翻訳と辞書 Words near each other ・ Sparse approximation ・ Sparse array ・ Sparse binary polynomial hashing ・ Sparse conditional constant propagation ・ Sparse distributed memory ・ Sparse file ・ Sparse graph code ・ Sparse grid ・ Sparse image ・ Sparse language ・ Sparse matrix ・ Sparse matrix converter ・ Sparse matrix-vector multiplication ・ Sparse network ・ Sparse PCA ・ Sparse ruler ・ Sparse voxel octree ・ Sparsely totient number ・ Sparsely-spotted stingaree ・ Sparsetooth dogfish ・ Sparsh ・ Sparsh (Festival) ・ Sparsh (film) ・ Sparsh (software) ・ Sparsh Khanchandani ・ Sparsh Shrivastav ・ Sparsha (film) ・ Sparsham ・ Sparsholt ・ Sparsholt College Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sparse ruler ： ウィキペディア英語版 Sparse ruler A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length $L$ with $m$ marks is a sequence of integers $a\_1,\; a\_2,\; ...,\; a\_m$ where . The marks $a\_1$ and $a\_m$ correspond to the ends of the ruler. In order to measure the distance $K$, with there must be marks $a\_i$ and $a\_j$ such that $a\_j-a\_i=K$. A ''complete'' sparse ruler allows you to measure any integer distance up to its full length. A complete sparse ruler is called ''minimal'' if there is no complete sparse ruler of length $L$ with $m-1$ marks. In other words, if any of the marks is removed one can no longer measure all of the distances, even if the marks could be rearranged. A complete sparse ruler is called ''maximal'' if there is no complete sparse ruler of length $L+1$ with $m$ marks. A sparse ruler is called ''optimal'' if it is both minimal and maximal. Since the number of distinct pairs of marks is $m(m-1)/2$, this is an upper bound on the length $L$ of any maximal sparse ruler with $m$ marks. This upper bound can be achieved only for 2, 3 or 4 marks. For larger numbers of marks, the difference between the optimal length and the bound grows gradually, and unevenly. For example, for 6 marks the upper bound is 15, but the maximal length is 13. There are 3 different configurations of sparse rulers of length 13 with 6 marks. One is . To measure a length of 7, say, with this ruler you would take the distance between the marks at 6 and 13. A Golomb ruler is a sparse ruler that requires all of the differences $a\_j-a\_i$ be distinct. In general, a Golomb ruler with $m$ marks will be considerably longer than an optimal sparse ruler with $m$ marks, since $m(m-1)/2$ is a lower bound for the length of a Golomb ruler. A long Golomb ruler will have gaps, that is, it will have distances which it cannot measure. For example, the optimal Golomb ruler has length 17, but cannot measure lengths of 14 or 15. == Wichmann rulers == Many optimal rulers are of the form W(r,s) = 1^r, r+1, (2r+1)^r, (4r+3)^s, (2r+2)^(r+1), 1^r, where a^b represents b segments of length a. Thus, if r = 1 and s = 2, then W(1,2) has (in order): 1 segment of length 1, 1 segment of length 2, 1 segment of length 3, 2 segments of length 7, 2 segments of length 4, 1 segment of length 1 That gives the ruler . The length of a Wichmann ruler is 4r(r+s+2)+3(s+1) and the number of marks is 4r+s+3. Note that not all Wichmann rulers are optimal and not all optimal rulers can be generated this way. None of the optimal rulers of length 1, 13, 17, 23 and 58 follow this pattern, but no optimal rulers with length greater than 68 are known that are not Wichmann rulers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sparse ruler」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.