Conclusive study of algebraic combinatorics

• Project/Area Number: 20540017
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: BANNAI Eiichi Kyushu University, 数理(科)学研究科(研究院), その他 (10011652)
• Project Period (FY): 2008 – 2012
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 代数的組合せ論 / アソシエーションスキーム / モジュラー形式 / デザイン / tightデザイン / 格子 / コード / Lehmer予想 / coherent configuration / tight design

Research Abstract

代表者の長年の代数的組合せ論の研究の総決算を行うと言う形で,いままでの研究の総まとめをしたいと言うのがこの研究の主旨である.置換群の多重可移という性質と,原始的であると言う性質の意味をもっと良く理解したいということが研究の元々の出発点であったが,それをより広げた形で,アソシエーションスキーム,あるいは球面上の代数的組合せ論と言う枠組みで,アソシエーションスキーム上の,あるいは球面上のt-デザインにおけるtの意味を良く理解したいと言うことであった.具体的には,アソシエーションスキーム自身の分類問題に加えて,アソシエーションスキームの部分集合である,コード,デザインの研究を行い,さらに球面上の良い有限集合の研究,およびその拡張であるユークリッドデザイン等の研究を行った.特に,ユークリッド格子の殻として出来る球面t-デザインの研究(Lehmer予想のtoy modelsの研究),ユークリッドデザインとcoherent configurationとの関係の研究,特にtightなユークリッドt-デザインの構成,分類問題に努力を集中して研究を行った

Research Products

• [Journal Article] A new Euclidean tight 6-design (2011)
  • Author(s): Bannai, Eiichi ; Bannai, Etsuko ; Shigezumi, Junichi
  • Journal Title: Annals of Combinatorics Volume: (印刷中)
  • NAID: 120003758418
  • Peer Reviewed
• [Journal Article] Tight 9-designs on two concentric spheres (2011)
  • Author(s): Bannai, Eiichi ; Bannai, Etsuko
  • Journal Title: J.of Math.Soc.Japan Volume: (印刷中)
  • NAID: 10030344913
  • Peer Reviewed
• [Journal Article] Toy models for D.H.Lehmer's conjecture (2010)
  • Author(s): Eiichi Bannai, Tsuyoshi Miezaki
  • Journal Title: J.of Math.Soc.Japan 62 Pages: 687-705
  • Peer Reviewed
• [Journal Article] Euclidean designs and coherent con-figurations (2010)
  • Author(s): Eiichi Bannai, Etsuko Bannai
  • Journal Title: Contemporary Mathematics(ed.Richard Brualdi) (American Math.Soc.) 531 Pages: 59-93
  • Peer Reviewed
• [Journal Article] 純粋数学としての組合せ論 (2010)
  • Author(s): 坂内英一
  • Journal Title: 数学 62 Pages: 433-452
  • Peer Reviewed
• [Journal Article] Toy models for D.H.Lehmer's conjecture (2010)
  • Author(s): Bannai, Eiichi ; Miezaki, Tsuyoshi
  • Journal Title: J.Math.Soc.Japan Volume: 62 Pages: 687-705
  • Peer Reviewed
• [Journal Article] 純粋数学としての組合せ論 (2010)
  • Author(s): 坂内英一
  • Journal Title: 数学 Volume: 62 Pages: 433-452
  • Peer Reviewed
• [Journal Article] Euclidean designs and coherent configurations (2010)
  • Author(s): Bannai, Eiichi ; Bannai, Etsuko
  • Journal Title: Contemporary Mathematics Volume: 531 Pages: 59-93
  • Peer Reviewed
• [Journal Article] Cubature formulas in numerical analysis and Euclidean tight designs (2010)
  • Author(s): Bannai, Eiichi, Bannai, Etsuko, Hirao, Masatake, Sawa, Masanori
  • Journal Title: Europ.J.Combinatorics 31 Pages: 423-441
  • Peer Reviewed
• [Journal Article] 純粋数学としての組合せ論 (2010)
  • Author(s): 坂内英一
  • Journal Title: 数学(日本数学会) 62(印刷中)
  • Peer Reviewed
• [Journal Article] A survey on spherical designs and algebraic Combinatorics on spheres (2009)
  • Author(s): Eiichi Bannai, Etsuko Bannai
