树指标m重Markov链转移矩阵的一个强极限定理

树模型近年来已引起物理学、概率论及信息论界的广泛兴趣.树指标随机过程已成为近年来发展起来的概率论的研究方向之一.在概率论的发展过程中,对强极限定理的研究一直占重要地位,强极限定理也一直是国际概率论界研究的中心课题之一.本文通过构造非负鞅,利用鞅论研究给出了非齐次树指标m重连续状态马氏链转移矩阵的一个强极限定理.     

概率论课程中蕴含的数学文化

数学文化是概率论课程的重要组成部分,因此在概率论课程教学中非常有必要展示所蕴含的数学文化,才能使概率论课程教学在培养学生的工程数学学习与应用技能的同时,又能形成良好的数学思维与素养.从概率论的发展历程、概率论的数学思想与方法,概率实际应用案例等几方面阐明概率论中蕴含的数学文化.概率论课程教学与数学文化的融合能够解读枯燥的概率知识,降低概率知识的抽象性,用数学文化自身的魅力吸引学生,培养学生的创新应用等综合素质.     

树指标马氏链关于泊松分布的一个强极限定理

树指标随机过程已成为近年来发展起来的概率论的研究方向之一.在概率论的发展过程中,对强极限定理的研究一直占重要地位,强极限定理也一直是国际概率论界研究的中心课题之一.本文通过引入滑动相对熵的概念和构造非负鞅,利用Doob鞅收敛定理研究给出了非齐次树指标马氏链关于泊松分布的一个强极限定理.     

树上非齐次马氏链转移矩阵的若干极限性质

树模型近年来已引起物理学、概率论及信息论界的广泛兴趣.树指标随机过程已成为发展起来的概率论的研究方向.在概率论的发展过程中,对强极限定理的研究一直占重要地位,强极限定理也一直是国际概率论界研究的中心课题之一.本文通过引入样本散度的概念,通过构造适当的非负鞅,将Doob收敛定理应用于几乎处处收敛的研究,给出了非齐次树上m重可列非齐次马氏链转移矩阵的若干极限性质.     

树指标马氏链的一个强极限定理

树指标随机过程已成为近年来发展起来的概率论的研究方向之一.强极限定理一直是国际概率论界研究的中心课题之一.研究给出了一类非齐次树上马氏链的一个强极限定理.     

《概率论与数理统计》中借助数学实验理解几个极限定理

立足于打破传统的教学模式,融入数学实验思想和方法,淡化严密形式,关注应用思维的数学教学指导思想.借助MATLAB设计实验,直观形象地展示《概率论与数理统计》中的泊松定理,棣莫弗—拉普拉斯中心极限定理等,从观念上、方法上解决了数学抽象性与人才培养特征之间的矛盾.     

杜布与随机过程

概述了数学家杜布的人生历程、分析其在概率理论方面的数学工作.重点阐明杜布选择数学领域概率论的背景、杜布建立鞅论的核心与意义,及其学术交往情况,借此理解杜布的概率论工作的意义与影响及20世纪中期的概率论与随机过程理论发展的状况.     

非齐次树上m重非齐次马氏链的若干强极限定理

树指标随机过程已成为近年来发展起来的概率论的研究方向之一.强极限定理一直是国际概率论界研究的中心课题之一.研究给出了一类非齐次树上m重非齐次马氏链的若干强极限定理.     

关于树指标可列非齐次马氏链的一个强极限定理

树指标随机过程已成为近年来发展起来的概率论的研究方向之一.强极限定理一直是国际概率论界研究的中心课题之一.利用Borel-Cantelli引理研究给出.了一个关于树指标可列非齐次马氏链的强极限定理.     

Z~2上相依渗流开簇的极限定理(英文)

中心极限定理, 大偏差定理和大数定律等极限定理在概率论中起着很重要的角色. 本文我们研究Z2上一类相依渗流模型. 对此模型, 我们不仅证明了其无穷开簇的存在唯一性, 而且得到了关于格点盒子类极大开簇的中心极限定理.     

Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation

In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

On a conjecture of D. B. Hunter

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare.     

Inequalities and Convergence Concepts of Fuzzy and Rough Variables

It is well-known that Markov inequality, Chebyshev inequality, Hölder's inequality, and Minkowski inequality are important and useful results in probability theory. This paper presents the analogous inequalities in fuzzy set theory and rough set theory. In addition, sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper presents four types of convergence concept of fuzzy/rough sequence: convergence almost surely, convergence in credibility/trust, convergence in mean, and convergence in distribution. Some mathematical properties of those new convergence concepts are also given.     

Logic and probabilistic systems

Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set of the sentences that get limit probability 1 is a theory, in fact can be a complete set. We prove incompleteness results for this setting, by showing for instance that for every there are true sentences that get limit probability less than . No set as above can contain the set of all true sentences, although we exhibit some containing all the true sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent. Received March 20, 1995     

Can the bounds in the multivariate Chebyshev inequality be attained?

Chebyshev’s inequality was recently extended to the multivariate case. In this paper we prove that the bounds in the multivariate Chebyshev’s inequality for random vectors can be attained in the limit. Hence, these bounds are the best possible bounds for this kind of regions.     

洛必达法则及斯铎兹定理的一种简便证法

针对∞/∞型的洛必达法则、数列极限∞/∞型的斯铎兹定理以及函数极限0/0型和∞/∞型的斯铎兹定理,分别建立相应的引理,为证明这些定理提供一种新的思路.对这些定理的传统证明是一种改进和补充.     

A generalized Chebyshev theory

In this paper we develop, without assuming the Haar condition, a generalized Chebyshev theory for Chebyshev approximation which is similar to the classical Chebyshev theory and contains it as a special case. The Project Supported by National Natural Science Foundation of China     

Approximation of a two-valued function by an algebraic polynomial

We consider the minimax model of a nonlinear structure for approximating a two-valued function by an algebraic polynomial. We establish optimality conditions as a strong generalization of P. L. Chebyshev alternance optimality conditions in approximation of a function by a polynomial.     

两类记录值之和的中心极限定理

该文给出了Ｌｏｇｉｓｔｉｃ分布纪录值序列部分和的中心极限定理；对于Ｐａｒｅｔｏ分布纪录值序列的部分和T_n，获得了ｌｎT_n的中心极限定理．这一工作不仅具有概率论的极限理论方面的研究价值，而且在金融、保险等领域也具有相当重要的应用前景．     

Chebyshev lattices,a unifying framework for cubature with Chebyshev weight function

We present a multivariate extension to Clenshaw-Curtis quadrature based on Sloan’s hyperinterpolation theory. At the centre of it, a cubature rule for integrals with Chebyshev weight function is needed. We introduce so called Chebyshev lattices as a generalising framework for the multitude of point sets that have been discussed in this context. This framework provides a uniform notation that extends easily to higher dimensions. In this paper we describe many known point sets as Chebyshev lattices.