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1answer
479 views

A question on gauge functions

In the second paragraph on Page 71 of the book Matrix Analysis by Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem III 4.4''. How can one get the inequality in Theorem III 4.4 from ...
0
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1answer
2k views

Why does sample standard deviation underestimate population standard deviation? [closed]

I am aware of Jensen's inequality where, given the concave square root function, the mean of the square root is lesser than the square root of the mean. However, I cannot figure out why the square ...
2
votes
2answers
437 views

Poincare inequality for the annulus

Assume that $A=A(r,1)=\{x: r<||x||<1\} \subset R^n$ is an annulus. Whether is known the constant of Poincare inequality for A or some its estimation (w.r.t. $L^2$): the constant $C$ in the ...
0
votes
1answer
94 views

Problem with making an estimate when values of many variables are unknown?

Hi all, I was wondering whether you could help me understand a part in a proof that is assumed to be an "obvious detail" by the author, but that I can't wrap my head around. Essentially, as far as I ...
7
votes
5answers
830 views

Rearrangement-style inequality with lots of terms and little evidence

This is another of the problems I designed for contests but wasn't able to solve on my own for years. Maybe AoPS is a better place for it, but let me try it here. [UPDATE: I have streamlined the ...
0
votes
1answer
312 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
0
votes
1answer
359 views

Help prove a maximal inequality

Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number. $S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$. I want to prove: $$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) ...
1
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1answer
715 views

On an eigenvalue inequality

Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $|\lambda_1 (\cdot)| ...
-1
votes
1answer
727 views

On an eigenvalue inequality [closed]

Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $|\lambda_1 (\cdot)| ...
4
votes
3answers
979 views

Factorial-inequalities

Let $n>15$ be an integer. Suppose also $n=\sum_{i=1}^n ic_i$, where $c_i$ are non-negative integers. Assume further that $c_1<4$. Is the following inequality true? ...
3
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0answers
206 views

Where can I find interpolation inequalities for derivatives of the following form?

Here they are: $$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$ and $$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ...
3
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0answers
220 views

Maximise $L^q$ norm of a vector, for fixed $L^1$ and fixed $L^p$ norms [closed]

Consider a vector $x \in \mathbb R_+^n$ and $p,q \in \mathbb R$ such that $1 < p < q$ We fix $\sum \limits_{i=1}^{n}|x_i| = 1$ and $ \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p} = ...
2
votes
2answers
876 views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
2
votes
1answer
232 views

Quotients of perfect powers separated by an integer

Let $a_n=\frac{(n+1)^{n+2}}{n^n}$ and $b_n=\frac{(n+2)^{(n+1)}}{(n+1)^{n-1}}$. Then it is easy to see that $a_n \leq b_n$ for all integers $n\geq 1$ (because the sequence $(1+\frac{1}{n})^n$ is ...
3
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4answers
472 views

Equidistribution Theorem: distance between solutions

Can please someone help me with the following problem. Say we have a sequence $nx \; \mathrm{mod} \; 1$, where $n$ is a whole number and $x$ is irrational. Now I need to solve the inequality $nx \; ...
3
votes
3answers
884 views

Poincare Metric on Hyperbolic Plane

as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$ by setting $$ d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2}, $$ ...
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5answers
1k views

Bounding the absolute sum of entries of the inverse of a 0-1 matrix

I have a non-singular square 0-1 matrix and I want to bound the sum of absolute values of its inverse as a function of n (or the vector 1-norm). Asymptotic results are also useful. Does anyone know ...
6
votes
1answer
1k views

Stochastic Matrix: Second largest eigenvalue and second largest absolute value of eigen value

Setup Let $A$ be a stochastic matrix. Let the eigenvalues of $A$ be $1 = \lambda_1 \geq \lambda_2 \geq \lambda_3 ... \geq -1$. Let $\lambda = \max_{x: x \perp 1} \frac{||Ax||}{|| x ||}$ Question: ...
4
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2answers
603 views

Proving a sequence of integrals increases (iterated minimax distributions)

I am trying to show that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = (1-(1-F_{n-1}(x))^c)^c$$ and $F_0(x) = x$ and $n$ and $c$ are integers, $n\geq 1$ and $c \geq 2$ Note that ...
4
votes
1answer
415 views

total variation distance of product of measures

Let $f, \hat{f}, g,$ and $\hat{g}$ be continuous probability densities. Define probability densities $p \propto fg$ and $\hat{p} \propto \hat{f}\hat{g}$. Is it true that \begin{align*} ||p - ...
2
votes
2answers
595 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
4
votes
2answers
1k views

bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?

Let $A_i$ with $i=1,\dots,N$ and $p$ be real $M\times M$ matrices. Further, let $p$ be positive definite, i.e., $p\succ 0$, with $Tr(p)=1$. Let $0< a_i<1$ and $\sum_{i=1}^N a_i = 1$. Claim: ...
2
votes
1answer
509 views

A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$, $s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the singular values of a $2\times2$ matrix. Is it true that ...
0
votes
2answers
555 views

Do you know this form of an uncertainty principle?

I hope this question is focused enough - it's not about real problem I have, but to find out if anyone knows about a similar thing. You probably know the Heisenberg uncertainty principle: For any ...
6
votes
0answers
509 views

Counting permutation matrices in 0,1,2 matrices

Let $M$ be a matrix with entries equal to 0, 1 or 2, such that all of the row and column sums are equal to $c$. The Van der Waerden bound gives (roughly) the following bound on the permanent of $M$: ...
4
votes
1answer
221 views

Extensions to the Golden-Thompson inequality?

