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We set : $\phi_1, \phi_2 : \mathbb{R} \to \mathbb{R}^M$ with compact support $\theta \in \mathbb{R}^M$ non-zero, $\theta_{p-1} = (\operatorname{sign}(\theta_i)|\theta_i|^{p-1})_{1 \leq i \leq M} \in \...

In the book Number Theory IV from Parshin, one can find this statement (precisely at p. 215) with the comment "it is easy to see that...": Let $a_{ij}$ ($1\le i,j\le m$) be complex numbers ...

$\newcommand\norm[1]{\lVert#1\rVert} \newcommand\opnorm[1]{\norm{#1}_{\text{op}}} \newcommand\Frnorm[1]{\norm{#1}_{\text F}}$Define \begin{align*} S_t=\{ (A,B)\in\mathbb{R}^{n\times n}\times\...

Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...

ORIGINAL QUESTION Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a $2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller eigenvalue of a $2\times2$ matrix. Is it true ...