关于勒贝格不可测集的再认识

利用勒贝格可测函数的本性化定理的性质,给出一个特征函数与狄里克莱函数有些相仿的勒贝格不可测集。     

关于勒贝格可测函数的再认识

由[a,b]上的勒贝格可测函数与几乎处处连续的函数几乎处处相等,给出勒贝格可测函数的等价定义及几个勒贝格可测函数的性质,提出一些关于勒贝格可测函数进入工科数学教学的设想。     

拉格朗日中值定理证明的辅助函数

讨论了拉格朗日中值定理证明中采用的各种辅助函数及其适用的教学对象,并为了使辅助函数的几 何意义更加明确,对其中的几何学角度出发建立的辅助函数作了改进。     

闭区间上有界可测函数的逼近定理(用多项式逼近)

本文借助于连续函数的多项式逼近与 L-可积函数的光滑化来处理有界可测函数的多项式逼近。特别是,它给出一系列有用的估计,按照这种估计,对这种函数,在几乎所有点处能给出一系列有用的估计,而这是以前不曾有过的。     

关于积分学基本公式的推广

积分学基本公式是计算定积分的一个重要公式，但它的使用条件较为苛刻，本文利用勒贝格测度，将定理的条件进行了改进，得到了定理3，并说明了定理3已不能再推广。     

两参数跳过程的Kolmogorov高阶微分方程组

给出了由三点转移函数P(s,t;x,y,z,A)产生的σ可加的集函数Ms;x,z(.)和可测函数fs;x,z(.),推出两者的两参数跳过程的Kolmogorov高阶微分方程组,由此得到了三点转移函数P(s,t;x,y,z,A)的Kolmogorov高阶微分方程组.     

关于本性函数积分定义的注

对照几乎处处连续的本性函数的积分定义,给出几乎处处连续的本性函数绝对可积的充要条件,证明几乎处处连续的本性函数积分是非绝对可积的,将无穷区间上几乎处处连续的本性函数的积分问题化为有穷区间的积分。     

有限覆盖定理与定积分充要条件的证明

利用函数间断度概念、有限覆盖定理结合定积分概念、达布和知识，证明了有界函数黎曼可积的充要条件为被积函数是几乎处处连续的。     

Helly定理的几何特征

线性赋范空间理论中的Ｈｅｌｌｙ定理的证明一般都是借助于深刻的开映射定理和线性泛函的Ｈａｈｎ—Ｂａｎａｃｈ扩张定理（几何形式）,而且,任意＞０的意义是不明确的．文中给出Ｈｅｌｌｙ定理一个简单、初等的证明（仅用到商空间的概念）．它具有显明的几何特征而且澄清了任意＞０的几何意义．     

关于Radom—Nikodym定理的证明

关于Radon—Nikodym定理,Van Neumann曾给出了一个精巧的证明,但不能应用到更一般的结果,本文根据Van Neumann的方法导出了关于一般结论的证明。 Van Neumann证明了:若u,v为σ—有限测度,满足vu,则存在f使得:对所有可测E,V(E)=∫_Efdu,本文推广这个结论,除掉v为σ—有限的限制. 引理设(X,B,U)为有限测度空间,v,u为测度满足vu,那么,X为不相交并YUZ使得在Y上,v为σ—有限,对每个EZ,U(E)>0有v(E)=∞。     

On the representation of linear system time‐varying half‐line maps

An expression is given for the most general continuous causal linear input–output map that takes the space of (Lebesgue) integrable functions into itself. This expression is a function‐space limit of an integral. As an application, a representation result is given for an important family of linear maps that take the space of bounded measurable functions into itself and satisfy a certain continuity condition. Some related material concerning engineering education is also given. Copyright © 2006 John Wiley & Sons, Ltd.     

The Convolution Theorem in Modern Analysis

In this paper, the convolution theorem is used as a fundamental tool to solve problems associated with frequency translation, amplitude modulation, sampling, and transformation of random variables. The four basic steps involved in the convolution of two functions are explained in detail using graphic techniques. It is concluded that the approach used makes possible the covering of a class of important problems at high level and in a short period of time.     

戴维南定理和诺顿定理在含受控源电路中的应用

介绍了戴维南定理和诺顿定理在含受控源电路中的应用 ,克服了两个定理在应用上的局限性 .     

Onsager's Theorem from Thermodynamics

The Onsager theorem pertains to symmetries inherent in two-dimensional current flow, through either a conducting or a semiconducting medium in the presence of an externally applied magnetic field oriented perpendicular to the current-carrying plane. Although the theorem can be derived on a statistical basis, as done originally by Onsager, in many courses dealing with thermoelectricity, it is necessary to state the theorem without proof, rather than take the time to develop the background properly. Therefore, it is of considerable pedagogical interest to have a derivation of the theorem based upon thermodynamics only. To that end, the following discussion has been found useful.     

Application of the Compensation Theorem in the Development of the Scattering Parameters

The compensation theorem relates the change in current when an impedance is added in a loop to the current that existed originally. It is shown that the voltage scattering variables may be defined in terms of the current change and the original current. The compensation theorem may then be used to determine the scattering matrix.     

The Application of the Compensation Theorem in the Proof of Thevenin's and Norton's Theorems

It has been found that, for linear networks, both Thevenin's and Norton's theorems are special cases of the compensation theorem. To illustrate the interdependence of these network theorems, Thevenin's theorem is proved using the compensation theorem. Norton's theorem can be derived by using a dual form of the compensation theorem.     

On the Presentation of Miller's Theorem

Miller's theorem is an important analysis tool. Its presentation in many introductory electronic circuits texts often leads to student misunderstandings with regard to its applicability. An alternative presentation that may improve students' understanding is suggested.     

On the Estimation of Frequency Response in Amplifiers Using Miller's Theorem

Conventional application of Miller's theorem in amplifiers often leaves a student with an erroneous impression that the approximation provides an incorrect estimate for the nondominant pole. The present paper clarifies that not only can a reasonable estimate for both dominant and nondominant poles be obtained through Miller's theorem but additional insight is offered into pole splitting, not afforded by conventional analysis.