstein Zeta Function (A Henn)On Putative q-Analogues of the Fano Plane (Th Honold & M Kiermaier)Integral Orthogonal Groups (A Krieg)The Role of Fourier Analysis in X-Ray Crystallography (F Rupp & J Scheurle)An Elementary Proof for Joint Continuity of Semiflows (S Schmitz)A Convergent String Method (H Schwetlick & J Zimmer)Variational Symmetries and Pluri-Lagrangian Systems (Y B Suris) Readership: Researchers in algebra and number theory, dynamical systems and analysis and differential equations. Key Features:This versatile book covers state-of-the art work in dynamical systems, analytic number theory and applied analysisIt appeals to a wide audience due to its broad range of topics, highlighting both the breadth and the depth of modern analytical work without losing sight of a common coreKeywords:Dynamical Systems;Evolution Equations;Number Theory;Differential Geometry

Moduli Spaces and Arithmetic Dynamics

Author: Joseph H. Silverman

Publisher: American Mathematical Soc.

ISBN: 0821885030

Category: Mathematics

Page: 140

View: 5317

Iteration of Rational Functions

Complex Analytic Dynamical Systems

Author: Alan F. Beardon

Publisher: Springer Science & Business Media

ISBN: 9780387951515

Category: Mathematics

Page: 280

View: 1045

This book makes available a comprehensive, detailed, and organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The material covered extends from the original memoirs of Fatou and Julia to the recent and important results and methods of Sullivan and Shishikura. Many of the details of the proofs have not occurred in print before. The theory of of dynamical systems and chaos has recently undergone a rapid growth in popularity, in part due to the spectacular computer graphics of Julia sets, fractals, and the Mandelbrot set. This text focuses on the specialized area of complex analytic dynamics, a subject that dates back to 1916 and is currently a very active area in mathematics.

Operator Theoretic Aspects of Ergodic Theory

Author: Tanja Eisner,Bálint Farkas,Markus Haase,Rainer Nagel

Publisher: Springer

ISBN: 3319168983

Category: Mathematics

Page: 628

View: 8657

Stunning recent results by Host–Kra, Green–Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory. Topics include: • an intuitive introduction to ergodic theory • an introduction to the basic notions, constructions, and standard examples of topological dynamical systems • Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand–Naimark theorem • measure-preserving dynamical systems • von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem • strongly and weakly mixing systems • an examination of notions of isomorphism for measure-preserving systems • Markov operators, and the related concept of a factor of a measure preserving system • compact groups and semigroups, and a powerful tool in their study, the Jacobs–de Leeuw–Glicksberg decomposition • an introduction to the spectral theory of dynamical systems, the theorems of Furstenberg and Weiss on multiple recurrence, and applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai,and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg–Sárközy) Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory

Ergodic Theory

with a view towards Number Theory

Author: Manfred Einsiedler,Thomas Ward

Publisher: Springer Science & Business Media

ISBN: 9780857290212

Category: Mathematics

Page: 481

View: 5139

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Dynamical Systems and Ergodic Theory

Author: Mark Pollicott,Michiko Yuri

Publisher: Cambridge University Press

ISBN: 9780521572941

Category: Mathematics

Page: 196

View: 9121

This book is an introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. The authors provide a number of applications, principally to number theory and arithmetic progressions (through Van der Waerden's theorem and Szemerdi's theorem). This text is suitable for advanced undergraduate and beginning graduate students.

Higher Order Fourier Analysis

Author: Terence Tao

Publisher: American Mathematical Soc.

ISBN: 0821889869

Category: Mathematics

Page: 187

View: 963

Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemeredi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems. This book, which is the first monograph in this area, aims to cover all of these topics in a unified manner, as well as to survey some of the most recent developments, such as the application of the theory to count linear patterns in primes. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one's knowledge.

Dynamics, Statistics and Projective Geometry of Galois Fields

Author: V. I. Arnold

Publisher: Cambridge University Press

ISBN: 1139493442

Category: Mathematics

Page: N.A

View: 5261

V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.

