Multidimensional continued fraction algorithm

• Project/Area Number: 22540037
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Toho University (2012-2013); Suzuka National College of Technology (2010-2011)
• Principal Investigator: YASUTOMI Shin-ichi 東邦大学, 理学部, 教授 (60230231)
• Co-Investigator(Kenkyū-buntansha): TAMURA Jun-ichi 津田塾大学, 付置研究所, 研究員 (90418905)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 連分数 / 単数 / 高次元連分数 / substitution / stepped surface / タイリング / 高次元連分数展開 / 基本単数 / 高次元連分数アルゴリズム / Farey分割 / 同時近似問題

Research Abstract

We were able to suggest a new class of multidimensional continued fraction algorithms, which improve our algorithms that we have studied. The numerical experiments support the extension of classical results of the Lagrange theorem
for the number field of degree less than 7 and the generation of fundamental units in number fields of degree less than 5. We have fundamental results about the relation between the class of algorithms and substitutions on two dimensional stepped surfaces.

Research Products

• [Journal Article] Some aspects of multidimensional continued fraction algorithms (2012)
  • Author(s): Jun-ichi Tamura, Shin-ichi Yasutomi
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 34 Pages: 463-475
  • Peer Reviewed
• [Journal Article] A p-adic phenomenon related to certain integer matrices, and p-adic values of a multidimensional continued fraction (2012)
  • Author(s): Jun-ichi Tamura
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 29 Pages: 1-40
  • Peer Reviewed
• [Journal Article] A NEW ALGORITHM OF CONTINUED FRACTIONS RELATED TO REAL ALGEBRAIC NUMBER FIELDS OF DEGREE≦5 (2011)
  • Author(s): Jun-ichi Tamura, Shin-ichi Yasutomi
  • Journal Title: INTEGERS Volume: 11B Pages: 1-21
  • URL: http://www.integers-ejcnt.org/vol11b.html
  • Peer Reviewed
• [Journal Article] A NEW ALGORITHM OF CONTINUED FRACTIONS RELATED TO REAL ALGEBRAIC NUMBER FIELDS OF DEGREE ≦5 (2011)
  • Author(s): Jun-ichi Tamura, Shin-ichi Yasutomi
  • Journal Title: INTEGERS Volume: 11B Pages: 1-21
  • Peer Reviewed
• [Journal Article] Algebraic Jacobi-Perron algorithm, for biquadratic numbers (2010)
  • Author(s): Jun-ichi Tamura, Shin-ichi Yasutomi
  • Journal Title: AIP Conf. Proc., Amer. Inst. Phys Volume: 1264 Pages: 139-149
  • Peer Reviewed
• [Journal Article] Algebraic Jacobi-Perron algorithm, for biquadratic numbers (2010)
  • Author(s): Tamura, Jun-ichi ; Yasutomi, Shin-ichi
  • Journal Title: AIP Conf. Proc.,Amer.Inst.Phys Volume: 1264 Pages: 139-149
• [Presentation] New classes of multidimensional continued fraction algorithms for algebraic number fields of higher degree and their applications (2014)
  • Author(s): 安冨真一, 田村純一
  • Organizer: Diophantine Analysis and Related Fields
  • Place of Presentation: 筑波大学(つくば市)
  • Year and Date: 2014-03-06
• [Presentation] New classes of algorithms of multidimensional continued fraction and the generation of the group of units of algebraic number fields of higher degree (2014)
  • Author(s): 田村純一
  • Organizer: 研究集会「数論とエルゴード理論」
  • Place of Presentation: 金沢大学サテライトキャンパス(金沢)
  • Year and Date: 2014-02-10
• [Presentation] Multidimensional continued fraction algorithms and the group of units/quartic number fields (2013)
  • Author(s): Jun-ichi Tamura
  • Organizer: Sino-Japanese Workshop on Fractals and Dynamic Systems
  • Place of Presentation: Morningside center of Mathematics(北京)
  • Year and Date: 2013-12-26
• [Presentation] 新しい高次元連分数アルゴリズムとその応用 (2013)
  • Author(s): 安冨真一, 田村純一
  • Organizer: 研究集会「代数的整数論とその周辺」
  • Place of Presentation: 京都大学数理解析研究所(京都)
  • Year and Date: 2013-12-09
• [Presentation] Algebraic units and multidimensional continued fraction algorithm for algebraic number fields of higher degree (1),(2) (2013)
  • Author(s): 田村純一, 安冨真一
  • Organizer: 日本応用数理学会2013年度年会
  • Place of Presentation: アクロス福岡(福岡)
  • Year and Date: 2013-09-10
• [Presentation] A new algorithm of continued fraction and the generation of stepped surface (2013)
  • Author(s): Jun-ichi Tamura
  • Organizer: subtile2013
  • Place of Presentation: CIRM(France)
  • Year and Date: 2013-01-13
• [Presentation] On the generation of a stepped surface by the modified Jacobi-Perron algorithm (2013)
  • Author(s): Shin-ichi Yasutomi
  • Organizer: subtile2013
  • Place of Presentation: CIRM(France)
  • Year and Date: 2013-01-13
