Let{Xα}be a family of random variables following a certain type of distributions with finite expectation E[Xα]and finite variance Var(Xα),where α is a parameter.Motivated by the recent paper of Hollom and Portier(arXiv:2306.07811v1),we study the anti-concentration function(0,∞)∋ y →infα P |Xα-E[Xα]|≥y√Var(Xα)and find its explicit expression.We show that,for certain familiar families of distributions,including the uniform,exponential,non-degenerate Gaussian and student's t-distributions,the anti-concentration function is not identically zero,which means that the corresponding families of random variables have some sort of anti-concentration property;while for some other familiar families of distributions,including the binomial,Poisson,negative binomial,hypergeometric,Gamma,Pareto,Weibull,log-normal and Beta distributions,the anti-concentration function is identically zero.
Let be a σ -finite complete measure space and let Y be a subspace of a Banach space X . Let φ be a generalized Φ-function on . Denote by and the Musielak-Orlicz spaces whose functions take values in Y and X , respectively. Firstly, let , we characterize the distance of f from . Then, if Y is weakly -analytic and proximinal in X , we show that is proximinal in . Finally, we give the connection between the proximinality of in and the proximinality of in .
In this paper, two temporal discrete numerical schemes of the modified phase field crystal equation are presented by introducing a Lagrange multiplier U and two auxiliary functions ψ, W, which reduce the order of the original equation in time and space. For these schemes, we prove their mass conservation and energy stability. Error estimates of the BDF2 scheme are given. In the end, some numerical experiments are used to verify the correctness of our theoretical results.
In this paper we study the influences of the base points of cubic pencils on the Mordell-Weil groups.Specifically,we investigate and classify the cubic pencils with 8,7 and 6 base points in general position,and give some applications.
1.本刊为向国内外公开发行的学术季刊,主要刊登数学领域的学术论文. 2稿件应当包括题目、作者、作者单位、摘要、关键词、正文、参考文献等内容.如稿件正文为中文文本,请在中文关键词后依次提供题目、作者、作者单位、摘要及关键词准确的英文翻译;如正文内容为英文,则在英文关键词后依次提供题目、作者、作者单位、摘要及关键词准确的中文翻译.
