Generalized Notions of Continued Fractions: Ergodicity and Number Theoretic Applications

Fernández Sánchez, Juan, López-Salazar Codes, Jerónimo, Seoane Sepúlveda, Juan B.

Generalized Notions of Continued Fractions: Ergodicity and Number Theoretic Applications
  • 出版商: CRC
  • 出版日期: 2023-07-20
  • 售價: $6,680
  • 貴賓價: 9.5 折 $6,346
  • 語言: 英文
  • 頁數: 142
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 103251678X
  • ISBN-13: 9781032516783

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相關主題

商品描述

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

商品描述(中文翻譯)

古代見證了連分數理論的起源。歷史上，數學天才如歐幾里德、阿耶巴塔、費波那契、邦貝利、華利斯、休謹斯或歐拉對這一著名理論的發展做出了重要貢獻，並且它至今仍在不斷演進，特別是作為連接數學不同領域的手段。

這本書的主要讀者是研究生和高級研究人員，它受到自1950年代以來確立的遞迴動力學理論和數論之間迷人的相互關係的啟發。它檢視了幾個對經典連分數的推廣和擴展，包括廣義的Lehner、簡單和Hirzebruch-Jung連分數。在推導出[0,1]上每個底層變換的不變遞迴測度後，證明了任何著名的公式，追溯到Khintchine和Levy，都可以適用於更一般的情況。作為補充，計算了變換的熵，並分析了動力系統對[0,1]2的自然擴展。

特點：
- 適合研究生和高級研究人員閱讀
- 由國際數論領域的資深專家撰寫
- 包含讀者可能需要事先了解的基本背景，包括一些初步結果，使其成為一本自成一冊的專書

作者簡介

Juan Fernández Sánchez earned his Ph.D. in mathematics from the University of Almería (Spain) in 2010. His research interests are in dependence modeling and copulas, dynamical systems, singular functions, and number theory.

Jerónimo López-Salazar Codes completed his doctoral work under the supervision of Professors José María Martínez Ansemil and Socorro Ponte at Universidad Complutense de Madrid (Spain) and obtained his Ph.D. degree in 2013. He currently works at Universidad Politécnica de Madrid (Spain). His research is mainly devoted to infinite dimensional holomorphy and lineability.

Juan B. Seoane Sepúlveda earned his first Ph.D. at the Universidad de Cádiz (Spain) jointly with Universität Karlsruhe (Germany) in 2005. His received his second Ph.D. at Kent State University (Kent, Ohio, USA) in 2006. His main interests include Real and Complex Analysis, Operator Theory, Number Theory, Mathematical Modeling, Mathematical Biology, Geometry of Banach spaces, History of Mathematics, and Lineability. He is the author of over 200 scientific publications, including several books. He is currently a professor at Universidad Complutense de Madrid, where he also holds the position of director of the Master's in Advanced Mathematics.

Wolfgang Trutschnig obtained his Ph.D. at the Vienna University of Technology, Austria, in 2006. He is currently the professor for stochastics and director of the IDA Lab at the Paris Lodron University Salzburg (PLUS) and mainly works in dependence modeling and nonparametric statistics with regular excursions to dynamical systems, fractals and ergodic theory.

作者簡介(中文翻譯)

Juan Fernández Sánchez於2010年在西班牙阿爾梅里亞大學獲得數學博士學位。他的研究興趣包括相依建模和copulas、動態系統、奇異函數和數論。

Jerónimo López-Salazar Codes在西班牙馬德里康普頓斯大學（Universidad Complutense de Madrid）在José María Martínez Ansemil教授和Socorro Ponte教授的指導下完成了博士研究，並於2013年獲得博士學位。他目前在西班牙馬德里理工大學（Universidad Politécnica de Madrid）工作。他的研究主要集中在無窮維全純性和線性。

Juan B. Seoane Sepúlveda於2005年在西班牙加的斯大學（Universidad de Cádiz）和德國卡爾斯魯厄大學（Universität Karlsruhe）聯合獲得第一個博士學位。他於2006年在美國俄亥俄州肯特州立大學（Kent State University）獲得第二個博士學位。他的主要研究領域包括實分析和復分析、算子理論、數論、數學建模、數學生物學、巴拿赫空間的幾何、數學歷史和線性。他是200多篇科學論文的作者，包括幾本書。他目前是馬德里康普頓斯大學的教授，同時擔任高等數學碩士課程的主任職務。

Wolfgang Trutschnig於2006年在奧地利維也納科技大學獲得博士學位。他目前是巴黎洛德隆大學薩爾茨堡分校（Paris Lodron University Salzburg，簡稱PLUS）的隨機過程教授和IDA實驗室主任，主要從事相依建模和非參數統計學的研究，並時常涉足動態系統、分形和遞歸理論。

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