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Mathematics for Non-commutative Harmonic Oscillators and Representation Theory of alpha-determinants

Research Project

Project/Area Number 21340011
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKyushu University

Principal Investigator

WAKAYAMA Masato  九州大学, マス・フォア・インダストリ研究所, 教授 (40201149)

Project Period (FY) 2009-04-01 – 2014-03-31
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥13,130,000 (Direct Cost: ¥10,100,000、Indirect Cost: ¥3,030,000)
Fiscal Year 2013: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2012: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2011: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2010: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2009: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Keywords非可換調和振動子 / ゼータ関数 / 表現論 / α行列式 / 不変式論 / 保型形式
Research Abstract

1.We described the special value at s=4 of the spectral zeta function of the non-commutative harmonic oscillator (NcHO) by introducing the notion of residual modular forms, which is a generalization of Eichler integrals. With this investigation, we introduced the periodic Eichler cohomology groups and determined the structure of certain congruence subgroups of SL_2(Z).
2.We showed a complete description of the eigenvalue problem for NcHO, which is defined by a matrix-valued self-adjoint ordinary differential operator, in terms of Heun's ordinary differential equations (Only the odd case was given by Ochiai in 2001). As a by-product, using the monodromy representation of Heun's ODEs, we proved that the multiplicity of the eigenvalue of NcHO is at most two.
3.We described a connection between the quantum Rabi model and NcHO, employing principal series of sl2 and confluent procedure of Heun ODE.

Report

(6 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (78 results)

All 2014 2013 2012 2011 2010 2009 Other

All Journal Article (17 results) (of which Peer Reviewed: 7 results) Presentation (47 results) (of which Invited: 1 results) Book (3 results) Remarks (11 results)

  • [Journal Article] Invariant differential operators on the Heisenberg group and Meixner-Pollaczek polynomials2013

    • Author(s)
      Jacques Faraut & Masato Wakayama
    • Journal Title

      Advances in Pure and Applied Mathematics

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Journal Article] Milnor-Selberg zeta functions and zeta regularizations2013

    • Author(s)
      Nobushige Kurokawa, Masato Wakayama & Yoshinori Yamasaki
    • Journal Title

      Journal of Geometry and Physics

      Volume: Volume 64 Pages: 120-145

    • DOI

      10.1016/j.geomphys.2012.10.015

    • NAID

      120005068800

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Non-linear algebraic differential equations satisfied by Certain family of elliptic functions2013

    • Author(s)
      M. Wakayama & K. Yamamoto
    • Journal Title

      The Ramanujan Journal

      Volume: 30(2) Issue: 2 Pages: 173-186

    • DOI

      10.1007/s11139-012-9425-3

    • NAID

      120005227064

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Journal Article] Spectrum of non-commutative harmonics oscillators and residual modular forms2013

    • Author(s)
      K. Kimoto & M. Wakayama
    • Journal Title

      Proceedings of the RIMS Thematic Year 2010 on Perspectives in Deformation Quantization and Noncommuntative Geometry

      Volume: Volume 1 Pages: 23-267

    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Journal Article] Multivariate Meixner-Pollaczek polynomials 若山正人and an application2013

    • Author(s)
      若山正人
    • Journal Title

      数理解析研究所講究録1825非現論と非可換調和解析の展望

      Volume: 1825巻 Pages: 197-204

    • Related Report
      2013 Final Research Report
  • [Journal Article] Simplicity of the lowest eigenvalue of non-commutative harmonic oscillators and the Riemann scheme of a certain Heun' s differential equation2013

    • Author(s)
      Masato Wakayama
    • Journal Title

      Proceedings of the Japan Academy

      Volume: 89巻 Issue: 6 Pages: 69-73

    • DOI

      10.3792/pjaa.89.69

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Invariant differential operators on the Heisenberg group and Meixner-Pollaczek polynomials(to appear)2013

