关于“近似商的取法”的教学

初中算术教材里关于小數除法所要研究的,就是小数除以整数的除法。如果这一法则得出了,这末,当小数除以小数时,只要把被除数和除数扩大同样的倍数,使除数变成整数,就成了小数(或整数)除以整数的除法,小数除以整数的除法中,有时需要取近似商。决定不足近似商与过剩近似商,二者的取舍的法则是:“在精确到指定的同一小数位的不足近似商和过剩近似商里,为了使所取的近似商的誤差小于这个指定的小数位上的單位的1/2,如果除到这个小数位,所得的余数(引者註:把它当整数看)小于除数的1/2,就取不足近似商;如果所得的余数大于除数的1/2,就取过剩近似商。”(初中算术§111)。由于这一教材內容对于初一学生来说,是較难理解的,因此每当教师教完这一节以后,常常会發現学生对这个法則只是机械地掌握了,而对它的理論根据却不甚了然,教师对于这一法則原理的教学,在效果上便常常流于形式主义,今根据个人的点滴体会談一談就正于大家。     

小数王国旅行记(二)

去小数除法村要宝鼎 小数除法村与小数乘法村仅一河之隔.因为除法村的村民既精通除法也精通乘法,而乘法村的村民只精通乘法,所以强大的小数除法村一心想吞并小数乘法村.乘法村不依,除法村就抢走了乘法村的镇村之宝--绿玉宝鼎,并扬言:我们留下路线图,有本事自己来取!贝卡他们告诉村长,一定替乘法村要回镇村之宝,乘村长感动得热泪盈眶.     

小数点后的“0”不能随意去掉

在实际应用中,珠算的结果往往不需要保留很多的小数位数,这时可以根据需要,用“四舍五入法”保留一定的小数位数,求出结果的近似值。在表示近似数的时候,小数末尾后的“0”不能随意去掉。我们在上课时强调过多次这个问题,但个别学生仍然没有注意这方面的问题,有的学生认为“0”就是表示没有,在小数后写不写效果一样,这种观点是不对的,这是一种概念错误,严格来讲这是一个严谨的科学态度问题。珠算作为     

低倍数速乘法

<正>珠算乘法从形式上划分,可分为三种乘法,第一种为小数乘小数,第二种为大数乘小数,第三种为大数乘大数。小数指:1、2、3、4大数指:5、6、7、8、9由于小数乘小数,人们很容易一口清,但对于大数乘小数及大数乘大数就不那么简单了。怎样扬长避短,比翼齐飞呢?采用低倍数速乘法就可以解决问题。首先,谈谈大数乘小数。     

小数到底是什么数?

<正>新初一的同学们在刚升入中学不久学习了有理数,整数和分数统称为有理数,有理数有下面的两种分类:正整数和零是自然数.这时有的同学有疑问了,为什么分类表里没有小学学过的"小数"呢?小数到底是什么数呢?有的同学说:"小数就是分数".小数是不是分数?小数和分数有什么关     

十进小数发展簡史

十进小数的出现,是数学史上的一件大事。美国数学史家卡约利(F.Cajori)就会认为十进小数是近代数学史上关于计算基础方面的三大发明之一,他说:“近代计算的异常势力是由于三大发明:印度计数法、十进分数和对数。”因此介绍一下十进小数的发展历史是有益处的。在西洋数学史上常常把十进小数的发明归功于欧洲人,特别归功于斯古文(S.Stevin,1548-1620)等人,这是不正确的。实际上,中国、印度和中亚都在欧洲人之前使用了十进小数。我国是使用十进小数最早的国家,我们从“九章算术”的刘徽注文中找到十进小数的思想。刘徽是我国第三世纪时最杰出的数学家,他于公元263年注解“九章算术”。当时对于奇另小数的记法,主要的是(1)化成分数,(2)或用十进制的数名法。刘徽在长度的记法中采用数名:丈、尺、寸、分、厘、毫、秒、忽,“忽”是最小数名,忽以下就没有专名,在计算中他把忽做为单位,以下那些没有专名的数就是小数。当时刘徽或者把它舍去,或者化成简单的分数,或者用十进小数表达。刘徽在“九章算术”注中有三个地方用到十进小数(以十进分数思想出现)。     

一类多重小数部分和的估计

研究了一类多重小数部分和渐近公式,并解决了Ovidiu Furdui的一个多重小数部分积分问题m=1的情形.     

幻和为24的三阶幻方到底有几个

<正>以正整数为元素(元素不重复),幻和为24的三阶幻方到底有几个?360网、新浪网、百度等网上都有类似的问题.笔者在文[1]论证得到制作三阶幻方的通法:"三阶幻方九宫数,一行中间最小数,二行中央中位数,三行最右二小数(第二小的数简称二小数),幻和中位三倍数(幻和是中位数的三倍),由此推出空格数."利用这一结论可以快速解决幻和为24的三阶幻方到底有几个的     

循环小数与无穷小数

假设有人提出一个问题:“循环小数是有穷小数还是无穷小数”? 每个读者听了,都会毫不迟疑地回答:“当然是无穷小数!”同时心中一定认为:提出这个问题的人或许对算术一无所知,或许把算术忘得干干净净了。的确,照今天数学的说法,毫无疑问地,循环小数是无穷小数,因为我们一向是把小数按照下表中左方的分类法而加以分类的: 尽管没有人对现在的分类法提出过异议,作者却想提出下列的问题:循环小数应该归入无穷小数去还是应该归入有穷小数去?     

