All Questions
2,253 questions
11
votes
0
answers
387
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
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
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 ...
- 380
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 ...
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 ...
- 380
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
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
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
23
votes
5
answers
2k
views
Axiomatic construction of trigonometric functions
I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties:
$\sin^2 x + \cos^2 x = 1$,
$\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \...
- 702
1
vote
0
answers
37
views
Inequality for function on Spinor bundle
I have a function $H(x,\psi)$ defined on the spinor bundle $\mathbb{S}$ with $H_\psi$ being the continuous derivative in fiber direction having the following properties:
(H-1) There exists $0<\...
- 11
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 &...
- 380
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
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:
$$
P \le \...
- 43
1
vote
1
answer
172
views
Ratio of the constants of the Marcinkiewicz–Zygmund inequality for p=1
The Marcinkiewicz–Zygmund inequality states that
$$
{\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
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.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
- 835
2
votes
1
answer
620
views
On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
- 101
2
votes
0
answers
313
views
Minimizing $P=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$ [closed]
Olympiad inequality. Let $a,b,c\ge 0: ab+bc+ca=1.$ Find the minimal value $P$ of $$f:=\frac{\sqrt{5a+8bc}}{8a+5bc}+\frac{\sqrt{5b+8ca}}{8b+5ca}+\frac{\sqrt{5c+8ab}}{8c+5ab}.$$
Note: Often Stack ...
- 129
2
votes
1
answer
877
views
Interpreting mincost flow dual variables
Consider the task of finding flow of size $b$ with minimum possible cost.
It may be formulated as linear programming in a following way:
$$\boxed{\begin{gather}
\min\limits_{f_{ij} \in \mathbb R} &...
- 1,171
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
8
votes
2
answers
642
views
Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...
- 518
19
votes
4
answers
3k
views
Strange result about convexity
$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$.
Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?
Source: AoPS
- 4,074
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 ...
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 $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
- 333
2
votes
0
answers
119
views
Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$
Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
2
votes
1
answer
138
views
Explicit upper and lower bounds for a certain support function
Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$,
$$
B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \...
- 380
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
8
votes
2
answers
440
views
Show $\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle $ for all $R,S \in \mathrm{SO}(3)$
$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(...
- 518
5
votes
1
answer
510
views
A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
0
votes
2
answers
309
views
Integer solutions of system of inequalities
I am trying to solve a problem in combinatorics and I came up with the following system of inequalities:
$0\leq x<y<z\leq n$ and $x+y<n$ and I am trying to count the number of integer ...
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
3
votes
1
answer
954
views
A geometric proof that there are infinitely many primes?
Let $e_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l_2(\mathbb{N})$.
Let $h(n) = J_2(n)$ be the second Jordan totient function, defined by:
$$J_2(n) = n^2 \prod_{p|n}(1-1/p^2)$$
...
- 3,054
7
votes
1
answer
1k
views
When does equality hold in a specific triangle inequality?
I'd like to know when the equality holds in the following inequality
$$
| x - y |^a \le | x - z |^a + | y - z |^a.
$$
More precisely, for which points $x = (x_1, x_2)$, $y = (y_1, y_2)$ and $z = (z_1, ...
- 191
5
votes
1
answer
474
views
An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 128k
5
votes
0
answers
169
views
Is there a sharper Golden–Thompson inequality?
For any two Hermitian matrices $A$ and $B$, the Golden–Thompson inequality
$$\mathrm{Tr} (e^A e^B) \geq \mathrm{Tr} \, e^{A + B}$$
holds, and it is known to be a strict inequality whenever $[A, B] \...
- 51
0
votes
1
answer
171
views
For which $p>p^*$ does the inequality $\cos^2(−π/4+π/p)>1/2+π/p^2$ hold?
I have the inequality $\cos^2(−\frac{\pi}{4}+\frac{\pi}{p})>\frac{1}{2}+\frac{π}{p^2}$ and have found that it holds for $p>p^*$, where $p^*$ is some positive number (around ~2.8). I'm looking ...
0
votes
1
answer
169
views
How to integrate an indicator function/constraint into the cost function of a linear program?
I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...
- 3
4
votes
3
answers
369
views
Non-negativity of a complicated function
Show that $f(x)\ge 0$ for $0\le x \le 1$, where:
$$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$$
The endpoints are $f(0)=\pi^2/4$ and $f(1)=0$. Plotting ...
- 391
1
vote
1
answer
186
views
Kolmogorov inequality for Bernoulli random variables
This question is also asked on math stackexchange. The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $...
- 33
0
votes
0
answers
69
views
Upper bound for an additional Product formula
We have three sequences of positive integers $l$, $p$ and $q$ such that:
$$
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 < ...
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
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
$$\dfrac{|x-y|^2}{t} + \dfrac{|y-z|^2}{s} = \dfrac{t+s}{ts} \left| y - \dfrac{sx+tz}{s+t} \right|^2 + \dfrac{|...
- 677
0
votes
0
answers
164
views
Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
- 129
3
votes
0
answers
219
views
Do these cousins of permanents satisfy the following inequality?
Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define
$$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...
- 5,215
3
votes
1
answer
345
views
Simple anticoncentration bound for binomially distributed variable
The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...
- 85
8
votes
1
answer
313
views
Lower bound $\int_0^1 \frac{|f'(x)|^2}{f} \,\mathrm{d} x$ by $\int_0^1 |f-1|^2\, \mathrm{d} x$
Assume that $f$ is a probability density on $x \in (0,1)$, I want to obtain a bound of the following form (if it is possible at all): $$ \int_0^1 \frac{|f'|^2}{f} \,\mathrm{d} x \geq C\,\int_0^1 |f-1|^...
- 730
7
votes
2
answers
418
views
A counterexample showing $BV_p \neq AC_p$
I am trying to work through a supposedly simple counterexample given in papers by Love and Gehring regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $p > 1$. ...
- 217
5
votes
1
answer
288
views
Maximal inequality of iid random variables $\{X_{ij}\}_{1\leqslant i,j \leqslant n}$
Suppose that $\{X_{ij}\}_{1\leqslant i,j\leqslant n}$ are iid random variables with $\mathbb{E}(X_{11})=0$ and $\mathrm{Var}(X_{11})=1$, does the following convergence hold:
$$
\max_{1\leqslant j\...
- 81
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:
$$
\sum_{i=1}^{...
- 29
43
votes
1
answer
2k
views
Is $\int_0^\infty{dx\over x^{x^{x^x}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^x}}}}}<\int_0^\infty{dx\over x^{x^{x^{x^{x^{x^{x^x}}}}}}}<\cdots$ true?
On MSE I asked whether each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ was less than $2$ and received answers on bounding ...
- 1,459
0
votes
1
answer
348
views
Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform
It is well-known that the Laplace transform, defined by $$\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