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.     

对工科概率论教材的看法和我们的改革尝试

<正> 1 对工科概率论教材的看法在从事工科概率论教学的实践中觉得目前国内不少工科(不是理科)概论率教材在基本概念上存在着以下三方面的问题: 1.关于概率空间和概率公理化定义。不少工科概率论教材既不讲概率空间,也不讲概率空间中的事件域(σ代数),因而产生了以下两方面的问题:一方面因为概率空间是概率论最     

浅析数学文化引导下的概率论与数理统计教学

数学文化是人类物质文明产物,也是精神文明产物,它来源于数学的形成过程,又影响数学的发展;它是数学的灵魂,将原本生硬的数学教学过程变得明晰、简洁、充满人文气息.本文通过对文化、数学文化及概率论与数理统计教学的诠释,从人类物质文明产物和精神文明产物两个方面系统阐述数学文化的特点;通过对工科概率论与数理统计教学特点的审视,阐述数学文化在概率与数理统计教学中的必要性,讨论如何在工科专业学开展数学文化引导下的概率与数理统计教学及其发展趋势.     

随机事件与概率的教学研究

随机事件与概率是概率论的入门知识,必须牢固掌握,才能为概率论的学习铺平道路。由于概率论是研究随机现象的统计规律的科学。它是数学中最活跃的分支,与描述确定性现象的数学相比,显然有它的特点。因此,我们在教     

由条件概率看概念教学在学生素养培养中的重要作用北大核心

1新增条件概率的背景分析 条件概率是概率论中一个非常重要的的概念,概率研究和生产实践中很多问题都涉及条件概率．在普通高中数学“课标教材”中（人教社新课标教材A版·普通高中课程标准实验教科书,下同）,条件概率属新增内容,从知识形成的顺序结构和逻辑层面上分析,它上联古典概型、几何概型,涉及事件、事件空间、事件条件、事件的关系,下联积事件概率、独立重复试验、二项分布,起着承上启下的作用,是与概率概念的综合运用．     

用概率模型解基础数学中问题的方法

利用概率计算公式和性质以及均值性质等概率论的方法,可以来解决基础数学中的一些问题。     

浅谈概率论与数理统计的教学

概率论与数理统计跟其它的数学分支课程相比,有其特殊的思维模式．本文主要从激发学生学习兴趣、平行概念类比教学、锻炼概率思维,N重视“辨误”数学四个方面阐述了如何搞好概率统计课的教学．     

《概率论》教学基本要求(讨论稿)

<正> 概率论是研究随机现象客观规律性的数学学科,是高等工业学校中一门重要的基础课。通过本课程的教学,使学生掌握概率论的基本概念,了解它的基本理论和方法。从而,使学生初步掌握处理随机现象的基本思想和方法,培养运用概率论的方法去分析和解决实际问题的能力。一、随机事件与概率     

第四届全国概率论会议

中国概率统计学会于1984年12月3日至8日在长沙举行了第四届全国概率论学术会议。这是继“中日统计讨论会”和“时间序列分析学术会议”之后,概率统计界在1984年召开的又一全国性大型学术活动。165名代表参加了会议,提供了245篇论文,在大会和分组会报告了150篇,显示了我国概率论队伍近几年来的成长和壮大。从向大会提供的论文和综合报告来看,近几年来通过与国外的学术交流,包括邀请专家讲     

P. L. Chebyshev的概率新思想溯源

切贝舍夫是俄国十九世纪最伟大的数学家之一,他第一个主张概率论的极限定理应该严格的证明,并尽可能精确地确定偏离极限的估计量,这在方法论上引起了很大变化.其创立的切贝舍夫不等式和切贝舍夫大数定律是概率论极限理论的基础,从而使概率论成为严密数学分支.切贝舍夫的概率思想是在一定数学文化背景产生的,尤其是法国数学文化对其发展的形成有着深刻影响.     

Quantum Probability: New Perspectives for the Laws of Chance

The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.     

Energy discriminant analysis, quantum logic, and fuzzy sets

In this paper, we show that quantum logic of linear subspaces can be used for recognition of random signals by a Bayesian energy discriminant classifier. The energy distribution on linear subspaces is described by the correlation matrix of the probability distribution. We show that the correlation matrix corresponds to von Neumann density matrix in quantum theory. We suggest the interpretation of quantum logic as a fuzzy logic of fuzzy sets. The use of quantum logic for recognition is based on the fact that the probability distribution of each class lies approximately in a lower-dimensional subspace of feature space. We offer the interpretation of discriminant functions as membership functions of fuzzy sets. Also, we offer the quality functional for optimal choice of discriminant functions for recognition from some class of discriminant functions.     