    • Author(s)
      Jacques Faraut & Masato Wakayama
    • Journal Title

      Advances in Pure and Applied Mathematics

      Volume: -

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Multivariate Meixner-Pollaczek polynomials and an application2013

    • Author(s)
      若山正人
    • Journal Title

      数理解析研究所講究録1825 非現論と非可換調和解析の展望

      Volume: 1825 Pages: 197-204

    • Related Report
      2012 Annual Research Report
  • [Journal Article] Spectrum of non-commutative harmonics oscillators and residual modular forms (to appear in 2012)2012

    • Author(s)
      K.Kimoto, M.Wakayama
    • Journal Title

      Noncommutative Geometry and Physics, World Scientific

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Higher regularizations for zeros of cuspidal automorphic L-functions of GL_d2011

    • Author(s)
      Masato Wakayama, Yoshinori Yamasaki
    • Journal Title

      J. Theor

      Volume: 23巻 Pages: 751-767

    • Related Report
      2013 Final Research Report
  • [Journal Article] Remarks on geodesics for multivariate normal models2011

    • Author(s)
      Takuro Imai, Akira Takaesu and Masato Wakayama
    • Journal Title

      Journal of Math-for-Industry

      Volume: 2011巻 Pages: 125-130

    • NAID

      40019161283

    • Related Report
      2013 Final Research Report
  • [Journal Article] Higher regularizations for zeros of cuspidal automorphic L-functions of GL_d2011

    • Author(s)
      Masato Wakayama, Yoshinori Yamasaki
    • Journal Title

      J.Theor.Nombres Bordeaux

      Volume: 23 Pages: 751-767

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on geodesics for multivariate normal models2011

    • Author(s)
      Takuro Imai, Akira Takaesu, Masato Wakayama
    • Journal Title

      Journal of Math-for-Industry

      Volume: 2011 Pages: 125-130

    • NAID

      40019161283

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Over the zeta regularized numeration2011

    • Author(s)
      Masato Wakayama
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: (to appear)

    • Related Report
      2010 Annual Research Report
  • [Journal Article] Alpha-determinant cyclic modules and Jacobi polynomials2009

    • Author(s)
      Kazufumi Kimoto, Sho Matsumoto and Masato Wakayama
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 361巻 Pages: 6447-6473

    • Related Report
      2013 Final Research Report
  • [Journal Article] Alpha-determinant cyclic modules and Jacobi polynomials2009

    • Author(s)
      Kazufumi Kimoto, Sho Matsumoto, Masato Wakayama
    • Journal Title

      Trans. Amer. Math. Soc. 361

      Pages: 6447-6473

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spectrum of non-commutative harmonics oscillators and residual modular forms

    • Author(s)
      K. Kimoto, M. Wakayama
    • Journal Title

      Noncommutative Geometry and Physics, World Scientific

    • Related Report
      2013 Final Research Report
  • [Presentation] An application of Lie theory to computer graphics via spherical harmonics2014

    • Author(s)
      若山正人
    • Organizer
      Berliner Colloquium fur wissenschaftliche Visualisierung
    • Place of Presentation
      Zuse-Institut Berlin
    • Year and Date
      2014-02-28
    • Related Report
      2013 Final Research Report
  • [Presentation] An application of Lie theory to computer graphics via spherical harmonics2014

    • Author(s)
      Masato Wakayama
    • Organizer
      Berliner Colloquium fur wissenschaftliche Visualisierung
    • Place of Presentation
      Zuse-Institut Berlin (ZIB)
    • Related Report
      2013 Annual Research Report
  • [Presentation] Spectrum of non-commutative harmonic oscillators - Modular forms2013

    • Author(s)
      若山正人
    • Organizer
      representation theory and the Rabi model, colloquium
    • Place of Presentation
      Mathematical Institute, Leiden University
    • Year and Date
      2013-11-29
    • Related Report
      2013 Final Research Report
  • [Presentation] Residual modular forms and curves, the quantum Rabi model, and discrete series representations from the study of spectral zeta functions of the NcHO2013

