Let $\{X_{\alpha}\}$ be a family of random variables following a certain type of distributions with finite expectation $\E[X_{\alpha}]$
and finite variance $\Var(X_{\alpha})$, where $\alpha$ is a parameter. Motivated by the recent paper of Hollom and Portier (arXiv: 2306.07811v1), we study the anti-concentration function
$(0, \infty)\ni y\to \inf_{\alpha}\P\left(|X_{\alpha}-\E[X_{\alpha}]|\geq y \sqrt{\Var(X_{\alpha})}\right)$ 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.
This paper is concerned with a class of
multi-dimensional reflected stochastic partial differential equations with elliptic operators, whose solutions are constrained on a bounded convex domain.
The aim of this paper is to establish the existence and uniqueness theorem of solutions for the reflected stochastic partial differential equations with the penalization method.
In this paper we study the $L^p$-convergence rate of the tamed Euler scheme for scalar stochastic differential equations (SDEs) with piecewise continuous drift coefficient. More precisely, under the assumptions that the drift coefficient is piecewise continuous and polynomially growing and that the diffusion coefficient is Lipschitz continuous and non-zero at the discontinuity points of the drift coefficient, we show that the SDE has a unique strong solution and the $L^p$-convergence order of the tamed Euler scheme is at least 1/2 for all $p \in [1,\infty)$. Moreover, a numerical example is provided to support our conclusion.
In this paper we consider the existence of solutions to the nonhomogeneous quasilinear Kirchhoff- Schr\"{o}dinger-Poisson system with $ p $-Laplacian:
\begin{equation*}\label{1ip}
\begin{cases}\displaystyle
- \Big(a-b \int_{\mathbb{R}^3}|\nabla u|^p{\rm d}x \Big)\Delta_p u+|u|^{p-2}u+\lambda\phi_uu= |u|^{q-2}u+h(x), &\text{ }x\in \mathbb{R}^3,\\
-\Delta\phi=u^2, &\text{ }x\in \mathbb{R}^3,\\
\end{cases}
\end{equation*}
where $ a,b>0 $, $ \frac{4}{3} < p < \frac{12}{5} $, $ p < q < p^* =\frac{3p}{3-p} $, $ \lambda > 0 $. Under suitable assumptions on $ h(x) $, the existence of multiple solutions of the system is obtained by using the Ekeland variational principle and the Mountain pass theorem.
In this paper, we study the (distinct) positive integer solution of the equation
\begin{equation*}\label{eq12}\frac{k}{n} = \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_t}\end{equation*} with $n>k\geq 2$ and $ t\geq 2$.
We show that the above equation has at least one distinct positive integer solution if it has a positive integer solution.
When $k=5$, we show the above equation has at least one distinct positive integer solution for all $n\geq 3$
except possibly when $n\equiv 1, 5041, 6301, 8821, 13861, 15121(\mbox{mod } 16380)$ with $t=3$,
and for all $n\geq 3$ except possibly when $n\equiv 1, 81901(\mbox{mod } 163800)$ with $t=4$.
Furthermore, we point out that the above equation has at least one distinct positive integer solution for all $n(>k)$
when $t\geq k\geq 2$.
Let ${\cal S}$=$({\cal P},{\cal B})$
be a nontrivial $3$-$(v,7,\lambda)$ design with $\lambda\geq 2$. In this paper we show that if $G$ is a block-transitive automorphism group of $\cal S$, then $G$ is point-primitive of affine
or almost simple type. In addition, the case that $G$ is a finite almost simple group with an alternating group socle $A_n$ is considered as well.
