函数振幅的性质与应用

函数振幅是微积分中最基本的概念之一，它的性态决定了函数许多解析性质．本文中扩充了函数振幅的定义．并利用它定义了函数振动性强弱的概念．作为这一概念的应用，建立起判定函数连续性、一致连续性、可积性、有界变差性的比较判别法，并给出了一些应用实例．     

问题驱动下的微分与全微分教学探索

微分概念是高等数学中的核心概念,通过问题教学法和几何图形演示,将一元函数的微分教学和二元函数的全微分教学统一起来,使学生更深刻理解微分这一概念的核心本质,并认识到从一元函数到多元函数许多结论要发生变化,量变引发质变的变化过程.     

关于初等函数

0引言初等函数是数学中的一个基本概念,但围绕这一概念的讨论始终没有停止过.究其原因就是对初等函数的概念认识还不够.甚至有些人对初等函数不很恰当的理解导致了一些不很恰当的结论.特别是对分段函数、积分上限函数的认识更是参差不齐.     

发挥学生的主体性 优化课堂教学——对《函数的奇偶性》的教学案例分析

在江苏省高中数学学科骨干教师提高培训活动中,听了一位老师的“函数的奇偶性”的课后,对“函数的奇偶性”这一概念的引入、概念的形成及学生主体地位的体现颇有感触,现将课堂内容整理如下．     

我是怎样讲术数列的极限的

极限概念是数学中重要概念之一,如果学生在中学阶段透彻地领会了这一概念,不仅为将来进入高等学校学习函数极限时奠定了良好的基础,同时还能够使学生更加自觉的接受与极限有关的中学教材(如圆周长、圆面积、旋转体的表面积及体积求法等等).由经验可知,几乎所有的教师在为如何使学生明白地接受这一概念上都感到很大的困     

浅谈复合函数的教学

在现行教材中,复合函数的概念是安排在高三《微积分初步》中才作正式介绍.由于这部分内容属于较高要求,目前多数学校已不予讲授,这样就使学生在整个高中阶段未能学到这一概念,因而造成对函数种类的认识模糊不清及滥用基木初等函数的性质去讨论有关复合函数的问题等错误.例如学生误将函数y=2~(1/x)称为指数函数,并认为它在定义域内是增函数等等.另一方面,在高一代数课本中有     

二重反常积分概念的学习与教学

重述二重反常积分的概念,采用具体例题解释这一概念的本质;建立二重反常积分在一定条件下的计算方法,并举例说明其应用..     

分部积分法的巧用

分部积分法作为积分学的基本方法之一有着重要的作用,它不但解决了许多常见的积分问题,而且在有些情况下可以发挥意想不到的效果．本文将结合例子来说明分部积分法在改善被积函数的性质、判别广义积分的致散性及证明积分不等式方面的巧用．分析该题由于被积函数在点不连续,因此不能直接应用对积分上限求导的公式,这里将用分部积分法将被积函数改善成连续的,从而使问题得到解决．由于是的可去间断点,故只须补充定义则在连续数在x＝0处可导且导数为零（可根据定义）,故有例2证明广义积分因为所以绝对收敛,因此广义积分因为所以绝对收…     

双连续n次积分C余弦函数的逼近定理

基于双连续半群概念,引入一致双连续半群序列概念,借助Laplace变换和Trotter-Kato定理,考察双连续n次积分C余弦函数与C-预解式之间的关系,得到逼近定理的稳定性条件,进而得出双连续n次积分C余弦函数逼近定理.从而对Banach空间强连续半群逼近定理和双连续半群逼近定理进行了推广,为相应抽象的Cauchy问题提供了解决方案.     

模糊积分变换与模糊Choquet积分的一致连续性

在一般非负单调函数空间 m[0 ,a]上引入模糊积分变换与距离的概念 ,证明了这种模糊积分变换与模糊 Choquet积分在 m[0 ,a]上关于这种距离是一致连续的 ,从而说明当 m[0 ,a]上两个函数变化不大时 ,不会使相应的模糊积分变换与模糊 Choquet积分产生较大的变化 .     

