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0
votes
0answers
7 views

Probability Inequality when X > Y > 0

I want to know whether the following statement is true or not, and the proof. Let X, Y be random variable, satisfying X > Y > 0, and have finite variance, $Var(X) < \infty$ and $Var(Y) < ...
2
votes
0answers
51 views

Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
-4
votes
0answers
21 views

Asymptotic Analysis with Inequalities [on hold]

I have a problem understanding how the following inequalities highlighted in red were derived for this asymptotic analysis problem. Could someone explain the nature of these inequalities and how they ...
3
votes
0answers
118 views

An inequality with a sum of integrals

Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad ...
4
votes
2answers
142 views

Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that : $\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge ...
-1
votes
0answers
14 views

Upper estimate between an original function and its sup-convolution under a limitation

My setting maybe look rather special but I'm glad if you give some answers. Let $f:[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function and $f^{\varepsilon}:[0,1]\to\mathbb{R}$ be $f$'s ...
1
vote
0answers
14 views

Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models. Consider a hidden Markov model (HMM) with ...
11
votes
1answer
370 views

An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
3
votes
1answer
322 views

inequality with exponents

We are given a graph $G$, each vertex $v$ has an assigned value $\gamma_v\in [0,1]$, and it happens that for every $v$ we have $\gamma_v+\sum_{u\in \delta(v)} \gamma_u = 1$. Assume that $\sum_v ...
30
votes
2answers
2k views

An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
7
votes
2answers
165 views

concentration inequality for entropy from sample

Consider a measure $\mu$ on a finite set, and let $x_1, \ldots, x_n$ be i.i.d samples from $\mu$. Then the expression $S_n = -\frac{1}{n} \sum_{i=1}^n \log \mu(x_i)$ converges by a.s. to the entropy ...
0
votes
1answer
73 views

Estimate $\left|\sum_{n,m}a_n \bar b_m\right|\leq C \left(\sum_n|a_n|^2\right)^{1/2} \left(\sum_n|b_n|^2\right)^{1/2}$ [closed]

It is well-known the Hilbert's inequality for double sum: $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. I am ...
5
votes
1answer
134 views

Rate of Convergence of Borwein Algorithm for computing Pi

In a book "Pi and the AGM" in 1987, authors, Jonathan Borwein and Peter Borwein, introduced a magical algorithm to compute $\pi$. However there is a problem that I couldn't understand and couldn't ...
2
votes
0answers
84 views

Logarithmic bound for Diophantine equation

Let $a_1 \geq a_2 \geq a_3$ be given positive integers and let $N(a_1,a_2,a_3)$ be the number of solutions $(x_1,x_2,x_3)$ of the equation $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = ...
7
votes
2answers
309 views

regular polygon and constant potential function

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
0
votes
0answers
68 views

Conditional version of martingale difference concentration inequality

Let $M_n$ be a $\mathscr{F}_n=\sigma(\eta_m,\theta_m, m\leq n)$ measurable martingale difference sequence. Then is it possible to find a exponential tail bound for the following $$P(|M_{n+1}| > ...
87
votes
17answers
8k views

Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?

I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
2
votes
1answer
70 views

Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha ...
2
votes
0answers
84 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ ...
1
vote
0answers
68 views

Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron $$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$ Notice that $\mathcal P_X$ is symmetric about the origin. ...
1
vote
0answers
54 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
2
votes
2answers
572 views

Poincaré inequality for the annulus

Assume that $A=A(r,1)=\{x: r<||x||<1\} \subset R^n$ is an annulus. Whether is known the constant of Poincaré inequality for A or some its estimation (w.r.t. $L^2$): the constant $C$ in the ...
2
votes
2answers
342 views

A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality: $$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$ I have just ...
0
votes
0answers
179 views

Can we find an upper bound?

