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Geometry Of Continued Fractions


Geometry Of Continued Fractions
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Geometry Of Continued Fractions


Geometry Of Continued Fractions
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Author : Oleg Karpenkov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-08-15

Geometry Of Continued Fractions written by Oleg Karpenkov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-08-15 with Mathematics categories.


Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.



Geometry Of Continued Fractions


Geometry Of Continued Fractions
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Author : Oleg N. Karpenkov
language : en
Publisher: Springer Nature
Release Date : 2022-05-28

Geometry Of Continued Fractions written by Oleg N. Karpenkov and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-28 with Mathematics categories.


This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The second edition now includes a geometric approach to Gauss Reduction Theory, classification of integer regular polygons and some further new subjects. Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.



Geometry Of Continued Fractions


Geometry Of Continued Fractions
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Author : Oleg Karpenkov
language : en
Publisher: Springer
Release Date : 2013-11-27

Geometry Of Continued Fractions written by Oleg Karpenkov and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-27 with Mathematics categories.


Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.



Cubic Fields With Geometry


Cubic Fields With Geometry
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Author : Samuel A. Hambleton
language : en
Publisher: Springer
Release Date : 2018-11-07

Cubic Fields With Geometry written by Samuel A. Hambleton and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-07 with Mathematics categories.


The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics.



Analytic Theory Of Continued Fractions


Analytic Theory Of Continued Fractions
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Author : Hubert Stanley Wall
language : en
Publisher: Courier Dover Publications
Release Date : 2018-05-16

Analytic Theory Of Continued Fractions written by Hubert Stanley Wall and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-16 with Mathematics categories.


One of the most authoritative and comprehensive books on continued fractions, this monograph presents a unified theory correlating certain parts and applications of the subject within a larger analytic structure. 1948 edition.



Neverending Fractions


Neverending Fractions
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Author : Jonathan Borwein
language : en
Publisher: Cambridge University Press
Release Date : 2014-07-03

Neverending Fractions written by Jonathan Borwein and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-03 with Mathematics categories.


This introductory text covers a variety of applications to interest every reader, from researchers to amateur mathematicians.



History Of Continued Fractions And Pad Approximants


History Of Continued Fractions And Pad Approximants
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Author : Claude Brezinski
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

History Of Continued Fractions And Pad Approximants written by Claude Brezinski and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.


The history of continued fractions is certainly one of the longest among those of mathematical concepts, since it begins with Euclid's algorithm for the great est common divisor at least three centuries B.C. As it is often the case and like Monsieur Jourdain in Moliere's "Ie bourgeois gentilhomme" (who was speak ing in prose though he did not know he was doing so), continued fractions were used for many centuries before their real discovery. The history of continued fractions and Pade approximants is also quite im portant, since they played a leading role in the development of some branches of mathematics. For example, they were the basis for the proof of the tran scendence of 11' in 1882, an open problem for more than two thousand years, and also for our modern spectral theory of operators. Actually they still are of great interest in many fields of pure and applied mathematics and in numerical analysis, where they provide computer approximations to special functions and are connected to some convergence acceleration methods. Con tinued fractions are also used in number theory, computer science, automata, electronics, etc ...



Orthogonal Polynomials And Continued Fractions


Orthogonal Polynomials And Continued Fractions
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Author : S. V. Khrushchev
language : en
Publisher:
Release Date : 2008

Orthogonal Polynomials And Continued Fractions written by S. V. Khrushchev and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Continued fractions categories.


"This new and exciting historical book tells how Euler introduced the idea of orthogonal polynomials and how he combined them with continued fractions, as well as how Brouncker's formula of 1655 can be derived from Euler's efforts in Special Functions and Orthogonal Polynomials. The most interesting applications of this work are discussed, including the great Markoff's Theorem on the Lagrange spectrum, Abel's Theorem on integration in finite terms, Chebyshev's Theory of Orthogonal Polynomials, and very recent advances in Orthogonal Polynomials on the unit circle. As continued fractions become more important again, in part due to their use in finding algorithms in approximation theory, this timely book revives the approach of Wallis, Brouncker and Euler and illustrates the continuing significance of their influence. A translation of Euler's famous paper 'Continued Fractions, Observation' is included as an Addendum."--Publisher's description.



Multidimensional Continued Fractions


Multidimensional Continued Fractions
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Author : Fritz Schweiger
language : en
Publisher: Oxford University Press, USA
Release Date : 2000

Multidimensional Continued Fractions written by Fritz Schweiger and has been published by Oxford University Press, USA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.


The book gives an up to date overview of various aspects of multidimensional continued fractions, which are here defined through iteration of piecewise fractional linear maps. This includes the algorithms of Jacobi-Perron, Güting, Brun, and Selmer but it also includes continued fractions on simplices which are related to interval exchange maps or the Parry-Daniels map. New classes of subtractive algorithms are also included and the metric properties of these algorithms can be therefore investigated by methods of ergodic theory. The recent connection between multiplicative ergodic theory and Diophantine approximation presented, as well as several results on convergence and Perron's approach to periodicity, which has never appeared in book despite being published in 1907. Further chapters include the basic properties of continued fractions in the complex plane, connections with Hausdorff dimension and the Kuzmin theory for multidimensional maps.



Elements Of Number Theory


Elements Of Number Theory
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Author : I. M. Vinogradov
language : en
Publisher: Courier Dover Publications
Release Date : 2016-01-14

Elements Of Number Theory written by I. M. Vinogradov and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-01-14 with Mathematics categories.


Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1949 edition.