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数学 大学生・専門学校生・社会人

多様体を構成するために、位相空間に完全アトラスを導入するところで質問です。 完全アトラスを導入するメリットとして、この文章の下線部を「異なる座標系を用いたのに同じ計算ができてしまうという問題が解消される」解釈したのですが、そこがよくわかりません。座標系を変えて計算する... 続きを読む

1 Two n-dimensional coordinate systems & and ŋ in S overlap smoothly provided the functions on¯¹ and ŋo §¯¹ are both smooth. Explicitly, if : U → R" and ŋ: R", then ŋ 1 is defined on the open set ε (ur) → ° (UV) V and carries it to n(u)—while its inverse function § 4-1 runs in the opposite direction (see Figure 1). These functions are then required to be smooth in the usual Euclidean sense defined above. This condition is con- sidered to hold trivially if u and do not meet. Č (UV) R" Ĕ(U) n(UV) R" S n(v) Figure 1. 1. Definition. An atlas A of dimension n on a space S is a collection of n-dimensional coordinate systems in S such that (A1) each point of S is contained in the domain of some coordinate system in, and (A2) any two coordinate systems in ✅ overlap smoothly. An atlas on S makes it possible to do calculus consistently on all of S. But different atlases may produce the same calculus, a technical difficulty eliminated as follows. Call an atlas Con S complete if C contains each co- ordinate system in S that overlaps smoothly with every coordinate system in C. 2. Lemma. Each atlas ✅ on S is contained in a unique complete atlas. Proof. If has dimension n, let A' be the set of all n-dimensional coordinate systems in S that overlap smoothly with every one contained in A. (a) A' is an atlas (of the same dimension as ✅).

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英語 高校生

上から5行目の And~easily. の文構造を教えて頂きたいです。justが形容詞でSVCではないのでしょうか?usの位置とthat節のはたらきが分からないです… また、下から2行目のrightの訳がよく分かりません。in the scientific literatu... 続きを読む

S V <なぜ> ~するために 名の~倍形だ。 倍数の表し方 ~times as 形 as ⑧ Fear takes an exposure time (of 250 mill seconds) (to recognize 125 times as long as a smile), makes absolutely no sense, evolutionarily speaking", Martinez says. 66 " which 以上 ☆2分のことを対比して表現するときに用いる whileは2つの意味を持つ!①~の間、②~だけれども≒though など Recognizing fear is fundamental to survival, while a smile isn't necessarily so, but that's how we are wired!" Studies have shown that smiling faces are judged as more familiar than neutral ones.> 名詞節をつくる And it's not just us that can recognize smiles more easily. 66 This is true both for humans and for machines" says Martinez. Although scientists have been studying smiles for about 150 years, they are still (at the stage of trying to categorize types) of smile among the millions) (of possible facial expressions). 63 many One of the fundamental questioness in the scientific literature right now is, how expressions do we actually produce)?" facial 疑問詞も名詞節をつくる 66 says Martinez. Nobody knows, a

解決済み 回答数: 2