All Questions
2,253 questions
2
votes
2
answers
235
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
- 128k
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
0
answers
162
views
Taylor coefficients of the integral of the ordered exponential
Let $A$ be a continuous $2\times 2$ matrix-valued function on $[0,1]$. Define $X_A$ as the solution of
$$
X_A'(t) = A(t) X_A(t), \qquad X(0) = I.
$$
In other words $X_A$ is the ordered exponential of $...
- 577
0
votes
1
answer
135
views
On polynomial equation of fourth order depending on two parameters and bound on a maximal root
I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:
\begin{eqnarray}
F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...
- 371
0
votes
0
answers
39
views
Max-flow modeling with unified vehicle and commodity variables
I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...
- 101
1
vote
0
answers
75
views
Integral inequality related to the (mixed?) moments of two functions
For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set
$$
S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}.
$$
Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce
$$
...
- 577
2
votes
1
answer
184
views
Estimating ${\left(\sum_{i=j}^k {x_i}\right)^2} \times \left\lvert\sum_{i=j}^k {a_i}\right\rvert$
Given two sets; $X = \{x_i : x_i \geq 0; i \in [\sqrt{n}]\}$ and $A = \{a_i : |a_i| \leq 1; i \in [\sqrt{n}]\}$ of size $n^{\frac{1}{2}}$ each, with the following properties
\begin{equation}\label{...
- 23
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
0
votes
1
answer
103
views
Demonstrating bound on integral
I found this question on another forum and it got me interested because of how tight the bound is: prove that
$$\int_0^\infty \frac{\arctan(x)}{x^2 + 4}\,dx > \frac{\pi}{4}.$$
The difference ...
- 689
0
votes
0
answers
69
views
Upper bound for an additional Product formula
We have three sequences of positive integers $l$, $p$ and $q$ such that:
$$
p_1 \geq p_2 \geq \cdots \geq p_k\text{ and } q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \text{ where: } k < ...
1
vote
1
answer
261
views
An inequality of Huygens
I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then
$$
\pi\ge \frac{\pi}{x}\sin(x)\bigl(20+51\cos(x)-2\cos^2(x)+6\cos^3(...
- 371
0
votes
0
answers
30
views
Application of greedy approach for optimization
I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
- 23
0
votes
0
answers
211
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
- 101
4
votes
1
answer
146
views
Bounding distance to an intersection of polyhedra
Let $P$ and $Q$ be polyhedra in ${\mathbb R}^m$ with a non-empty intersection. I believe there should exist a constant $C_{PQ}>0$ such that for any point $x\in {\mathbb R}^m$ the following ...
- 1,579
1
vote
1
answer
109
views
Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
- 3,065
2
votes
0
answers
114
views
Poincare inequality on the hemisphere
Background:
Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
- 537
3
votes
2
answers
205
views
Bounding distance to a polyhedron
I need to estimate the Euclidean distance from a point $x\in {\mathbb R}^m$ to a polyhedron $P\subset {\mathbb R}^m$ in terms of distances from $x$ to the tangent hyperplanes which define $P$. By a ...
- 1,579
0
votes
0
answers
22
views
Solve $S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{km} \right) \geq 0$ for $x_{kj}$
Let $S_{nm} \in \{\pm1\}^{N\times M}$ be a matrix of signs. Can we find real numbers $x_{kj} \in \mathbb{R}^{N\times K}$, such that:
$$S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{...
- 884
2
votes
0
answers
120
views
Information inequality for Renyi divergences
Let $X^1 \ldots X^n$ be random variables on $\mathbb{R}^d$ with an arbitrary joint probability distribution $\mu$ on $\mathbb{R}^{n \times d}$. Let $\nu = \nu^1 \times \ldots \times \nu^n$ be a ...
7
votes
1
answer
556
views
A variation on the Borel–Cantelli lemma theme
Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let
\begin{equation*}
E:=\bigcap_{n\ge0}B_n,
\end{equation*}
where
\begin{equation*}
B_n:=\...
- 128k
1
vote
1
answer
66
views
Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?
Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
- 835
3
votes
1
answer
493
views
A strange condition of convexity?
During my research, I come across this question.
Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$.
Is it true that $\forall x \in \mathbb R, f''(x) \...
- 4,074
2
votes
1
answer
168
views
Has the function $F_s(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(k+1)^s}$ been studied before?
While studying an application of Grönwall's inequality I found that the function
$$
F_s(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(k+1)^s}
$$
for $s\geq0$ in some cases provides a sharper bound.
I had a ...
2
votes
1
answer
209
views
Prove inequality: $2\Big[ \sum_{k=0}^{n} (k+1) a_k \Big]^2 -\Big[1+ \sum_{k=0}^{n} a_k \Big]\Big[ \sum_{k=0}^{n} k(k+1)a_k \Big] \geq 0.$
Suppose $a_0\geq\dots \geq a_n \geq 0$ is a sequence of non-negative numbers, where $n+1\leq \sum_{k=0}^{n} a_k \leq n+2$. Then, I want to prove that the following statement is true,
$2\Big[ \sum_{k=0}...
