The tag has no wiki summary.

learn more… | top users | synonyms (1)

-1
votes
2answers
325 views

On a inequality [closed]

I hope the following kind of inequality holds: let $a_i,b_i\in R$ with $b_i>0$, $\sum _{i=1}^mt_i=1$ with $t_i>0$, then ...
0
votes
1answer
175 views

a simple probability inequality

For independent Rademacher random variables $\epsilon_i, i=1,2, \cdots, n$, i.e. $P(\epsilon_i=-1)=P(\epsilon_i=1)=\frac{1}{2}$, do we have $$max_{0\le a_i\le b, ...
6
votes
2answers
255 views

Inequality involving the weak second moment

I want to ask the following probability inequality: Is it true that for any random variable $X\ge 0$, we have $$ \sup_{t>0}(t\mathbb E(X\mathbf 1_{X\ge t})) \le 2\sup_{t>0}(t^2 \mathbb P(X ...
3
votes
0answers
291 views

A curious inequality

Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that $|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that ...
1
vote
0answers
134 views

A Multiplicative version of McDiarmid's Inequality like the one of Chernoff-Hoeffding Bounds

McDiarmid's Inequality basically says the following: Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the ...
1
vote
1answer
149 views

An upper bound on a simple sum

Hi, I am trying to put a bound on a sum. Given $\omega=\exp(2\pi i/3)$ and $n$ positive real numbers $ 0=\tau_0 < \tau_1 < \tau_2 < ...\tau_{n-1}< \tau_n=1 $ such that ...
4
votes
1answer
270 views

Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf you can find a trajectorial version of Doob's inequality. It is given by: ...
0
votes
1answer
225 views

Inequalities Involving Wedge Product (Reference Request)

Hello, I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The ...
11
votes
2answers
499 views

Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the ...
1
vote
2answers
259 views

A Gronwall-type inequality.

I want to derive a Gronwall-type inequality from the inequality below. Here all the functions are nonnegative, continuous and if you need some assumptions you may use that. $$ f^2(t) \leqslant g^2(t) ...
4
votes
1answer
271 views

Upper bound on joint Renyi entropy

Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by \begin{equation} H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha. \end{equation} ...
11
votes
1answer
592 views

A delicate elementary inequality

The following "piecewise-quadratic" inequality emerged in a joint work of Rom Pinchasi and myself. The inequality is surprisingly delicate, and all our attempts to simplify it made it false. By the ...
4
votes
0answers
399 views

System of Equations Upper Bound

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here: For ...
5
votes
4answers
620 views

An inequality involving sums of powers

I asked this question at Stack Exchange but received no answer. The origins of the question are unclear, as I came across it rummaging through old notebooks from highschool, in one of which it was ...
1
vote
1answer
228 views

Does this variable have an upper bound?

Let $x$ be a positive scalar variable whose time derivative satisfies $$|\dot{x}(t)|\leq \exp \left\{\left(-\int_{0}^{t}\frac{1}{x(\tau)} \mathrm{d} \tau \right)\right\},$$ where $|\cdot|$ denotes the ...
5
votes
1answer
373 views

Trace matrix inequality

Hello all, I come across the following problem. Is it true that for a positive definite matrix $X^{n\times n}$, the following holds $\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$, ...
5
votes
1answer
391 views

Hadamard-like inequalites for positive definite symmetric matrices

Let $S$ be any positive semi-definite symmetric matrix (Hermitian psd matrices work as well). The Hadamard inequality is that $$\det S\le\prod_{i=1}^n s_{ii}.$$ My question is whether there are some ...
0
votes
1answer
79 views

Continuous variant of KFG Inequality

Context Let $t_1, .., t_n$ be jointly independent boolean random variables. Let $X, Y$ be monotone functions (i.e. $\forall i: t_i \geq t'_i$ implies $X(t_1, ... t_i ..., t_n) \geq X(t'_1, ... t'_i, ...
1
vote
1answer
111 views

Approximating Moment of Sum of RVs

Given $X_i$ are independent random variables. $|X_i| < 1$ $E[X_i] = 0$ $X = \sum_i^n X_i$ $var(X)=\sigma$ Prove: $$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p Things I've tried: ...
5
votes
2answers
133 views

Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...
1
vote
1answer
151 views

Unitary matrix and matrix inequality [closed]

Dear all, Suppose U and V are unitary matrix, A and B are positive definite, Does: $UAU^{-1} < VBV^{-1}$ implies $A< B$ and vice versa?
0
votes
1answer
204 views

Inequality with even powers of trigonometric functions

For $m>0$, $0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
3
votes
3answers
281 views

Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$?

What is the asymptotic behaviour or an upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$? Or any good reference for tools to tackle this question? ...
1
vote
2answers
246 views

how to solve a singular integral equation involving the kernel $1/x$

Dear all, Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that $$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$ ...
15
votes
4answers
752 views

Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'?

My colleagues and I are working on a project related to an old paper of C. Borell and we have boiled it down to the following problem: Show, for all integers $1 \leq i \leq k$, that the univariate ...
2
votes
2answers
299 views

Is there a software to prove or deduce symbolic inequalities?