Introduction to Applied Nonlinear Dynamical Systems and Chaos

Author: Stephen Wiggins

Publisher: Springer Science & Business Media

ISBN: 1475740670

Category: Mathematics

Page: 672

View: 6514

This volume is an introduction to applied nonlinear dynamics and chaos. The emphasis is on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about their behavior. The new edition has been updated and extended throughout, and contains an extensive bibliography and a detailed glossary of terms.

Introduction to Mathematical Modeling and Chaotic Dynamics

Author: Ranjit Kumar Upadhyay,Satteluri R. K. Iyengar

Publisher: CRC Press

ISBN: 1439898863

Category: Mathematics

Page: 363

View: 6195

Introduction to Mathematical Modeling and Chaotic Dynamics focuses on mathematical models in natural systems, particularly ecological systems. Most of the models presented are solved using MATLAB®. The book first covers the necessary mathematical preliminaries, including testing of stability. It then describes the modeling of systems from natural science, focusing on one- and two-dimensional continuous and discrete time models. Moving on to chaotic dynamics, the authors discuss ways to study chaos, types of chaos, and methods for detecting chaos. They also explore chaotic dynamics in single and multiple species systems. The text concludes with a brief discussion on models of mechanical systems and electronic circuits. Suitable for advanced undergraduate and graduate students, this book provides a practical understanding of how the models are used in current natural science and engineering applications. Along with a variety of exercises and solved examples, the text presents all the fundamental concepts and mathematical skills needed to build models and perform analyses.

Dynamical Systems and Population Persistence

Author: Hal L. Smith,Horst R. Thieme

Publisher: American Mathematical Soc.

ISBN: 082184945X

Category: Mathematics

Page: 405

View: 2034

"The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term. It applies to infinite-dimensional as well as to finite-dimensional dynamical systems, and to discrete-time as well as to continuous-time semiflows. This monograph provides a self-contained treatment of persistence theory that is accessible to graduate students. The key results for deterministic autonomous systems are proved in full detail such as the acyclicity theorem and the tripartition of a global compact attractor. Suitable conditions are given for persistence to imply strong persistence even for nonautonomous semiflows, and time-heterogeneous persistence results are developed using so-called 'average Lyapunov functions'. Applications play a large role in the monograph from the beginning. These include ODE models such as an SEIRS infectious disease in a meta-population and discrete-time nonlinear matrix models of demographic dynamics. Entire chapters are devoted to infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat."--Publisher's description.

Modern Fourier Analysis

Author: Loukas Grafakos

Publisher: Springer

ISBN: 1493912305

Category: Mathematics

Page: 624

View: 4132

This text is aimed at graduate students in mathematics and to interested researchers who wish to acquire an in depth understanding of Euclidean Harmonic analysis. The text covers modern topics and techniques in function spaces, atomic decompositions, singular integrals of nonconvolution type and the boundedness and convergence of Fourier series and integrals. The exposition and style are designed to stimulate further study and promote research. Historical information and references are included at the end of each chapter. This third edition includes a new chapter entitled "Multilinear Harmonic Analysis" which focuses on topics related to multilinear operators and their applications. Sections 1.1 and 1.2 are also new in this edition. Numerous corrections have been made to the text from the previous editions and several improvements have been incorporated, such as the adoption of clear and elegant statements. A few more exercises have been added with relevant hints when necessary.

Combinatorics of Coxeter Groups

Author: Anders Bjorner,Francesco Brenti

Publisher: Springer Science & Business Media

ISBN: 3540275967

Category: Mathematics

Page: 366

View: 8128

Includes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

An Introduction to Number Theory

Author: G. Everest,Thomas Ward

Publisher: Springer Science & Business Media

ISBN: 1852339179

Category: Mathematics

Page: 297

View: 1806

Includes up-to-date material on recent developments and topics of significant interest, such as elliptic functions and the new primality test Selects material from both the algebraic and analytic disciplines, presenting several different proofs of a single result to illustrate the differing viewpoints and give good insight