- 23
1
vote
1
answer
106
views
Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$
Let $\alpha \in (0, 1)$ and $\psi : \mathbb R^d \to \mathbb R^d$ be a $C^\infty$-diffeomorphism such that $\|\nabla \psi\|_\infty + \|\nabla \psi^{-1}\|_\infty < + \infty$. Let
$$
I (t) := \sup_{x \...
- 835
2
votes
1
answer
213
views
Is matrix B obtained from matrix A?
Assuming a matrix $\mathbf{A} \in \mathbb{R}^{4096 \times 4096}$ sampled from a standard normal distribution $N(0, 1)$, and another matrix $\mathbf{B} \in \mathbb{R}^{4096 \times 4096}$ either sampled ...
- 49
1
vote
1
answer
136
views
Inequality with convolution
I have some troubles with the following problem:
A definition
Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian ...
- 393
1
vote
0
answers
141
views
$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$
I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
- 155
12
votes
2
answers
704
views
Polynomial inequalities of the form $\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \,dx$
Let $P_n$ denote all (real or complex) polynomials $f(x)=\sum_{k=0}^n a_k x^k$. I'm interested in inequalities of the form
$$
\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \, dx, \quad \text{for ...
- 161
0
votes
0
answers
36
views
ILPs with square constraint matrix
Given the Integer Linear Programming ($\text{ILP}$) problem
\begin{array}{ll}
\text{minimize} & c^T x \\
\text{subject to}& \mathbf{A}^T x \ge b \\
\text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
- 13.2k
43
votes
3
answers
2k
views
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$
Let
\begin{...
- 4,270
3
votes
2
answers
645
views
Upper bound for complex integral
I am interested in obtaining a good upper bound for the absolute value of the following integral
$$
\left| \int_{0}^{\pi/3} e^{-itn} \left( 1-e^{it} \right)^{k} dt \right|,
$$
when $n>k>0$ are ...
- 43
0
votes
0
answers
26
views
Monotony of enforced subtour merging
Is it true that for a symmetric TSP instance in the sequence of edges generated by successively:
calculating the optimal 2-factor
adding cardinality constraints on the edgesets of the 2-factor's ...
- 13.2k
0
votes
0
answers
171
views
Solve NP-hard type problems with linear programming
I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force.
I ask this ...
0
votes
0
answers
64
views
Alternatives to McCormick Envelope
I have an optimization problem for which I have the optimal solution obtained by the ILP.
However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
- 3
2
votes
0
answers
119
views
Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$
Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
1
vote
2
answers
188
views
Approximation of the Gaussian Mills' ratio
Let $R(t) = \frac{1 - \Phi(t)}{\varphi(t)}$ where $\Phi, \varphi$ represent the CDF and PDF of the standard Normal distribution, respectively.
I am interested in approximations of the function for $t &...
- 460
1
vote
1
answer
182
views
Inequalities involving entropy: quantum discord and mutual information
My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
- 9,330
1
vote
1
answer
131
views
Optimal constant comparing $f(1/2)$ and $\|f\|_2$ when $f$ is $t$-Hölder?
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 460
3
votes
2
answers
269
views
Is the passage in argument of existence solution of PDE correct?
The passage below is from a fixed point argument in Wang's paper enter link description here pg 566. At the end of the $I_1$ calculation, it somehow makes the following estimate
$$C\|\phi\|_X (e^{-|x|^...
- 677
2
votes
2
answers
223
views
Relating function value to $L^2$ norm in Holder space
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 460
4
votes
1
answer
427
views
Inequality of inclusion-exclusion term
This question was initially posted on math.stackexchange.com but did not receive any answers for half a week.
While analyzing the properties of an algorithm I am working on (I'm a computer scientist), ...
- 45
0
votes
1
answer
169
views
How to integrate an indicator function/constraint into the cost function of a linear program?
I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...
- 3
3
votes
1
answer
245
views
What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?
This question was posted in MSE but is still open hence posting in MO.
The area of the largest triangle that can be inscribed in a circle of raidus $1$ is $\displaystyle \frac{3 \sqrt{3}}{4}$ for a ...
- 5,460
1
vote
1
answer
186
views
Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
- 33
5
votes
0
answers
169
views
Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
- 51
3
votes
1
answer
151
views
Derive distributional inequalities from pointwise estimates
My question is how to prove the following claim:
Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous
on $\mathbb{R}^n$. If ...
- 33
16
votes
1
answer
770
views
Find a special integer coefficients polynomial which takes small absolute value on [0,4]
The question is easy to state: Is there a non-constant $f\in\mathbb{Z}[x]$ such that for all $x\in [0,4]$, we have $|f(x)|\leq 1$? I do not know where to find a useful reference for it.
I did a few ...
- 287
2
votes
0
answers
313
views
Minimizing $P=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$ [closed]
Olympiad inequality. Let $a,b,c\ge 0: ab+bc+ca=1.$ Find the minimal value $P$ of $$f:=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$$
Note: Often Stack ...
- 129