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Graph cut that cuts nothing?

We have a graph $G = (V, E)$. Is it possible to have a cut (S, V-S) that cuts the graph G such that S is empty? That is, a cut that cuts nothing, like the image below shows. The reason that I am asking is that I am reading Introduction to Algorithms, 3rd edition. […]

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Some question on ideal and quotient ring

Let $S,T$ be subrings with unity of a big commutative ring with unity $R$. Let $I,J$ be ideals of $S$ and $T$ respectively. Then I am wodering whether quotient ring $IT/I \cap J$ can be embedded into $T/J$. Does this hold? Any comments will be appreciated!

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Use $\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$ and $\mathbb{P}(\vert \hat{s}_n-s_n\vert > x)\leq b(n,x)$ to bound $\vert s_n – s\vert$

Let $s, s_n\in\mathbb{R}$ and $\hat{s}_n$ be a random variable. I have two concentration inequalities: $$\mathbb{P}(\vert \hat{s}_n-s\vert > x)\leq a(n,x)$$ for all $n\geq1$ and $x>0$; and $$\mathbb{P}(\vert \hat{s}_n-s_n\vert > x)\leq b(n,x)$$ for all $n\geq1$ and $x>0$. Is there a way to bound $\vert s_n – s\vert$?

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Equivalence of Weak/Strong Uniform Glivenko-Cantelli Classes

Let ($S, \sum, P)$ be a probability space and $H$ a class of real valued measurable functions on $S$. Then we say that $H$ is a weak uniform Glivenko-Cantelli class if for any $\epsilon > 0$ $$ \lim_{n \rightarrow \infty} \sup_P P^n \{(s_1,…,s_n) \in S^n: ||P_n – P||_H^* > \epsilon\} = 0 $$ and a […]

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Metric on Power Set

Let (S, d) be a metric space. Can one always define a metric d# on the power set P(S) of $S$ such that d# ({x} , {y}) = d(x, y) for every x, y ∈ S?

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Spectrum of product of self-adjoint operators contained in $\mathbb{R}$

Let $S,T$ be self-adjoint bounded operators on a complex Hilbert space. In this post, it is shown that $\sigma(ST)\subset\mathbb{R}$. The answerer uses that $\sigma(ST)\cup\{0\}=\sigma(TS)\cup\{0\}$ and that $\sigma(U)=\sigma(U^{*})^{*}$ for any operator $U$. I know that these results are indeed true. But how does he use these results to conclude that $\sigma(ST)\subset\mathbb{R}$? If I play around with […]