I have a bunch of inequalities, and I'm trying to see if another inequality can be deduced from the first bunch. For example, assuming that $a \leq b$ and $c \leq d$, we can deduce that $a + c \leq b ...
0
votes
1answer
578 views

Difficult Inequality [closed]

Suppose $x,y\geq 0$ and $b,c,d\geq 1$ are integers. Prove or find a counterexample to the following inequality $$ \frac{1}{1+\frac{b+d}{\sqrt{x^{2}+c^{2}}}} + ...
0
votes
2answers
166 views

inequality of a function

Hello everyone, could someone gives the conditions for $\lambda$ that the following inequality is correct for any $0\leq x\leq\alpha$ $$\lambda x-1+e^{-\lambda x}\leq\lambda^2x\sqrt{\alpha x}$$ ...
0
votes
1answer
140 views

Strengthening an inequality

Let $k$ be an integer. The following inequality is standard. $$ (a+b)^{k+1} - b^{k+1} \leq (k+1)a(a+b)^k $$ for $a,b > 0$. However, does the following inequality still hold $$ (a+b)^{k+1} - ...
7
votes
1answer
397 views

Shapiro inequality for $n=23$

Is the Shapiro inequality for $n=23$ an open problem? The reason why I am asking is I have two contradictory pieces of information from two different articles. The first article titled "The validity ...
5
votes
1answer
209 views

Convexity of a specific semialgebraic set

I have an engineering problem which maybe resolved with semi-definite programming optimization. I have a set which I would like to know if is convex: Being $m \in \mathbb{R}^+$ a positive real ...
1
vote
1answer
125 views

inequality for a symmetric nonnegative matrix

Given $A$ symmetric and semidefinite positive, for each $x$ $$ x'Ax \geq \frac{1}{\Vert A\Vert} \Vert Ax \Vert^2 $$ This inequality appears at page 24 of "Introduction to Optimization" from Boris T. ...
1
vote
1answer
176 views

Mapping multivariate polynomial inequalities system to subspace

What I will ask, more than a solution, is better mathematical definition of my problem and directions to find the solution. I have a set of linear equations, e.g.: \begin{align} d_1 &= L_1 - ...
3
votes
2answers
332 views

anyone help me with this inequality

I'm have some difficulties in bounding the following inequality: I want to find a c as small as possible s.t. $$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq ...
1
vote
2answers
253 views

An inequality with $\ell_p$ norm

I encounter the following claim in my research for which I couldn't get a solution for a long time. I asked a more general version of the question at math.stackexchange which did not attract much ...
0
votes
1answer
272 views

A Polynomial Inequality Proof

Given $x_1,x_2, \ldots, x_n \ge 0, \alpha \ge 1$, show that $\sum_{i}\alpha x_i(\sum_{j \le i}{x_j})^{\alpha-1} \ge (\sum_{i}x_i)^\alpha$ We're pretty sure the ineuqality holds for the given ...
6
votes
2answers
640 views

(sharp)Garding's inequality and inequality with lower bounds

The origin of Garding's inequality was an effort to solve Dirichlet's problem for linear elliptic operators of high even order.Let $$P(x,D)= \sum a_{\alpha}(x)D^{\alpha}$$ with principal part ...
3
votes
1answer
341 views

Can these logarithmic inequalities show existence of a prime between (x-1)^2 and x^2

SOME PROPERTIES OF THE SERIES OF COMPOSED NUMBERS, p2 gives the bounds $$l(x)=\frac{x}{\log(x)-28/29}<\pi(x) < u(x)=\frac{x}{\log(x)-1.12} \qquad (1)$$ for $x \geq 3299$. TWO GENERALIZATIONS ...
1
vote
1answer
288 views

sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste $\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $ wtih $Ric\ge-(n-1)K$, for ...
10
votes
2answers
889 views

Proving a messy inequality

EDIT: After much work I was able to reduce the inequality to a single variable function which I need to show is non-positive. That function is (for $0\leq p\leq\frac{1}{2}$) $$\frac{p^2(\log(p))^2 - ...
2
votes
0answers
264 views

Does this inequality of negative relative entropy and quantum relative entropy hold?

Hello, everyone! Question I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices ...
4
votes
1answer
300 views

Is it possible to extend this inequality about Euclidean distance &Frobenius norm to more general Bregman divergence such as relative entropy & von Neumann divergence?

Motivation- A Special Case Supposing $A,B\in\mathbb{S}^{m\times m}$ are symmetric positive semi-definite (SPD) matrices and $\mathbf{x}\in\mathbb{R}^m$ is a unit vector where $\|\mathbf{x}\|=1$, we ...
1
vote
0answers
115 views

Boundedness of Integral

Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point. Define the integral $$ Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta $$ and ...
1
vote
0answers
166 views

Given a jointly convex function $f$, what is the bound of $f\left(\sum_ip_i^2x_i,\sum_jq_j^2y_j\right)$if $\mathbf{p},\mathbf{q}$ are constrained in a manifold?

Suppose there is a jointly convex function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $\mathbf{x},\mathbf{y}\in\mathbb{R}^m$ and ...
4
votes
1answer
2k views

Determinant of a sum of two matrices (one dominating the other)

Let $A$ and $B$ be two $n \times n$ real matrices such that: $\forall i, j: a_{ij} \geq 0, b_{ij} \geq 0$ let $a_\max$ be the largest entry of $A$ and $b_\min$ be the smallest nonzero entry of $B$; ...
4
votes
1answer
321 views

A Hölder like inequality

If $0< a_1\le a_2\le \cdots \le a_n\le a_{n+1}$ and $p>1$, is it true that ...
9
votes
2answers
449 views

A property of unimodal sequences

It is well-known that $(-1)^j \sum_{i=0}^j (-1)^i\binom{n}{i} \geq 0$. This inequality can be used to prove Bonferroni's inequalities for example. Recently I noticed that a similar inequality applies ...
7
votes
2answers
309 views

Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...
1
vote
0answers
94 views

Inequality involving BV norm and a regularizing kernel

In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ...