球面上のランダム幾何とその応用

前原, 濶; Maehara, Hiroshi

WEKO3

lat lon distance

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Now, let F denote a family of N random caps all of the same size (4\u03c0c/N)log N. If c\u003e1/2, then the probability that G(F) is 2-connected tends to 1 as N tends to infinity. If c\u003c1/4, 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 \u2220AOB=\u03c9. 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 \u0027visual\u0027 size of the angle \u2220AOB is called the random visual size and denoted by \u0398(\u03c9). By a joint study with Yoich Maeda (Tokai univ.), we proved that the expected value of \u0398(\u03c9) is equal to \u03c9, and derived a formula to calculate the variance of \u0398(\u03c9).", "subitem_description_type": "Other"}, {"subitem_description": "\u672a\u516c\u958b\uff1aP.7\uff5e71\uff08\u8ad6\u6587\u5225\u5237\u306e\u305f\u3081\uff09", "subitem_description_type": "Other"}, {"subitem_description": "\u7814\u7a76\u5831\u544a\u66f8", "subitem_description_type": "Other"}]}, "item_1617186643794": {"attribute_name": "Publisher", "attribute_value_mlt": [{"subitem_1522300295150": "ja", "subitem_1522300316516": "\u524d\u539f\u6ff6"}]}, "item_1617186702042": {"attribute_name": "Language", "attribute_value_mlt": [{"subitem_1551255818386": "jpn"}]}, "item_1617186783814": {"attribute_name": "Identifier", "attribute_value_mlt": [{"subitem_identifier_type": "HDL", "subitem_identifier_uri": "http://hdl.handle.net/20.500.12000/9309"}]}, "item_1617186920753": {"attribute_name": "Source Identifier", "attribute_value_mlt": [{"subitem_1522646500366": "NCID", "subitem_1522646572813": "BA64192299"}]}, "item_1617187056579": {"attribute_name": "Bibliographic Information", "attribute_value_mlt": [{"bibliographicIssueDates": {"bibliographicIssueDate": "2003-03", "bibliographicIssueDateType": "Issued"}, "bibliographicPageStart": "none"}]}, "item_1617258105262": {"attribute_name": "Resource Type", "attribute_value_mlt": [{"resourcetype": "research report", "resourceuri": "http://purl.org/coar/resource_type/c_18ws"}]}, "item_1617265215918": {"attribute_name": "Version Type", "attribute_value_mlt": [{"subitem_1522305645492": "VoR", "subitem_1600292170262": "http://purl.org/coar/version/c_970fb48d4fbd8a85"}]}, "item_1617605131499": {"attribute_name": "File", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_access", "download_preview_message": "", "file_order": 0, "filename": "13640126-2.pdf", "future_date_message": "", "is_thumbnail": false, "mimetype": "", "size": 0, "url": {"objectType": "fulltext", "url": "https://u-ryukyu.repo.nii.ac.jp/record/2004460/files/13640126-2.pdf"}, "version_id": "f31f2177-682d-4c06-a91f-537796bb7c1e"}, {"accessrole": "open_access", "download_preview_message": "", "file_order": 1, "filename": "13640126-1.pdf", "future_date_message": "", "is_thumbnail": false, "mimetype": "", "size": 0, "url": {"objectType": "fulltext", "url": "https://u-ryukyu.repo.nii.ac.jp/record/2004460/files/13640126-1.pdf"}, "version_id": "16aea07e-72f9-4b78-bdc4-79a32818e468"}]}, "item_title": "\u7403\u9762\u4e0a\u306e\u30e9\u30f3\u30c0\u30e0\u5e7e\u4f55\u3068\u305d\u306e\u5fdc\u7528", "item_type_id": "15", "owner": "1", "path": ["1642838403123", "1642838405037"], "permalink_uri": "http://hdl.handle.net/20.500.12000/9309", "pubdate": {"attribute_name": "PubDate", "attribute_value": "2009-03-19"}, "publish_date": "2009-03-19", "publish_status": "0", "recid": "2004460", "relation": {}, "relation_version_is_last": true, "title": ["\u7403\u9762\u4e0a\u306e\u30e9\u30f3\u30c0\u30e0\u5e7e\u4f55\u3068\u305d\u306e\u5fdc\u7528"], "weko_shared_id": -1}

