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Questions tagged [isometry]

等轴测图是保留距离的度量空间之间的映射。该标签用于有关等距的问题。

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Frenet基础和扭转,有助于此证明。

如何显示: 3维空间中的等轴测图保留了扭转的值(大小),而无需使用扭转公式。 找到一个可以改变扭转符号(在特定点上)的等距图。 一世 ...
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1回答
54浏览

等面积内表面之间的每个等距浸没都具有内射性吗?

Let $M,N$ be smooth connected, compact two-dimensional Riemannian manifolds, such that $M$ has a non-empty Lipschitz boundary. Suppose that $\operatorname{Vol}(M)=\operatorname{Vol}(N)$. 题: ...
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47浏览

等轴测图在局部嵌入了lipschitz

文本中的詹姆斯·蒙克雷斯"流形分析" 给出以下定义。 定义 Let $h:\Bbb R^n\rightarrow\Bbb R^n$. We say that $h$ is a (euclidean) isometry if $$ ||h(x)-h(y)|| ...
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29浏览

等距的推广

度量空间同构中的等轴测图,意味着它保留了度量空间的所有属性。 是否有Isometry的概括,它讨论直到...
3
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26浏览

Embedding of $\sqrt{|i-j|}$ distance into $(\mathbb{R}^n,\lVert \cdot\rVert_2)$

Consider the metric space $(X=\{1,\ldots,n\},d)$ such that: $$d(i,j)=\sqrt{|i-j|}$$ Can $(X,d)$ be isometrically embedded in $(\mathbb{R}^n,\lVert \cdot\rVert_2)$? If that is the case, can we find ...
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42浏览

欧氏等式的离散阿贝尔亚群

Suppose $\Gamma$ is a discrete, cocompact Abelian subgroup of the Lie group $\mathrm{E}(n) = \mathbb R^n \rtimes \mathrm{O}(n)$ of Euclidean isometries, where $$(b,A)(b',A') = (b+Ab', AA') \quad \...
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25浏览

关于内积空间上的反射运算符的问题

Let $V$ be an $n$-dimensional real inner product space. Suppose $u \in V$ has norm 1, and define the reflection in the direction of $u$ as $$r_u(v) = v - 2 \langle v, u \rangle u = v - 2proj_{span(u)}(...
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11浏览

证明范数相等(Hardy空间上的合成算子)

For $\lambda\in \mathbb{D}$ fixed, let $C_{\phi}$ denote the composition operator on Hardy space $H^2(\mathbb{D})$ (I.e. $C_{\phi}f:=f(\phi)$) with symbol $\phi=\frac{\lambda-z}{1-\bar{\lambda}z}$ (a ...
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176浏览

If there is an into isometry from $(\mathbb{R}^m,\|\cdot\|_p)$ to $(\mathbb{R}^n, \|\cdot\|_q)$ where $m\leq n$, then $p=q$?

Let $p,q\in [1,\infty)$. Note that $p,q\neq\infty$. Let $m\geq 2$ be a natural number. The paper Isometries of Finite-Dimensional Normed Spaces by Felix and Jesus asserts that if $(\mathbb{R}^m,\|\...
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12浏览

等距和等距同构定义中的等效项。

Given two vector spaces $X$ and $Y$ equipped with norm $|| \cdot||_X$ and $|| \cdot||_Y$, respectively. 我想知道以下语句是否等效。 (1)。 $ X $和$ Y $是等轴测图。 (...
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40浏览

关于等量表示作为反射产物的证明的困惑。

I am reading a proof of reflection representations of isometries that fix the origin in $\mathbb{R}^n$. It is a simple induction on the dimension $n$. For $f(0)=0$ we have some $v\neq w=f(v)$. The ...
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19浏览

关于证明特征局部等距的问题

考虑以下文本片段"墨菲的$ C ^ * $-代数和算子理论": Could someone explain why the marked step is true? I don't see how this follows from $\Vert u(x) \...
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77浏览

If $T:(\mathbb{R}^2,\|\cdot\|_p) \to (\mathbb{R}^2,\|\cdot\|_q)$ is an onto linear isometry, then must it be $p=q$?

Question: Let $p,q\in [1,\infty)$ and suppose that that $T:(\mathbb{R}^2,\|\cdot\|_p) \to (\mathbb{R}^2,\|\cdot\|_q)$ is an onto linear isometry. Must it be $p=q$? 我认为等轴测是真的...
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41浏览

If a Hilbert space is linearly isometric to a Banach space, will the Banach space be a Hilbert space? [duplicate]

Question: Given two Banach spaces $X$ and $Y$, if there exists an onto linear isometry $T:X\to Y$ and $X$ is a Hilbert space, is it true that $Y$ is a Hilbert space? 直觉上这对我来说是真的。
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33浏览

Let $X$ be a Banach space and $E$ a sublinear subspace . Show there exists a surjective isometry $\phi : E^* \rightarrow \overline{ E}^* $

Let $X$ be a Banach space and $E$ a sublinear subspace of $X$ . Show there exists a surjective isometry $\phi : E^* \rightarrow \overline{ E}^* $ 我认为这可能是戈尔斯廷的衍生事实...

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