Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.     

Sobolev functions on infinite-dimensional domains

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.     

How Do Cultural Classes Emerge from Assimilation and Distinction? An Extension of the Cucker-Smale Flocking Model

When cultural tastes are not neutral but hierarchically matched to social status, people assimilate themselves to higher status by consuming cultural goods while distinguishing themselves from lower status by developing new tastes. Extending the Cucker-Smale model for mutual influence among agents, we examine when and how many cultural classes emerge from continuous distributions of tastes and what conditions those classes satisfy, through the assimilation-distinction mechanism. We simulate the models with different initial distributions of tastes (uniform, normal, and chi-square), given various ranges of 2 parameters: (a) the strength and (b) the range of distinction relative to assimilation. Tastes are flocking and cultural classes emerge when the range of assimilation is much larger than that of distinction. The number of classes increases with the strength of distinction, whereas the distance between classes equals the range of distinction. Some properties of emergent classes are mathematically proved. First, in a two-class system, the stronger distinction, the larger the upper class. Second, in a three-class system, the middle class is necessarily larger than the lower class and likely larger than the upper class. Third, a 3-class system cannot emerge if distinction is weaker than assimilation. These properties are universal and do not depend on the initial distribution of cultural tastes. This independence predicts homogeneous cultural classes emerging across different social conditions. Also, the cultural middle class as the largest group may explain why subjective class consciousness is often higher than objective position. Unless assimilating efforts can reach an infinite range, there emerges a cultural outcast at the lowest end of the cultural hierarchy.     

具有较少非循环子群共轭类的有限群

设G是有限群,用δ(G)表示群G的非循环子群的共轭类数,πr(G)表示整除|G|的素因子的集合.本文主要研究满足条件δ(G)≤|π(G)|+1的有限群,得到这类群可解,并给出它们的同构分类进一步证明,δ(G)=|π(G)|+2的有限非可解群必同构于A_5或SL(2,5).     

Global unique continuation from a half space for the Schrödinger equation

We obtain a global unique continuation result for the differential inequality |(i_∂t+Δ)u|?|V(x)u|$|(i{\partial }_t+\mathrm{\Delta })u|?|V(x)u|$ in Rⁿ⁺¹${\mathbb{R}}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials V(x)$V(x)$ in Rⁿ${\mathbb{R}}^n$. Our method is based on a new type of Carleman estimates for the operator i_∂t+Δ$i{\partial }_t+\mathrm{\Delta }$ on Rⁿ⁺¹${\mathbb{R}}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.     

共轭可交换性及其在集合形式单调类定理中的应用

引入了集列的极限运算与二元集运算之间的共轭可交换性的概念,讨论了常见的和重要的集列的极限运算与集代数运算之间所具有的共轭可交换性质,在一般的情形下证明了共轭可交换性是集类的极限运算生成类对集代数运算具有封闭性的一个充分条件,使得集合形式的单调类定理的理论得到了统一的解释,并且证明更简单,思路更清晰,最后还给出了一个新的集合形式的单调类定理.     

基于洛伦兹曲线几何特征的中等收入优化模型

中等收入人口的确定对于分析一个社会的经济发展,稳定性具有重要的意义.洛伦兹曲线是一种分析中等收入的经典工具.本文提出一种具有更高拟合精度的洛伦兹曲线模型.并基于其几何特征,考虑社会的贫富程度与贫富分化因素,建立了收入空间法与人口空间法的优化模型.     

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

On the invariant faces associated with a cone-preserving map

For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.