相关论文: The best constant in the Khinchine inequality for …

We provide an explicit necessary condition to have that no extremal for the best constant in the Caffarelli-Kohn-Nirenberg inequality is radially symmetric.

Let $X$ be a real-valued random variable with distribution function $F$. Set $X_1,\dots, X_m$ to be independent copies of $X$ and let $F_m$ be the corresponding empirical distribution function. We show that there are absolute constants…

We obtain the best possible upper bounds for the moments of a single order statistic from independent, non-negative random variables, in terms of the population mean. The main result covers the independent identically distributed case.…

This paper concerns the problem of determining the optimal constant in the Montgomery--Vaughan weighted generalization of Hilbert's inequality. We consider an approach pursued by previous authors via a parametric family of inequalities. We…

We prove an optimal lower bound for the best constant in a class of weighted anisotropic Poincar\'e inequalities

Suppose $n$ independent random variables $X_1, X_2, \dots, X_n$ have zero mean and equal variance. We prove that if the average of $\chi^2$ distances between these variables and the normal distribution is bounded by a sufficiently small…

We study further Mumford's notion of local semistability and, in particular, show that semistable singularities are log canonical under mild assumptions. We provide many new examples of semistable and unstable singularities. More generally,…

Let m_j denote the jth moment of a random variable X. One of the results is the inequality m_3 \leq c m_4^{3/4} with the best possible factor c over all random variables X with a non-positive mean.

Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these…

One classical measure of the quality of an interpolating function is its Lipschitz constant. In this paper we consider interpolants with additional smoothness requirements, in particular that their derivatives be Lipschitz. We show that…

In this paper both we establish the best constants for the Nash inequalities on the standard unit sphere $\mathbb{S}^n$ of $\mathbb{R}^{n+1}$ and we give answers on the existence of extremal functions on the corresponding problems. Also we…

In this article we present a Bernstein inequality for sums of random variables which are defined on a spatial lattice structure. The inequality can be used to derive concentration inequalities. It can be useful to obtain consistency…

The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of…

We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter.…

An inequality is derived for the correlation of two univariate functions operating on symmetric bivariate normal random variables. The inequality is a simple consequence of the Cauchy-Schwarz inequality.

We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…

As part of the search for the value of the smallest upper bound of the best constant for the famous Grothendieck inequality, the so-called Grothendieck constant (a hard open problem - unsolved since 1953), we provide a further approach,…

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely we prove that the best constant is achieved by a non-constant solution of the…

We establish formulas for the constant factor in several asymptotic estimates related to the distribution of integer and polynomial divisors. The formulas are then used to approximate these factors numerically.