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0
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0answers
11 views

Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
3
votes
0answers
309 views

Question on a proof by Solonnikov,Ladyzhenskaya,Ural'tseva

I have already asked this question on Mathematics SE, because I suppose that it is not research level. But I haven't got an answer, possibly here someone can answer. Let $G(t,x)$ be the fundamental ...
5
votes
0answers
136 views

Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
10
votes
2answers
484 views

Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (The question was first asked at math.SE, ...
-2
votes
1answer
159 views

using jensen's inequality

Suppose we have an expression f(x, h(x,y)), for some function f and h, and x, y are random variables, now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's ...
2
votes
1answer
53 views

Local fractional Sobolev inequality

If $\mathcal{X}$ is a smooth cutoff near 0 in $\mathbb{R}^n$, then $M_0 = \mathcal{X}(-\Delta+Id)\mathcal{X}$ is a self-adjoint operator in $L^2(\mathbb{R}^n)$. Because $M_0$ is semi-positive and the ...
0
votes
0answers
57 views

How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
9
votes
1answer
249 views

Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)

Is there a "Cauchy-Schwarz proof" of the following inequality? Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has $$ \int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) ...
4
votes
1answer
69 views

$2$-D Hlawka inequality

The classical counter-example to Hlawka inequality $$|a+b|+|b+c|+|c+a|\le|a+b+c|+|a|+|b|+|c|$$ is the $l^\infty$-norm in dimension $3$, with vectors $$a=\begin{pmatrix} 1 \\ 1 \\0 \end{pmatrix},\quad ...
0
votes
0answers
26 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
2
votes
3answers
195 views

Characterizing orthants with polynomials

Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$. Can one find a polynomial $p$ (of arbitrary degree) in the coordinates of $x$ such that $p(x)\geq 0$ if and only if $x$ is an element of the positive orthant ...
5
votes
2answers
217 views

Poincare-like inequality on compact Riemannian manifolds

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of $$ ...
3
votes
0answers
90 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
0
votes
0answers
69 views

Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?

By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the ...
0
votes
0answers
60 views

Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...
1
vote
1answer
131 views

How does Azuma's Inequality result from Pinelis Inequality?

According to [1] Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let $S(\mathcal{X})$ denote the class of all sequences $f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...
6
votes
0answers
236 views

Is the heat kernel more spread out with a smaller metric?

Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
4
votes
3answers
505 views

Is $\lceil \frac{n}{\sqrt{3}} \rceil > \frac{n^2}{\sqrt{3n^2-5}}$ for all $n > 1$?

An equivalent inequality for integers follows: $$(3n^2-5)\left\lceil n/\sqrt{3} \right\rceil^2 > n^4.$$ This has been checked for n = 2 to 60000. Perhaps there is some connection to the ...
3
votes
1answer
109 views

Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$): $$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$ where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...
2
votes
4answers
162 views

Simple bound for generalized geometric series

Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound $$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$ where $c>0$ is independent from $m$. Is there a simple way of proving this ...
3
votes
1answer
169 views

Is this inequality true?

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...
2
votes
0answers
58 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...
1
vote
1answer
136 views

trigonometric sum and inequalities

let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq ...
3
votes
2answers
207 views

Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold? $$ \int_{k + 1/2}^{k + 3/2} \frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}} ...
4
votes
2answers
195 views

Operator norm versus Hlawka inequality

Let $E$ be a finite dimensional normed vector space. If $E$ is $\ell^1$-embeddable, then the norm satisfies Hlawka inequality ...
4
votes
0answers
160 views

Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...
9
votes
1answer
179 views

Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property $$f(\lambda x)=|\lambda|\,f(x).$$ ...
6
votes
1answer
72 views

Summability of ratios of moments a weight

Recently, I encounter the following problem: Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e., $$m_k=\int_0^1t^kw(t)dt.$$ Under what condition can we have ...
1
vote
0answers
73 views

Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if $$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall ...
1
vote
2answers
121 views

investigating positivity/negativity of a function [closed]

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function ...
11
votes
1answer
213 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 ($C=\bf 0$) ...
3
votes
1answer
116 views

Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following: If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and ...
3
votes
2answers
219 views

Generalized Hardy-Littlewood-Sobolev Inequality

The Hardy-Littlewood-Sobolev Inequality says that $$\text{for $p,q,r\in (1,+\infty)$ such that }\quad 1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$} $$ $$ \exists C, \forall u\in L^p(\mathbb ...
0
votes
0answers
60 views

Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity \begin{equation*} ...
1
vote
1answer
229 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 ...
3
votes
1answer
124 views

Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
2
votes
2answers
340 views

Does this simple inequality have a name?

Let $x_{1},\ldots,x_{n}$ be nonnegative numbers such that $m \leq x_{i} \leq M$. Let $$ S=\sum_{i=1}^{n}{x_{i}} $$ and $$ Q=\sum_{i=1}^{n}{x_{i}^{2}}. $$ Then $$ Q \leq S(M+m)-nMm. $$ This has ...
4
votes
3answers
383 views

Inequality of arithmetic, geometric and harmonic means

Let $a_1,\dots,a_n$ be positive numbers, does the following inequality holds? $$\frac{a_1+a_2+\cdots+a_n}{n}-\sqrt[n]{a_1a_2\cdots a_n}\geq\sqrt[n]{a_1a_2\cdots ...
2
votes
1answer
139 views

Inequality for the tail of normal distribution function

Let $ Ф(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-t^2/2} \, dt $ be the cumulative distribution function of the standard normal distribution. Numerical calculations suggest the following ...
9
votes
0answers
217 views

Determinant inequality involving Hermitian, positive definite matrices

Question: Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question ...
0
votes
0answers
88 views

An inequality for moments of a random variable

I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy an inequality of the type $$ (1) \qquad E|\xi|^p \leq F(E|\xi|^2), $$ where $p>2$, $F$ is a certain ...
0
votes
0answers
65 views

Bounding multiplications of PSD random matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
0
votes
1answer
59 views

angle inequality in n-dimensional vector space [closed]

Does anyone has answer for the following doc product problem? Let A,B,C be three vectors of magnitude of 1. Let A*B = Cos(x) ( * means dot product) B*C = Cos(y) ...
2
votes
1answer
443 views

An inequality involving traces and matrix inversions

The following question kept me wondering for some time: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
5
votes
0answers
129 views

Maximal inequalities for square of partial sums

Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...
1
vote
1answer
92 views

Positive Definiteness of a certain function

Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries $$a_{ij} = ...
2
votes
0answers
84 views

Variant form of the gronwall inequality

I know the following statement for gronwall inequality: Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have, $f' \leq \phi f$ and $f(0)=0$ then $f=0$ Now is ...
15
votes
1answer
294 views

A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$ I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
0
votes
1answer
271 views

A number theoretic inequality

Is this inequality true? : $$\displaystyle \prod_{i\le \left\lfloor{\frac{n-1}{2}}\right\rfloor}\left\lfloor{\frac{\left\lfloor{\frac{n-1}{2}}\right\rfloor}{i}}\right\rfloor\le ...
21
votes
0answers
431 views

Are there any nontrivial near-isometries of the $n$-dimensional cube?

Consider the $n$-dimensional Hamming cube, $C = \{-1,1\}^n$. Given an $n \times n$ orthogonal matrix $O$, I'll measure "how close $O$ is to being an isometry of $C$" by the following scoring ...