关于Mahler超越小数的无理性度量

本文在一般条件下研究了Mahler超越小数的无理性度量,给出无理性指数的上、下界估计及某些特殊小数的无理性指数的精确值．     

具有奇点的Laplace方程边值问题的原始能量-有限元结合法

在用变分原理离散数理方程时,可取函数在部分求解区域内,取成解析函数或奇异函数类;而在其余求解区域内取成分片低阶插值多项式.在两区域交界的结点上,可取函数连续.我们称此法为原始能量一有限元结合法.[3,p135]中提到过这种方法,但缺少理论分析.[4]曾给出初步的分析.本文与[5,6]就此方法给出较严格的理论分析.     

Newton-Type Methods for Quasidifferentiable Equations

In this paper, we present two Newton-type methods for solving quasidifferentiable equations in the sense of Demyanov and Rubinov (Ref. 1). Method I is well defined and is a natural extension of the classical Newton method, based on a generalized Kakutani fixed-point theorem. Method II is a simplified version and requires less computation than Method I. Under some mild assumptions, we establish a locally quadratic convergent theorem for Method I and prove a semilocal convergence theorem for Method II.     

吴消元法与四元术

在解多项式方程组的过程中,吴消元法的核心是用对多项式约化求余式的方法消元.研究中发现,清代沈钦裴四元消法的三条法则均系互乘对消,都可以写成除法变换的形式.从而找到吴消元法与四元术的内在联系.得出吴消元法是四元术的直接继承,吴消元法是四元术现代化发展的结论.     

The Multiplier Method for optimal control problems of parabolic systems

In this paper the application of the Multiplier Method (also known as Augmented Lagrangian Method or Penalty Shifting Method) to quadratic optimal control problems for linear parabolic distributed systems is studied and the convergence of the method on appropriate Hilbert spaces is proved. Numerical examples are reported.This work was partially supported by the Consiglio Nazionale delle Ricerche.     

A tale of three eras: The discovery and rediscovery of the Hungarian Method

In the Fall of 1953, a translation of a paper of Jen? Egerváry from Hungarian into English combined with a result of Dénes K?nig provided the basis of a good algorithm for the Linear Assignment Problem. To honor the Hungarian mathematicians whose ideas had been used, it was called the Hungarian Method. In 2005, Francois Ollivier discovered that the posthumous papers of Jacobi contain an algorithm that, when examined carefully, is essentially identical to the Hungarian Method. Since Jacobi died in 1851, this work was done over a hundred years prior to the publication of the Hungarian Method in 1955. This paper will provide an account for the mathematical, academic, social and political worlds of Jacobi, K?nig, Egerváry, and Kuhn. As sharply different as they were (Prussian monarchy, Hungary under the Nazis and the Communists, and the post-war USA), they produced the same mathematical result. The paper is self-contained, assuming little beyond the duality theory of linear programing. The Hungarian Method and Jacobi’s algorithm will be explained at an elementary level and will be illustrated by an example, solved both by the Hungarian Method and by Jacobi’s Method.     

动态资产份额定价理论模型

运用倒向随机微分方程数学方法 ,建立了动态资产份额定价理论模型 .这一模型是资产份额定价法的改进 .求解模型得到动态资产份额定价理论公式 ,并得出结论 :资产份额定价公式完全可以作为特例 ,以离散时间意义和在不考虑动态投资的情况下 ,由动态资产份额定价理论公式得到 .     

Approximate traveling wave solution for shallow water wave equation

Reconstruction of Variational Iteration Method (RVIM) is used for computing the coupled Whitham-Broer-Kaup shallow water. Then RVIM solution is verified against exact one and is compared with powerful approximate solutions, the Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). The existent error of the methods is computed and convergence of the RVIM solution has been presented. Results obtained expose effectiveness and capability of this method to solve the nonlinear systems in mechanics, analytically.     

浅析大M法与两阶段法的一致性

分析了大M法与两阶段法在思想方法、辅助线性规划问题的构造、初始可行基、初始单纯形表、最优性检验和算法步骤等方面的一致性.     

C_0复杂度的改进及其在证券市场中的应用

系统地分析了Lemple-Ziv复杂性度量方法的应用过程中,将实际信号(时间序列)转变成符号序列的诸多方法中存在的一些问题,提出了更合理兼容法.该方法可以有效地刻划各种时间序列的复杂度.文章最后动态地分析了中国证券市场的复杂性.     

ON BLOCK MATRICES ASSOCIATED WITH DISCRETE TRIGONOMETRIC TRANSFORMS AND THEIR USE IN THE THEORY OF WAVE PROPAGATION

Block matrices associated with discrete Trigonometric transforms （DTT＇s） arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT＇s are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are unitary and offer simultaneous diagonalizations of specific matrix algebras. The main tool for the investigation of the aforementioned wave applications is the efficient inversion of such types of block matrices. To this direction, in this paper we develop an efficient algorithm for the inversion of matrices with U-diagonalizable blocks （U a fixed unitary matrix） by utilizing the U- diagonalization of each block and subsequently a similarity transformation procedure. We determine the developed method＇s computational complexity and point out its high efficiency compared to standard inversion techniques. An implementation of the algorithm in Matlab is given. Several numerical results are presented demonstrating the CPU-time efficiency and accuracy for ill-conditioned matrices of the method. The investigated matrices stem from real-world wave propagation applications.