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Author : Andrew Mansfield Rockett
Release : 1977
Genre : Continued fractions
Kind : eBook
Book Rating : /5 ( reviews)
Book Synopsis The Metrical Theory of Continued Fractions to the Nearest Integer by : Andrew Mansfield Rockett
Download or read book The Metrical Theory of Continued Fractions to the Nearest Integer written by Andrew Mansfield Rockett. This book was released on 1977. Available in PDF, EPUB and Kindle. Book excerpt:
Author : M. Iosifescu
Release : 2014-03-14
Genre : Mathematics
Kind : eBook
Book Rating : 412/5 ( reviews)
Book Synopsis Metrical Theory of Continued Fractions by : M. Iosifescu
Download or read book Metrical Theory of Continued Fractions written by M. Iosifescu. This book was released on 2014-03-14. Available in PDF, EPUB and Kindle. Book excerpt: 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 : M. Iosifescu
Release : 2013-06-29
Genre : Mathematics
Kind : eBook
Book Rating : 408/5 ( reviews)
Book Synopsis Metrical Theory of Continued Fractions by : M. Iosifescu
Download or read book Metrical Theory of Continued Fractions written by M. Iosifescu. This book was released on 2013-06-29. Available in PDF, EPUB and Kindle. Book excerpt: 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 : A. M. Rockett
Release : 1992-08-01
Genre : Mathematics
Kind : eBook
Book Rating : 526/5 ( reviews)
Book Synopsis Continued Fractions by : A. M. Rockett
Download or read book Continued Fractions written by A. M. Rockett. This book was released on 1992-08-01. Available in PDF, EPUB and Kindle. Book excerpt: 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
Release : 2006-03-01
Genre : Mathematics
Kind : eBook
Book Rating : 438/5 ( reviews)
Book Synopsis Continued Fractions by : Doug Hensley
Download or read book Continued Fractions written by Doug Hensley. This book was released on 2006-03-01. Available in PDF, EPUB and Kindle. Book excerpt: 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.
