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## Tagged Questions

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### the following inequality is true，but I can’t prove it

The inequality is \begin{equation*} \sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right) \end{equation*} for all integer $d\geq 1$. I use com …
Hi, What is the maximum of the following function?: $f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - … 0answers 49 views ### Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, … 0answers 62 views ### concavity of the function$(g\circ f)/f$, where$g$is concave and$f$is decreasing and convex Suppose we have functions$g\colon [0,1]\mapsto \mathbb{R}$, which is concave, vanishes at the origin and fullfills condition $$g(xy)/xy\leq g(x)/x+g(y)/y$$ for any$x,y\in[0,1]$a … 1answer 365 views ### Combinatorial Inequality Consider a set of$2^n-1$non negative integers$S= ${$ a_{i,j}|1\le i\le n; 1\le j\le 2^{i-1} } such that: \begin{align}{} 1.\ \ &a_{i,j}\le 2^{n+1-i} \\ 2.\ \ &a_{i,j}\ … 1answer 163 views ### concentration inequality for averages of dependent random variables LetX \in R^n$be a random vector such that $$P(|X_i| > \epsilon) \le e^{-\epsilon^2}$$ What is a tight bound on $$P(\sum_{i=1}^n |X_i| > \epsilon)$$ and on $$P(\max_{1\le … 0answers 66 views ### Multivariate polynomial with positive coefficients This question was originally asked at stack exchange (http://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive an … 1answer 161 views ### probability inequality I want to ask the following probability inequality: Is it true that for any random variable X\ge 0, we have$$ \sup_{t>0}(t\mathbb E(X\mathbf 1_{X\ge t})) \le 2\sup_{t>0}(t … 1answer 162 views ### a simple probability inequality For independent Rademacher random variables$\epsilon_i, i=1,2, \cdots, n$, i.e.$P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have $$max_{0\le a_i\le b, i=1,2,\cdots,n}P … 1answer 229 views ### Trajectorial version of Doob’s L^2 inequality In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf you can find a trajectorial version of Doob's inequality. It is given by:$$\bar{s}^2_T+4\su … 2answers 283 views ### On a inequality [closed] I hope the following kind of inequality holds: let$a_i,b_i\in R$with$b_i>0$,$\sum _{i=1}^mt_i=1$with$t_i>0$, then $$\frac{t_1a_1+\cdot+t_ka_k}{t_1b_1+\cdot+t_kb_k}\le\frac{a … 2answers 197 views ### A Gronwall-type inequality. I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that.$$ f^2( … 1answer 125 views ### An upper bound on a simple sum Hi, I am trying to put a bound on a sum. Given$\omega=\exp(2\pi i/3)$and$n$positive real numbers$ 0=\tau_0 < \tau_1 < \tau_2 < ...\tau_{n-1}< \tau_n=1 $such th … 0answers 90 views ### Upper bound on integrals of Legendre polynomials Hi, If$P_n(x) $is unnormalized shifted Legendre polynomial, and$g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $then what is the upper bound$ |g_{n,m} (x)|_{max} , x\in (0,1) …
Dear all, Suppose U and V are unitary matrix, A and B are positive definite, Does: $UAU^{-1} < VBV^{-1}$ implies $A< B$ and vice versa?