Generalized Notions of Continued Fractions: Ergodicity and Number Theoretic Applications
暫譯: 廣義連分數的概念：遍歷性與數論應用

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: $6,690
  • 貴賓價: 9.5 折 $6,356
  • 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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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。隨著時間的推移，數學天才如歐幾里得（Euclid）、阿里雅巴塔（Aryabhata）、費波那契（Fibonacci）、邦貝利（Bombelli）、華利斯（Wallis）、惠更斯（Huygens）和歐拉（Euler）等對這一著名理論的發展做出了重要貢獻，並且這一理論至今仍在不斷演變，特別是在連結數學不同領域的方式上。

本書的主要讀者為研究生和高級研究人員，受到隨著1950年代以來的遍歷理論（ergodic theory）與數論（number theory）之間迷人相互關係的啟發。它探討了古典連分數的幾個推廣和擴展，包括廣義的Lehner連分數、簡單連分數和Hirzebruch-Jung連分數。在推導出每個基礎變換在[0,1]上的不變遍歷測度後，顯示出任何著名的公式，追溯至Khintchine和Levy，都可以轉移到更一般的情境中。補充這些結果，還計算了變換的熵，並分析了動態系統對[0,1]²的自然擴展。

特色
- 適合研究生和高級研究人員
- 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一些基本背景和初步結果，使其成為一本自足的著作

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)

胡安·費爾南德斯·桑切斯於2010年在西班牙阿爾梅里亞大學獲得數學博士學位。他的研究興趣包括依賴模型與聯結、動態系統、奇異函數和數論。

赫羅尼莫·洛佩斯-薩拉薩爾·科德斯在西班牙馬德里康普頓斯大學（Universidad Complutense de Madrid）在何塞·瑪麗亞·馬丁內斯·安塞米爾教授和索科羅·龐特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限維全純性和線性性。

胡安·B·塞奧安·塞普爾維達於2005年在西班牙卡迪斯大學（Universidad de Cádiz）與德國卡爾斯魯厄大學（Universität Karlsruhe）共同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。他的主要興趣包括實數與複數分析、算子理論、數論、數學建模、數學生物學、巴拿赫空間幾何、數學史和線性性。他是200多篇科學出版物的作者，包括幾本書籍。他目前是馬德里康普頓斯大學的教授，並擔任高級數學碩士課程的主任。

沃爾夫岡·特魯茨尼希於2006年在奧地利維也納科技大學獲得博士學位。他目前是薩爾茨堡巴黎洛德龍大學（Paris Lodron University Salzburg, PLUS）的隨機過程教授及IDA實驗室主任，主要從事依賴模型和非參數統計的研究，並定期研究動態系統、分形和遍歷理論。

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