    • Author(s)
      若山正人
    • Organizer
      Geometric zeta functions and related topics
    • Place of Presentation
      Saga University
    • Year and Date
      2013-10-31
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] A Lie theoretic proposal on algorithms for spherical harmonic lighting2013

    • Author(s)
      若山正人
    • Organizer
      Mathematical Progress in Expressive Image Synthesis
    • Place of Presentation
      Centennial Hall Kyushu University School of Medicine, Fukuoka
    • Year and Date
      2013-10-22
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] Quantum Interaction Models and Number Theory2013

    • Author(s)
      若山正人
    • Organizer
      Zeta Functions in OKINAWA 2013
    • Place of Presentation
      沖縄コ ンベンションセンター
    • Year and Date
      2013-10-19
    • Related Report
      2013 Final Research Report
  • [Presentation] Spectrum of non-commutativa harmonic oscillators-number theory2013

    • Author(s)
      若山正人
    • Organizer
      representation theory and Fuchsian Equations, Mathematical Quantum Field Theory and Related Topics, RIMS
    • Place of Presentation
      Kyoto University
    • Year and Date
      2013-10-03
    • Related Report
      2013 Final Research Report
  • [Presentation] Problems around NcHO2013

    • Author(s)
      若山正人
    • Organizer
      表現論がつなぐ数学
    • Place of Presentation
      ひめぎんホール別館・第20会議室
    • Year and Date
      2013-09-28
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] Analysis and Number Theory for Non-commutative Harmonic Oscillator2013

    • Author(s)
      若山正人
    • Organizer
      Distinguished Lecture Series
    • Place of Presentation
      Department of Mathematics, Indiana University, Bloomington
    • Year and Date
      2013-04-22
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] Analysis and arithmetic of non-commutative harmonic oscillators2013

    • Author(s)
      若山正人
    • Organizer
      BCAM Seminar
    • Place of Presentation
      Basque Center of Applied Mathematics, Spain
    • Year and Date
      2013-03-01
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] Introduction to Mathematics for Industry2013

    • Author(s)
      若山正人
    • Organizer
      Seminar in Grenoble
    • Place of Presentation
      Univ. Joseph Fourier, France
    • Year and Date
      2013-03-01
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] Spectrum of non-commutativa harmonic oscillators-number theory,representation theory and Fuchsian equations2013

    • Author(s)
      Masato Wakayama
    • Organizer
      Mathematical Quantum Field Theory and Related Topics, RIMS
    • Place of Presentation
      Kyoto University
    • Related Report
      2013 Annual Research Report
  • [Presentation] Quantum Interaction Models and Number Theory2013

    • Author(s)
      Masato Wakayama
    • Organizer
      Zeta Functions in OKINAWA 2013
    • Place of Presentation
      沖縄コンベンションセンター
    • Related Report
      2013 Annual Research Report
  • [Presentation] Spectrum of non-commutative harmonic oscillators - Modular forms, representation theory and the Rabi model2013

    • Author(s)
      Masato Wakayama
    • Organizer
      colloquium
    • Place of Presentation
      Mathematical Institute, Leiden University
    • Related Report
      2013 Annual Research Report
  • [Presentation] Multivariate Meixner-Pollaczek Polynomials2012

    • Author(s)
      若山正人
    • Organizer
      Sala de Seminarios
    • Place of Presentation
      Centro de Modelamiento Matematico, Chile
    • Year and Date
      2012-06-06
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] カスプ表現に付随するL-関数の零点に関するMilnor型ゼータ正規化積について2012

    • Author(s)
      山崎 義徳(愛媛大理) , 若山正人(九大 IMI)
    • Organizer
      日本数学会2012年度会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-29
    • Related Report
      2013 Final Research Report 2011 Annual Research Report
  • [Presentation] Remarks on geodesics for multivariate normal models2012

