All Questions
2,253 questions
1
vote
1
answer
171
views
Ratio of the constants of the Marcinkiewicz–Zygmund inequality for p=1
The Marcinkiewicz–Zygmund inequality states that
$$
{\displaystyle A_{p}E\left(\left(\sum _{i=1}^{n}\left\vert X_{i}\right\vert ^{2}\right)_{}^{p/2}\right)\leq E\left(\left\vert \sum _{i=1}^{n}X_{i}\...
- 13
0
votes
1
answer
264
views
A question about the prime counting function
I was playing around with the prime counting function and came across something that seemed correct to me, maybe it's already been proven but I don't know so I decided to ask here.
maybe a stupid ...
- 61
5
votes
1
answer
304
views
Recover unknown vector through shifted argmax queries
$\DeclareMathOperator*{\argmax}{arg\,max}$
I am interested in finding an efficient algorithm for the following problem:
Let $x \in [0,1]^n$ be some vector, with $x_n = 1$.
We want to recover $x$, ...
- 167
4
votes
1
answer
211
views
Erdős–Rényi random graphs — reproducing 2 inequalities
In Erdős and Renyi's 1959 paper On random graphs I
, I'm trying to reproduce, starting from Eq.\eqref{1} in their paper, the two inequalities that appear in Eq.\eqref{2}.
Eq.\eqref{1} is:
$$
P \le \...
- 43
0
votes
0
answers
164
views
Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
- 129
2
votes
0
answers
181
views
How to prove this weighted sum inequality with non-increasing sequences?
Problem
I have two non-increasing sequences, $X = (x_1, x_3, x_5, \ldots, x_{n-1})$ and $Y = (y_1, y_3, y_5, \ldots, y_{n -1})$, $n$ is an even integer. I want to prove this inequality:
$$
\sum_{i=1}^{...
- 29
6
votes
1
answer
372
views
Maximizing a sum minus its maximal summand
This is a followup to a question that appeared on m.SE:
Maximize $\displaystyle f(\pi)=\left(\sum_{i=1}^{n}{i\pi_i}\right)-\max_{1\le i\le n}{(i\pi_i)}$ over permutations $\pi\in S_n$.
The problem ...
- 1,190
2
votes
1
answer
138
views
Explicit upper and lower bounds for a certain support function
Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$,
$$
B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \...
- 460
4
votes
2
answers
485
views
How to get this inequality in Santambrogio's book about optimal transport?
Let $\hat{\varrho}, \tilde{\varrho}$ be probability density functions on $\mathbb R^d$ where $\tilde{\varrho} \in L^{\infty} (\mathbb R^d)$. For $\varepsilon \in [0, 1]$, we define $\varrho_{\...
- 835
0
votes
1
answer
69
views
Inequality for extremal values of product of Hermitian matrices
I am looking for a reference to verify the following inequality, where $X$ and $Y$ are Hermitian positive semidefinite matrices:
$$
\lambda_n(X^{1/2}YX^{1/2}) = \lambda_n(XY) \leq \lambda_n(X)\...
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 128k
52
votes
24
answers
11k
views
Most elementary proof showing that exponential growth wins against polynomial growth
This question is motivated by teaching : I would like to see a completely elementary proof showing for example that for all natural integers $k$ we have eventually $2^n>n^k$.
All proofs I know rely ...
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
0
votes
1
answer
171
views
For which $p>p^*$ does the inequality $\cos^2(−π/4+π/p)>1/2+π/p^2$ hold?
I have the inequality $\cos^2(−\frac{\pi}{4}+\frac{\pi}{p})>\frac{1}{2}+\frac{π}{p^2}$ and have found that it holds for $p>p^*$, where $p^*$ is some positive number (around ~2.8). I'm looking ...
1
vote
1
answer
123
views
Friedrich's second inequality for functions with zero average
Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
- 31
6
votes
2
answers
241
views
Does "perpendicular phase incoherence" satisfy the triangle inequality?
I asked this question at https://math.stackexchange.com/q/4783968/222867, but even after a 200-point bounty, no solution was provided, only some thoughts regarding possible directions. So I'm now ...
- 698
0
votes
2
answers
280
views
Bounds tighter than the additive Chernoff
Additive Chernoff
Suppose $X_1, \ldots, X_n$ are i.i.d. random variables, taking values in $\{0,1\}$. Let $p=\mathrm{E}\left[X_i\right]$ and $\varepsilon>0$.
\begin{gather*}
\operatorname{Pr}\left(\...
- 105
0
votes
1
answer
28
views
Calculating vertex potentials from optimal matchings
Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...
- 13.2k
8
votes
3
answers
595
views
Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
- 1,459
0
votes
1
answer
99
views
Is it possible obtain an identity of the type $\dfrac{|x-y|}{t} + \dfrac{|y-z|}{s} \approx a(t,s) | y - b(t,x,s,z)| + c(t,s)|x-z|$?
using the Euclidean inner product, and by completing squares it is possible to prove that
$$\dfrac{|x-y|^2}{t} + \dfrac{|y-z|^2}{s} = \dfrac{t+s}{ts} \left| y - \dfrac{sx+tz}{s+t} \right|^2 + \dfrac{|...
- 677
1
vote
1
answer
120
views
estimate involving Gaussian data
Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^2} \right)^{\frac{...
