| Co-Investigator(Kenkyū-buntansha) |
TOKUSHIGE Norihide Univ.of the Ryukyus, Faculty of Education, Associate Professor, 教育学部, 助教授 (00217481)
MATSUMOTO Shuichi Univ.of the Ryukyus, Faculty of Education, Professor, 教育学部, 教授 (20145519)
KATO Mitsuo Univ.of the Ryukyus, Faculty of Education, Professor, 教育学部, 教授 (50045043)
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| Research Abstract |
1. A cyclic sequence of nonoverlapping unit balls in R^3 in which each consecutive balls are tangent, is called a necklace of pearls. We show that to make a knotted necklace of pearls, 15 unit balls are sufficient. To make a knotted necklace that can be inscribed between a pair of parallel planes with distance 2+√<2> apart, 16 unit balls are necessary, and the trefoil is the unique knot that can be made by 16 unit balls.
2. A chain is a finite sequence of balls in which each consecutive pair of balls are tangent. Make a graph by representing vertices by balls, and edges by chains connecting two vertex-balls. Let b_n be the minimum number of balls necessary to make a complete graph of n vertices. Then we got the bound c_1n^3<b_n<c_2n^3 log n. A similar bound is also obtained when we use balls all sitting on a fixed table.
3. For a family F of balls in d-dimensional space R^d, let λ= λ(F)=(the max. radius) / (the min. radius). We proved that for any family of n balls in R^d, there is a direction such that any line with this direction intersects at most O (√<(1+logλ)n log n>) balls. On the otherhand, for n【greater than or equal】d, there is a family of nonoverlapping n balls in R^d such that for any direction, there is a line with this direction that intersects at least n-d+1 balls. For a family of balls sitting on a fixed table in R^3, we also got an upper bound of the average number of balls pierced by a vertical line meeting the table.
4. If a family of nonoverlapping balls in R^3 satisfies that logλ=o ((n/log n)^<1/3>), then there is a plane both sides of which contain n/2-o (n) intact balls.
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