Let $f\in C^1(\mathbb R)$ with $f(0)=0$ and $|f'(x)|\le m$, where $m\in (1,2]$. Let $x(0)\in\mathbb R$ be arbitrary, and define $x(n),y(n)$ recursively by $$ x(n+1)=f(x(n)) , \quad\quad ...
17
votes
3answers
681 views

Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove the following deceptively simple inequality: $$(*)\qquad E(X^a) E(X^{a+1}\log X) > E(X^{a+1})E(X^a\log X) $$ where $X>e$ has ...
2
votes
1answer
146 views

Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1

Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...
2
votes
0answers
56 views

Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$ $$G = ...
4
votes
1answer
118 views

concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...
4
votes
2answers
269 views

Extending inequality for $\ell^p$ from integer $p$ to real $p$

Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are two decreasing sequences of positive numbers such that $a_1<b_1$ and $$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ ...
6
votes
1answer
293 views

Inequality for the maximum of Gaussian variables

Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix $\Gamma_Y-\Gamma_X$ is positive definite. Is it ...
14
votes
3answers
594 views

An inequality improvement on AMM 11145

I have asked the same question in math.stackexchange, I am reposting it here, looking for answers: How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$: ...
0
votes
0answers
71 views

Brunn-Minkowski Inequality : References request or A Particular Example of a 2 dimensional set

I had this question on Mathematics StackExchange unanswered for a month or so. Hence, it came here which it seems is a more natural place. I have an inequality - $(R_{C+D})^{2} \geqslant (R_{C})^{2} ...
2
votes
1answer
81 views

Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem: Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ ...
1
vote
2answers
220 views

Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...
8
votes
0answers
73 views

Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...
1
vote
2answers
284 views

Mapping multivariate polynomial inequalities system to subspace

What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution. I have a set of linear equations, e.g.: \begin{align} d_1 &= L_1 - ...
9
votes
2answers
625 views

Is this ergodic inequality true?

Is anything similar to the following inequality true, $\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$ where $A_n f = ...
12
votes
1answer
353 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...
2
votes
3answers
380 views

Hard inequality and number theory [closed]

Let $m,n \in \mathbb{N}$ and $m \geq n \geq 2$ and $x_1,x_2,...x_n \in \mathbb{N}_{\geq 1}$ such as $x_1+x_2+...+x_n=m$. Find $\min P$ with $P= \sum_{i=1}^{n} x_i^2.$
6
votes
1answer
176 views

Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M det(A)$ and the determinant of its ...
0
votes
1answer
124 views

For a set of positive integers, is this inequality always true?

The input consists of a set of positive integers $\{b_1,...,b_2\}$ such that $$\sum_{i=1}^nb_i=CK,$$ with $C$ and $K$ two positive integers. The question is the following, is there $k\in\{1,...,n\}$ ...
2
votes
1answer
72 views

Proof of Binet-Cauchy identity through the polarization transformation [closed]

This questions is motivated by exercise 3.7 in Steele's "The Cauchy-Schwarz Master Class." This is not a homework (I am trying to learn some math by myself) and I have already posted the question on ...
3
votes
1answer
659 views

Number of solutions in a sum of squares Diophantine equation

Let $n$ be an integer. I'm interested in upper bounding the number of integer solutions of \begin{equation*} x_1^2+x_2^2+\ldots+x_p^2=y_1^2+y_2^2+\ldots+y_p^2, \end{equation*} where for all ...
2
votes
0answers
91 views

Intuitive (?) inequality extremal inequality

Consider $N$ pairs of random variables $(X_i, Y_i)$. $X_i$ are iid, with $EX_i=0$ and $EX_i^2=1$. The same conditions hold for $Y_i$. Moreover all $X_i$ are independent of all $Y_j$. It seems very ...
0
votes
0answers
90 views

Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have $$(f(t))^4\le ...
3
votes
1answer
500 views

concentration inequality for averages of dependent random variables

Let $X \in R^n$ be a random vector such that $$P(|X_i| > \epsilon) > e^{-\epsilon^2}$$ 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| ...
-3
votes
1answer
611 views

How i show this beautiful inequality :$\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}\geq \frac{3} {2}(\frac{1}{\sqrt{3}})^{n-m}$? [closed]

This question accross to this question from SE which there some answers but they r n't enough to me hop to see MO what can they say about it . let $m,n$ be integers, show that if $ n>m\geq 0 $ : ...
0
votes
0answers
142 views

Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments. Maybe you guys can help. ...
2
votes
1answer
68 views

interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
2
votes
1answer
111 views

maximal inequalities for dependent random variables

I want to know literature about maximal inequalities for dependent random variables i.e. upper bound for $P(\max_{n\ge k\ge 1}\sum_{i=1}^{k}X_i > \delta)$ where $X_i$ are dependent random ...