Tagging - 2017-12-17

Some more new tags.

The tags lyapunovtheory and detectability were created here (by Urgent):

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Suppose $E,A$ are $m\times n$ and $C$ is $p\times n$ matrix. it is given that a pair of matrices $(A,C)$ is detectable i.e as by definition: $rank\begin{bmatrix} \lambda E-A\\C\end{bmatrix}=n\forall \lambda\in\bar{\mathbb{C}}^+=\{\lambda=a+ib\in: a\ge 0\}$ Suppose $N=A-KC$ and $\dot e=Ne\dots \d...

The tag fractional-sobolev-spaces was created by Guy Fsone, including the tag-excerpt and the tag-wiki.

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Q: Fractional Sobolev Spaces $ W^{s,\infty}(\Omega)$ on a bounded domain

I have two questions concerning Fractional Sobolev Spaces. Let $W^{s,p}(\Omega) := \{u\in L^p(\Omega) |\frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}} \in L^p(\Omega \times \Omega)\}$ resp. $W^{s,\infty}(\Omega) := \{u\in L^\infty(\Omega) |\frac{|u(x)-u(y)|}{|x-y|^s} \in L^\infty(\Omega \times \Omeg...

sobolev-spaces fractional-sobolev-spaces

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Q: $W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : \|(\langle{\xi}\rangle^{s}\widehat{f})^{\vee}\|_{L^{p}}<\infty\right\}$$, equipped with norm $$\|f\|_{W^{s,p}}=\|(\l...

functional-analysis fourier-analysis sobolev-spaces littlewood-paley-theory fractional-sobolev-spaces

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Q: Comparaison of two version of Fractional sobolev spaces: what do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?

There are two version of Fractional sobolev spaces . Definition1: (Via Galiardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and $\Omega\subseteq \mathbb{R}^n$ an open set. The fractional Sobolev space $W^{s,p}(\Omega)$ is defined to be $$ W^{s,p}(\Omega) = \left\{ u\in L^p(\Omega) : \frac{|u(...

functional-analysis banach-spaces sobolev-spaces fractional-sobolev-spaces

There is also question by the same useron MO:

1

Q: Comparaison of two version of Fractional sobolev spaces: what do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?

There are two version of Fractional sobolev spaces . Definition1: (Via Galiardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and $\Omega\subseteq \mathbb{R}^n$ an open set. The fractional Sobolev space $W^{s,p}(\Omega)$ is defined to be $$ W^{s,p}(\Omega) = \left\{ u\in L^p(\Omega) : \frac{|u(...

fa.functional-analysis banach-spaces sobolev-spaces fractional-calculus

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$Mathematics$ Tagging