All Questions
2,253 questions
3
votes
0
answers
213
views
A family of polynomials related to integer partitions
For a positive integer $n$, let $p(n)$ be the number of partitions of $n$.
For $1\le k\le n$, let $p(n,k)$ denote the number of partitions of $n$ having exactly $k$ terms; in other words, $p(n,k)$ is ...
- 15.6k
3
votes
1
answer
247
views
A non-standard inequality for univalent functions
Related to my other question, here is an inequality from Rakhmanov's paper upon which the proof hinges.
Let $F(z) = z + a_0 + \mathcal O(z^{-1})$ be analytic and univalent on $|z|>1$, continuous up ...
user528012
4
votes
1
answer
96
views
On the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$.
Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$?
Remark 1: A numerical experiment suggests that $...
- 128k
2
votes
1
answer
99
views
What does it mean when one says the inequality must be understood in the barrier sense, when necessary?
What does it mean when one says the inequality must be understood in the barrier sense, when necessary?
I encountered this notion when I was reading Perelman's paper "The entropy formula for the ...
9
votes
1
answer
847
views
Huygens' final unproved inequality
The analytic statement of Proposition XX in Huygens' "Inventa" is: If $x> 0$, and less than $\frac{\pi}{2}$, then
$$x>\sin x +\frac{10(4\sin^2\frac{x}{2}-\sin^2 x)}{12\sin\frac{x}{2}+9\...
- 371
5
votes
0
answers
183
views
On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
3
votes
1
answer
327
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 43.2k
1
vote
0
answers
181
views
+50
A question relates to edge chromatic-polynomial
Properly colored graph (edge has color) means that any two adjacent edges have distinct colors.
The edge chromatic polynomial $ech(G, k)$ gives the number of proper edge coloring of the $G$ with $k$ ...
- 91
2
votes
0
answers
76
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
2
votes
2
answers
127
views
Optimizing a matrix quadratic form with respect to Loewner order
Fix integers $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times k}$ be such that $P^T P$ has full rank.
Let $\mathcal{X}$ denote the set of unit trace, real $n \times n$ symmetric positive ...
- 380
0
votes
1
answer
150
views
Inequality $(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$
Conjecture: Let $a_1, a_2, \cdots , a_n>0$ and $y \ge x $ then
$$(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$$
Equality iff $x=y$
Is the conjecture right? Have you ever seen this ...
- 1,776
2
votes
1
answer
144
views
Concentration inequality for double sum
I am looking for a concentration inequality of a double sum….
Let $X_1,\dots, X_n$ be iid r.v. and also let $Y_1,\dots ,Y_n$ be iid such that even $X_i$ and $Y_j$ are independent.
I am looking for a ...
- 51
1
vote
3
answers
562
views
Huygens' trigonometric inequality
Prove that $$1-(4/3(\sin^3 \theta/2))/(\theta-\sin\theta)<(1-\cos\theta/2)(3/5-(3/1400)\pi^2/n^2)$$ holds for $0\le\theta\le\pi/2.$ Here $n$ is an integer greater than or equal to two.
This an ...
- 371
43
votes
3
answers
2k
views
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that
$$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$
Let
\begin{...
- 4,280
1
vote
0
answers
80
views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
8
votes
2
answers
492
views
A trig inequality
Let $m$ be an odd and $n$ an even positive integers. I need to estimate the maximum value of $\left|\sin(m\theta)\sin(n\theta)\right|$:
$$
c_{m,n}:=\max_{\theta\in[0,\pi]} \left| \sin(m\theta) \sin(n\...
- 697
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
2
votes
0
answers
96
views
System of quadratic inequations
I have variables $x_1,...,x_n>0$, and fixed parameters $a_1,a_2,...a_n >0$.
I compute quantities $s_1,s_2,...,s_n$, each with one of the following types of equation:
$s_i =a_i(x_i - x_j x_k)$
$...
- 1,341
4
votes
0
answers
165
views
Szegő's inequality
I know Erdős-Lax's inequality and a couple of proofs. It states that:
If $P(z)=\sum_{v=0}^{n} a_{v} z^{v}$ is a complex polynomial of degree $n$ having no zeros in $|z|<1$, then
$$
\max _{|z|=1}\...
- 2,829
1
vote
1
answer
153
views
How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
2
votes
1
answer
173
views
Maximizing a quadratic form involving a trace-bounded positive definite matrix?
$\newcommand{\tr}{\mathrm{tr}}$Suppose $P, Q$ are two real, symmetric positive definite matrices and $v$ a nonzero unit vector.
Consider
$$
f(X) = v^T(P + X^{-1})^{-1} v + v^T(Q + X^{-1})^{-1} v.
$$
...
- 380
5
votes
1
answer
428
views
Lower bound for the rank of the sum of $n$ matrices
I found a mathematical note by George Marsaglia entitiled "Bounds for the rank of the sum of two matrices", where he proves the following result.
Let $A_1$ and $A_2$ be two complex matrices ...
- 5,215
6
votes
2
answers
225
views
On a trigonometric inequality by Huygens
The following inequality, ascribed to Huygens, appeared in this post:
\begin{equation*}
1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta}
>(1-\cos\theta/2)\Big(\frac35-\frac3{1400}\frac{\...