    • Author(s)
      今井卓郎(九大数理) , 若山正人(九大 IMI) , 高江洲旭(NEC)
    • Organizer
      日本数学会, 2012年度会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-28
    • Related Report
      2013 Final Research Report
  • [Presentation] Hermitian symmetric spaces of tube type and multivariate Meixner Pollaczek polynomials2012

    • Author(s)
      若山正人(九大 IMI) , J. Faraut (Paris6)
    • Organizer
      日本数学会2012年度会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-28
    • Related Report
      2013 Final Research Report 2011 Annual Research Report
  • [Presentation] 非可換調和振動子のスペクトルゼータ値に現れるΤ(2)-モジュラー性2012

    • Author(s)
      木本一史(琉球大理) , 若山正人(九大 IMI)
    • Organizer
      日本数学会2012年度会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-28
    • Related Report
      2013 Final Research Report
  • [Presentation] Remarks on geodesies for multivariate normal models2012

    • Author(s)
      今井卓郎(九大数理)、若山正人(九大IMI、高江洲旭(NEC)
    • Organizer
      日本数学会2012年度会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-28
    • Related Report
      2011 Annual Research Report
  • [Presentation] 非可換調和振動子のスペクトルゼータ値に現れるT(2)-モジュラー性2012

    • Author(s)
      木本一史(琉球大理)、若山正人(九大IMI)
    • Organizer
      日本数学会2012年度会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-28
    • Related Report
      2011 Annual Research Report
  • [Presentation] Number theoretical approach to spectrum for non-commutative harmonic oscillators, "Avoided ? Crossing of Eigenvalue Curves"2012

    • Author(s)
      若山正人
    • Organizer
      Non-commutative Harmonic Oscillator, Special Functions and Number Theory
    • Place of Presentation
      九州大学
    • Year and Date
      2012-03-07
    • Related Report
      2013 Final Research Report
  • [Presentation] Number theoretical approach to spectrum for non-commutative harmonic oscillators2012

    • Author(s)
      Masato Wakayama
    • Organizer
      "Avoided? Crossing of Eigenvalue Curves"--Non-commutative Harmonic Oscillator, Special Functions and Number Theory-
    • Place of Presentation
      九州大学(招待講演)
    • Year and Date
      2012-03-07
    • Related Report
      2011 Annual Research Report
  • [Presentation] Residual modular forms, Eichler cohomology and spectrum of non-commutative harmonic oscillators2012

    • Author(s)
      若山正人
    • Organizer
      Institute for Advanced study, New Zealand Colloquium
    • Place of Presentation
      Massey University
    • Year and Date
      2012-02-27
    • Related Report
      2013 Final Research Report
  • [Presentation] Residual modular forms, Eichler cohomology and spectrum of non-commutative harmonic oscillators2012

    • Author(s)
      Masato Wakayama
    • Organizer
      Institute for Advanced study, New Zealand Colloquium
    • Place of Presentation
      New Zealand, Massey University(招待講演)
    • Year and Date
      2012-02-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] Remarks on geodesics for multivariate normal models2012

    • Author(s)
      若山正人
    • Organizer
      WIAS+MATHON Colloquium
    • Place of Presentation
      WIAS, Berlin, Germany
    • Year and Date
      2012-01-09
    • Related Report
      2013 Final Research Report
  • [Presentation] Remarks on geodesies for multivariate normal models2012

    • Author(s)
      Masato Wakayama
    • Organizer
      WIAS+MATHON Colloquium
    • Place of Presentation
      WIAS, Berlin, Germany(招待講演)
    • Year and Date
      2012-01-09
    • Related Report
      2011 Annual Research Report
  • [Presentation] 表現論と非可換調和解析の展望2012

    • Author(s)
      若山正人
    • Organizer
      Multivariate Meixner-Pollaczek polynomials and an application
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Final Research Report
  • [Presentation] Oscillator representations and Meixner-Pollaczek polynomials2012

