Generalized Notions Of Continued Fractions

Author : Juan Fernández Sánchez,Jerónimo López-Salazar Codes,Juan B. Seoane Sepúlveda,Wolfgang Trutschnig
Publisher : CRC Press
Page : 154 pages
File Size : 52,6 Mb
Release : 2023-07-20
Category : Mathematics
ISBN : 9781000907582

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Generalized Notions of Continued Fractions by Juan Fernández Sánchez,Jerónimo López-Salazar Codes,Juan B. Seoane Sepúlveda,Wolfgang Trutschnig Pdf

Ancient times witnessed the origins of the theory of continued fractions. Throughout time, mathematical geniuses such as Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, or Euler have made significant contributions to the development of this famous theory, and it continues to evolve today, especially as a means of linking different areas of mathematics. This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed. Features Suitable for graduate students and senior researchers Written by international senior experts in number theory Contains the basic background, including some elementary results, that the reader may need to know before hand, making it a self-contained volume

Author : A. M. Rockett,Peter Szsz
Publisher : World Scientific
Page : 202 pages
File Size : 48,8 Mb
Release : 1992-08-01
Category : Mathematics
ISBN : 9810210523

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Continued Fractions by A. M. Rockett,Peter Szsz Pdf

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,Haakon Waadeland
Publisher : atlantis press
Page : 321 pages
File Size : 42,9 Mb
Release : 2008
Category : Mathematics
ISBN : 9789078677079

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Continued Fractions by Lisa Lorentzen,Haakon Waadeland Pdf

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 : Aleksandr I?Akovlevich Khinchin,Herbert Eagle
Publisher : Courier Corporation
Page : 114 pages
File Size : 42,8 Mb
Release : 1997-05-14
Category : Mathematics
ISBN : 9780486696300

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Continued Fractions by Aleksandr I?Akovlevich Khinchin,Herbert Eagle Pdf

Elementary-level text by noted Soviet mathematician offers superb introduction to positive-integral elements of theory of continued fractions. Clear, straightforward presentation of the properties of the apparatus, the representation of numbers by continued fractions, and the measure theory of continued fractions. 1964 edition. Prefaces.

Author : Oleg Karpenkov
Publisher : Springer Science & Business Media
Page : 405 pages
File Size : 51,5 Mb
Release : 2013-08-15
Category : Mathematics
ISBN : 9783642393686

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Geometry of Continued Fractions by Oleg Karpenkov Pdf

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 : Graham Everest, Alf van der Poorten,Igor Shparlinski,Thomas Ward
Publisher : American Mathematical Soc.
Page : 318 pages
File Size : 54,7 Mb
Release : 2015-09-03
Category : Electronic
ISBN : 9781470423155

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Recurrence Sequences by Graham Everest, Alf van der Poorten,Igor Shparlinski,Thomas Ward Pdf

Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.

Author : Hubert Stanley Wall
Publisher : Courier Dover Publications
Page : 449 pages
File Size : 48,8 Mb
Release : 2018-05-16
Category : Mathematics
ISBN : 9780486823690

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Analytic Theory of Continued Fractions by Hubert Stanley Wall Pdf

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.

Author : Claude Brezinski
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 53,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642581694

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History of Continued Fractions and Padé Approximants by Claude Brezinski Pdf

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 : Carl Douglas Olds
Publisher : Mathematical Association of America (MAA)
Page : 186 pages
File Size : 46,6 Mb
Release : 1963
Category : Mathematics
ISBN : PSU:000027139950

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Continued Fractions by Carl Douglas Olds Pdf

Author : S. Clement Cooper
Publisher : CRC Press
Page : 402 pages
File Size : 42,8 Mb
Release : 2020-12-17
Category : Mathematics
ISBN : 9781000154146

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Continued Fractions and Orthogonal Functions by S. Clement Cooper Pdf

This reference - the proceedings of a research conference held in Loen, Norway - contains information on the analytic theory of continued fractions and their application to moment problems and orthogonal sequences of functions. Uniting the research efforts of many international experts, this volume: treats strong moment problems, orthogonal polynomials and Laurent polynomials; analyses sequences of linear fractional transformations; presents convergence results, including truncation error bounds; considers discrete distributions and limit functions arising from indeterminate moment problems; discusses Szego polynomials and their applications to frequency analysis; describes the quadrature formula arising from q-starlike functions; and covers continued fractional representations for functions related to the gamma function.;This resource is intended for mathematical and numerical analysts; applied mathematicians; physicists; chemists; engineers; and upper-level undergraduate and agraduate students in these disciplines.