QUASI-MODULUS OF CONTINUITY AND BEST APPROXIMATIONS

In this paper a concept of the quasi-modulus of continuity of the functions of L^2(T) on the unit circle T is introduced, and the relations between it and modulus of continuity are discussed. And we give a sufficient condition such that the best uniform approxi marion of the continuous functions on T is continuous.     

Effective continuities on effective topological spaces

We extend a notion of effective continuity due to Mori, Tsujii and Yasugi to real-valued functions on effective topological spaces. Under reasonable assumptions, Type-2 computability of these functions is characterized as sequential computability and the effective continuity. We investigate effective uniform topological spaces with a separating set, and adapt the above result under some assumptions. It is also proved that effective local uniform continuity implies effective continuity under the same assumptions.     

Uniform operator continuity of the stationary Riccati equation in Hilbert space

In this paper the continuity in the uniform operator topology of the solution of the stationary Riccati equation in Hilbert space as a function of parameters is verified. The assumptions for this verification are the uniform operator continuity of the uncontrolled semigroup with respect to parameters, the uniform finiteness of the infimum of the quadratic cost functionals over the admissible controls, and uniform detectability. Some families of semigroups are described that satisfy the condition of continuity in the uniform operator topology with respect to parameters. The uniform operator continuity of the solution of the stationary Riccati equation with respect to parameters is important for applications to problems in adaptive control of stochastic evolution systems.This research was partially supported by NSF Grant ECS-8718026.     

On the properties of sequences of fuzzy-valued Choquet integrable functions

In this paper, we focus on investigating the properties of sequences of fuzzy-valued Choquet (for short, (C)-) integrable functions. Firstly, the concept of uniform (C)-integrabiliy and other new concepts like uniform absolute continuity and uniform boundedness for sequences of fuzzy-valued (C)-integrable functions are introduced and then the relations among them are discussed. As the applications of these concepts, we also present several convergence theorems for sequences of fuzzy-valued (C)-integrable functions by using uniform (C)-integrability.     

Fragmentability and Continuity of Semigroup Actions

fragmentability in the sense of Jayne and Rogers and its natural uniform generalizations play a major role in this paper. Our applications show that problems concerning the continuity of induced actions have satisfactory solutions for Asplund Banach spaces X (without additional restrictions, if S is a topological group) and, moreover, for a new locally convex version of Asplund spaces introduced in the paper. The starting point of this concept was the characterization of Asplund spaces due to Namoika and Phelps in terms of fragmentability.     

Uniform continuity and growth of real continuous functions

The purpose of this paper is to understand whether there exists any link between the uniform continuity of a real function defined on an unbounded interval and its growth at infinity. The primary objective is to present some results from teaching experience which help in the comprehension of this notion and yield some classroom techniques. It is well known that a uniformly continuous function has a monomial growth; it will be proved that there does not exist another growth of positive order. After introducing three kinds of growth, some results are recalled in connection with the behaviour near infinity of a uniformly continuous function. Using a series of counterexamples, it is shown that the uniform continuity of a function cannot be described by its asymptotic behaviour near infinity. Finally, some useful properties of the averaging convergence are reviewed, and how this is related to uniform continuity is investigated.     

Strong continuity implies uniform sequential continuity

Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic.     

Bolzano and uniform continuity

It has often been thought that the distinction between pointwise and uniform continuity was a relatively late arrival to real analysis, due to the mathematicians associated with Weierstrass. In this note, it is argued that Bolzano, in his work on real function theory dating from the 1830s, had grasped the distinction and stated two key theorems concerning uniform continuity.     

Uniform continuity on unbounded intervals

We present a teaching approach to uniform continuity on unbounded intervals which, hopefully, may help to meet the following pedagogical objectives:

1. To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous;

2. To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions.

Assembling some well-known facts and refining the resulting statement, we establish a useful asymptotic coincidence test for the uniform continuity on unbounded intervals. That test is the core of the present note and yields an easily applicable technique. In particular, one of its immediate consequences is the elementary fact that continuity and existence of horizontal or oblique asymptotes imply uniform continuity.     

A shape optimization approach for a class of free boundary problems of Bernoulli type

We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincaré inequality for variable domains.