随机整数划分的概率分析（英文）

令n≥1是自然数,它的一个整数划分是一列自然数λ_1,λ_2,…,λ_l(l≥1)使得λ_1+λ_2+…+λ_l=n.令P_n表示n的所有整数划分的集合,P表示所有整数划分的集合,通过赋予每个划分λ一个概率测度可以得到概率空间.特别地,有趣的概率测度包括均匀测度和Plancherel测度以及乘性测度.这些整数随机划分的概率极限理论内容非常丰富.本文综述一些研究成果和进展,包括极限形以及波动.为叙述简单起见,主要集中在均匀划分和Plancherel划分,并且重点介绍两个模型中所用到条件概率的论证思想和方法,大多结果不加以详细证明.     

半群K(n,r)中的幂等生成元

设Singn是由一个n元集上的所有奇异变换所构成的奇异变换半群，I是由Singn中一些亏数为1的幂等元组成的集合，Howie利用有向图证明了：I是Singn的一个生成集当且仅当与其相应的有向图D（I）是强连通的完全图，本文利用多重有向图将这一结果推广到Singn的每个理想K（,r)上。     

极大S2NS阵的分支数与非零元个数

一个实方阵A称为是S^2NS阵,若所有与A有相同符号模式的矩阵均可逆,且它们的逆矩阵的符号模式都相同.若A是S^2NS阵且A中任意一个零元换为任意非零元后所得的矩阵都不是S2NS阵,则称A是极大S^2NS阵.论文证明了当n≥5时,所有n阶极大S^2NS阵的分支个数所成之集合Fn为{1,…,n}／{2},而所有n阶极大S^2NS阵的非零元个数所成之集合S（n）,除去2n＋1到3n-4间的一段外,也得到了完全确定.     

关于数列{Γ(n+1/2)/√nΓ(n/2)}+∞n=1的极限

用单调有界定理证明了数列{Γ(n+1/2)/√nΓ(n/2)}+∞n=1的奇子列和偶子列极限的存在性,并给出了该数列的极限为1/√2.本文所得结果对帮助学生更好理解概率统计论中t分布密度函数的极限函数的证明有一定指导作用.     

关于数列{[Γ（n＋1）/2]/√nΓ（n/2）}n=1^＋∞

用单调有界定理证明了数列{[Γ（n＋1）/2]/√nΓ（n/2）}n=1^＋∞的奇子列和偶子列极限的存在性，并给出了该数列的极限为1/√2本文所得结果对帮助学生更好理解概率统计论中t分布密度函数的极限函数的证明有一定指导作用．     

Cauchy运动和极小集合

<正> 如果R是n维欧氏空间或任意的度量空间时,则运动f(p,t)的极限集合有比较复杂的拓扑结构.在一般情况下,极限集合的性质依赖于f(p,t)的运动特征.因此B.B.提出这样问题:如何由极限集合附近的轨道的运动状态来研究这些集合的内部结     

随机事件与随机变量概率分布的拓展研究

给出了随机事元的拓展概率以及随机事元可拓集的概念.运用可拓集合、可拓变换与可拓推理等可拓学的理论与方法,对随机事件发生的概率与随机变量概率分布的变化作了初步的拓展研究.     

关于极大$S^2NS$阵的一个注记

一个实方阵A称为是S2NS阵,若所有与A有相同符号模式的矩阵均可逆,且它们的逆矩阵的符号模式都相同．若A是S2NS阵且A中任意一个零元换为任意非零元后所得的矩阵都不是S2NS阵,则称A是极大S2NS阵．设所有n阶极大S2NS阵的非零元个数所成之集合为S(n),Z4(n)={1/2n(n-1) 4,…,1/2n(n 1)-1},除了2n 1到3n一4间的一段和Z4(n)外,S(n)得到了完全确定．本文将用图论方法证明Z4(n)∩S(n)=(?)．     

