Metrical Theory Of Continued Fractions

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Metrical Theory Of Continued Fractions

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Author : M. Iosifescu
language : en
Publisher: Springer
Release Date : 2014-03-14

Metrical Theory Of Continued Fractions written by M. Iosifescu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-03-14 with Mathematics categories.

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.

Metrical Theory Of Continued Fractions

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Author : M. Iosifescu
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29

Metrical Theory Of Continued Fractions written by M. Iosifescu 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 2013-06-29 with Mathematics categories.

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.

Metrical And Ergodic Theory Of Continued Fraction Algorithms

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Author : Gabriela Ileana Sebe
language : en
Publisher: Springer Nature
Release Date : 2025-04-22

Metrical And Ergodic Theory Of Continued Fraction Algorithms written by Gabriela Ileana Sebe and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-22 with Mathematics categories.

This monograph presents the work of the authors in metrical theory of continued fractions in the last two decades. The monograph cuts a particular path through this extensive theory and describes the theory in its current form for three families of continued fractions, namely, θ-continued fractions, N-continued fractions, and generalized Rényi continued fractions. The book systematically lays out the required preliminaries, making the book easy to read. This monograph provides a solid introduction into the theory of continued fractions. The book is intended for researchers in metrical theory, as well as advanced graduate students and mathematicians interested in this field.

The Metrical Theory Of Continued Fractions To The Nearest Integer

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Author : Andrew Mansfield Rockett
language : en
Publisher:
Release Date : 1977

The Metrical Theory Of Continued Fractions To The Nearest Integer written by Andrew Mansfield Rockett and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1977 with Continued fractions categories.

The Metrical Theory Of Jacobi Perron Algorithm

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Author : F. Schweiger
language : en
Publisher: Springer
Release Date : 2006-11-15

The Metrical Theory Of Jacobi Perron Algorithm written by F. Schweiger and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.

Continued Fractions

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Author : Andrew M Rockett
language : en
Publisher: World Scientific Publishing Company
Release Date : 1992-08-08

Continued Fractions written by Andrew M Rockett and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-08-08 with Mathematics categories.

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.

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.

Generalized Notions Of Continued Fractions

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Author : Juan Fernández Sánchez
language : en
Publisher: CRC Press
Release Date : 2023-07-20

Generalized Notions Of Continued Fractions written by Juan Fernández Sánchez and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-20 with Mathematics categories.

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

Continued Fractions

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Author : A. M. Rockett
language : en
Publisher: World Scientific
Release Date : 1992-08-01

Continued Fractions written by A. M. Rockett and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-08-01 with Mathematics categories.

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.

Continued Fractions

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Author : Doug Hensley
language : en
Publisher: World Scientific
Release Date : 2006

Continued Fractions written by Doug Hensley and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.

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.