Fractions To Be Continued

Author : Andrew J. Simoson
Publisher : American Mathematical Soc.
Page : 480 pages
File Size : 54,8 Mb
Release : 2021-04-30
Category : Education
ISBN : 9781470461287

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Exploring Continued Fractions: From the Integers to Solar Eclipses by Andrew J. Simoson Pdf

There is a nineteen-year recurrence in the apparent position of the sun and moon against the background of the stars, a pattern observed long ago by the Babylonians. In the course of those nineteen years the Earth experiences 235 lunar cycles. Suppose we calculate the ratio of Earth's period about the sun to the moon's period about Earth. That ratio has 235/19 as one of its early continued fraction convergents, which explains the apparent periodicity. Exploring Continued Fractions explains this and other recurrent phenomena—astronomical transits and conjunctions, lifecycles of cicadas, eclipses—by way of continued fraction expansions. The deeper purpose is to find patterns, solve puzzles, and discover some appealing number theory. The reader will explore several algorithms for computing continued fractions, including some new to the literature. He or she will also explore the surprisingly large portion of number theory connected to continued fractions: Pythagorean triples, Diophantine equations, the Stern-Brocot tree, and a number of combinatorial sequences. The book features a pleasantly discursive style with excursions into music (The Well-Tempered Clavier), history (the Ishango bone and Plimpton 322), classics (the shape of More's Utopia) and whimsy (dropping a black hole on Earth's surface). Andy Simoson has won both the Chauvenet Prize and Pólya Award for expository writing from the MAA and his Voltaire's Riddle was a Choice magazine Outstanding Academic Title. This book is an enjoyable ramble through some beautiful mathematics. For most of the journey the only necessary prerequisites are a minimal familiarity with mathematical reasoning and a sense of fun.

Author : Aleksandr I?Akovlevich Khinchin,Herbert Eagle
Publisher : Courier Corporation
Page : 116 pages
File Size : 51,6 Mb
Release : 1997-05-14
Category : Mathematics
ISBN : 0486696308

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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 : Bowen Kerins,Darryl Yong,Al Cuoco,Glenn Stevens
Publisher : American Mathematical Soc.
Page : 126 pages
File Size : 45,7 Mb
Release : 2021-09-17
Category : Education
ISBN : 9781470462932

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Fractions: To Be Continued by Bowen Kerins,Darryl Yong,Al Cuoco,Glenn Stevens Pdf

This is the eighth book in the Teacher Program Series. Each book includes a full course in a mathematical focus topic. The topic for this book is the study of continued fractions, including important results involving the Euclidean algorithm, the golden ratio, and approximations to rational and irrational numbers. The course includes 14 problem sets designed for low-threshold, high-ceiling access to the topic, building on one another as the concepts are explored. The book also includes solutions for all the main problems and detailed facilitator notes for those wanting to use this book with students at any level. The course is based on one delivered at the Park City Math Institute in Summer 2018.

Author : Andrew M Rockett,Peter Szüsz
Publisher : World Scientific Publishing Company
Page : 196 pages
File Size : 45,7 Mb
Release : 1992-08-08
Category : Mathematics
ISBN : 9789813103412

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Continued Fractions by Andrew M Rockett,Peter Szüsz 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. Request Inspection Copy

Author : Annie A.M. Cuyt,Vigdis Petersen,Brigitte Verdonk,Haakon Waadeland,William B. Jones
Publisher : Springer Science & Business Media
Page : 430 pages
File Size : 41,9 Mb
Release : 2008-04-12
Category : Mathematics
ISBN : 9781402069499

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Handbook of Continued Fractions for Special Functions by Annie A.M. Cuyt,Vigdis Petersen,Brigitte Verdonk,Haakon Waadeland,William B. Jones Pdf

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 : Jonathan Borwein,Alf van der Poorten,Jeffrey Shallit,Wadim Zudilin
Publisher : Cambridge University Press
Page : 223 pages
File Size : 40,8 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 : M. Iosifescu,Cor Kraaikamp
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 44,6 Mb
Release : 2002-09-30
Category : Mathematics
ISBN : 1402008929

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

The book is essentially based on recent work of the authors. In order to unify and generalize the results obtained so far, new concepts have been introduced, e.g., an infinite order chain representation of the continued fraction expansion of irrationals, the conditional measures associated with, and the extended random variables corresponding to that representation. Also, such procedures as singularization and insertion allow to obtain most of the continued fraction expansions related to the regular continued fraction expansion. The authors present and prove with full details for the first time in book form, the most recent developments in solving the celebrated 1812 Gauss' problem which originated the metrical theory of continued fractions. At the same time, they study exhaustively the Perron-Frobenius operator, which is of basic importance in this theory, on various Banach spaces including that of functions of bounded variation on the unit interval. The book is of interest to research workers and advanced Ph.D. students in probability theory, stochastic processes and number theory.

Author : Tomas Sauer
Publisher : Springer Nature
Page : 275 pages
File Size : 48,9 Mb
Release : 2021-09-06
Category : Mathematics
ISBN : 9783030843601

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Continued Fractions and Signal Processing by Tomas Sauer Pdf

Besides their well-known value in number theory, continued fractions are also a useful tool in modern numerical applications and computer science. The goal of the book is to revisit the almost forgotten classical theory and to contextualize it for contemporary numerical applications and signal processing, thus enabling students and scientist to apply classical mathematics on recent problems. The books tries to be mostly self-contained and to make the material accessible for all interested readers. This provides a new view from an applied perspective, combining the classical recursive techniques of continued fractions with orthogonal problems, moment problems, Prony’s problem of sparse recovery and the design of stable rational filters, which are all connected by continued fractions.

Author : C. D. Olds
Publisher : Unknown
Page : 162 pages
File Size : 55,6 Mb
Release : 2013
Category : Electronic
ISBN : OCLC:1120890215

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Continued Fractions by C. D. Olds Pdf

Author : Oleg Karpenkov
Publisher : Springer Science & Business Media
Page : 409 pages
File Size : 41,8 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 : A. M. Rockett,Peter Szsz
Publisher : World Scientific
Page : 202 pages
File Size : 47,5 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 : Doug Hensley
Publisher : World Scientific
Page : 261 pages
File Size : 40,7 Mb
Release : 2006
Category : Mathematics
ISBN : 9789812774682

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Continued Fractions by Doug Hensley Pdf

This book is the first authoritative and up-to-date survey of the history of Iraq from earliest times to the present in any language. It presents a concise narrative of the rich and varied history of this land, drawing on political, social, economic, artistic, technological, and intellectual material. It also includes excerpts from works of ancient, medieval, and modern literature written in Iraq, some of which are translated for the first time into English. The final chapters provide an introduction to the history of archaeology in Iraq, set in the wider context of the development of archaeology into a scientific discipline. A special section highlights selected objects from the Iraq Museum, with emphasis on their cultural significance and current status in the aftermath of the looting in April 2003. The last chapter offers a unique guide to the complex international and national legal regimes for the protection of cultural heritage. The American-led invasion and occupation of Iraq are a turning point in Iraq's modern history, with important cultural consequences for all periods of its past. For all who seek to understand more fully the current situation, this book includes discussion of cultural and legal issues of the war and occupation, placing recent events in their full context.

Author : Claude Brezinski
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 47,8 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 : Fritz Schweiger
Publisher : Oxford University Press, USA
Page : 250 pages
File Size : 52,5 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 : Lisa Lorentzen,Haakon Waadeland
Publisher : atlantis press
Page : 321 pages
File Size : 48,9 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.