Geometry Of Continued Fractions
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| Author | : Oleg Karpenkov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 409 |
| Release | : 2013-08-15 |
| Genre | : Mathematics |
| ISBN | : 3642393683 |
Download Geometry of Continued Fractions Book in PDF, Epub and Kindle
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.
| Author | : Oleg N. Karpenkov |
| Publisher | : Springer Nature |
| Total Pages | : 462 |
| Release | : 2022-05-28 |
| Genre | : Mathematics |
| ISBN | : 3662652773 |
Download Geometry of Continued Fractions Book in PDF, Epub and Kindle
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.
| Author | : Doug Hensley |
| Publisher | : World Scientific |
| Total Pages | : 261 |
| Release | : 2006-03-01 |
| Genre | : Mathematics |
| ISBN | : 9814479438 |
Download Continued Fractions Book in PDF, Epub and Kindle
The Euclidean algorithm is one of the oldest in mathematics, while the study of continued fractions as tools of approximation goes back at least to Euler and Legendre. While our understanding of continued fractions and related methods for simultaneous diophantine approximation has burgeoned over the course of the past decade and more, many of the results have not been brought together in book form. Continued fractions have been studied from the perspective of number theory, complex analysis, ergodic theory, dynamic processes, analysis of algorithms, and even theoretical physics, which has further complicated the situation.This book places special emphasis on continued fraction Cantor sets and the Hausdorff dimension, algorithms and analysis of algorithms, and multi-dimensional algorithms for simultaneous diophantine approximation. Extensive, attractive computer-generated graphics are presented, and the underlying algorithms are discussed and made available.
| Author | : Claude Brezinski |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 556 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642581692 |
Download History of Continued Fractions and Padé Approximants Book in PDF, Epub and Kindle
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 ...
| Author | : A. M. Rockett |
| Publisher | : World Scientific |
| Total Pages | : 202 |
| Release | : 1992-08-01 |
| Genre | : Mathematics |
| ISBN | : 9789810210526 |
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This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of A. Ya. Khintchine's classic of the same title. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included (the continued fraction of e, Ostrowski representations and t-expansions, period lengths of quadratic surds, the general Pell's equation, homogeneous and inhomogeneous diophantine approximation, Hall's theorem, the Lagrange and Markov spectra, asymmetric approximation, etc). Suitable for upper level undergraduate and beginning graduate students, the presentation is self-contained and the metrical results are developed as strong laws of large numbers.
| Author | : Lisa Lorentzen |
| Publisher | : atlantis press |
| Total Pages | : 321 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 9078677074 |
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Continued Fractions consists of two volumes -- Volume 1: Convergence Theory; and Volume 2: Representation of Functions (tentative title), which is expected in 2011. Volume 1 is dedicated to the convergence and computation of continued fractions, while Volume 2 will treat representations of meromorphic functions by continued fractions. Taken together, the two volumes will present the basic continued fractions theory without requiring too much previous knowledge; some basic knowledge of complex functions will suffice. Both new and advanced graduate students of continued fractions shall get a comprehensive understanding of how these infinite structures work in a number of applications, and why they work so well. A varied buffet of possible applications to whet the appetite is presented first, before the more basic but modernized theory is given.This new edition is the result of an increasing interest in computing special functions by means of continued fractions. The methods described in detail are, in many cases, very simple, yet reliable and efficient.
| Author | : Annie A.M. Cuyt |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 430 |
| Release | : 2008-04-12 |
| Genre | : Mathematics |
| ISBN | : 1402069499 |
Download Handbook of Continued Fractions for Special Functions Book in PDF, Epub and Kindle
Special functions are pervasive in all fields of science and industry. The most well-known application areas are in physics, engineering, chemistry, computer science and statistics. Because of their importance, several books and websites (see for instance http: functions.wolfram.com) and a large collection of papers have been devoted to these functions. Of the standard work on the subject, the Handbook of mathematical functions with formulas, graphs and mathematical tables edited by Milton Abramowitz and Irene Stegun, the American National Institute of Standards claims to have sold over 700 000 copies! But so far no project has been devoted to the systematic study of continued fraction representations for these functions. This handbook is the result of such an endeavour. We emphasise that only 10% of the continued fractions contained in this book, can also be found in the Abramowitz and Stegun project or at the Wolfram website!
| Author | : Frances D'Ann Fuquay |
| Publisher | : |
| Total Pages | : 102 |
| Release | : 1974 |
| Genre | : Continued fractions |
| ISBN | : |
Download The Soʹs Geometry of Continued Fractions Book in PDF, Epub and Kindle
| Author | : Jonathan Borwein |
| Publisher | : Cambridge University Press |
| Total Pages | : 223 |
| Release | : 2014-07-03 |
| Genre | : Mathematics |
| ISBN | : 0521186498 |
Download Neverending Fractions Book in PDF, Epub and Kindle
This introductory text covers a variety of applications to interest every reader, from researchers to amateur mathematicians.
| Author | : Hubert Stanley Wall |
| Publisher | : Courier Dover Publications |
| Total Pages | : 449 |
| Release | : 2018-05-16 |
| Genre | : Mathematics |
| ISBN | : 0486830446 |
Download Analytic Theory of Continued Fractions Book in PDF, Epub and Kindle
One of the most authoritative and comprehensive books on the subject of continued fractions, this monograph has been widely used by generations of mathematicians and their students. Dr. Hubert Stanley Wall presents a unified theory correlating certain parts and applications of the subject within a larger analytic structure. Prerequisites include a first course in function theory and knowledge of the elementary properties of linear transformations in the complex plane. Some background in number theory, real analysis, and complex analysis may also prove helpful. The two-part treatment begins with an exploration of convergence theory, addressing continued fractions as products of linear fractional transformations, convergence theorems, and the theory of positive definite continued fractions, as well as other topics. The second part, focusing on function theory, covers the theory of equations, matrix theory of continued fractions, bounded analytic functions, and many additional subjects.
