On modified Runge-Kutta trees and methods

Modified Runge-Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge-Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested.     

在Mathematica环境下实现交互式矩阵初等变换

在Mathematica7.0环境下,实现了交互式矩阵初等变换。使用时只需选择矩阵初等变换的类型,并输入相应的变换参数,即可完成相应的初等变换并以可视化的方式显示矩阵初等变换的步骤及结果。在线性代数的教学中采用此交互式矩阵初等变换,可以在课堂中随时改变实例并进行演示来提高多媒体教学内容的灵活性。     

组合数学趣题的Mathematica算法

列举的几个组合数学难题,有的来自传统的计算机竞赛,有的经常出现在考验智力的杂志上。要用计算机来解决这些问题,传统上是通过C语言或Pascal语言编程来实现解题算法。通过使用Mathematica来实现这些算法,并对各个算法进行了比较和分析。     

利用Mathematica绘制二元函数图形的方法与技巧

介绍了Mathematica软件中二元函数图形绘制的基本方法,进而研究了曲面相交所围成的空间立体图形的绘制方法和技巧,利用Mathematica软件绘制出了牟合方盖、球面与圆柱面相交、马鞍面与平面相交的立体图形.     

1~5轮LBlock的多项式表示及完全性分析

用统计测试方法分析密码算法的完全性存在误差。为此,利用符号计算软件Mathematica 7.0,以明文和密钥比特为自变量,得到LBlock分组密码第1轮~第5轮输出的多项式表达式,结果显示,LBlock第5轮输出的任何比特至多与45个明文比特、49个密钥比特有关,说明5轮LBlock还未达到完全性。     

《高等数学B》课程建设的认识和实践

依据我校定位，结合国内高等职业院校数学课程改革情况，对我校文科专业的《高等数学B》课程建设与教学改革作了初步分析，论述了《高等数学B》教学改革的重要性，并相应进行了有益的实践探索，取得了一定的成效，提出了一些建设性建议。     

计算间接测量误差的Mathematica程序包设计

利用Mathematica的符号计算功能，设计出了一个能计算间接测量误差的通用Mathematica程序包，它把间接量的函数形式作为输入，通用性比一般的数值型误差计算程序要好。     

随机微分方程的计算机代数研究

将计算机代数的思想应用在随机微分方程理论中，借助数学软件系统Mathematica编制了运算程序包，它可以完成不同意义下的随机微分方程之间的转化，也可按Ito微分法则对随机变量函数进行微分，推导其转移概率密度函数的FPK方程，对于高维的随机微分方程组，以上步骤的计算量是很大的，本文提供的程序包可以快速准确地完成以上工作。     

Conditions for the normal rectilinear congruences whose focal surfaces are minimal

The present work deals with the study of special categories of nonnal rectilinear congruences whose focal surfaces are minimal and the solution to the problem is in the form of the membrane surface [3], but also, of the way of its supporting, at points beyond its perimeter, that exist. The method applied deals with fundamental elements of the Differential Geometry and more precisely, with the theory of normal rectilinear congruence C.