Econophysics and Quantum Statistics

In this paper, we associate notions of entropy, temperature, free energy, and Hamiltonian occuring in problems in probability theory with a system of identical objects so that modern methods of quantum statistics can be applied to problems in mathematical finance.     

Statistics and probability theory in renewable energy: Teaching and research

In this paper, the key-role and utility of statistics and probability theory in the field of renewable energy are emphasized and illustrated via specific examples. It is demonstrated that renewable energy is a very suitable field to effectively teach and implement many statistical and probabilistic concepts and techniques. From a research point of view, statistical and probabilistic methods have been successfully employed in evaluating renewable energy systems. These methods will continue to be of core interest for the renewable energy sector in the future, as new and more complex renewable energy systems are developed and installed. In this context, some future research directions in relation to the evaluation of renewable energy systems are also presented.     

An application of bilinear integration to quantum scattering

Scattering theory has its origin in Quantum Mechanics. From the mathematical point of view it can be considered as a part of perturbation theory of self-adjoint operators on the absolutely continuous spectrum. In this work we deal with the passage from the time-dependent formalism to the stationary state scattering theory. This problem involves applying Fubini's Theorem to a spectral measure integral and a Lebesgue integral of functions that take values in spaces of operators. In our approach, we use bilinear integration in a tensor product of spaces of operators with suitable topologies and generalize the results previously stated in the literature.     

A probabilistic model for the Solar System planetary distribution

The discovery of several extrasolar systems, each one characterized by its own planetary distribution around a central star, made the scientific interest addressed to the analysis of models permitting to predict, or at least estimate, the orbital features of the extrasolar planets. The main purpose of this work is to describe a mathematical model, inspired by quantum mechanics, able to provide a probability distribution of planets placing in a star system, mainly driven by the central star mass. More in detail, for any given eigenvalue of the model discrete spectrum, a distinct probability distribution with respect to the central star distance can be built. As per the Solar System, it has been possible to prove that both inner and outer planets belongs to two different spectral sequences, each one originated by the minimum angular momentum owned by silicate/carbonate and icy planetesimals respectively. In both sequences, the peak of the probability distributions almost precisely coincided with the average planets distance from Sun; furthermore, the eigenvalue spectrum of the inner planets thickens in an accumulation point corresponding to the asteroids belt, thus showing a striking similarity to the real matter distribution in the Solar System. From this point of view, the Titius–Bode law for the Solar System planets distribution is nothing but an exponential interpolation of the eigenvalues of both inner and outer sequences.     

Using the wigner function for quantum transport in device simulation

The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the cost of dealing with generalized functions in place of proper functions.     

谈谈概率论与其他学科的若干交叉

近一二十年以来，概率论获得了很大发展，特别是与其他学科交叉融合，形成了一些新的学科分支和学科生长点．我们首先从2002年国际数学家大会（ICM2002）所反映的情况予以说明．作为这种交融的一个侧面，也概述我们研究群体的三项成果．最后介绍取得这些成果的一种数学工具及其与线性规划和非线性偏微分方程等学科的联系．     

Quantum stochastic calculus

An introduction to quantum stochastic calculus in symmetric Fock spaces from the point of view of the theory of stochastic processes. Among the topics discussed are the quantum Itô formula, applications to probability representation of solutions of differential equations, extensions of dynamical semigroups. New algebraic expressions are given for the chronologically ordered exponential functions generated by stochastic semigroups in classical probability theory.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 3–28, 1990.     

An analogue of Hunt's representation theorem in quantum probability

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of, instruments on groups and the associated semigroups of probability operators, which now are defined on spaces of functions with values in a von Neumann algebra. We consider a semigroup of probability operators with a continuity property weaker than uniform continuity, and we succeed in characterizing its infinitesimal generator under the additional hypothesis that twice differentiable functions belong to the domain of the generator. Such hypothesis can be proved in some particular cases. In this way a partial quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. Our result provides also a closed characterization of generators of a new class of not norm continuous quantum dynamical semigroups.