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An ergodic study of algorithms

Research Project

Project/Area Number 21340027
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKeio University

Principal Investigator

NAKADA Hitoshi  慶應義塾大学, 理工学部, 教授 (40118980)

Co-Investigator(Kenkyū-buntansha) TAMURA Akihisa  慶應義塾大学, 理工学部, 教授 (50217189)
ATSUJI Atsushi  慶應義塾大学, 経済学部, 教授 (00221044)
TAMURA Yozo  慶應義塾大学, 理工学部, 教授 (50171905)
NATSUI Rie  日本女子大学, 理学部, 准教授 (60398633)
KATSURA Takeshi  慶應義塾大学, 理工学部, 准教授 (50513298)
MIYAZAKI Takuya  慶應義塾大学, 理工学部, 准教授 (10301409)
HAMA Masaki  文京学院短期大学, 英語科, 准教授 (90389875)
ISHIKAWA Shiro  慶應義塾大学, 理工学部, 准教授 (10051913)
FUKUYAMA Katsushi  神戸大学, 大学院・理学研究科, 教授 (60218956)
Co-Investigator(Renkei-kenkyūsha) YOSHIDA Masamichi  大阪市立大学, 大学院・理学研究科, 准教授 (60264793)
EI Hiromi  弘前大学, 大学院・理学研究科, 助教 (60333051)
MORITA Takehiko  大阪大学, 大学院・理学研究科, 教授 (00192782)
INOUE Kae  慶應義塾大学, 薬学部, 専任講師 (90621011)
Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥10,400,000 (Direct Cost: ¥8,000,000、Indirect Cost: ¥2,400,000)
Fiscal Year 2011: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2010: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2009: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Keywordsエルゴード理論 / 連分数 / ユークリッドアルゴリズム / Rauzy induction / Diophantus近似 / 国際研究者交流 / オランダ:フランス:韓国:イスラエル / 非斉次ディオファンタス近似 / 国際情報交換 / オランダ:フランス:韓国:イスラエル:イタリア
Research Abstract

We studied some properties of algorithms, mostly appeared in number theoretic algorithms, as measurable dynamical systems. In particular, we focused problems related to the projects in below and got a number of new results. The projects we focused are the

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (52 results)

All 2012 2011 2010 2009 Other

All Journal Article (24 results) (of which Peer Reviewed: 24 results) Presentation (28 results)

  • [Journal Article] A metric discrepancy result for lacunary sequences2012

    • Author(s)
      Katusi Fukuyama, Tetsujin Watada
    • Journal Title

      Proceedings of American Mathematical Society

      Volume: vol. 49 Issue: 3 Pages: 52-78

    • DOI

      10.1090/s0002-9939-2011-10940-7

    • NAID

      120005980789

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Metric discrepancy results for Erdos-Fortet sequence2012

    • Author(s)
      Katusi Fukuyama
    • Journal Title

      Studia Scientiarum Mathematicarum Hungarica

      Volume: vol. 140 Issue: 1 Pages: 749-754

    • DOI

      10.1556/sscmath.2011.1186

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Invariant measures for subshifts arising from substitutions of some primitive components2011

    • Author(s)
      Masaki Hama, Hisatoshi Yuasa
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: vol. 40 Pages: 279-312

    • URL

      http://hmj2.math.sci.hokudai.ac.jp/1370

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the Existence of a Darling-Kac Set for the Renormalized Rauzy map2011

    • Author(s)
      Kae Inoue, Hitoshi Nakada
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: vol. 34 Issue: 2 Pages: 289-302

    • DOI

      10.3836/tjm/1327931385

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On Schweiger's problems on fully subtractive algorithms2011

    • Author(s)
      Robbert Fokkink, Cor Kraaikamp, Hitoshi Nakada
    • Journal Title

      Israel Journal of Mathematics

      Volume: vol. 186 Issue: 1 Pages: 285-302

    • DOI

      10.1007/s11856-011-0140-3

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Metric inhomogeneous Diophantine approximations on the field of formal Laurent series2011

    • Author(s)
      Dong Han Kim, Hitoshi Nakada
    • Journal Title

      Acta Arithmetica

      Volume: vol. 150 Issue: 2 Pages: 129-142

    • DOI

      10.4064/aa150-2-3

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the Legendre constant of α-continued fractions2011

