The inequalities tag has no wiki summary.

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### 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**

votes

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599 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 ...

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**1**answer

399 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 - ...

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**2**answers

586 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 ...

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**2**answers

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:
...

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votes

**1**answer

506 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
...

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votes

**2**answers

550 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 ...

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**0**answers

503 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 $M$:
...

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votes

**1**answer

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

**1**answer

877 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 ...

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votes

**2**answers

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 $\gamma_k$.
-- ...

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**1**answer

498 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 ...

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votes

**1**answer

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 $$
...

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**1**answer

417 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 ...

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**2**answers

598 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 ...

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**4**answers

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 ...

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votes

**0**answers

173 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 ...

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**0**answers

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 ...

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vote

**1**answer

824 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 ...

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**1**answer

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 ...

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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 ...

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votes

**5**answers

929 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 ...

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vote

**1**answer

735 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 ...

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votes

**2**answers

765 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 ...

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**0**answers

307 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 ...

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**6**answers

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 ...

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vote

**1**answer

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 ...

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538 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 = ...

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**5**answers

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 $n$ ...

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votes

**1**answer

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 ...

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**1**answer

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 ...

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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

**0**answers

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 ...

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votes

**1**answer

389 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 ...

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votes

**1**answer

484 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 ...

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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.
$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,
...

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**3**answers

386 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 ...

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**1**answer

794 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 ...

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votes

**0**answers

202 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 ...

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votes

**3**answers

671 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 $v_i$, $F \geq 0$?

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947 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 $\sum_na_n^2=1$. ...

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**1**answer

539 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) $?

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608 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:
...

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**1**answer

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 ...

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**4**answers

682 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
...

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votes

**5**answers

521 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 ...

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**6**answers

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 ...

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votes

**1**answer

926 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} ...

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vote

**1**answer

326 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 ...

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votes

**2**answers

772 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 ...