The inequalities tag has no wiki summary.

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### 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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### 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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**1**answer

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### 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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### 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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### 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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### 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

389 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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### 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 ...

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### 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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383 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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### 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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### 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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367 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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### 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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### 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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### 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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### 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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### “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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### 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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### 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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### 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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### 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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### 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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**1**answer

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### 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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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $y$. ...

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### Why does this inequality hold?

Hi people. Can you help me realize why this is true? I can tell you that $P_i$ and $P_j$ are probabilities, i.e. $0 \leq P_i, P_j \leq 1$.
$\displaystyle \sum_{i=1}^\infty \sum_{j=1}^\infty ijP_iP_j ...

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### Little conjecture about sums of reciprocals

Given a finite list $x_i$ of $N$ positive reals, it seems that $\sum_{i=1}^N x_i = \sum_{i=1}^N x_i {}^{-1} \Rightarrow \sum_{i=1}^N x_i \geq N$. Can anyone give me a proof?

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### subadditive implies concave

Let $f:R_+\to R_+$ be smooth on $(0,\infty)$, increasing, $f(0)=0$ and
$\lim_{x\to\infty}=\infty$. Assume also that $f$ is subadditive:
$f(x+y)\le f(x)+f(y)$ for all $x,y\ge 0$. Must $f$ be concave? ...

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### Maximal inequality over two indices

In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like:
P[$\max_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$]
in the background of ...

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### Inequality concerning absolute value of a polynomial

Let
$$f(z) = (1-1/t) z^w + z/t - 1$$
with integers $t\geq2$ and $w\geq2$.Let $r=1+1/(tw^3)$. How do I show
$$\left\lvert f(r e^{i\varphi}) \right\rvert \geq \left\lvert f(r) \right\rvert$$
for any ...

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### Proving inequalities over algebraic structures

I've been looking at proof techniques in formal systems like Coq and Agda recently, and encountered the newring tactic described here for proving equalities over ...

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### Analogues of the Golden-Thompson inequality

Are there any analogues of the Golden-Thompson Inequality for moment generating functions? The Golden-Thompson Inequality asserts the following: If $A$ and $B$ are two $n \times n$ Hermitian matrices, ...

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### Denominators in the solution to Hilbert's XVII

Hilbert's seventeenth problem asks to prove that every positive semidefinite form can be written as the sum of squares of rational functions. Currently we don't seem to have a good understanding of ...

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### Inequality involving probability measures [closed]

I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck.
An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...

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### Positivity of a finite sum

Let $i$, $k$ be integers such that $2 \leq i \leq k$. I would like to show that the sum
$$
\sum_{j=1}^{i-1} \frac{(-1)^{j-1}(i-j)^k}{(i-j)! (j-1)!}
$$
is positive. I have carried out extensive ...

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### Positivity of sequences via generating series

There are different ways of showing that a given sequence $a_0,a_1,a_2,\dots$
of integers, say, is nonnegative. For example, one can show that $a_n$ count
something, or express $a_n$ as a (multiple) ...

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### Inequality constraints, probability distributions, and integer partitions

I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus,
$a + b + c = 10$
...

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### A plausible positivity

After getting stuck with the
previous positivity
(it probably sounds too complex),
I would like to give a version of the problem which is of most interest to me.
Consider a sequence of real numbers
...

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### Positivity of “harmonic” summation

The settings for the problem are as follows. Given
a real number $\alpha\in[0,1]$, consider
a sequence of real (positive, negative and zero) numbers
$a_1,a_2,\dots,a_n,\dots$ satisfying
(1) $a_1=1$,
...

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### Salem Inequality

I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" ...

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### Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?

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 ...

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### Another plausible inequality.

I come across the following problem in my study.
Consider in the real field. Let $ 0\le x\le1 $, $a_1^2+a_2^2=b_1^2+b_2^2=1$.Is it true
$ ...

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### Linear algebra inequality

I'm wondering (hoping) if an inequality is true. Please can anyone help me?
Let $V$ be a complex vector space $dim_{\mathbb{C}}(V)=n$
with a hermitian scalar product $h$.
Let $v,a, b \in V$.
Is it ...

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### Cyclic order relation in Zn

The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?