• [Presentation] A new multidimensional slow continued fraction algorithm and its application (2012)
  • Author(s): Shin-ichi Yasutomi
  • Organizer: Numeration and substitution 2012
  • Place of Presentation: 数理解析研究所(京都)
  • Year and Date: 2012-06-06
• [Presentation] Pell-Fermat equations over composite fields (2011)
  • Author(s): Jun-ichi Tamura
  • Organizer: Fractal geometry and Ergodic Thoery
  • Place of Presentation: 北京morning side center
  • Year and Date: 2011-12-29
• [Presentation] On multidimensional continued fraction algorithm (2011)
  • Author(s): Shin-ichi Yasutomi
  • Organizer: Fractal geometry and Ergodic Thoery
  • Place of Presentation: 北京morning side center
  • Year and Date: 2011-12-29
• [Presentation] Multidimensional Farey type algorithm (2011)
  • Author(s): Yasutomi, Shin-ichi
  • Organizer: Workshop「数論とエルゴード理論」
  • Place of Presentation: 金沢大学サテライト・プラザ
  • Year and Date: 2011-02-19
• [Presentation] Some aspects of a multicontinued fraction algorithm (2010)
  • Author(s): Yasutomi, Shin-ichi ; Tamura, Jun-ichi
  • Organizer: Functions in Number Theory and Their Probabilistic Aspects
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-12-17
• [Presentation] New classes of multidimensional continued fraction algorithms for algebraic number fields of higher degree and their applications
  • Author(s): 田村 純一, 安冨 真一
  • Organizer: Diophantine Analysis and Related Fields 2014
  • Place of Presentation: 筑波大学（茨城県つくば市）
• [Presentation] New classes of algorithms of multidimensional continued fraction and the generation of the group of units of algebraic number fields of higher degree
  • Author(s): 田村 純一
  • Organizer: 研究集会「数論とエルゴード理論」
  • Place of Presentation: 金沢大学サテライトキャンパス（石川県金沢市）
• [Presentation] Algebraic condition on a generation of the 2- dimensional stepped surface via modified Jacobi-Perron algorithm
  • Author(s): 安冨 真一
  • Organizer: 研究集会「数論とエルゴード理論」
  • Place of Presentation: 金沢大学サテライトキャンパス（石川県金沢市）
• [Presentation] On the genaration of stepped surfaces via modified Jacobi-Perron algorithm
  • Author(s): Shin-ichi YASUTOMI
  • Organizer: Sino-Japanese Workshop on Fractals and Dynamic Systems
  • Place of Presentation: Morningside center of Mathematics(北京)
• [Presentation] Multidimensional continued fraction algorithms and the group of units/quartic number fields
  • Author(s): Jun-ichi Tamura
  • Organizer: Sino-Japanese Workshop on Fractals and Dynamic Systems
  • Place of Presentation: Morningside center of Mathematics(北京)
• [Presentation] 新しい高次元連分数アルゴリズムとその応用
  • Author(s): 安冨 真一, 田村 純一
  • Organizer: 研究集会「代 数的整数論とその周辺」
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
• [Presentation] Algebraic condition on a generation of the 2-dimensional stepped surface via modified Jacobi-Perron algorithm
  • Author(s): 安冨 真一, 古門 麻貴, 伊藤 俊次
  • Organizer: タイル張り力学系とその周辺
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
• [Presentation] Algebraic units and multidimensional continued fraction algorithm for algebraic number fields of higher degree (1)
  • Author(s): 田村 純一, 安冨 真一
  • Organizer: 日本応用数理学会2013年度年会
  • Place of Presentation: アクロス福岡（福岡県福岡市）
• [Presentation] Algebraic units and multidimensional continued fraction algorithm for algebraic number fields of higher degree (1)
  • Author(s): 安冨 真一, 田村 純一
  • Organizer: 日本応用数理学会2013年度年会
  • Place of Presentation: アクロス福岡（福岡県福岡市）
• [Presentation] A new algorithm of continued fraction and the generation of stepped surface
  • Author(s): Jun-ichi Tamura
  • Organizer: Subtile2013
  • Place of Presentation: CIRM(France)
• [Presentation] On the generation of a stepped surface by the modified Jacobi-Perron algorithm
  • Author(s): Shin-ichi Yasutomi
  • Organizer: Subtile2013
  • Place of Presentation: CIRM(France)
• [Presentation] Norm equations of certain biquadratic number fields, and simultaneous approximation of three quadratic irrationalities
  • Author(s): Jun-ichi Tamura
  • Organizer: Numeration and substitution 2012
  • Place of Presentation: 数理解析研究所（京都）
• [Presentation] A new multidimensional slow continued fraction algorithm and its application
  • Author(s): Shin-ichi Yasutomi
  • Organizer: Numeration and substitution 2012
  • Place of Presentation: 数理解析研究所（京都）
• [Remarks]
  • URL: http://www.lab2.toho-u.ac.jp/sci/c/math/yasutomi/mfarey.html
• [Remarks] MultiFarey