微分中值定理的应用

介绍了使用罗尔定理和拉格朗日中值定理的一些常见方法，讨论了它们在证明存在性问题、判定级数的敛散性、证明不等式和求极限等方面的应用，并给出一些推论．     

Lagrange定理的逆问题的一些探讨

总结了有限交换群和一般有限群情形下Lagrange定理的逆问题的探讨和结论,并分别给出了例子。特别地,对于有限非交换群情形下,Lagrange定理的逆问题一般有否定答案的情况给出例子——24阶置换群S4的所有偶置换的集形成的4次交代群不存在6阶子群来加以说明。     

How Large is a Large Sample?

When teaching students about the Central Limit Theorem, it is difficult to give simple guidelines about when its use is justified. We demonstrate that the sample size needed depends to a great extent on the parent distribution and propose a method, based on the use of microcomputers, for helping students to appreciate the theorem and when it can safely be used.     

RSA在高性能DSP上实现的算法分析及优化

RSA是公开密码体系中应用最广泛的算法，通过对算法结构的分析，运用中国剩余定理、M-ary乘方以及Montgomery约化等方法对RSA的运算过程进行了重新设计，提出了一种减少预计算数的M-ary算法；并结合TI公司TMS320C6201高性能DSP的流水特点，对模幂运算中用到最频繁的模平方运算进行了优化，提高了计算的速度。     

浅谈构造函数法在高等数学解题中的应用

构造适当地辅助函数，利用函数的单调性、定积分的定义、罗尔定理解决高等数学中不等式、等式的证明、方程根的讨论等数学问题。     

Unpacking Rouché’s Theorem

Abstract

Rouché’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouché’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouché’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.     

拉氏中值定理的两种推广形式及其统一结构

文章对拉格朗日中值定理的推广形式——高阶拉格朗日中值定理提出了另一种证法,并提出了从中间点的个数上推广的拉朗日中值定理及从阶数和中间点的个数上同时推广的拉格朗日中值定理。     

具p-Laplacian算子二阶多点边值问题正解的存在性

通过利用锥上,不动点定理研究一类具p-Laplacian算子二阶微分方程多点边值问题正解的存在性,得到了正解的存在性定理。     

Mathematical derivation of the impact factor distribution

Experimental data [Mansilla, R., Köppen, E., Cocho, G., & Miramontes, P. (2007). On the behavior of journal impact factor rank-order distribution. Journal of Informetrics, 1(2), 155–160] reveal that, if one ranks a set of journals (e.g. in a field) in decreasing order of their impact factors, the rank distribution of the logarithm of these impact factors has a typical S-shape: first a convex decrease, followed by a concave decrease. In this paper we give a mathematical formula for this distribution and explain the S-shape. Also the experimentally found smaller convex part and larger concave part is explained. If one studies the rank distribution of the impact factors themselves, we now prove that we have the same S-shape but with inflection point in μ, the average of the impact factors. These distributions are valid for any type of impact factor (any publication period and any citation period). They are even valid for any sample average rank distribution.     

中国剩余定理与同余式组

“中国剩余定理”是初等数论中一个很重要的定理,同时在抽象代数中占有很重要的地位。最近,匡正从组合学的角度给出了两个模的情形下的“中国剩余定理”一个证明。作者利用这个方法证明了一般情形下（即k（k≥3）个模的情形）的“中国剩余定理”,同时给出了一次同余式组的一种较为简捷易懂的解法。