- 128k
0
votes
0
answers
43
views
Monotonicity of averages for positive-definite kernels
Let $\kappa:\mathbb{R}^d\times \mathbb{R}^d \to\mathbb{R}$ be a positive semidefinite kernel. If $A,B \subseteq \mathbb{R}^d$ are disjoint sets of equal, finite measure, then applying the definition ...
- 775
6
votes
1
answer
287
views
Determinantal inequality for difference of substochastic matrices
Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if
$$
\forall i,\ \ \sum_j A_{ij}\le 1\ .
$$
In the course of my ...
3
votes
1
answer
232
views
Bounds on relative entropy for MLE in Bernoulli coin tosses
In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...
- 63
1
vote
1
answer
148
views
An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
- 35
0
votes
1
answer
158
views
Techniques for bounding the operator norm of the expectation of random matrix?
Let $\mu$ be a distribution on the unit sphere in $\mathbb{R}^n$. Let $u \sim \mu$ and consider the random matrix
$$
A = I_n - uu^T.
$$
Question: What techniques are available to provide (reasonably ...
- 380
1
vote
2
answers
231
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 ...
- 371
2
votes
0
answers
103
views
A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 1,759
3
votes
1
answer
176
views
A generalization of Barrow's inequality
More than seven years ago. I posted this problem in stackexchange:
Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$,...
- 1,776
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
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
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
2
votes
0
answers
94
views
A surprisingly simple and difficult problem on sums of upper bounds
Let $T$ be a large integer, and $C$ be a positive real constant.
Consider a sequence $\{p_t\}_{T\geq t\geq 1}$ of real numbers in $[0,1]$. The sequence $\{b_t\}_{T\geq t\geq 1}$ can be defined as ...
- 353
2
votes
1
answer
216
views
Hypergeometric function ratio inequality
Related to a recent question (Concavity of hypergeometric function ratio), I actually only need the following inequality:
$$
\frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\...
- 391
2
votes
1
answer
309
views
Concavity of hypergeometric function ratio
I would like to show that the function,
$$
f(x) = \frac{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c+1\,;x\big)}{{}_2\mathrm{F}_1\big(\frac{1}{2},\frac{1}{2};c\,;x\big)}
$$
is concave for $0 < x &...
- 391
2
votes
1
answer
105
views
Inequality for Gaussian measures
Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
- 197
2
votes
0
answers
54
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 380
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 990
12
votes
2
answers
704
views
Polynomial inequalities of the form $\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \,dx$
Let $P_n$ denote all (real or complex) polynomials $f(x)=\sum_{k=0}^n a_k x^k$. I'm interested in inequalities of the form
$$
\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \, dx, \quad \text{for ...
- 161
3
votes
1
answer
70
views
Multiplicative approximation for a negative moment of the binomial distribution
Let $X$ be a binomial random variable with parameters $n,p$.
Define the function
$f(n, p, t) = E\frac{1}{1 + t X},
$
where $t > 0$.
Question: Can we find an elementary function $F(n, p, t)$ such ...
- 380
0
votes
1
answer
92
views
Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
- 215
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
2
votes
1
answer
106
views
Sharp approximation to expectation of a ratio of a Gaussian vector
Let $g =(g_1, ..., g_n)$ denote a sequence of standard Gaussian variables. Let $p = (p_1, ..., p_n)$ denote a vector in the simplex $\mathcal{P}_n$, given by
$$
\mathcal{P}_n = \{p \in \mathbb{R}^n : ...
- 380
1
vote
1
answer
120
views
Does Gaussian heat kernel ensure $\int_{\mathbb R^d} (1+|x|) \sqrt{\ell_{t_0} (x)} \, \mathrm d x < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\int_{\bR^d} (1+|x|) \sqrt{\ell (x)} \diff x < ...
- 835
3
votes
1
answer
272
views
An inequality about factorial function
Let $d,s,k$ be integers such that $d<s+2$, $s=o(k)$. For sufficiently large integer $k$, is the following inequality right?
$$\frac{(k-2d+1)^{k+s-d}}{(k-d)!\cdot (k-2)_s} \ge 1$$
We write $(k)_s = ...
- 91
2
votes
2
answers
191
views
Gronwall's inequality in discretized time
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
2
votes
1
answer
172
views
Inequality with matrix exponentials. Is it true that $x^\top e^{-xx^\top - AA^\top} x \leq x^\top e^{-xx^\top} x$?
As in the object, I'm looking at the case where $x \in \mathbb R^d$ is a generic vector, and $AA^\top \in \mathbb R^{d \times d}$ is a p.s.d. matrix.
I'm investigating the following inequality
$x^\top ...
- 69
1
vote
1
answer
93
views
Bound measure of difference of advected sets by norm of difference of vector fields
Consider two smooth vector fields $v$ and $u$ in $\mathbb{R}^n$, and a smooth set $\Omega$. Consider the flow of $\Omega$ via $v$ and $u$ for a time $T$, namely let
$$ \Omega_v =\{x(T, x_0) | x \text{ ...
- 697