相关论文: The best constant in the Khinchine inequality for …
The upper bound inequality for variance of weighted sum of correlated random variables is derived according to Cauchy-Schwarz's inequality, while the weights are non-negative with sum of 1. We also give a novel proof with positive…
For a sequence $\{X_{n}, \, n \geqslant 1 \}$ of random variables satisfying $\mathbb{E} \lvert X_{n} \rvert < \infty$ for all $n \geqslant 1$, a maximal inequality is established, and used to obtain strong law of large numbers for…
We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant…
Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies…
We extend some sharp inequalities for martingale-differences to general multiplicative systems of random variables. The key ingredient in the proofs is a technique reducing the general case to the case of Rademacher random variables without…
We determine the explicit value of the optimal constant in the trace inequality for functions of bounded variations in the case the domain has a particular class of singularities.
We study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their…
A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple…
We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schr\"odinger operators, which significantly improve the so far existing bounds.
We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2\pi}\gamma^pw^2\le C\int_0^{2\pi}\gamma^qw'^2 \] where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such…
The purpose of this note is to recall one remarkable theorem of Khinchin about the special role of the Gaussian distribution. This theorem allows us to give a new interpretation of the Lindeberg condition: it guarantees the uniform…
Let $X=\sum \epsilon_n x_n$ be a Rademacher series with vector-valued coefficients. We obtain an approximate formula for the distribution of the random variable $||X||$ in terms of its mean and a certain quantity derived from the…
We prove an optimal error estimate for the flux variable for a stabilized unfitted Nitsche finite element method applied to an elliptic interface problem with discontinuous constant coefficients. Our result shows explicitly that this error…
We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.
We study finite sections of weighted Carleman's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant.
We discuss the problem of finding the best measurement strategy for estimating the value of a quantum system parameter. In general the optimum quantum measurement, in the sense that it maximizes the quantum Fisher information and hence…
An inequality refining the lower bound for a periodic (Breitenberger) uncertainty constant is proved for a wide class of functions. A connection of uncertainty constants for periodic and non-periodic functions is extended to this class. A…
We prove the optimality of a criterion of Koksma (1953) in Khinchin's conjecture, settling a long standing open problem in analysis. Using this result, we also give a near optimal condition for the a.e.\ convergence of series…
We provide a systematic approach to deal with the following problem. Let $X_1,\ldots,X_n$ be, possibly dependent, $[0,1]$-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than…
In this article, for the radiative transport equation, we study inverse problems of determining a time independent scattering coefficient or total attenuation by boundary data on the complementary sub-boundary after making one time input of…