All Questions
2,253 questions
152
votes
18
answers
24k
views
Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
62
votes
7
answers
26k
views
Is the Jaccard distance a distance?
Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any ...
- 1,355
53
votes
7
answers
51k
views
Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
- 541
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 ...
51
votes
2
answers
5k
views
A strengthening of the Cauchy-Schwarz inequality
Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
- 6,351
47
votes
3
answers
6k
views
Absolute value inequality for complex numbers
I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality
$$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$
...
- 953
46
votes
3
answers
5k
views
Putnam 2020 inequality for complex numbers in the unit circle
The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020:
Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
- 33.8k
44
votes
7
answers
4k
views
The missing link: an inequality
I've been working on a project and proved a few relevant results, but got stuck on one tricky problem:
Conjecture. If $2\leq n\in\mathbb{N}$ and $0<x<1$ is a real number, then
$$F_n(x)=\...
- 43.2k
43
votes
3
answers
2k
views
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that
$$\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
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
42
votes
11
answers
6k
views
Important (but not too well known) inequalities
After seeing the question Important formulas in combinatorics, I thought it might be of interest to have a similar list of inequalities, although not restricted to combinatorics. As with that list, ...
41
votes
4
answers
5k
views
The sum of squared logarithms conjecture
I am searching for the first proof of (or counterexample to) the following conjecture.
(The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
- 421
38
votes
4
answers
3k
views
Binomial again, and again
Let $\lceil a\rceil=$ the smallest integer $\geq a$, otherwise known as the ceiling function. When the arguments are real, interpret $\binom{a}b$ using the Euler's gamma function, $\Gamma$.
Recently, ...
- 43.2k
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
36
votes
3
answers
4k
views
the following inequality is true，but I can't prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...
- 363
35
votes
4
answers
5k
views
Why are optimization problems often called "programs"?
Why are optimization problems often called programs?
linear programming
geometric programming
convex programming
Integer programming
...
- 461
35
votes
3
answers
4k
views
A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
- 1,748
31
votes
1
answer
1k
views
Finding a path through real rooted polynomials
This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true.
Let $z^n+a_{n-1} z^{n-1} + \cdots + ...
- 156k
30
votes
1
answer
4k
views
What goes wrong for the Sobolev embeddings at $k=n/p$?
For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{...
- 17.5k
29
votes
3
answers
2k
views
Wanted: Positivity certificate for the AM-GM inequality in low dimension
I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
Can one write the LHS as a sum
$\,\...
- 489
29
votes
8
answers
6k
views
Is there a good reason why $a^{2b} + b^{2a} \le 1$ when $a+b=1$?
The following problem is not from me, yet I find it a big challenge to give a nice (in contrast to 'heavy computation') proof. The motivation for me to post it lies in its concise content.
If $a$ and ...
- 1,858
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 391
28
votes
4
answers
3k
views
Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$
Let $x>0$ and $n$ be a natural number. Prove that:
$$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$
This question is very similar to many contests problems, but ...
- 1,054
28
votes
3
answers
1k
views
Inequality for hook numbers in Young diagrams
Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
- 17k
28
votes
1
answer
794
views
Are there any nontrivial near-isometries of the $n$-dimensional cube?
Consider the $n$-dimensional Hamming cube, $C = \{-1,1\}^n$. Given an $n \times n$ orthogonal matrix $O$, I'll measure "how close $O$ is to being an isometry of $C$" by the following scoring function:...
- 9,823
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,121
27
votes
5
answers
2k
views
Is this inequality on sums of powers of two sequences correct?
Let $a_1,\dots,a_n$ and $b_1,\dots,b_n$ be two sequences of non negative numbers such that for every positive integer $k$,
$$ a_1^k+\cdots+a_n^k \leq b_1^k+\cdots+b_n^k,$$
and
$$a_1+\cdots+a_n = ...
- 4,484
27
votes
2
answers
2k
views
How often are irrational numbers well-approximated by rationals?
Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...
- 13.1k
27
votes
1
answer
2k
views
Does a proof of Selberg's 3.2 inequality exist?
A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi \...
- 13k
26
votes
3
answers
3k
views
An $L^0$ Khintchine inequality
Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...