一类数列极限的概率方法求解

数列√2πn（a＋1）[n（a＋1）]^na＋ne^-n（a＋1）/Г（na＋n＋1）的极限仅靠基础方法很难求得，但利用Г-分布、中心极限定理、矩收敛问题等概率方法可完成该极限求解．     

n元集合的子集有多少个？

<正>大数据时代,计数问题在学习和生活中处处可见.本文讨论一个计数问题,源自于高中数学《选修》2-3第一章,探究与发现.问题n元集合的A={a_1,a_2,…,a}子集有多少个?1利用乘法原理探究一要得到集合A的一个子集T,可以分个步骤:第1步,考察a_1是否属于T,由集合元素     

Matchings in superpositions of (n,n)-bipartite trees

Pick two trees from among all “bipartite trees” with a fixed (n, n) two-coloring. We estimate the probability that the superposition of these two trees contains a perfect matching. As n →∞, this probability approaches 1. © 1994 John Wiley & Sons, Inc.     

Changes of leadership in a random graph process

Let be a random graph process in which in each step we add to a graph a new edge, chosen at random from all available pairs. Define the leader of G(n, M) as either the unique largest component or, if G(n, M) contains many components of the maximum size, the one from the largest components which emerged first during the process. We show that the longest period between two changes of leaders in the random graph process is, with probability tending to 1 as n →∞, of the order of n/log log n/log n. © 1994 John Wiley & Sons, Inc.     

Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

A family of probability measures on the unit ball in generates a family of generalized Steiner (GS-)points for every convex compact set in . Such a “rich” family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.      

單葉函數的係數(Ⅱ)

<正> §1.引言 在[1]文中,作者曾經對於單葉函數的係數進行討論,在本文中,將繼續之.在本文中所用的記號與[1]相同,不再一一重複定義,例如:記在單位圓|z|<1中k次對稱正则單葉函數的全體為S_k,特別記S_1,=S等等.     

The Bunce-Deddens algebras as crossed products by partial automorphisms

We describe both the Bunce-DeddensC ^*-algebras and their Toeplitz versions, as crossed products of commutativeC ^*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy of the set of natural numbers , fitted together in such a way that is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on . From this we deduce, by taking quotients, that the Bunce-DeddensC ^*-algebras are isomorphic to the (classical) crossed product of the algebra of continuous functions on the Cantor set by the odometer map.     

人口方程的总和生育率的控制

研究了年龄递进的人口分布参数控制方程,当控制变量β(t)发生改变时,可以找到两个阶梯函数βn(t)和βn(t)同时逼近β(t),并得出当n→∞时,Pn(r,t)和Pn(r,t)也同时逼近P(r,t).     

复lp（Cn）空间中单位球的体积及其渐近性质

令Bp（Cn）={x∈Cn｜｜x｜｜p≤1}为n维复lp^n空间中的单位球,1≤p≤＋∞,主要得到其体积公式,并讨论当n→∞,p→∞时其体积的某些渐近性质.     

正拼挤流形的F-调和映射

设M~n(n≥3)是R~(n+1)中紧致凸超曲面,本文证明了:若F″≤0且M的n个主曲率λ_i满足0<λ_i<1/2∑_(j=1)~nλ_j,则M~n和任何紧致黎曼流形之间的稳定F-调和映射必为常值映射.     

基样条插值的极限性质

本文证明了当m→∞时‖s^(k)mf-f^(k)‖p→0（1＜p＜∞,k=0,1,2）的充要条件是f∈Bπ,p,其中Bπ,p＝Bπ∩Lp(R),Bπ表示指数π型的整函数在R上限制为有界函数所构成的集合,而Smf是在整数点对f 插值的唯一确定的m-1次基样条,进而得一了函数类Bπ,p的三个等价的特征刻划。     

Joint distributions of runs in a sequence of higher-order two-state Markov trials

Consider a time homogeneous {0, 1}-valued m-dependent Markov chain . In this paper, we study the joint probability distribution of number of 0-runs of length and number of 1-runs of length in n trials. We study the joint distributions based on five popular counting schemes of runs. The main tool used to obtain the probability generating function of the joint distribution is the conditional probability generating function method. Further a compact method for the evaluation of exact joint distribution is developed. For higher-order two-state Markov chain, these joint distributions are new in the literature of distributions of run statistics. We use these distributions to derive some waiting time distributions.