  • Journal Title: Europ.J.Combinatorics 30 Pages: 1392-1425
  • Peer Reviewed
• [Journal Article] A survey on spherical designs and algebraic combinatorics on spheres (2009)
  • Author(s): Bannai, Eiichi, Bannai, Etsuko
  • Journal Title: Europ.J.Combinatorics 30 Pages: 1392-1425
  • Peer Reviewed
• [Journal Article] The Mathematics of Donald Gordon Higman (2009)
  • Author(s): Bannai, Eiichi, Griess, Robert L.Jr., Praeger, Cheryl E., Scott, Leonard
  • Journal Title: Michigan Mathematical Journal 58 Pages: 3-30
  • Peer Reviewed
• [Journal Article] On antipodal spherical $t$-designs of degree $s$ with $t\geq 2s-3$ (2009)
  • Author(s): Bannai, Eiichi, Bannai, Etsuko
  • Journal Title: Journal of Combinatorics, Information & System Sciences 34 Pages: 33-50
  • Peer Reviewed
• [Journal Article] Finite Euclidean graphs and Ramanujan graphs (2009)
  • Author(s): Bannai, Eiichi, Shimabukuro, Osamu, Tnanaka, Hajime
  • Journal Title: Discrete Mathematics 309 Pages: 6126-6134
  • Peer Reviewed
• [Journal Article] Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets (2009)
  • Author(s): Abduthalikov, Kanat, Bannai, Eiichi, Suda, Sho
  • Journal Title: J. of Combinatorial Theory (A) 116 Pages: 434-448
  • Peer Reviewed
• [Journal Article] Spherical designs and Euclidean designs (2009)
  • Author(s): Bannai, Eiichi, Bannai, Etsuko
  • Journal Title: Recent Developments in Algebra and Related Areas, ALM 8, Higher Education Press and International Press, Beijing-Boston 8 Pages: 1-37
  • Peer Reviewed
• [Journal Article] On the zeros of Hecke type Faber polynomials (2008)
  • Author(s): Banaai, Eiichi, Kojima, Koji, Miezaki, Tsuyoshi
  • Journal Title: Kyushu J. Math. 62 Pages: 15-61
• [Journal Article] Uniqueness of certain association schemes (2008)
  • Author(s): Bannai, Eiichi, Bannai, Etsuko, Bannai Hideo
  • Journal Title: Europ. J. Combinatorics (spedal issue in honor of 80th birthday of Professor ZheXian Wan) 29 Pages: 1379-1305
  • Peer Reviewed
• [Journal Article] Tight 9-designs on two concentric spheres
  • Author(s): Eiichi Bannai, Etsuko Bannai
  • Journal Title: J.of Math.Soc.Japan 印刷中
  • NAID: 10030344913
  • Peer Reviewed
• [Journal Article] A new example of Euclidean tight 6-design
  • Author(s): Eiichi Bannai, Etsuko Bannai, Junichi Shigezumi
  • Journal Title: Annals of Combinatorics 印刷中
  • Peer Reviewed
• [Journal Article] Cubature formulas in numerical analysis and Euclidean tight designs
  • Author(s): Bannai, Eiichi, Bannai, Etsuko, Hirao, Masatake, Sawa, Masanori
  • Journal Title: Special issue in honor of Michel Deza (=Europ. J. Combinatorics). 掲載決定
  • Peer Reviewed
• [Journal Article] A survey on spherical designs and algebraic combinatorics on spheres
  • Author(s): Bannai, Eiichi, Bannai, Etsuko
  • Journal Title: Europ. J. Combinatorics. 掲載決定
  • Peer Reviewed
• [Journal Article] On antipodal spherical t-designs of degree s with t〓2s-3
  • Author(s): Bannai, Eiichi, Bannai, Etsuko
