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1
vote
1answer
107 views

Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step ...
3
votes
1answer
153 views

How to use Gronwall's inequality? [closed]

I am just trying to understand the role of Grownwall's Lemma to show global wellposedness results, in the paper I have been reading. And So I hope this is OK for MO. Let $u\in C(\mathbb R, L^{2})$...
2
votes
1answer
101 views

Sufficient conditions for weak majorization

Suppose $x_1\ge x_2\ge \cdots \ge x_n\ge 0$ and $y_1\ge y_2\ge\cdots\ge y_n\ge0$ be reals such that for any positive integer $p$, $$ \sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p. $$ Question: Is ...
1
vote
0answers
60 views

Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} $-algebra

Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: \...
0
votes
0answers
68 views

Extremum of the cyclic sum of polynomial ratios (same degree)

I've noticed a few times (probably nothing new) that cyclic sums (assuming $x, y, z > 0$) like: $\frac{x^2+y^2}{yz} + \frac{y^2+z^2}{zx} + \frac{z^2+x^2}{xy}$, where in each of the 3 ratios, all ...
0
votes
0answers
733 views

Probabilistic Modeling Parameters Request

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
8
votes
2answers
485 views

A log inequality for positive definite trace-one matrices

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...
1
vote
1answer
76 views

On generalization of rearrangement inequality

In fact, I'm intrested in such a question: If $\sigma$ and $\pi$ are two permutations of $\{1,2,\ldots,n\}$ what is necessary and sufficient condition to be the following statement true. For all $...
13
votes
2answers
458 views

A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that $\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
5
votes
1answer
372 views

An elementary inequality for a recursive double sequence

Here is what looks like (but is not) an Olympiad problem. Is it really that tough, or am I overlooking a simple solution? I have a system of sequences $\sigma_0(m),\sigma_1(m),\ldots\,$ defined for ...
0
votes
0answers
67 views

Refined versions of Azuma's inequality

Is there any version of Azuma's inequality where the bound $c_k$ as mentioned in https://en.wikipedia.org/wiki/Azuma's_inequality comes in the numerator of the fraction in the negative expoential.
0
votes
2answers
70 views

Level sets and integral of functions of two variables

Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...
0
votes
0answers
57 views

Maximal inequality for Markov process

For a Markov process $\{X_n\}$ is there any inequality available for $$ E[\sup_{0 \leq n \leq k} X_{n}]$$ in terms of moments of $E[X_n], 0 \leq n \leq k$
4
votes
0answers
62 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
2
votes
0answers
76 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?
6
votes
0answers
220 views

An inequality which involves 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
167 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 \frac{AC}{...
1
vote
0answers
17 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 ...
7
votes
2answers
199 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 $...
11
votes
1answer
472 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 ...
0
votes
1answer
79 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 ...
3
votes
0answers
145 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 $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = 1$$...
5
votes
1answer
150 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 ...
0
votes
0answers
70 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}| > u|\...
2
votes
0answers
131 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+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...
2
votes
1answer
83 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
77 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ $$\sum_{k=...
1
vote
0answers
73 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. ...
13
votes
2answers
462 views

regular polygon question

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
185 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 y(n)=\frac{x(n+...
2
votes
2answers
358 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 ...
2
votes
1answer
155 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 ...
17
votes
3answers
729 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
0answers
58 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
125 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
275 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 $$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ ...
14
votes
3answers
626 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 $n \ge 3$: $$\sum_{k=1}^{n}\...
6
votes
1answer
359 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 ...
2
votes
1answer
87 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 $ a_{ij}\...
0
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0answers
74 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} +...
1
vote
2answers
261 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
94 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}+ AABB+BBAA+B^{...
2
votes
3answers
421 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 $P= \sum_{i=1}^{n} x_i^2.$
0
votes
1answer
130 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 $k\in\{1,...,n\}$ ...
8
votes
1answer
312 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 ...
2
votes
1answer
114 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 ...
2
votes
0answers
99 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
148 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. http://math.stackexchange.com/questions/1440931/...
2
votes
1answer
75 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
139 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 ...