    • Author(s)
      Rie Natsui
    • Journal Title

      Journal of Number Theory

      Volume: vol. 131 Issue: 3 Pages: 487-507

    • DOI

      10.1016/j.jnt.2010.09.008

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] A metric discrepancy result for a lacunary sequence with small gap2011

    • Author(s)
      Katusi Fukuyama
    • Journal Title

      Monatshefte fur Mathematik

      Volume: vol. 162 Issue: 3 Pages: 277-288

    • DOI

      10.1007/s00605-009-0185-0

    • NAID

      120005980780

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] A central limittheorem for trigonometric series with bounded gaps2011

    • Author(s)
      Katusi Fukuyama
    • Journal Title

      Probability Theory and Related Fields

      Volume: vol. 149 Issue: 1-2 Pages: 139-148

    • DOI

      10.1007/s00440-009-0245-3

    • NAID

      120005980785

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the Existence of a Darling-Kac Set for the Renormalized Rauzy Map2011

    • Author(s)
      Kae Inoue
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 34 Pages: 289-302

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Invariant measures for subshifts arising from substitutions of some primitive components2011

    • Author(s)
      Masaki Hama
    • Journal Title

      Hokkaido Mathemtical Journal

      Volume: 40 Pages: 279-312

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the Legendre constant of α-continued fractions2011

    • Author(s)
      夏井利恵
    • Journal Title

      Journal of Number Theory

      Volume: 131 Pages: 487-507

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A central limit theorem for trigonometric series with bounded gaps2011

    • Author(s)
      福山克司
    • Journal Title

      Probability Theory and related Fields

      Volume: 149 Pages: 139-148

    • NAID

      120005980785

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A metric discrepancy result for a lacunary sequence with small gaps2011

    • Author(s)
      福山克司
    • Journal Title

      Monatshefte fur Mathematik

      Volume: 162 Pages: 277-288

    • NAID

      120005980780

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Pure Gaussian limit distributions of trigonometric series with bounded gaps2010

    • Author(s)
      Katusi Fukuyama
    • Journal Title

      Acta Mthematica Hungarica

      Volume: vol. 129 Issue: 4 Pages: 303-313

    • DOI

      10.1007/s10474-010-0034-6

    • NAID

      120005980772

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] A law of the iterated logarithm for discrepancies: non-constant lim sup2010

    • Author(s)
      Katusi Fukuyama
    • Journal Title

      Monatshefte fur Mathematik

      Volume: vol. 160 Issue: 2 Pages: 141-149

    • DOI

      10.1007/s00605-008-0062-2

    • NAID

      120005980779

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the number ofomitted values by a meromorphic finctionof finite energy and heat diffusions2010

    • Author(s)
      Katusi Fukuyama
    • Journal Title

      Journal of Geometric Analysis

      Volume: vol. 20 Issue: 4 Pages: 1008-1025

    • DOI

      10.1007/s12220-010-9131-6

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Generalized Brjuno functions associated to α-continued fractions2010

    • Author(s)
      Laura Luzzi, Stefano Marmi, Hitoshi Nakada, Rie Natsui
    • Journal Title

      Journal of Approximation Theory

      Volume: vol. 162 Issue: 1 Pages: 24-41

    • DOI

      10.1016/j.jat.2009.02.004

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Large deviation for stochastic line integral as L^<p->-recurrents2010

    • Author(s)
      Shigeo Kusuoka, Kazumasa Kuwada, Yozo Tamura
    • Journal Title

      Probability Theory and Related Fields

      Volume: vol. 147 Issue: 3-4 Pages: 649-674

    • DOI

      10.1007/s00440-009-0219-5

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] A law of the iterated logarithm for discrepancies : non-constant limsup2010

    • Author(s)
      福山克司
    • Journal Title

      Monatshefte fur Mathematik

      Volume: 160 Pages: 141-149

    • NAID

      120005980779

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Pure gaussian limit distributions of trigonometric series with bounded gaps2010

    • Author(s)
      福山克司
    • Journal Title

      Acta Mathematica Hungarica

      Volume: 129 Pages: 303-313

    • NAID

      120005980772

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the number of omitted values by a meromorphic function of finite energy and heat diffusions2010

    • Author(s)
      厚地淳
    • Journal Title

      Journal of Geometric Analysis

      Volume: 20 Pages: 1008-1025

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Generalized Brjuno functions associated to α-continued fractions2010

    • Author(s)
      Luzzi, Laura
    • Journal Title

      Journal of Approximation Theory 162

      Pages: 24-41

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Schweiger'sproblem on fully subtractive algorithms