  • Journal Title: a special volume of Journal of Combinatorics, Information and System Sciences, honoring the 75th birthday of Prof. D. K. Ray-Chaudhuri. 掲載決定
  • Peer Reviewed
• [Journal Article] The Mathematics of Donald Gordon Higman
  • Author(s): Bannai, Eiichi, Griess, Robert L. Jr., Praeger, Cheryl E. Scott, Leonard
  • Journal Title: Michigan Mathematical Journal 掲載決定
  • Peer Reviewed
• [Presentation] Introduction on spherical designs and Euclidean designs (2011)
  • Author(s): 坂内英一
  • Organizer: 近畿大学談話会
  • Place of Presentation: 近畿大学
  • Year and Date: 2011-01-18
• [Presentation] Introduction to spherical designs and Euclidean designs (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Workshop om Algebraic Combinatorics
  • Place of Presentation: Hebei Normal University, Shijiazhuang, China
  • Year and Date: 2010-11-27
• [Presentation] A survey on spherical designs and Euclidean designs (2010)
  • Author(s): Eiichi Bannai
  • Organizer: The International Conference "Geometry, Topology, Algebra and Number Theory, and Applications (Delone120)
  • Place of Presentation: Steklov Inst.Math.Moscow
  • Year and Date: 2010-08-16
• [Presentation] On the concept of t-designs on the sphere, Euclidean space, and the real hyperbolic space (2010)
  • Author(s): Eiichi Bannai
  • Organizer: International Conference on Algebraic and Geometric Combinatorics(AGC2010)
  • Place of Presentation: Kyonju, Korea
  • Year and Date: 2010-07-12
• [Presentation] Several remarks on the concepts of t-designs (2010)
  • Author(s): 坂内英一
  • Organizer: 第27回代数的組合せ論シンポジウム
  • Place of Presentation: 高知大学
  • Year and Date: 2010-06-21
• [Presentation] Euclidean designs and cubature formulas in Euclidean spaces (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Colloquium at Department of Mathematics, Pusan National University
  • Place of Presentation: Pusan National University, Korea
  • Year and Date: 2010-05-28
• [Presentation] Euclidean designs and cubature formulas for spherical symmetric measures on R^n (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Optimal Configurations on the Sphere and Other Manifolds
  • Place of Presentation: Vanderbilt Univ, Nashville, USA(30 min)
  • Year and Date: 2010-05-19
• [Presentation] Euclidean designs and cubature formulas for spherical symmetric measures on $R^n$ (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Optimal Configurations on the Sphere and Other Manifolds
  • Place of Presentation: Vanderbilt Univ, Nashville, USA
  • Year and Date: 2010-05-19
• [Presentation] 球の詰め込み問題について (2010)
  • Author(s): 坂内英一
  • Organizer: 大阪大学理学部学生に対する特別講義、理学への招待
  • Place of Presentation: 大阪大学
  • Year and Date: 2010-05-06
• [Presentation] Spherical designs and toy models for D.H.Lehmer's conjecture in number theory (2010)
  • Author(s): Eiichi Bannai
  • Organizer: University of Rome 1 (La Sapienza)
  • Place of Presentation: (60 min.)