Author : Charles Godat Moore
Publisher : Unknown
Page : 112 pages
File Size : 45,8 Mb
Release : 1964
Category : Mathematics
ISBN : UOM:49015000671934

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An Introduction to Continued Fractions by Charles Godat Moore Pdf

Author : M. Iosifescu,Cor Kraaikamp
Publisher : Springer Science & Business Media
Page : 397 pages
File Size : 46,7 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789401599405

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Metrical Theory of Continued Fractions by M. Iosifescu,Cor Kraaikamp Pdf

This monograph is intended to be a complete treatment of the metrical the ory of the (regular) continued fraction expansion and related representations of real numbers. We have attempted to give the best possible results known so far, with proofs which are the simplest and most direct. The book has had a long gestation period because we first decided to write it in March 1994. This gave us the possibility of essentially improving the initial versions of many parts of it. Even if the two authors are different in style and approach, every effort has been made to hide the differences. Let 0 denote the set of irrationals in I = [0,1]. Define the (reg ular) continued fraction transformation T by T (w) = fractional part of n 1/w, w E O. Write T for the nth iterate of T, n E N = {O, 1, ... }, n 1 with TO = identity map. The positive integers an(w) = al(T - (W)), n E N+ = {1,2··· }, where al(w) = integer part of 1/w, w E 0, are called the (regular continued fraction) digits of w. Writing . for arbitrary indeterminates Xi, 1 :::; i :::; n, we have w = lim [al(w),··· , an(w)], w E 0, n--->oo thus explaining the name of T. The above equation will be also written as w = lim [al(w), a2(w),···], w E O.

Author : Jonathan Borwein,Alf van der Poorten,Jeffrey Shallit,Wadim Zudilin
Publisher : Cambridge University Press
Page : 223 pages
File Size : 45,5 Mb
Release : 2014-07-03
Category : Mathematics
ISBN : 9780521186490

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Neverending Fractions by Jonathan Borwein,Alf van der Poorten,Jeffrey Shallit,Wadim Zudilin Pdf

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

Author : Fritz Schweiger
Publisher : Oxford University Press, USA
Page : 250 pages
File Size : 44,8 Mb
Release : 2000
Category : Mathematics
ISBN : 0198506864

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Multidimensional Continued Fractions by Fritz Schweiger Pdf

Mathematician Fritz Schweiger, whose academic affiliation is not provided, provides an introduction to a field of research that has seen remarkable progress in recent decades, concentrating on multidimensional continued fractions which can be described by fractional linear maps or equivalently by a set of (n + 1) x (n + 1) matrices. Addressing the question of periodicity, he refines the problem of convergence to the question of whether these algorithms give "good" simultaneous Diophantine approximations. He notes that these algorithms are not likely to provide such "good" approximations which satisfy the n-dimensional Dirichlet property. Also studied are the ergodic properties of these maps. Annotation copyrighted by Book News Inc., Portland, OR

Author : L. Lorentzen,H. Waadeland
Publisher : North Holland
Page : 636 pages
File Size : 40,6 Mb
Release : 1992-11-08
Category : Computers
ISBN : UOM:39076001255129

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Continued Fractions with Applications by L. Lorentzen,H. Waadeland Pdf

This book is aimed at two kinds of readers: firstly, people working in or near mathematics, who are curious about continued fractions; and secondly, senior or graduate students who would like an extensive introduction to the analytic theory of continued fractions. The book contains several recent results and new angles of approach and thus should be of interest to researchers throughout the field. The first five chapters contain an introduction to the basic theory, while the last seven chapters present a variety of applications. Finally, an appendix presents a large number of special continued fraction expansions. This very readable book also contains many valuable examples and problems.