    • Author(s)
      若山正人
    • Organizer
      表現論から見える数学の諸相
    • Place of Presentation
      サンパレス 球陽館(OKINAWA)
    • Related Report
      2013 Final Research Report
  • [Presentation] Remarks on parity of eigenfunctions of NcHO via Heun DE2012

    • Author(s)
      若山正人
    • Organizer
      Zetas and Limit Laws in OKINAWA 2012
    • Place of Presentation
      Festone, Ginowan, OKINAWA
    • Related Report
      2013 Final Research Report 2012 Annual Research Report
  • [Presentation] Eichler cohomology groups for spectrum of NcHO' s2012

    • Author(s)
      若山正人
    • Organizer
      Spectral analysis of non-commutative harmonic oscillators and quantum devices
    • Place of Presentation
      Kyushu University
    • Related Report
      2013 Final Research Report
  • [Presentation] Eichler cohomology groups for spectrum of NcHO's2012

    • Author(s)
      Masato Wakayama
    • Organizer
      Spectral analysis of non-commutative harmonic oscillators and quantum devices
    • Place of Presentation
      Kyushu University
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Oscillator representations and Meixner-Pollaczek polynomials2012

    • Author(s)
      Masato Wakayama
    • Organizer
      表現論から見える数学の諸相
    • Place of Presentation
      サンパレス球陽館(OKINAWA)
    • Related Report
      2012 Annual Research Report
  • [Presentation] 表現論と非可換調和解析の展望2012

    • Author(s)
      Masato Wakayama
    • Organizer
      Multivariate Meixner-Pollaczek polynomials and an application(招待)
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Annual Research Report
  • [Presentation] Spectrum of Non-commutative Harmonic Oscillators and Modular Forms2011

    • Author(s)
      若山正人
    • Organizer
      RMS研究集会"Spectral and Scattering Theory and Related Topics"
    • Place of Presentation
      RIMS, 京都大学
    • Year and Date
      2011-12-15
    • Related Report
      2013 Final Research Report
  • [Presentation] Spectrum of Non-commutative Harmonic Oscillators and Modular Forms2011

    • Author(s)
      Masato Wakayama
    • Organizer
      RMS研究集会"Spectral and Scattering Theory and Related Topics"
    • Place of Presentation
      RIMS,京都大学(招待講演)
    • Year and Date
      2011-12-15
    • Related Report
      2011 Annual Research Report
  • [Presentation] Negative Weight Modular Forms2011

    • Author(s)
      若山正人
    • Organizer
      Zetas and Limit Laws in OKINAWA 2011
    • Place of Presentation
      Festone, 沖縄
    • Year and Date
      2011-11-27
    • Related Report
      2013 Final Research Report
  • [Presentation] Negative Weight Modular Forms2011

    • Author(s)
      Masato Wakayama
    • Organizer
      Zetas and Limit Laws in OKINAWA 2011
    • Place of Presentation
      Festone,沖縄(招待講演)
    • Year and Date
      2011-11-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] ゼータ正規化積による保型形式の構成2011

    • Author(s)
      若山正人
    • Organizer
      研究集会"表現論がつなぐ数学の展望"
    • Place of Presentation
      ホテルサンルート名古屋
    • Year and Date
      2011-08-29
    • Related Report
      2013 Final Research Report
  • [Presentation] ゼータ正規化積による保型形式の構成2011

    • Author(s)
      Masato Wakayama
    • Organizer
      研究集会"表現論がつなぐ数学の展望"
    • Place of Presentation
      ホテルサンルート名古屋(招待講演)
    • Year and Date
      2011-08-29
    • Related Report
      2011 Annual Research Report
  • [Presentation] Non commutative Harmonic Oscillators and Number Theory2010

    • Author(s)
      若山正人
    • Organizer
      Non Commutative Geometry and Number Theory RIMS Special Year on Non Commutative Geometry
    • Place of Presentation
      京都大学理学部 数学教室
    • Year and Date
      2010-11-26
    • Related Report
      2010 Annual Research Report
  • [Presentation] Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials2010

