两类解析函数族的一些性质

研究两类解析函数族的性质，推广了Chichra[4]、Mocanu[5]和Obradovic[6]的一些结果。另外，也指出了S.Owa[8]的一个错误。     

R^n中Banaeh值函数的MeShane积分（I）

R．A．Gordon在[1]中定义了从R^1到Banach空间抽象函数的McShane积分，证明了当X不含C0时，如果，在[a，b]上MeShane f可积，则在[a．b]上Pettis可积．在这篇文章中，我们定义了从R^n到Banaach空间抽象函数的Mcshane积分。证明了f McShane可积，则f是Pettis可积．于是我们推广了[1]的结果。     

过一点作三次函数图像切线条数的完备结论

文[1]回答了过哪些点可以作三次函数图像的三条切线．受文[1]启发，一个自然的问题是：过哪些点可以作三次函数Y图像的一条切线、两条切线？本文在文[1]的基础上给出过一点所作三次函数图像切线条数的完备结论．     

一类非均匀分形插值函数的可微性

本文研究一类分形插值函数的可微性问题，通过构造一迭代函数系，利用迭代函数系的唯一吸引子。给出了一类分形插值函数。并获得了此类分形插值函数在[0，1]区间上几乎处处可微和在[0，1]区间上某一点不可微判定的充分条件，推广了文献[2]的结论。     

截断样本下的强率函数和密度函数核估计的渐进正态性

本文用[1]发展的计数过程去研究截断样本下强率函数核估计的渐进正态性．在弱于[7]和[10]的条件下，得到了更一般的结果．接着我们将这种方法运用到密度函数核估计，在较弱的条件下，得到了截断样本下密度函数核估计的渐进正态性．     

再谈配方法求一类二元函数的最值

文[1]、文[2]运用配方法求形如g（x，y）=ax^2＋bxy＋cy^2＋dx＋ey＋f的二元函数最值，配方成两个一次式的平方和加上一个常数的形式，美中不足的是，文[1]、文[2]所举范例配方中的两个一次式均出现了分式或分数，这就加大了配凑系数的难度，不够自然流畅．     

含有解析函数与亚纯函数的一些不等式

指出文献 [2 ]和文献 [4]中的错误 ,并且得到含有单位圆盘内接解析函数与亚纯函数的几个不等式 .     

高维频谱有限函数的外推

在实轴 R~1上的限制.因此若已知 f(t)在 t∈[-T,T]上的值,则 f(t)在[-T,T]外的值就被唯一确定了.作为猜测,J.L.C.Sanz 和 T.S.Huang 于1983年提出了一种外推 f(t)在[-T,T]外的值的方法(见[1]).文献[2]和[3]从不同角度证实了这     

B值随机元和的完全收敛性

在B值随机元随机有界于一非负随机变量的情况下．讨论了B值独立随机元序列非随机足标和的完全收敛性，作为应用，得到了随机足标和完全收敛性的相应结果，将[5]中的一些结果推广到B值独立随机无情形，同时使[3]（d=1），[4]，[6]的相应结果成为特例．     

有关三次函数图像与正方形关系的两个结论

众所周知，任一三次函数的图像都有唯一的对称中心（参见文[1]-[3]）．与此相关的两个有趣问题是：以三次函数图像的对称中心为中心，且四个顶点都在此三次函数图像上的正方形是否存在？若存在，其个数如何？本文将圆满解决这两个问题．     

随机模糊集与随机集

本文研究了三个方面的工作：一是定义了一种模糊集上的可测结构，从而定义了随机模糊集，这些定义都与论域Ｘ上的拓扑结构无关。将通常意义下的集合看成特殊模糊集得到的通常集合上的超可测结构与文（３）中的定义一致；二是给出了随机模糊集、随机集的一些等价条件；三是研究了随机模糊集、随机集的分布与其有限维落影族的关系。     

Calculation of the Distance betweeen Two Random Sets

Two independent random sets are formed by finite systems of independent random variables uniformly distributed on the unit circle. A formula for the distribution of the distance between the sets is derived.     

On a Class of Random Variational Inequalities on Random Sets

We study a class of random variational inequalities on random sets and give measurability, existence, and uniqueness results in a Hilbert space setting. In the special case where the random and the deterministic variables are separated, we present a discretization technique based on averaging and truncation, prove a Mosco convergence result for the feasible random set, and establish norm convergence of the approximation procedure.     

Laws of Large Numbers for Exchangeable Random Sets in Kuratowski-Mosco Sense

Abstract

Most of the results for laws of large numbers based on Banach space valued random sets assume that the sets are independent and identically distributed (IID) and compact, in which Rådström embedding or the refined method for collection of compact and convex subsets of a Banach space plays an important role. In this paper, exchangeability among random sets as a dependency, instead of IID, is assumed in obtaining strong laws of large numbers, since some kind of dependency of random variables may be often required for many statistical analyses. Also, the Hausdorff convergence usually used is replaced by another topology, Kuratowski-Mosco convergence. Thus, we prove strong laws of large numbers for exchangeable random sets in Kuratowski-Mosco convergence, without assuming the sets are compact, which is weaker than Hausdorff sense.     

Topologies of Nodal Sets of Random Band-Limited Functions

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band-limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular, the results apply to random monochromatic waves and to random real algebraic hypersurfaces in projective space. © 2018 Wiley Periodicals, Inc.     

随机模糊集及其落影大数定理

本文给出随机模糊集的定义,并证明了随机模糊集的落影大数定理,为模糊集值统计提供了理论依据。     

On the Geometric Densities of Random Closed Sets

Abstract

In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper we suggest to cope with these difficulties by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. In this last one we analyze mean generalized densities, and relate them to densities of the expected values of the relevant measures. Many models of interest in material science and in biomedicine are based on time dependent random closed sets, as the ones describing the evolution of (possibly space and time inhomogeneous) growth processes; in such a situation, the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the local relevant kinetic parameters of birth and growth. In this context connections with the concepts of hazard function, and spherical contact distribution function are offered.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

Countable Random Sets: Uniqueness in Law and Constructiveness

The first part of this article deals with theorems on uniqueness in law for σ-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: first, the study of generators for σ-fields used in this context and, secondly, the analysis of hitting functions. The last section of this paper deals with the notion of constructiveness. We prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.     

关于随机紧凸集的更新定理

该文在讨论了多维更新定理的基础上,重点研究了随机紧凸集的Minkowski和的更新定理,得到了一系列重要结论.