球面上のランダム幾何とその応用

http://hdl.handle.net/20.500.12000/9309

	名前 / ファイル	ライセンス	アクション
	13640126-2.pdf
	13640126-1.pdf

Item type

デフォルトアイテムタイプ（フル）(1)

公開日

2009-03-19

タイトル

球面上のランダム幾何とその応用

言語

作成者

前原, 濶
Maehara, Hiroshi

アクセス権

open access

アクセス権URI

http://purl.org/coar/access_right/c_abf2

主題

言語

主題Scheme

Other

主題

漸近分布

言語

主題Scheme

Other

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最小距離の分布

言語

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Other

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ランダム球帽系

言語

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Other

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交グラフ

言語

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Other

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片側球帽

言語

主題Scheme

Other

主題

見かけの角

内容記述

内容記述タイプ

Other

内容記述

科研費番号: 13640126

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Other

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科学研究費補助金(基盤研究(C)(2))研究成果報告書

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Other

内容記述

研究概要 : 1.辺数がNの任意のグラフGに対して、その頂点集合をd次元単位球面上にランダムに配置する。Gにおいて辺で結ばれているような頂点対に対応する球面上の点対の間の球面距離の最小値をDとする。N→∞のとき、ND^dの分布は平均がdB(1/2,d/2)の指数分布に収束する。ここでB(p,q)はベータ関数である。2.2次元の単位球面上の有限個の球帽系F={C_1,C_2,...,C_N}に対して、その交グラフをG(F)で表わす。球帽系Fにおいて球帽C_iと交わる球帽の中心がすべてC_iの中心を通るある大円の一方の側にあるとき、C_iは片側球帽という。また、半球面より小さい球帽はプロパーであるという。Fに片側球帽が存在しなければ、交グラフG(F)は連結となる。また、片側球帽が存在せず、さらに、球帽がすべてプロパーなら、G(F)は2連結となる。(3次元以上の球面では、これと類似な結果は成立しない。)これを応用して、ランダム球帽系の交グラフに関する次の漸近的な結果が得られる。こんどは、F={C_1,C_2,...,C_N}を2次元単位球面上のすべて同じ大きさ(4πc/N)log Nのランダムな球帽の系とする。c>1/2なら、N→∞のとき、G(F)が2連結になる確率は1に収束し、c<1/4なら、G(F)が連結になる確率は0に収束する。3.次元空間内の三角形ΔAOBの∠AOBの大きさをωとする。点Pから∠AOBを見るときその見かけの大きさは、 ΔAOBを直線POに垂直な平面に正射影して得られる三角形ΔA'O'B'の∠A'O'B'の大きさに等しい。点PがOを中心とする単位球面上のランダムな点のとき、∠A'O'B'の大きさΘ(ω)を∠AOBのランダムな見かけの角という。東海大の前田陽一氏との共同研究により、確率変数Θ(ω)の平均が ωに等しいことを示し、また、その分散が容易に数値計算できる2重積分の式を得た。

内容記述タイプ

Other

内容記述

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).

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Other

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未公開：P.7～71（論文別刷のため）

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Other

内容記述

研究報告書

出版者

言語

出版者

前原濶

言語

jpn

資源タイプ

research report

資源タイプ識別子

http://purl.org/coar/resource_type/c_18ws

出版タイプ

VoR

出版タイプResource

http://purl.org/coar/version/c_970fb48d4fbd8a85

識別子

http://hdl.handle.net/20.500.12000/9309

識別子タイプ

HDL

収録物識別子

収録物識別子タイプ

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収録物識別子

BA64192299

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p. none, 発行日 2003-03

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2022-01-27 08:05:54.845196

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球面上のランダム幾何とその応用

× 前原, 濶

× Maehara, Hiroshi

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