半圆区域的包含测度

本文研究了半圆区域内定长线段的包含测度问题.利用广义支撑函数和限弦函数,得到了这类区域的包含测度.关于非多边形凸域的包含测度的现有结果不多,本文的结果丰富了凸体包含测度的具体结果.     

正测度Cantor集与正测度Cantor函数

本文利用类比的方法,将Cantor集上定义的Cantor函数进行了推广.首先给出了正测度Cantor集及正测度Cantor函数的定义;然后通过严格的证明得到了正测度Cantor函数的一些性质,并给出了正测度Cantor函数的一些应用;最后通过实例说明,由于正测度Cantor函数构造的特殊性,可以用来作为一些命题的反例.     

凸域的外平行集的包含测度

本文研究了凸域内定长线段的包含测度问题,利用广义支持函数和限弦函数,得到了一类特殊凸域的包含测度.     

P-可测函数的构造性质

可测函数的构造性质是定义它关于测度μ的积分的理论基础.为了在P-测度空间上定义P-积分,借鉴可测函数的构造性质,引入了P-示性函数、P-简单函数、P-初等函数以及P-可测函数的概念,在此基础上系统地研究了P-实可测函数、有界P-实可测函数和非负P-可测函数与P-简单函数序列及P-初等函数序列的收敛关系;找出了P-实可测的充分必要条件;证明了实P-可测函数正部和负部都是非负P-实可测函数,最终得出任何P-实可测函数均可以表示为二非负P-可测函数之差,为定义P-积分提供了理论依据.     

乘积空间上 Carleson 测度的一点注记及其应用

本文通过对 Carleson 测度的进一步研究,讨论了极大函数及平方函数之分布函数不等式,进而给出了 Str(o|¨)mberg H_φ 空间的极大函数及平方函数特征.新方法的优点在于它不依赖于欧氏空间的平移及伸缩结构,可用于某些一般情形.     

P-可测空间及其可测函数

经典集合理论认为集合就是具有一定属性的对象所构成的整体,当一个普通集合的属性发生改变时,由此生成的新的集合称为P-集合.在测度空间上研究P-集合时所生成的新的空间称为P-测度空间.由于任何测度空间均可转化为概率空间,首先利用随机数的产生研究了随机P-集合的产生.然后借助P-可测空间提出了内P-可测映射、内P-可测函数和外P-可测映射、外P-可测函数及P-可测,给出了其有关性质.     

属性识别预警系统(AFS)——一种新的信用评价方法

在属性集和属性测度函数的基础上建立了属性测度预警系统.利用我国深、沪两交易所2000年公布的106家上市公司作为训练样本,计算出“差”企业和“好”企业的属性测度均值,构造出属性测度函数.进而对2001年公布的13家预亏公司进行预警分析,用置信度准则及最大属性测度准则判别,预警准确率分别达到92.31%和100%.因此,属性识别预警系统在信用评价方面具有良好的应用前景.     

关于Baernstein星函数

若函数g(θ)在区间[-π,π]上是可积的.称g在测度为2θ的各子集上的积分的最大值g^*(θ)为g的星函数.本文考察了星函数的某些性质.     

统计测度及它在概率空间的应用

设=2 N,一实值函数μ:→[0,1]称为统计测度.如果若A,B∈,A∩B=,则μ(A∪B)=μ(A)+μ(B)且μ(k)=0.本文得到了统计测度的表示定理及统计测度在概率空间的应用.     

μ—不变测度的Q—预解式刻划

用Q－预解式函数给出了μ－不变测度的另一种定义,并讨论了μ－不变测度几种定义之间的关系。     

凸域内矩形的运动测度

本文研究了凸域内矩形的运动测度,通过对凸域内定长线段运动测度的推广,建立了包含在凸域内且长、宽都确定的矩形运动测度的一般公式,利用此公式得到了圆域和矩形域内此类矩形的运动测度,并以此为基础得到了推广后的Buffon投针问题的一些结果.     

Attractors of nonlinear dynamical systems with a weakly monotone measure

This paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure. It is assumed that the measure of a set of points in the phase space is increasing along the flow. We prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions. For a system of autonomous ordinary differential equations, we present attractivity conditions in terms of the divergence with a density function. Unlike previous results in the literature, our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure. As an example, the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions.     

A Translative Integral Formula for Absolute Curvature Measures

Absolute curvature measures are certain positive variants of curvature measures which are defined by means of the kinematic measure of flats of a given dimension touching the examined body. A kinematic flat section formula for absolute curvature measures was proved by Baddeley and Rother and Zähle. We present here a translative version of this formula.     

A simple proof of the kinematic formula

We give a short proof of the kinematic fundamental formula in geometric probability theory, using a few basic properties of the Haar measure and some convexity.     

A generalized multiplicative directional distance function for efficiency measurement in DEA

For measuring technical efficiency relative to a log-linear technology, a generalized multiplicative directional distance function (GMDDF) is developed using the framework of multiplicative directional distance function (MDDF). Furthermore, a computational procedure is suggested for its estimation. The GMDDF serves as a comprehensive measure of efficiency in revealing Pareto-efficient targets as it accounts for all possible input and output slacks. This measure satisfies several desirable properties of an ideal efficiency measure such as strong monotonicity, unit invariance, translation invariance, and positive affine transformation invariance. This measure can be easily implemented in any standard DEA software and provides the decision makers with the option of specifying preferable direction vectors for incorporating their decision-making preferences. Finally, to demonstrate the ready applicability of our proposed measure, an illustrative empirical analysis is conducted based on real-life data set of 20 hardware computer companies in India.     

ON GENERALIZED BUFFON NEEDLE PROBLEM FOR LATTICES

Two new concepts, the generalized support function and restricted chord function, both referring to a convex set, were introduced in [1]. General formulae to yield the kinematic measure of a segment of fixed length in a convex set were established based on these concepts. In this article , using the partial intersection method, we consider the generalized Buffon problem for three kinds of lattices. We determine the probability of intersection of a body test needle of length l, l < a.     

Rényi’s residual entropy: A quantile approach

In the present paper, we introduce a quantile based Rényi’s entropy function and its residual version. We study certain properties and applications of the measure. Unlike the residual Rényi’s entropy function, the quantile version uniquely determines the distribution.     

A SUFFICIENT CONDITION FOR ONE CONVEX BODY CONTAINING ANOTHER

A sufficient condition for one convex body to be able to contain another in 3-dimensionalEuclidean space has been derived by estimating the kinematic measure of one ccnvexbody moving in another. The equality of estimating inequality holds if and only ii twoconvex bodies are congruent balls.     

Quantization dimension function and Gibbs measure associated with Moran set

In this paper we consider the Gibbs measure on the one-sided shift dynamical system and determine the quantization dimension function for the image measure supported on a Moran set. A relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is also established.