query: Prove that $|||A||| \leq |||I|||$ for every doubly stochastic matrix $A \in M_n$
positive_passages.docid: 16779
positive_passages.text: Why does $\left\| {\left| A \right|} \right\| \le \left\| {\left| I \right|} \right\|$, for every doubly stochastic matrix $A \in M_n$?
negative_passages.docid: 39409
negative_passages.text: $X_1,...,X_n$ be independent RVs and $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$. Show that $\sigma (X_1,...,X_n) \perp \mathcal F$
negative_passages.docid: 45585
negative_passages.text: Proof of Steinitz Theorem
negative_passages.docid: 42366
negative_passages.text: Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
negative_passages.docid: 63124
negative_passages.text: Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
negative_passages.docid: 110222
negative_passages.text: Maximal inequality for a sequence of partial sums of independent random variables
negative_passages.docid: 89325
negative_passages.text: $a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
negative_passages.docid: 36374
negative_passages.text: Exercise 1.6.3 from Alon & Spencer's *The Probabilistic Method*: prove that $Pr[|X-Y| \leq 2] \leq 3 Pr[|X-Y| \leq 1]$ for i.i.d. real RVs $X$ and $Y$
negative_passages.docid: 15851
negative_passages.text: the converse of Schur lemma
negative_passages.docid: 28565
negative_passages.text: Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
negative_passages.docid: 349
negative_passages.text: Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
negative_passages.docid: 40930
negative_passages.text: Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$
negative_passages.docid: 117640
negative_passages.text: Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
negative_passages.docid: 144382
negative_passages.text: Proving $\lVert x \rVert_3$ is smaller than euclidean norm
negative_passages.docid: 98631
negative_passages.text: Postive-semidefiniteness of matrix with entries $1/(a_i+a_j)$
negative_passages.docid: 122326
negative_passages.text: Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$