    • Author(s)
      Fokkink, Robbert
    • Journal Title

      Israel Journal of Mathematics (掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] On the dual of Rauzy induction2012

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Workshop on Infinite Ergodic Theory
    • Place of Presentation
      University of Surrey 英国
    • Year and Date
      2012-05-30
    • Related Report
      2011 Final Research Report
  • [Presentation] Another approach to Nevanlinna theory from diffusion processes2012

    • Author(s)
      Atsushi Atsuji
    • Organizer
      International Conference in Navanlinna Theory and Complex Geometry
    • Place of Presentation
      University of Notre Dame 米国
    • Year and Date
      2012-03-17
    • Related Report
      2011 Final Research Report
  • [Presentation] Another approach to Nevanlinna theory from diffusion processes2012

    • Author(s)
      厚地淳
    • Organizer
      International Conference in Nevanlinna Theory and Complex Geometry
    • Place of Presentation
      University of Notre Dame (USA)(招待講演)
    • Year and Date
      2012-03-17
    • Related Report
      2011 Annual Research Report
  • [Presentation] On costs of some Euclidean type algorithms over F_q[X]^32012

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Dynamics of complex systems 2012
    • Place of Presentation
      北海道大学理学部
    • Year and Date
      2012-03-06
    • Related Report
      2011 Final Research Report
  • [Presentation] On costs of some Euclidean type algorithms over F_q[X]^32012

    • Author(s)
      仲田均
    • Organizer
      Dynamics of complex systems 2012
    • Place of Presentation
      北海道大学(招待講演)
    • Year and Date
      2012-03-06
    • Related Report
      2011 Annual Research Report
  • [Presentation] On the dual algorithm of Rauzy induction1, 2, 32012

    • Author(s)
      Hitoshi Nakada
    • Organizer
      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
      2012-02-18
    • Related Report
      2011 Final Research Report
  • [Presentation] On the dual algorithm of Rauzy induction 1, 2, 32012

    • Author(s)
      仲田均
    • Organizer
      数論とエルゴード理論
    • Place of Presentation
      金沢大学(招待講演)
    • Year and Date
      2012-02-18
    • Related Report
      2011 Annual Research Report
  • [Presentation] On the dual algorithm of Rauzy induction2011

    • Author(s)
      Hitoshi Nakada
    • Organizer
      エルゴード理論とその周辺
    • Place of Presentation
      大阪大学理学部
    • Year and Date
      2011-12-08
    • Related Report
      2011 Final Research Report
  • [Presentation] On the dual algorithm of Rauzy induction2011

    • Author(s)
      仲田均
    • Organizer
      エルゴード理論とその周辺
    • Place of Presentation
      大阪大学
    • Year and Date
      2011-12-08
    • Related Report
      2011 Annual Research Report
  • [Presentation] Bit complexity and some Eucleidean type algorithm over F_q[X]^32011

    • Author(s)
      Rie Natsui
    • Place of Presentation
      大阪大学理学部
    • Year and Date
      2011-12-07
    • Related Report
      2011 Final Research Report
  • [Presentation] Bit complexity of some Euclidean type algorithms2011

    • Author(s)
      夏井利恵
    • Organizer
      エルゴード理論とその周辺
    • Place of Presentation
      大阪大学
    • Year and Date
      2011-12-07
    • Related Report
      2011 Annual Research Report
  • [Presentation] Eucleidean algorithm over F_q[X] and its cost functions2010

    • Author(s)
      Rie Natsui
    • 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
    • Related Report
      2011 Final Research Report
  • [Presentation] Euclidean algorithm over F_q[X] and its cost functions2010

    • Author(s)
      夏井利恵
    • Organizer
      Functions in Number Theory and Their Probabilistic Aspects
    • Place of Presentation
      京都大学数理解析研究所(京都)(招待講演)
    • Year and Date
      2010-12-17
    • Related Report
      2010 Annual Research Report
  • [Presentation] On the dynamics of fully subtractive algorithm2010

    • Author(s)
      Hitoshi Nakad
    • Organizer
      Japan-Korea Workshop on Number Theory and Ergodic Theory
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-07-28
    • Related Report
      2011 Final Research Report
  • [Presentation] On the dynamics of fully subtractive algorithm2010