- 17.1k
26
votes
3
answers
17k
views
Hölder's inequality for matrices
I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...
- 837
26
votes
1
answer
1k
views
Finding the closest matrix to $\text{SO}_n$ with a given determinant
$\newcommand{\GLp}{\operatorname{GL}_n^+}$
$\newcommand{\SLs}{\operatorname{SL}^s}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\Sig}{\Sigma}$
$\newcommand{\id}{\text{Id}}$
$\newcommand{\...
- 6,741
26
votes
2
answers
1k
views
Symmetric strengthening of the Cauchy-Schwarz inequality
In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have
\begin{align*}
\|v^2\| \, \|w^2\| - \langle ...
- 12.5k
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 363
25
votes
3
answers
2k
views
Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
- 33.8k
24
votes
6
answers
7k
views
Applications of Hardy's inequality
Every so often I would encounter Hardy's inequality:
Theorem 1 (Hardy's inequality). If $p>1$, $a_n \geq 0$, and $A_n=a_1+a_2+\cdots+a_n$, then
$$\sum_{n=1}^\infty \left(\frac{A_n}{n}\right)^p ...
24
votes
4
answers
2k
views
Curious inequality satisfied by $g(x)=\sum_{k=0}^{\infty}1/(x^{2k+1}+1)$
Set
$$
g(x)=\sum_{k=0}^{\infty}\frac{1}{x^{2k+1}+1} \quad \text{for} \quad x>1.
$$
Is it true that
$$
\frac{x^{2}+1}{x(x^{2}-1)}+\frac{g'(x)}{g(x)}>0 \quad \text{for}\quad x>1?
$$
The ...
- 3,946
24
votes
2
answers
1k
views
Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?
$\newcommand\abs[1]{\lvert#1\rvert}\newcommand\Abs[1]{\left\lvert#1\right\rvert}$Question: Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Does the following ...
- 4,270
24
votes
4
answers
2k
views
A reinterpretation of the $abc$ - conjecture in terms of metric spaces?
I hope it is appropriate to ask this question here:
One formulation of the abc-conjecture is
$$ c < \text{rad}(abc)^2$$
where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
user6671
24
votes
4
answers
2k
views
This inequality why can't solve it by now (Only four variables inequality)?
I asked a question at Math.SE last year and later offered a bounty for it, only johannesvalks give Part of the answer; A few months ago, I asked the author(Pham kim Hung) in Facebook, he said that ...
- 4,270
24
votes
4
answers
1k
views
show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$
I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....
- 4,270
24
votes
1
answer
1k
views
A Rademacher ‘root 7’ anti-concentration inequality
Let $r_1,r_2,r_3,\dotsc$ be an IID sequence of Rademacher random variables, so that $\mathbb P(r_n=\pm1)=1/2$, and $a_1,a_2,\dotsc$ be a real sequence with $\sum_na_n^2=1$. For $S=\sum_na_nr_n$, does ...
- 17.1k
23
votes
2
answers
2k
views
Are such functions differentiable?
In my recent researches, I encountered functions $f$ satisfying the following functional inequality:
$$
(*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}.
$$
Since $f$ is convex (because $\...
- 995
23
votes
3
answers
3k
views
Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)
Is there a "Cauchy-Schwarz proof" of the following inequality?
Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has
$$
\int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...
- 986
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
23
votes
1
answer
1k
views
Symmetric polynomial inequality arising from the fixed-point measure of a random permutation
A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows.
Given $n$ non-negative reals $a_1, ...
- 543
21
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
21
votes
3
answers
2k
views
Trigonometric inequality
For odd and coprime positive integers $p$ and $q$, the following inequality holds:
$$\sum_{m=1}^{p} \sum_{n=1}^{q} \frac{2}{\cos(\frac{2m\pi}{p})+\cos(\frac{2n\pi}{q})} \le pq(|p-q|+1)$$
Unfortunately,...
- 683
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
1
answer
2k
views
Is there a combinatorial proof of Cauchy-Schwarz?
I've only played with this a little for the past day or so, and haven't thought about it too hard, so it might be obvious. Obviously it's not fair to ask for a "combinatorial proof" of an inequality ...
- 12.6k