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0
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1answer
243 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
523 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
270 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
316 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
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1answer
618 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
408 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
698 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
229 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
411 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
432 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
84 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
114 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
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2answers
137 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
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1answer
163 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
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1answer
207 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
294 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
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2answers
267 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 $$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$ ...
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4answers
773 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 ...
3
votes
2answers
326 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
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1answer
619 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
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2answers
170 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
151 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
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1answer
424 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
218 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
137 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
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1answer
201 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
340 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
262 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
286 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 ...
7
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2answers
786 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
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1answer
345 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
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1answer
314 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 ...
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2answers
1k 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
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0answers
283 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
325 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
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0answers
117 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
170 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
337 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 ...
10
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2answers
470 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 ...
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2answers
320 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 ...
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0answers
98 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 ...
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2answers
619 views

Bounding Euler products (or almost) by products of zeta functions

Let $s_1, s_2 \in (1/2,1\rbrack$. I would like to bound the product $$A=\prod_p \left(1 + \frac{p^{-s_1} p^{-s_2}}{(1-p^{-s_1}+p^{-1}) (1-p^{-s_2}+p^{-1})}\right)$$ Now, I am almost positive that ...
2
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0answers
190 views

Proving that an increasing iterative sequence increases at a decreasing rate

In this question Proving a sequence of integrals increases (iterated minimax distributions) Pietro Majer proved that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...
3
votes
1answer
344 views

Inequality involving perimeter and area

I am studying an article: The parametric problem of capillarity: the case of two and three fluids, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this: ...
3
votes
1answer
702 views

Ratio sum comparison on operators

It is known by the Lidskii inequality, that $\sum_{i=1}^n \left|s_i(S)-s_i(T)\right|\le\sum_{i=1}^n s_i(S+T)$, where $s_i(S)$ is the $i$-th singular value of $S$. How would one prove that ...
4
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2answers
219 views

Bounding the series of the geometric means of the terms of a given positive series

Let $ \{ a _ k \} _{k\in\mathbb{N} _ +} $ be a sequence of non-negative numbers, and let $MG(a_1,\dots,a_n)$ denote the geometric mean of the first $n$ terms. Then, the inequality $$ \sum _ {n\ge ...
1
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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 ...