Generalized Notions Of Continued Fractions
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| Author | : Juan Fernández Sánchez |
| Publisher | : CRC Press |
| Total Pages | : 154 |
| Release | : 2023-07-20 |
| Genre | : Mathematics |
| ISBN | : 1000907589 |
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 | : Oleg Karpenkov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 409 |
| Release | : 2013-08-15 |
| Genre | : Mathematics |
| ISBN | : 3642393683 |
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 |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 338 |
| Release | : 2015-09-03 |
| Genre | : Mathematics |
| ISBN | : 1470423154 |
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 | : Fritz Schweiger |
| Publisher | : Oxford University Press, USA |
| Total Pages | : 250 |
| Release | : 2000 |
| Genre | : Mathematics |
| ISBN | : 9780198506867 |
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 | : Tomas Sauer |
| Publisher | : Springer Nature |
| Total Pages | : 275 |
| Release | : 2021-09-06 |
| Genre | : Mathematics |
| ISBN | : 3030843602 |
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 | : Jonathan M. Borwein |
| Publisher | : CRC Press |
| Total Pages | : 372 |
| Release | : 2004-04-12 |
| Genre | : Mathematics |
| ISBN | : 1439864195 |
New mathematical insights and rigorous results are often gained through extensive experimentation using numerical examples or graphical images and analyzing them. Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments. The authors address the role of
| Author | : Jonathan Borwein |
| Publisher | : Cambridge University Press |
| Total Pages | : 223 |
| Release | : 2014-07-03 |
| Genre | : Mathematics |
| ISBN | : 0521186498 |
This introductory text covers a variety of applications to interest every reader, from researchers to amateur mathematicians.
| Author | : William B. Jones |
| Publisher | : Cambridge University Press |
| Total Pages | : 0 |
| Release | : 2009-02-19 |
| Genre | : Mathematics |
| ISBN | : 9780521101523 |
This is an exposition of the analytic theory of continued fractions in the complex domain with emphasis on applications and computational methods.
| Author | : Claude Brezinski |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 556 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642581692 |
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 | : Hubert Stanley Wall |
| Publisher | : Courier Dover Publications |
| Total Pages | : 449 |
| Release | : 2018-05-16 |
| Genre | : Mathematics |
| ISBN | : 0486830446 |
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.