    • Author(s)
      若山正人
    • Organizer
      Analysis, Geometry and Group Representations for Homogeneous Spaces
    • Place of Presentation
      Lorentz Center, Leiden University
    • Year and Date
      2010-11-22
    • Related Report
      2010 Annual Research Report
  • [Presentation] 非可換調和振動子の数論2010

    • Author(s)
      若山正人
    • Organizer
      2010年度表現論シンポジウム
    • Place of Presentation
      静岡県伊豆の国市おおとり荘
    • Year and Date
      2010-11-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] Ramanujan's Delta and zeta regularizations2010

    • Author(s)
      若山正人
    • Organizer
      第10回ゼータ研究集会
    • Place of Presentation
      沖縄コンベンションセンター
    • Year and Date
      2010-11-07
    • Related Report
      2010 Annual Research Report
  • [Presentation] ゼータ式計算を巡って2010

    • Author(s)
      若山正人
    • Organizer
      解析数論-複素関数の値の分布と性質を通して-
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-10-05
    • Related Report
      2010 Annual Research Report
  • [Presentation] ゼータ正規化積についての一考察2010

    • Author(s)
      若山正人
    • Organizer
      表現論がつなぐ数論・解析学・組合せ論
    • Place of Presentation
      愛媛大学
    • Year and Date
      2010-08-25
    • Related Report
      2010 Annual Research Report
  • [Presentation] Hecke L-関数の零点とhigher depth regularizde product2010

    • Author(s)
      山崎義徳
    • Organizer
      日本数学会2010年度会
    • Place of Presentation
      慶応義塾大学矢上キャンパス
    • Year and Date
      2010-03-26
    • Related Report
      2009 Annual Research Report
  • [Book] 可視化の技術と現代幾何学2010

    • Author(s)
      若山正人
    • Total Pages
      240
    • Publisher
      岩波書店
    • Related Report
      2013 Final Research Report 2009 Annual Research Report
  • [Book] Casimir Operators and the Riemann Hypothesis--Mathematics for Innovation in Industry and Science2010

    • Author(s)
      若山正人
    • Total Pages
      286
    • Publisher
      Walter De Gruyter Inc, Casimir Force
    • Related Report
      2013 Final Research Report
  • [Book] Casimir Force, Casimir Operators and the Riemann Hypothesis-Mathematics for Innovation in Industry and Science-2010

    • Author(s)
      Ed.by Dijk, Gerrit van/Wakayama, Masato
    • Total Pages
      286
    • Publisher
      Walter De Gruyter Inc
    • Related Report
      2010 Annual Research Report
  • [Remarks] 若山正人ホームページ

    • URL

      http://imi.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2013 Final Research Report
  • [Remarks] 九州大学研究者情報(若山正人)

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K000521/

    • Related Report
      2013 Final Research Report
  • [Remarks] 若山正人ホームページ

    • URL

      http://imi.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2013 Annual Research Report
  • [Remarks] 九州大学研究者情報(若山正人)

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K000521/

    • Related Report
      2013 Annual Research Report
  • [Remarks] 若山正人ホームページ

    • URL

      http://imi.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2012 Annual Research Report
  • [Remarks] 九州大学研究者情報(若山正人)

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K000521/

    • Related Report
      2012 Annual Research Report
  • [Remarks]

    • URL

      http://imi.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://hyoka.ofc.kyushu-u.ac.jp/search/details/K000521/index.html

    • Related Report
      2011 Annual Research Report
  • [Remarks] 2011年3月31日迄

    • URL

      http://www2.math.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2010 Annual Research Report
  • [Remarks] 2011年4月1日以降

    • URL

      http://imi.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://www2.math.kyushu-u.ac.jp/~wakayama/

    • Related Report
      2009 Annual Research Report

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Published: 2009-04-01   Modified: 2019-07-29  

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