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Random Geometry on the Sphere and its Applications

Research Project

Project/Area Number 13640126
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionUniversity of the Ryukyus

Principal Investigator

MAEHARA Hiroshi  University of the Ryukyus, Faculty of Education, Professor, 教育学部, 教授 (60044921)

Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordsasymptotic distribution / minimum distance / random caps / intersection graph / extremal cap / visual angle / 漸近確率 / 最小球面距離 / 指数分布 / 球面上のランダムグラフ
Research Abstract

1. For a graph G with N edges, put its vertices on the d-dimensional unit sphere. Let D denote the minimum spherical distance between a pair of points that correspond to a pair of adjacent vertices in G. Then, it was proved that the distribution of ND^d tends to exponential distribution with mean dB(1/2,d/2) as N tends to infinity, where B(p,q) denotes the beta function.
2. Let F={C_1,C_2,…,C_N} be a family of caps on the two dimensional unit sphere. A cap C_i is called extremal if the centers of those caps that intersect C_i are all contained in the same side of a great circle passing through the center of C_i. A cap that is smaller than a hemisphere is called proper. It was proved that if F has no extremal cap then the intersection graph G(F) of F is connected. If furthermore, all caps in F are proper then G(F) is 2-connected. For higher dimensional sphere, the similar result never holds. Applying this the following asymptotic result was proved. Now, let F denote a family of N random caps all of the same size (4πc/N)log N. If c>1/2, then the probability that G(F) is 2-connected tends to 1 as N tends to infinity. If c<1/4, then the probability that G(F) is connected tends to 0 as N tends to infinity.
3. Let AOB be a triangle in the 3-space with angle ∠AOB=ω. When we look at this angle from a viewpoint P, this angle looks as though the angle of the orthogonal projection of AOB on a plane perpendicular to the line PO. And its size changes according to the location of the viewpoint P. If P is a random point on a unit sphere centered at O, then the 'visual' size of the angle ∠AOB is called the random visual size and denoted by Θ(ω). By a joint study with Yoich Maeda (Tokai univ.), we proved that the expected value of Θ(ω) is equal to ω, and derived a formula to calculate the variance of Θ(ω).

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] H.Maehara, A.Oshiro: "Piercing a set of disjoint balls by a line"Journal of Combinatorial Theory (A). 94. 393-398 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara, N.Tokushige: "When does a planar bipartite framework admit a continuous deformatin?"Theoretical Computer Science. 263. 345-354 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "On acute triangulation of quadrilaterals"Proceedings of JCDCG2000 (LINCS). 2698. 237-243 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "On the total edge-length of a tetrahedron"American Mathematical Monthly. 108. 967-969 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara, A.Oshiro: "Cutting a bunch of grapes by a plane"European Journal of Combinatirocs. 22. 847-853 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Hosono, H.Maehara, K.Matuda: "A pair in a crowd of unit balls"European Journal of Combinatorics. 22. 1083-1092 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "Isoperimetric theorem for spherical polygon and the problem of 13 spheres"Ryukyu Mathematical Journal. 14. 41-57 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "Acute triangulations of polygons"European Journal of Combinatorics. 23. 45-55 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara, A.Oshiro: "Arranging solid balls to represent a graph"Graphs and Combinatorics. 18. 343-365 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "The length of the shortest edge of a graph on a sphere"European Journal of Combinatorics. 23. 713-717 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "An inequality on the size of a set in a Cartesian product"European Journal of Combinatirocs. 23. 1055-1059 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara: "A simple proof of the existence of non-obtuse triangulation for polygons"Ryukyu Mathematical Journal. 15. 43-48 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara and A. Oshiro: "Piercing a set of disjoint balls by a line"Journal of Combinatorial Theory (A). 94. 393-398 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara and N. Tokushige: "When does a planar bipartite framework admit a continuous deformation?"Theoretical Computer Science. 263. 345-354 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "On acute triangulation of quadrilaterals"Proceedings of JCDCG2000 (LNCS). 2698. 237-243 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "On the total edge-length of a tetrahedron"American Mathematical Monthly. 108. 967-969 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara and A. Oshiro: "Cutting a bunch of grapes by a plane"European Journal of Combinatorics. 22. 847-853 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Hosono, H. Maehara, K. Matsuda: "A pair in a crowd of unit balls"European Journal of Combinatorics. 22. 1083-1092 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "Isoperimetric theorem for spherical polygon and the problem of 13 spheres"Ryukyu Mathematical Journal. 14. 41-57 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "Acute triangulations of polygons"European Journal of Combinatorics. 23. 45-55 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara and A. Oshiro: "Arranging solid balls to represent a graph"Graphs and Combinatorics. 18. 343-365 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "The length of the shortest edge of a graph on a sphere"European Journal of Combinatorics. 23. 713-717 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "An inequality on the size of a set in a Cartesian product"European Journal of Combinatorics. 23. 1055-1059 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Maehara: "A simple proof of the existence of non-obtuse triangulation for polygons"Ryukyu Mathematical Journal. 15. 43-48 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Maehara, A.Oshiro: "Arranging solid balls to represent a graph"Graphs and Combinatorics. 18. 343-365 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Maehara: "The length of the shortest edge of a graph on a sphere"European Journal of Combinatorics. 23. 713-717 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Maehara: "An inequality on the size of a set in a Cartesian product"European Journal of Combinatorics. 23. 1055-1059 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Maehara: "A simple proof of the existence of non-obtuse triangulation for polygons"Ryukyu Mathematical Journal. 15. 43-48 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Maehara: "Acute triangulations of polygons"European Journal of Combinatorics. 23. 45-55 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Maehara, A.Oshiro: "Piercing a set of disjoint balls by a line"Journal of Combinatorial Theory (A). 94. 393-398 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Maehara, N.Tokushige: "When does a planar bipartite framework admit a continuous deformation ?"Theoretical Computer Science. 263. 345-354 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Maehara: "On acute triangulation of quadrilaterals"Proc. JCDCG2000(LNCS 2098). 237-243 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Maehara: "On the total edge-length of a tetrahedron"American Mathematical Monthly. 108. 967-969 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Maehara, A.Oshiro: "Cutting a bunch of grapes by a plane"European Journal of Combinatorics. 22. 847-853 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Maehara, K.Hosono, K.Matsuda: "A pair in a crowd of unit balls"European Journal of Combinatorics. 22. 1083-1092 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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