  • Year and Date: 2010-04-22
• [Presentation] Spherical designs and toy models for D.H.Lehmer's conjecture in number theory (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Seminar talk at University of Rome 1
  • Place of Presentation: University of Rome 1(La Sapienza)
  • Year and Date: 2010-04-22
• [Presentation] A survey on spherical designs and algebraic combinatorics on spheres (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Seminar talk at University of Rome 3
  • Place of Presentation: University of Rome 3
  • Year and Date: 2010-04-19
• [Presentation] A survey on spherical designs, with emphasis on the aspects related to fnite groups (2010)
  • Author(s): Eiichi Bannai
  • Organizer: ISCHIA GROUP THEORY 2010
  • Place of Presentation: Ischia, Italy
  • Year and Date: 2010-04-16
• [Presentation] $t$-designs in real hyperbolic spaces (2010)
  • Author(s): 坂内英一
  • Organizer: 代数的組合せ論ミニ集会
  • Place of Presentation: 神戸学院大学、神戸市
  • Year and Date: 2010-03-17
• [Presentation] On the concept of t-designs on the sphere, Euclidean space, and the real hyperbolic space (2010)
  • Author(s): 坂内英一
  • Organizer: ISM Symposium on Stochastic Models and Discrete Geometry
  • Place of Presentation: 統数研、立川市
  • Year and Date: 2010-03-01
• [Presentation] Spherical designs and D.H.Lehmer's conjecture (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Focused Week on Integral Lattices, University of Florida
  • Place of Presentation: Gainesville, Florida, USA.(50 minutes)
  • Year and Date: 2010-02-16
• [Presentation] Spherical designsand D.H.Lehmer's conjecture (2010)
  • Author(s): 坂内英一
  • Organizer: Focused Week on Integral Lattices
  • Place of Presentation: University of Florida, Gainesville, Florida, USA
  • Year and Date: 2010-02-16
• [Presentation] More on Euclidean tight $t$-designs (2010)
  • Author(s): 坂内英一
  • Organizer: The 8th Korea-Japan Workshop on Algebra and Combinatorics
  • Place of Presentation: Postech, Pohang, Korea
  • Year and Date: 2010-02-05
• [Presentation] A survey on spherical designs and Euclidean designs (2010)
  • Author(s): Eiichi Bannai
  • Organizer: Geometry, Topology, Algebra and Number Theory,Applications (Delone-120Conference)
  • Place of Presentation: Steklov Math.Inst., Moscow, Russia(plenary talk)
• [Presentation] Examples and classifications of certain Euclidean $t$-designs (2009)
  • Author(s): 坂内英一
  • Organizer: Metric Graph Theory
  • Place of Presentation: 東横ホテル、金沢市
  • Year and Date: 2009-11-11
• [Presentation] Spherical designs and toy models of D.H.Lehmer's conjecture in number theory (2009)
  • Author(s): 坂内英一
  • Organizer: 金沢大学数学教室談話会
  • Place of Presentation: 金沢大学、金沢市
  • Year and Date: 2009-11-04
• [Presentation] Spherical designs and toy models of D.H.Lehmer's conjecture in number theory (2009)
  • Author(s): 坂内英一
  • Organizer: 近畿大学数学教室談話会
  • Place of Presentation: 近畿大学、東大阪市
  • Year and Date: 2009-10-30
• [Presentation] Euclidean designs and coherent configurations (2009)
  • Author(s): 坂内英一
  • Organizer: 26th Algebraic Combinatoric
  • Place of Presentation: 遊学館、山形市
  • Year and Date: 2009-06-26
• [Presentation] Toy models of D.H.Lehmer's conjecture (2009)
  • Author(s): Eiichi Bannai
  • Organizer: The analytic theory of modular forms at the 65th birthday of Roelof Bruggeman
  • Place of Presentation: Woudschoten, The Netherlands(60 minutes)
  • Year and Date: 2009-06-18
• [Presentation] Toy models of D.H.Lehmer's conjecture (2009)
  • Author(s): 坂内英一
  • Organizer: The analytic theory of modular forms at the 65th birthday of Roelof Bruggeman
  • Place of Presentation: Woudschoten, The Nether lands
  • Year and Date: 2009-06-18
• [Presentation] Toy models of D.H.Lehmer's conjecture (2009)
  • Author(s): 坂内英一
  • Organizer: Discrete Geometry and Statistics of Configurations
