翻訳と辞書 Words near each other ・ Equity Weekly ・ Equity-indexed annuity ・ Equity-linked note ・ Equity-linked savings scheme ・ EquityNet ・ EquityZen ・ Equivalence ・ Equivalence (formal languages) ・ Equivalence (measure theory) ・ Equivalence (trade) ・ Equivalence class ・ Equivalence class (music) ・ Equivalence group ・ Equivalence of categories ・ Equivalence of direct radiation ・ Equivalence of metrics ・ Equivalence partitioning ・ Equivalence point ・ Equivalence principle ・ Equivalence principle (geometric) ・ Equivalence relation ・ Equivalent (chemistry) ・ Equivalent air depth ・ Equivalent airspeed ・ Equivalent annual cost ・ Equivalent average ・ Equivalent carbon content ・ Equivalent circuit ・ Equivalent concentration ・ Equivalent Concrete Performance Concept 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 Equivalence of metrics ： ウィキペディア英語版 Equivalence of metrics In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or equivalent. In the following, $X$ will denote a non-empty set and $d\_$ and $d\_$ will denote two metrics on $X$. ==Topological equivalence== The two metrics $d\_$ and $d\_$ are said to be topologically equivalent if they generate the same topology on $X$. The adjective "topological" is often dropped.〔Bishop and Goldberg, p. 10.〕 There are multiple ways of expressing this condition: * a subset $A\; \backslash subseteq\; X$ is $d\_$-open if and only if it is $d\_$-open; * the open balls "nest": for any point $x\; \backslash in\; X$ and any radius $r\; >\; 0$, there exist radii $r\text{'},\; r\text{'}\text{'}\; >\; 0$ such that :$B\_\; (x;\; d\_)\; \backslash subseteq\; B\_\; (x;\; d\_)$ and $B\_\; (x;\; d\_)\; \backslash subseteq\; B\_\; (x;\; d\_).$ * the identity function $I\; :\; X\; \backslash to\; X$ is both $(d\_,\; d\_)$-continuous and $(d\_,\; d\_)$-continuous. The following are sufficient but not necessary conditions for topological equivalence: * there exists a strictly increasing, continuous, and subadditive $f:\backslash mathbb\_\; \backslash to\; \backslash mathbb$ such that $d\_\; =\; f\; \backslash circ\; d\_$.〔Ok, p. 127, footnote 12.〕 * for each $x\; \backslash in\; X$, there exist positive constants $\backslash alpha$ and $\backslash beta$ such that, for every point $y\; \backslash in\; X$, :$\backslash alpha\; d\_\; (x,\; y)\; \backslash leq\; d\_\; (x,\; y)\; \backslash leq\; \backslash beta\; d\_\; (x,\; y).$ 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「Equivalence of metrics」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.