Supports of Weighted Equilibrium Measures and Examples

We analyze the supports of weighted equilibrium measures in ? ⁿ . We give explicit examples of families of compact sets which arise as the support of a weighted equilibrium measure for some admissible weight w. These examples also give new constructions of plurisubharmonic functions in the Lelong class. We also include a list of open problems on the support of extremal measures which are related to solutions of Monge–Ampère equations.     

On the Supports of Vector Equilibrium Measures in the Angelesco Problem with Nested Intervals

A vector logarithmic-potential equilibrium problem with the Angelesco interaction matrix is considered for two nested intervals with a common endpoint. The ratio of the lengths of the intervals is a parameter of the problem, and another parameter is the ratio of the masses of the components of the vector equilibrium measure. Two cases are distinguished, depending on the relations between the parameters. In the first case, the equilibrium measure is described by a meromorphic function on a three-sheeted Riemann surface of genus zero, and the supports of the components do not overlap and are connected. In the second case, a solution to the equilibrium problem is found in terms of a meromorphic function on a six-sheeted surface of genus one, and the supports overlap and are not connected.     

Convergence of Weighted Empirical Measures

This article considers a system with infinitely many interacting particles, starting with a system of n interacting particles that is described by a system of n stochastic differential equations for the time-varying locations and weights. Any particle in the system interacts with others through the weighted empirical measure Uⁿ formed by the sum of weighted Dirac measures on the n particles. Weak convergence of the weighted empirical measure is studied under suitable conditions, such as bounded initial values, and linear growth of drift and diffusion coefficients. Thereafter, the limit of the weighted empirical measures is identified to be a martingale solution of the infinite interacting system.     

Balayage Ping-Pong: A Convexity of Equilibrium Measures

In this paper we establish that the density of the equilibrium measure of finitely many intervals for both logarithmic and Riesz potentials is convex. The main tool is a balayage ping-pong technique. A?similar result is obtained for finitely many arcs on the unit circle. Applications to external field problems and constrained energy problems are presented.     

Complete Continuity of Eigen-Pairs of Weighted Dirichlet Eigenvalue Problem

In this paper, we will study the dependence of eigen-pairs \((\lambda _k(\rho ), \varphi _k(x,\rho ))\) of weighted Dirichlet eigenvalue problem on weights \(\rho \). It will be shown that \(\lambda _k(\rho )\) and \(\varphi _k(x,\rho )\) are completely continuous (CC) in \(\rho \). Precisely, when \(\rho _n\) is weakly convergent to \(\rho \) in some Lebesgue space, \(\lambda _k(\rho _n)\) is convergent to \(\lambda _k(\rho )\). As for the convergence of eigenfunctions, since eigenvalues may have multiple eigenfunctions, it will be shown that the distance from \(\varphi _k(x,\rho _n)\) to the eigen space \(V_k(\rho )\) of \(\lambda _k(\rho )\) is tending to zero. As applications, the CC dependence of solutions of linear inhomogeneous equations and the CC dependence of the heat kernels on coefficients will be given.     

Complete Convergence for Weighted Sums of WOD Random Variables

In this article, we study the complete convergence for weighted sums of widely orthant dependent random variables. By using the exponential probability inequality, we establish a complete convergence result for weighted sums of widely orthant dependent ran-dom variables under mild conditions of weights and moments. The result obtained in the paper generalizes the corresponding ones for independent random variables and negatively dependent random variables.     

亚正常加权移位算子的谱刻画

在这篇文章里，我们给出了亚正常单侧与双侧加权移们算子的谱及其各部分的完全刻画，推广了已有文献中的相关结果。     

NOD随机变量随机加权部分和的完全收敛性

本文研究了NOD随机变量双下标随机加权部分和的完全收敛性,获得了一些完全矩收敛结果和完全收敛结果,从而获得了Marcinkiewicz-Zygmund型强大数律.我们的结果推广了目前已有的一些结论.进一步,我们给出一些数据模拟工作来展示收敛性结果.     

两两NQD列加权和的安全收敛性

本文研究两两NQD系加权和的完全收敛性，证明了一般双下标加权系数的加权部分和的完全收敛性，改进了吴群英（2002）的结果。     

两两NQD阵列加权和的完全收敛性

本文研究行两两NQD阵列加权和的完全收敛性,证明了一般双下标加权系数的加权部分和的完全收敛性,推广了文献[8]的结果.     

Weighted Logarithmic Sobolev Inequalities for Sub-Gaussian Measures

In this short paper, We establish the weighted Logarithmic Sobolev inequalities for sub-Gaussian measure of high dimension with explicit constants via phase transition and the well-known Bakry-émery criterion.     

Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables

In this paper,the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables are investigated.Some sufficient conditions for the convergence are provided.In addition,the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables is obtained.The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

A Characterization of Complete Bounded Domain

1 IntroductionThis paper is concerned with biholomorphic mappings between two bounded domains D andG both in C". Consequently, an important question is whether the domain D is biholomorphicto G? We give an answer for this question under a very weak condition.Let D be a bounded domain and Bn the unit ball in Cn. Let T(D) be the holomorphictangent bundle of D. We will identify T(D) with D × Cn. Let H(D1, D2) be the family ofholomorphic mappings from D1 to D2. We introduce the followin…     

A New Characterization of Weighted Peetre K-Functionals

The purpose of this paper is to present a characterization of certain types of generalized weighted Peetre K-functionals by means of a modulus of smoothness. This new modulus is based on the classical one taken on a certain linear transform of the function. A new modulus of smoothness which describes the best algebraic approximation is introduced.     

B值随机变量阵列加权和的完全收敛性

设{X_ni:1≤i≤n,n≥1}为行间独立的B值r.v.阵列,g(z)是指数为1/p的正则变化函数,r>0,{ani 1≤t≤n,n≥1}为实数阵列,本文得到了使(?)成立的条件,推广并改进了Stout及Sung等的著名结论.     

Some Supports of Fourier Transforms of Singular Measures are not Rajchman

The notion of Riesz sets tells us that a support of Fourier transform of a measure with non-trivial singular part has to be large. The notion of Rajchman sets tells us that if the Fourier transform tends to zero at infinity outside a small set, then it tends to zero even on the small set. Here we present a new angle of an old question: Whether every Rajchman set should be Riesz.     

B值随机元阵列加权和的完全收敛性

在随机元阵列随机有界的条件下,得到了行间独立B值随机元阵列加权和的完全收敛性成立的充分条件,延伸了文献[9]的主要结果和文献[6]的部分结果.     

不同分布NA序列加权和的完全收敛性

本文讨论了不同分布NA随机变量序列加权和的完全收敛性，获得了较[7]中的定理1及定理A更为一般的安全收敛性，并得到了完全收敛速度与矩条件之间的等价关系。     

Complete Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables

Let $\{X_n,n\geq1\}$ be a sequence of negatively superadditive dependent (NSD, in short) random variables and $\{a_{nk}, 1\leq k\leq n, n\geq1\}$ be an array of real numbers. Under some suitable conditions, we present some results on complete convergence for weighted sums $\sum_{k=1}^na_{nk}X_k$ of NSD random variables by using the Rosenthal type inequality. The results obtained in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.