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1
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0answers
115 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, but I think it is ...
5
votes
1answer
285 views

A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and $$ K^* := \{y \in \mathbb{R}^n : \langle y, x ...
7
votes
1answer
471 views

A spectral inequality for positive-definite matrices

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues $$ \lambda_1 \leq \cdots \leq \lambda_n , $$ is there a sharp upper bound for the product $\lambda_2 \cdots ...
2
votes
0answers
53 views

Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &=& ...
0
votes
1answer
186 views

Inequality of Partial Taylor Series

Hi, For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} ...
12
votes
0answers
337 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
3
votes
1answer
247 views

Chernoff-Hoeffding bound for complex values

Consider the Chernoff-Hoeffding bound, stated as follows: Let $X_1, \dots, X_K$ be i.i.d. real-valued random variables with expectation value $\mu$ and satisfying $|X_i| \le b$. Let $\epsilon > 0$. ...
17
votes
1answer
881 views

What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...
1
vote
1answer
352 views

for what arguments the function reaches maximum?

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 - \frac{w_{i}}{ ...
9
votes
2answers
482 views

The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a ...
12
votes
5answers
782 views

Understanding Gibbs's inequality

Short version Gibbs's inequality is a simple inequality for real numbers, usually understood information-theoretically. In the jargon, it states that for two probability measures on a finite set, ...
2
votes
1answer
266 views

concentration inequality for averages of dependent random variables

Let $X \in R^n$ 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 i\le n} |X_i| ...
0
votes
1answer
243 views

An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function: \begin{equation} g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k. \end{equation} Prove that $\exists C>0$ and $\phi(s)$ such ...
2
votes
1answer
447 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}\le a_{i-1,j} \\\\ ...
4
votes
1answer
342 views

Inequality on Trigonometric polynomials

My question comes from trying to understand a technical step in this paper by Bourgain. Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. ...
1
vote
0answers
90 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 any feedback for more ...
1
vote
2answers
289 views

Spherical Bessel functions

I wish to show that $|j_n(x)| < \frac{1}{\sqrt{x}}$ for $n=0,1,2,\ldots$ and $x>0$, where $j_n$ is the spherical Bessel function of the first kind. Experimenting with Matlab I am sure that ...
0
votes
2answers
340 views

Efficient algorithm finding 'a' solution of system of linear inequalities

I'm working on rational number field $\mathbb{Q}$. Is there an efficient algorithm finding a solution of system of linear inequalities? In many computer algebra systems like Sage or Maple, there ...
2
votes
1answer
217 views

another diameter-perimeter-area inequality

Recently I learnt that $$ \inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no ...
3
votes
2answers
262 views

a diameter-perimeter-area inequality for convex figures

Is the following inequality known? I believe it's true, but I could find no reference. For any convex body $C$ in the plane we have $$\left(4-\frac{8}{\pi}\right)area(C)\leq > ...
2
votes
1answer
253 views

on an inequality of Brezis-Lieb

In their 1983 JFA paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their inequality ((3.14) ...
16
votes
2answers
397 views

What is $A+A^T$ when $A$ is row-stochastic ?

This is motivated by this MO question. If $A\in{\bf M}_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum_j a_{ij}=1$ for all $i$), then $M:=A+A^T$ is symmetric, entrywise ...
3
votes
2answers
114 views

a monotone relation for s-numbers

Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one ...
0
votes
1answer
132 views

Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ such that $$\sum_{i=1}^n (x_i - x') = x - x'.$$ Is it ...
0
votes
1answer
378 views

How to solve this optimization with the orthogonal constraint?

Problem Supposing that $A$ is a symmetric real matrix and $\{\mathbf{w}\_i\}_{i=1}^n$ is any orthogonal basis on $\mathbb{R}^n$ such that $W^\top W=WW^\top=\mathbf{I}_n$ where ...
3
votes
1answer
203 views

A Wirtinger-like inequality involving two functions

Let $f(t)$ and $g(t)$ be periodic functions on $t\in[0,2\pi]$. By using the Fourier series of the two functions, we can easily prove the inequality $$\left|\int_0^{2\pi}f(t)g'(t)dt\right|= ...
3
votes
0answers
137 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) $ as a function ...
-1
votes
2answers
330 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 ...
0
votes
1answer
177 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, ...
6
votes
2answers
259 views

Inequality involving the weak second moment

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^2 \mathbb P(X ...
3
votes
0answers
301 views

A curious inequality

Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that $|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that ...
1
vote
0answers
166 views

A Multiplicative version of McDiarmid's Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following: Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...
1
vote
1answer
150 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 that ...
4
votes
1answer
272 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: ...
0
votes
1answer
238 views

Inequalities Involving Wedge Product (Reference Request)

Hello, I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The ...
11
votes
2answers
518 views

Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the ...
1
vote
2answers
267 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(t) \leqslant g^2(t) ...
4
votes
1answer
300 views

Upper bound on joint Renyi entropy

Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by \begin{equation} H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha. \end{equation} ...
11
votes
1answer
610 views

A delicate elementary inequality

The following "piecewise-quadratic" inequality emerged in a joint work of Rom Pinchasi and myself. The inequality is surprisingly delicate, and all our attempts to simplify it made it false. By the ...
4
votes
0answers
403 views

System of Equations Upper Bound

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here: For ...
5
votes
4answers
663 views

An inequality involving sums of powers

I asked this question at Stack Exchange but received no answer. The origins of the question are unclear, as I came across it rummaging through old notebooks from highschool, in one of which it was ...
1
vote
1answer
228 views

Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies $$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$ where $|\cdot|$ denotes the ...
5
votes
1answer
396 views

Trace matrix inequality

Hello all, I come across the following problem. Is it true that for a positive definite matrix $X^{n\times n}$, the following holds $\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$, ...
5
votes
1answer
415 views

Hadamard-like inequalites for positive definite symmetric matrices

Let $S$ be any positive semi-definite symmetric matrix (Hermitian psd matrices work as well). The Hadamard inequality is that $$\det S\le\prod_{i=1}^n s_{ii}.$$ My question is whether there are some ...
0
votes
1answer
83 views

Continuous variant of KFG Inequality

Context Let $t_1, .., t_n$ be jointly independent boolean random variables. Let $X, Y$ be monotone functions (i.e. $\forall i: t_i \geq t'_i$ implies $X(t_1, ... t_i ..., t_n) \geq X(t'_1, ... t'_i, ...
1
vote
1answer
112 views

Approximating Moment of Sum of RVs

Given $X_i$ are independent random variables. $|X_i| < 1$ $E[X_i] = 0$ $X = \sum_i^n X_i$ $var(X)=\sigma$ Prove: $$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p Things I've tried: ...
5
votes
2answers
134 views

Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...
1
vote
1answer
161 views

Unitary matrix and matrix inequality [closed]

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?
0
votes
1answer
206 views

Inequality with even powers of trigonometric functions

For $m>0$, $0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
3
votes
3answers
290 views

Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$? Or any good reference for tools to tackle this question? ...