    • Author(s)
      仲田均
    • Organizer
      Japan-Korea Workshop on Number Theory and Ergodic Theory
    • Place of Presentation
      京都大学数理解析研究所(京都)(招待講演)
    • Year and Date
      2010-07-28
    • Related Report
      2010 Annual Research Report
  • [Presentation] Eucleidean algorithm over F_q[X]2010

    • Author(s)
      Rie Natsui
    • Organizer
      apan-Korea Workshop on Number Theory and Ergodic Theory
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-07-27
    • Related Report
      2011 Final Research Report
  • [Presentation] Euclidean algorithm over F_q[X]2010

    • Author(s)
      夏井利恵
    • Organizer
      Japan-Korea Workshop on Number Theory and Ergodic Theory
    • Place of Presentation
      京都大学数理解析研究所(京都)(招待講演)
    • Year and Date
      2010-07-27
    • Related Report
      2010 Annual Research Report
  • [Presentation] On continued fraction mixing2010

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Workshop on Infinite Ergodic Theory and Related Fields
    • Place of Presentation
      Weizmann Institute of Science イスラエル
    • Year and Date
      2010-06-28
    • Related Report
      2011 Final Research Report
  • [Presentation] On continued fraction mixing2010

    • Author(s)
      仲田均
    • Organizer
      Workshop on Infinite Ergodic Theory and Related Fields
    • Place of Presentation
      Weizmann研究所(Israel)(招待講演)
    • Year and Date
      2010-06-28
    • Related Report
      2010 Annual Research Report
  • [Presentation] On the Eucleidean algorithm and Farey map for formal power series2009

    • Author(s)
      Rie Natsui
    • Organizer
      エルゴード理論の最近の話題
    • Place of Presentation
      日本大学文理学部
    • Year and Date
      2009-11-06
    • Related Report
      2011 Final Research Report
  • [Presentation] On the Euclidean algorithm and Farey map for formal power series2009

    • Author(s)
      夏井利恵
    • Organizer
      研究集会「エルゴード理論の最近の話題
    • Place of Presentation
      北海道大学理学部(札幌市)
    • Year and Date
      2009-11-06
    • Related Report
      2009 Annual Research Report
  • [Presentation] Renormalized Rauzy map に対するDarling-Kac set の存在について2009

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Dynamics of complex systems 2009
    • Place of Presentation
      北海道大学理学部
    • Year and Date
      2009-08-31
    • Related Report
      2011 Final Research Report
  • [Presentation] Euclidean algorithm in positive characteristic and associated Farey map2009

    • Author(s)
      Rie Natsui
    • Organizer
      Dynamics of complex systems 2009
    • Place of Presentation
      北海道大学理学部
    • Year and Date
      2009-08-31
    • Related Report
      2011 Final Research Report
  • [Presentation] Renormalized Rauzy map に対する Darling-Kac set の存在について2009

    • Author(s)
      仲田均
    • Organizer
      研究集会「Dynamics of complex systems 2009」
    • Place of Presentation
      北海道大学理学部(札幌市)
    • Year and Date
      2009-08-31
    • Related Report
      2009 Annual Research Report
  • [Presentation] Euclidean algorithm in positive characteristic and associated Farey map2009

    • Author(s)
      夏井利恵
    • Organizer
      研究集会「Dynamics of complex systems 2009」
    • Place of Presentation
      北海道大学理学部(札幌市)
    • Year and Date
      2009-08-31
    • Related Report
      2009 Annual Research Report
  • [Presentation] Ergodicity of a group extension of Jacobi-Perron algorithm2009

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Workshop on Measurable and Topological Dynamical Systems in Asia
    • Place of Presentation
      Ajou University 韓国
    • Year and Date
      2009-06-18
    • Related Report
      2011 Final Research Report
  • [Presentation] Non-monotonicity of entropy of alpha continued fractiontransformations2009

    • Author(s)
      Rie Natsui
    • Organizer
      Workshop on Measurable and Topological Dynamical Systems in Asia
    • Place of Presentation
      Ajou University 韓国
    • Year and Date
      2009-06-18
    • Related Report
      2011 Final Research Report
  • [Presentation] Ergodicity of a group extension of Jacobi-Perron algorithm2009

    • Author(s)
      仲田均
    • Organizer
      Workshop on Measurable and Topological Dynamical Systems in Asia
    • Place of Presentation
      Ajou University(水原、韓国)
    • Year and Date
      2009-06-18
    • Related Report
      2009 Annual Research Report

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Published: 2009-04-01   Modified: 2016-04-21  

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