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2
votes
1answer
499 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
545 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
497 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
votes
1answer
869 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
210 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
vote
1answer
496 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
votes
1answer
308 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
413 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
595 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
votes
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
172 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
204 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
815 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
188 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
vote
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
920 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
728 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
756 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
votes
0answers
305 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 ...
14
votes
6answers
9k 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
342 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
510 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
2k 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
277 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
390 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
100 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 ...
7
votes
1answer
386 views

Characterizing intersection of zero sets of elementary symmetric polynomials on R^n

Stated simply, the question is: Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb ...
7
votes
1answer
483 views

Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$: $\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...
5
votes
0answers
443 views

Any similar inequality in literature?

I got the following inequality: $B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...
2
votes
2answers
369 views

About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$. I am considering the generalized question for ...
3
votes
1answer
784 views

Cauchy-like inequality for Kronecker (tensor) product

General question first: upper/lower bound a sum of Kronecker products by its components. More specifically, how is $$ \Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$ bounded by the operator ...
4
votes
0answers
201 views

Applying the amplification trick + probabilistic method on connected graphs

First of all, I apologise if my question is not very specific. I am not really looking for a concrete answer but rather some hints and pointers what could be used/applied in this situation. A concrete ...
2
votes
3answers
669 views

a “reverse Hadamard inequality”

Is there an inequality of the form $|\det(A)| \geq F(v_1, \ldots, v_n)$ for a real $n\times n$-matrix $A$ with columns $v_i$, $F \geq 0$?
13
votes
1answer
941 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 $\sum_na_n^2=1$. ...
4
votes
1answer
538 views

“Less than” formula for complete theory of the rationals

Does anyone know the name/author of the model theory paper where it's proven that you can define the $ < $ relation on $ (\mathbb{Q}, +, \cdot, 1, 0) $?
4
votes
3answers
602 views

Is the componentwise square-root of a positive-definite matrix also pos.-def.?

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and $B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$. Is $B$ positive-definite whenever $A$ is? In other words: ...
1
vote
1answer
1k views

How to estimate derivatives of multivariate polynomial near a manifold

I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value ...
2
votes
4answers
679 views

Proving triangle inequality for an affine invariant distance on $Sym_n^+$

In the paper Riemannian framework for tensor computing, by Pennec et al., on page 46 the authors state a "distance" function on the manifold of positive definite matrices $Sym_n^+$ given by ...
6
votes
5answers
519 views

Binary operations compatible with the usual order on the reals

An officemate passes along the following natural-seeming question: Say that a binary operation $\oplus$ is compatible with the usual order $\leq$ on $\mathbb{R}$ if for any $w, x, y, z$ in ...
16
votes
6answers
3k 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 ...
3
votes
1answer
914 views

Inequality on probability distributions

I would like to know if the following inequality is satisfied by all probability distributions (or at least some class of probability distributions) for all integer $n \geq 2$. $\int_0^{\infty} ...
1
vote
1answer
322 views

Statistical inequality

Let $X$ be a finite discrete variable and $X\ge0$. Is it true that $$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$ where ...
3
votes
2answers
757 views

Tight bound for sum of entries of the inverse of a nonnegative matrix

While playing around with certain non-negative matrices, I got stuck at the following question. Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...
5
votes
2answers
405 views

Sturm chain analogue for exponential polynomials?

I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form $f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real). My first question is: is there an algorithm for ...
0
votes
1answer
614 views

Probability space analogue of Cauchy-Schwarz inequality

Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that: $$2\sum_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum_{i \in A}\Pr[i]+ \sum_{j \in B}\Pr[j] +\sum_{i, j ...
7
votes
2answers
770 views

Generalization of the positive semidefinite Grothendieck inequality

In a recent paper, S. Khot and A. Naor show a natural generalization of the positive semidefinite Grothendieck's inequality. Grothendieck showed that there exists a constant $K > 0$ such that for ...
2
votes
5answers
3k views

What is the angle between two complex vectors?

Let $x, y\in R^n$ and $x, y$ are nonzero, it is well known $\frac{x^Ty}{\parallel x\parallel_2\parallel y\parallel_2}(\parallel x\parallel_2+\parallel y\parallel_2)\le \parallel x+y\parallel_2$. How ...
4
votes
2answers
584 views

Logarithm of AM/GM ratio: $\sqrt{\log((x+y)/(2\sqrt{xy}))}$

Recently, while playing around with infinite-divisibility, i arrived at the following metric: $$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$ defined for positive reals $x$ and $y$. ...