  • Place of Presentation: Steklov Institute of Math., Moscow, Russia
  • Year and Date: 2009-06-01
• [Presentation] Spherical designs and association schemes versus Euclidean designs and coherent configurations (2009)
  • Author(s): Eiichi Bannai
  • Organizer: IPM-International Workshop on Combinatorics 2009
  • Place of Presentation: IPM, Tehran, Iran(60 minutes)
  • Year and Date: 2009-05-16
• [Presentation] Spherical designs and association schemes versus Euclidean designs and coherent configurations (2009)
  • Author(s): 坂内英一
  • Organizer: IPM-International Workshop on Combinatorics 2009
  • Place of Presentation: IPM, Tehran, Iran
  • Year and Date: 2009-05-16
• [Presentation] Euclidean designs and coherent configurations (2009)
  • Author(s): 坂内 英一
  • Organizer: ISM Symposium : Stochastic Models and Discrete Geometry
  • Place of Presentation: 統数研(ISM, Tokyo )
  • Year and Date: 2009-02-19
• [Presentation] Toy models for D. H. Lehmer's conjecture (joint work with Tsuyohi Miezaki) (2009)
  • Author(s): 坂内 英一
  • Organizer: 6^<th> Korea-Japan Workshop on Algebra and Combinatorics
  • Place of Presentation: Pusan National Univ., Korea
  • Year and Date: 2009-02-10
• [Presentation] On the late work of D. G. Higman on Combinatorics and groups (2009)
  • Author(s): 坂内 英一
  • Organizer: Finite groups, Vertex operator algebras, and combinatorics
  • Place of Presentation: RIMS, Kyoto Univ.
  • Year and Date: 2009-01-09
• [Presentation] Review of spherical designs (2009)
  • Author(s): Eiichi Bannai
  • Organizer: CIMPA-UNESCO-PHILIPPINES School : Semidefinite Programming in Algebraic Com-binatorics
  • Place of Presentation: University of the Philippines, Manilla(five 50 minute talks)
• [Presentation] Review of spherical designs(three 50 minute talks) (2009)
  • Author(s): 坂内英一
  • Organizer: CIMPA-UNESCO-PHILIPPINES School : Semidefinite Programming in Algebraic Combinatoricss Symposium
  • Place of Presentation: University of the Philippines, Manilla, Philippines
• [Presentation] On spherical designs I, II(two 75 minute talks) (2009)
  • Author(s): 坂内英一
  • Organizer: Hebei Normal University
  • Place of Presentation: Shijiazhuang, China
• [Presentation] "Spherical designs in Euclidean designs" and "Spherical designs and toy models of D.H.Lehmer's conjecture in number theory(two 60 minute talks) (2009)
  • Author(s): 坂内英一
  • Organizer: 松江セミナー
  • Place of Presentation: 島根大学、松江市
• [Presentation] "Combinatorics as pure mathematics----Aims of Algebraic Combinatorics"(90 minutes), and "Spherical designs and toy models of D.H.Lehmer's conjecture in number theory",(60 minutes) (2009)
  • Author(s): 坂内英一
  • Organizer: Shanghai Jiao Tong University
  • Place of Presentation: Shanghai, China
• [Presentation] Invariants of finite groups and modular forms…a survey from a personal viewpoint (2008)
  • Author(s): 坂内 英一
  • Organizer: Codes, Invariants and Modular forms
  • Place of Presentation: Max-Plank Institute of Mathematics, Univ Bonn, Germany
  • Year and Date: 2008-07-10
• [Presentation] Study of Spherical designs and Euclidean designs from the viewpoint of algebraic combinatorics (2008)
  • Author(s): 坂内 英一
  • Organizer: International conference on Differential Equations and Topology dedicatedto Centenial of L. S. Pontryagin
  • Place of Presentation: Moscow State University
  • Year and Date: 2008-06-18
• [Presentation] Finite subsets on spheres and association schemes (2008)
  • Author(s): 坂内 英一
  • Organizer: 6th Shanghai Conference on Combinatorics (6SHCC)
  • Place of Presentation: Shanghai Jiao Tong University, Shaghai, China
  • Year and Date: 2008-05-24
• [Presentation] Finite subsets on spheres and association schemes (2008)
  • Author(s): 坂内 英一
  • Organizer: Workshop on Combinatorial geometry and Topology
  • Place of Presentation: Brownsville, Texas, USA
  • Year and Date: 2008-04-12