Let $A$ and $B$ be two Hermitian matrices. The famous Golden-Thompson inequality states that $$\text{tr}(e^{A+B}) \le \text{tr}(e^Ae^B)$$ However, for determinants we have equality $$\det(e^{A+B}) ...
7
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1answer
896 views

Multilinear generalization of Cauchy-Schwarz inequality

Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties: $(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the ...
3
votes
2answers
212 views

How to find a proper decay rate from an iterative inequality

Suppose we have the iterative inequality $\gamma_{k+1} \leq \gamma_k(1 - c \gamma_k^\alpha)$ with $c, \alpha \in (0, 1)$ and $(1 - c \gamma_k^\alpha)>0$ for all non-negative terms $\gamma_k$. -- ...
1
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1answer
503 views

Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$

p[i] is the i-th prime. $\pi(x)$ is prime counting function. Firstly, I think that this Prime gap inequality holds true, $ p[i+1] - p[i] <= i $ Prove:for any i>0, we can always find distinct ...
4
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1answer
310 views

Ask the validity of a scalar inequality

Let $a_i>0$, $x_i, y_i\in \mathbb{R}$ $i=1,\cdots, n$, such that $\sum\limits_{i=1}^nx_iy_i=0$, $\sum\limits_{i=1}^nx_i^2=\sum\limits_{i=1}^ny_i^2=1$. Is it true $$ ...
7
votes
1answer
428 views

What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
4
votes
2answers
609 views

Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello! Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...
12
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4answers
1k views

How to prove a known inequality from a book

The following inequality is from page 125 of D.S. Mitrinovic, J. Pecaric, A.M. Fink, Classical and new inequalities in analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. If ...
0
votes
0answers
177 views

Two-sample t-test variant?

I have a situation that is the opposite of the usual two-sample t-test for equal means: I'm trying to show that it's statistically unlikely that two samples are different. Is there a statistical test ...
0
votes
0answers
207 views

Square variation norm and non-negative, non-increasing sequences

I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual ...
1
vote
1answer
869 views

Sufficient and necessary conditions on equality of Cauchy-Schwarz inequality of determinants

Cauchy-Schwarz inequality of determinants: for $A_{n\times k}$, $B_{n\times k}$, and $B'B$ non-singular, we have $|A'B|^2\leq |A'A||B'B|$ I was wondering what's the sufficient and necessary ...
0
votes
1answer
190 views

An inequality for a continuous non-smooth function

Hello, I have a question about how to prove a lemma such as this one, For any $0<\alpha<1$ and $M_{0}>0$, there exists a $M_{1}>0$ such that $\left|z\right|^{\alpha}\leq ...
1
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0answers
67 views

Bound of polynomial on product space in terms of values on the diagonal

We work in the multivariate case so that $x$ stands for $(x_1,\ldots, x_n)$. Let $q(x,y)$ be a symmetric matrix representation of a homogeneous polynomial $f(x)$ of degree $d$. Explicitly, ($q$ is ...
4
votes
5answers
941 views

An inequality on concave functions

Could somebody help me to answer the following question? Let $f:R_+ \rightarrow R_+$ be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any ...
1
vote
1answer
743 views

Probability inequalities

Hi everyone, I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an ...
2
votes
2answers
796 views

How covariance-variance inequality implies Kantorovich inequality

Hi, I read this article OPERATOR INEQUALITIES RELATED TO CAUCHY-SCHWARZ AND H\"OLDER-McCARTHY INEQUALITIES, Nihonkia Math. J. 1997,117-122. In the introduction part, it states covariance-variance ...
0
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0answers
310 views

Divisor function inequality

I have been reading a paper on the Goldbach conjecture found at http://people.exeter.ac.uk/pt224/Goldbach.pdf. At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then $$\sum_{w\leq ...
17
votes
6answers
10k 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 ...
1
vote
1answer
344 views

Is there a complex analog of this sharpened Cauchy Inequality?

Let $x$ and $y$ be two points on the unit sphere $S^{n-1}$ in Euclidean space ${\mathbb{R}}^n$. Suppose that the angle $\theta$ between the points $x$ and $y$ is acute, so that the dot product $x\cdot ...
7
votes
2answers
540 views

Is this ergodic inequality true?

Is anything similar to the following inequality true, $\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$ where $A_n f = ...
11
votes
5answers
3k views

Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
0
votes
1answer
281 views

Complex version of Farkas' lemma

It is well known result of linear optimization theory that, if a finite set $S$ of linear inequalities in real variables $x_1,x_2, \ldots ,x_n$ implies a linear inequality $i$ in $x_1,x_2, \ldots ...
1
vote
1answer
391 views

Coefficient bounds of an inequality

Hello, Given positive integers $k$ and $n$. Are there upper bounds on coefficients $A$ and $B$ such that they depends only on $k$ (eg., $2 k^k$) and for all non-negative integer sequences ...
9
votes
2answers
1k views

Question on eigenvalue square root subadditivity

ORIGINAL QUESTION Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a $2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller eigenvalue of a $2\times2$ matrix. Is it true ...
1
vote
0answers
101 views

Linear complementarity problem - Classification

For Linear complementarity problems (LCP) like $\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$ $\mathbf{z} \ge \mathbf{0}$ $\mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q}) = 0$ there exists a vast amount ...