- 131
8
votes
0
answers
357
views
Bounding a sum of reciprocals of square-free integers
(Cross-posted from MSE, as the question did not get any clear answer)
Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
- 189
0
votes
1
answer
141
views
Integral inner product with exponential function
Suppose on some unknown interval $[0, I]$ we have non-negative functions $f, g : [0, I] \rightarrow \mathbb{R}^{\geq 0}$.
If we know that
\begin{aligned}
\int_0^I f & = c \\
\int_0^I e^f & = e^...
- 129
2
votes
1
answer
620
views
On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
- 101
3
votes
0
answers
219
views
Do these cousins of permanents satisfy the following inequality?
Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define
$$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...
- 5,215
4
votes
1
answer
320
views
Sub-Gaussian random variables and convex ordering
Suppose that $X$ is a $1$-sub-Gaussian real-valued random variable, i.e. for all $t \in \mathbf{R}$, it holds that $\log \mathbf{E} \exp \left( t X \right) \leqslant \frac{1}{2} t^2 $.
Does there ...
- 801
1
vote
1
answer
58
views
Possibility of bounding one functional by another functional (under certain constraints)
Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
- 730
1
vote
0
answers
76
views
Help with understanding a proof of existence of solutions
In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, can someone give a detailed ...
- 121
1
vote
0
answers
58
views
Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?
Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
11
votes
1
answer
4k
views
Understanding the application of two inequalities?
I am reading the paper "The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps" by Driss Kiouach and Yassine Sabbar.
I have two ...
- 185
2
votes
1
answer
235
views
Simplified upper bounds for moment-generating function of symmetrised random variable
Let $X$ be a nonnegative random variable such that $\mathbf{E} \left[ \exp X \right] < \infty$. For $\theta \leqslant 1$, an appropriate application of Jensen's inequality, yields that
\begin{align}...
- 801
0
votes
1
answer
348
views
Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform
It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
- 730
12
votes
4
answers
1k
views
Understanding the condition $\frac{1}{p} + \frac{1}{q} = 1$ in the estimate $xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$
I just read a proof of Holder's inequality in measure theory, which boils down to the following inequality:
$$xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$$
where $x,y\ge 0$ and $\frac{1}{p} + \frac{1}{q} = ...
- 7,527
1
vote
1
answer
143
views
$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
- 101
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
0
votes
1
answer
114
views
Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
2
votes
1
answer
252
views
A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book
This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...
- 187
13
votes
2
answers
2k
views
One specific inequality
How to prove this inequality $$\left(a+\frac{1}{2} \left(a b-\sqrt{a^2-1} \sqrt{b^2-1}\right)\right)^{3/4}-\frac{\sqrt{3} \cos\left[\frac{3 (\pi -t)}{4}\right]}{2 \left(\frac{1}{2}+b\right)^{1/4}}-(a+\...
- 333
1
vote
0
answers
117
views
Inequality concerning the imaginary parts of a recurrent sequence, Laplacian eigenvectors
Let $u=(u_1,\dots,x_n)\in\mathbb{C}^n$ be a sequence that satisfies the cyclic recurrence
$$
\lambda+1 =a_{i-1}\frac {u_{i-1}}{u_i} + (1-a_{i+1})\frac{ u_{i+1} }{u_i }
$$
with $a_i \in (0,1)$ and $\...
- 695
4
votes
2
answers
781
views
Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
- 185
1
vote
0
answers
101
views
Hardy's inequality can be improved?
Hardy's inequality in $L^{p}$-spaces, taking the form [3]
$${\displaystyle \left\|{\frac {f}{|x|}}\right\|_{L^{p}(R^{n})}\leq {\frac {p}{n-p}}\|\nabla f\|_{L^{p}(R^{n})},2\leq n,1\leq p<n,}$$
where
...
- 677
2
votes
1
answer
240
views
Is the problem of vertex enumeration from an H-representation of a polytope NP-hard?
According to the Wikipedia page on the issue, the vertex enumeration problem is NP-hard.
However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, ...
- 123
6
votes
1
answer
378
views
An inequality for a concave function $f(x)=x^{p/2}$
Assume that $p\in(1,2]$, $a,b\ge 1$, $b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
- 333
1
vote
0
answers
269
views
Monotone likelihood ratio of a kernel based on $\log(\cosh(x))$
Let $f(x) = \log(\cosh(x))$, and define the kernel density:
$$p_r(\phi;\theta) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta)\big) \Big)\hspace{0.5pt} \frac{\sin(2\phi)}{\sin(2\theta)}...
- 391
2
votes
0
answers
51
views
What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
- 283
1
vote
1
answer
151
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
- 391
1
vote
1
answer
194
views
Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
- 484
0
votes
1
answer
33
views
Sign Regularity of a Density Kernel with Convexity Properties
(Asking a final time in a new question because the previous version had insufficient conditions, as pointed out in the answer there.)
Define the densities:
$$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\...
- 391
3
votes
1
answer
233
views
Min problem on integers
Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that
$$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
4
votes
3
answers
368
views
Non-negativity of a complicated function
Show that $f(x)\ge 0$ for $0\le x \le 1$, where:
$$f(x) = \arccos(x)^2 -8x(5x^2-2) \sqrt{1-x^2}\arccos(x)+36 x^8-112 x^6+93 x^4-17 x^2$$
The endpoints are $f(0)=\pi^2/4$ and $f(1)=0$. Plotting ...
- 391