子列在研究极限方面的应用

给出了子列的几个重要性质,应用这些性质巧妙地给出致密性定理及数列收敛的柯西准则的证明,并利用子列讨论数列的发散问题.     

非线性方程迭代法近似求根及程序实现

迭代法就是用某种收敛于所给问题的精确解的极限过程来逐步逼近的一种计算方法，利用该方法可以用有限个步骤算出精确解的具有指定精度的近似解。由于计算机的普遍使用，迭代法的应用更为广泛。本文对几种迭代法求取非线性方程根的算法及实现作了对比介绍。     

斯铎兹(stolz)定理的推广与应用

斯铎兹定理的推广是联系斯铎兹定理与罗必达法则的重要桥梁。本首先给出了斯铎兹定理的推广并证明之，在此基础上证明了斯铎兹定理和罗必达法则，以及斯铎兹定理推广的其他应用。     

一类次线性三点边值问题正解的存在性

利用Krasnosel’skii不动点定理研究了一类次线性二阶非线性常微分方程三点边值问题正解的存在性问题,得到了正解存在的一个充分条件。     

计算非线性问题的分歧解

对一类二阶非线性微分方程的正解进行了进一步的研究,考虑正解为退化的奇异解时,利用非线性泛函分析知识和分歧理论方法,首次推导论证了一个积分公式E(α),并得出它的具体数值符号可以判断解曲线在相平面(λ,u)中的转向情况.     

Algebraic methods in plane geometry

In this article we examine the role of mappings in elementary geometry. After making some comments about the Erlangen programme initiated by Felix Klein in 1872, wherein he proposed a way of studying geometries based on the underlying transformation groups, we see how theorems like Von Aubel’s theorem and Napoleon’s theorem can be proved in an elegant manner using similarity mappings, and how some construction problems may be solved using isometries. At the end we present a recent proof by Alain Connes of “Morley’s Miracle”, based on affine transformations. Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject.     

The prime ordeal

Prime numbers have fascinated mankind through the ages. In fact, one may think that we know all about them. However, this is not so! One does not know the answers to many basic questions on primes. We shall concentrate here mainly on questions and discoveries whose statements are elementary and accessible. Right at the end, we mention a result whose statement is simple but whose proof uses rather sophisticated mathematics. Even here, we do not try to be exhaustive. The subject is too vast for that to be possible. After a long stint (1981–1999) at TIFR, Bombay, B Sury moved to ISI, Bangalore.     

替换定理的两个证明

替换定理是高等代数 (线性代数 )中向量空间理论的一个十分重要的定理 ,文章给出了两个新证明。     

拓扑空间中的KKM型定理及其应用

在具有性质（H）的拓扑空间中引入了广义RKKM映射的概念．给出了一些非空交定理和不动点定理,证明了一个极大极小不等式,并进行了推广．     

数学期望在高等数学教学中的应用

本文讨论了数学期望在高等数学教学过程中几个方面的应用,为学习高等数学提供了概率背景.首先,给出了积分第一中值定理以及推广的积分第一中值定理的概率解释.接着,证明了平面薄板的重心坐标实际上可以看作二维随机变量的数学期望.最后,用数学期望重新给出了凸、凹函数的定义以及詹森(Jensen)不等式.