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21
votes
0answers
430 views

Are there any nontrivial near-isometries of the $n$-dimensional cube?

Consider the $n$-dimensional Hamming cube, $C = \{-1,1\}^n$. Given an $n \times n$ orthogonal matrix $O$, I'll measure "how close $O$ is to being an isometry of $C$" by the following scoring ...
12
votes
0answers
363 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
9
votes
0answers
215 views

Determinant inequality involving Hermitian, positive definite matrices

Question: Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question ...
9
votes
0answers
313 views

Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$ Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...
7
votes
0answers
284 views

Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites. Let the ...
7
votes
0answers
455 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} ...
6
votes
0answers
234 views

Is the heat kernel more spread out with a smaller metric?

Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
6
votes
0answers
516 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 $M$: ...
5
votes
0answers
134 views

Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
5
votes
0answers
129 views

Maximal inequalities for square of partial sums

Let $S_n = \sum_{i \leq n} X_i$ be the partial sums of a nice sequence of random variables $X_i$. In my application, $X_i$ is a functional of a finite-state, irreducible, aperiodic Markov chain, so ...
5
votes
0answers
508 views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
5
votes
0answers
191 views

Maximizing Renyi entropy for a certain channel

The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
5
votes
0answers
449 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. $(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$, ...
4
votes
0answers
160 views

Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...
4
votes
0answers
109 views

Carleman estimates on monotonicity formulas

I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from $$\int_{B_r} ...
4
votes
0answers
143 views

Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all $n\geq3$, the function: ...
4
votes
0answers
409 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 ...
4
votes
0answers
206 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 ...
3
votes
0answers
230 views

Question on a proof by Solonnikov,Ladyzhenskaya,Ural'tseva

I have already asked this question on Mathematics SE, because I suppose that it is not research level. But I haven't got an answer, possibly here someone can answer. Let $G(t,x)$ be the fundamental ...
3
votes
0answers
90 views

Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added. This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
3
votes
0answers
100 views

Will relative entropy increase with majorization?

Suppose that three probability distributions have the relation $P\succ Q\succ R$, then do the following relations between their relative entropy valid? (assume that $p_1\geq p_2\geq\cdots \geq p_n$ ...
3
votes
0answers
92 views

On the comparison of Egyptian fractions of two kinds

I posted the question on MSE here but it did not get any answer. Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset ...
3
votes
0answers
185 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
3
votes
0answers
106 views

Comparison of Hardy's inequality and Sobolev's inequality

Both, Hardy's inequality and Sobolev's inequality, are estimates that compare the Laplacian of a function and to the function itself, admittedly in a slightly different fashion. Still they seem to be ...
3
votes
0answers
101 views

A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...
3
votes
0answers
244 views

Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053 Specifically, I can't derive the following inequality in (1.20): \begin{equation} ...
3
votes
0answers
144 views

Upper bound on integrals of Legendre polynomials

Hi, If $P_n(x) $ is unnormalized shifted Legendre polynomial, and $g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $ then what is the upper bound $ |g_{n,m} (x)|_{max} , x\in (0,1) $ as a function ...
3
votes
0answers
308 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 ...
3
votes
0answers
211 views

Where can I find interpolation inequalities for derivatives of the following form?

Here they are: $$||f||_{\infty} \leq C ||f||_q^{\frac {qk} {n+kq}} \left( \sum_{|\mu|=k} ||D^\mu f||_{BMO} \right)^{ \frac n {n+kq}}$$ and $$||f||_{Lip_\alpha} \leq C ||f||_q^{\frac {qk} {n+kq} \frac ...
2
votes
0answers
57 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...
2
votes
0answers
84 views

Variant form of the gronwall inequality

I know the following statement for gronwall inequality: Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have, $f' \leq \phi f$ and $f(0)=0$ then $f=0$ Now is ...
2
votes
0answers
88 views

Positive, Uni-modal, Log-concave Combinatorics

We define a sequence, $\{a_n\}_{n=0}^\infty$, to be a uni-modal sequence if for some $m$, $$a_0<a_1<\cdots<a_m,\ \ \ \ a_m>a_{m+1}>a_{m+2}>\cdots.$$ We define a sequence, ...
2
votes
0answers
87 views

Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

I duplicate here a question I asked on math.stackexchange. Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some ...
2
votes
0answers
209 views

An integral inequality

Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be bounded with derivative $g'$. I have shown that the following inequality holds for all $w\in\mathbb{R}$, ...
2
votes
0answers
156 views

An inequality for Lp-functions

I am interested in the following inequality: \begin{equation} \int\limits_\Omega \left\vert f_1 - f_2 \right\vert^p ~ d\mu \le C_p \left[ \int\limits_\Omega \left\vert f_1 \right\vert^p ~ d\mu + ...
2
votes
0answers
58 views

Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &=& ...
2
votes
0answers
286 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 ...
2
votes
0answers
191 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) = ...
1
vote
0answers
73 views

Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if $$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall ...
1
vote
0answers
110 views

Uniform bound for an alternating series of functions

I have mainly two questions, the first one being motivated by the second one. 1) Is there a way to prove that $F(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^{2k}}{(2k)!}$ is bounded on $\mathbb{R}_+$ ...
1
vote
0answers
106 views

about an inequality which looks like a Hardy inequality

I would like to know if the following inequality can be true : consider a double sequence $(u_{i,j})_{(i,j)\in (\mathbb{N}^\star)^2}$ of real numbers and a real $p\geq 1$, do we have $$ ...
1
vote
0answers
103 views

Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...
1
vote
0answers
214 views

Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$ where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...
1
vote
0answers
219 views

Bounding a sum of binomial coefficients in terms of 'the next one'

I need to bound a sum of a portion of binomial coefficients in terms of "the next one", and understand what is the best which can be said in this sense. Given a real number $t \geq 2$, call $P(t)$ ...
1
vote
0answers
128 views

Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...
1
vote
0answers
99 views

Multivariate polynomial with positive coefficients

This question was originally asked at stack exchange (http://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more ...
1
vote
0answers
194 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
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
173 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 ...
1
vote
0answers
101 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 ...