把握高考传统与创新题型 提高立体几何复习有效性

在新课标精神的指引下,高考题型也有了相应变化,笔者对高考立体几何题型进行综合分析. 一、三视图问题 三视图是高中新课标的新增内容.空间几何体的三视图能够使学生更好地把握几何体的形状和性质,可以培养学生的几何直观能力和空间想象能力.不仅可以单独考查,而且可以与其他知识交汇渗透考查;不仅可以考查选择、填空题,还可以考查解答题.考查的形式丰富,内容灵活,所以三视图问题也成为了近年高考的热点.     

多元函数极值的求法及其应用

在实际问题中,往往会遇到多元函数的最大值、最小值问题.多元函数的最大值、最小值问题与极大值、极小值有密切联系.求多元函数极值,一般可以利用偏导数来解决.与一元函数相类似,可以利用函数的极值来求函数的最大值和最小值.但是由于自变量个数的增加,应特别注意概念中的一些变化和计算复杂性.这里主要讨论二元函数,对于二元以上的函数极值可以类似加以解决.     

有限元方法(二)

§2. 剖分插值 本节讨论平面区域的几何剖分和相应的分片插值方法.对区域进行剖分时,基本单元可以取为三角形、矩形、四边形等等.插值函数可以取为一次(线性)或高次多项式等等.其中以三角剖分和相应的三顶点线性插值最简单,最常用,故主要讨论这一情况.为了尽快进入有限元离散化,可以只读这里的§§2.1-3而转至§3.     

一道开放题的编写经过

在日常教学中,开放性数学试题的编拟是一个困难问题.一般而言,编拟开放性试题的方法有:某些问题的联想、类比、拓广,某些问题的逆命题及其类比和拓广.其中“某些问题”可以是教学中遇到的、可以是自己思考所得、可以由成题改编而得,总之,来源是多元的.本文谈一道开放题的编写经过,供参考.     

三角函数在高中数学教学中的应用研究

在新课程改革下,三角函数可以广泛应用于解决几何形状的问题,教师引导学生利用正弦、余弦和正切函数来计算三角形的边长、角度和面积.熟练掌握函数概念可以帮助学生理解和计算各种图形的属性.同时,三角函数在物理学中也具有广泛的应用,在运动学中的平抛运动和斜抛运动,通过三角函数来描述分析和计算抛射角度.此外,三角函数还可以用于描述波动、振动、电路、光学等现象,也可以应用于统计学和数据分析中,计算周期性数据的分析,通过正弦函数和余弦函数进行建模和预测,转换和调整数据的规律、趋势和周期性变化,有助于学生深入理解数学的实际应用领域,培养学生的数学建模和解决实际问题的综合素养能力.     

从高斯公式到格林公式和牛顿-莱布尼茨公式

本文讨论高等数学课程中,高斯公式、格林公式和牛顿-莱布尼兹公式之间的内在联系,指出格林公式和牛顿-莱布尼茨公式可以分别看作一维和二维欧氏空间中的高斯公式.实际上,n维欧氏空间中的高斯公式可以看作微积分基本定理在高维欧氏空间中的表述形式.利用高斯公式还可以导出定积分、二重积分和任意n重积分的分部积分公式.     

数学机械化与数学道理化

以中国传统文化为底蕴、机械化和道理化为特色的中国传统数学,曾经取得光辉的成就.将这些思想用于计算机科学、孤子理论和量子场论,不仅可以系统地产生已有结果,还可以开辟新的研究方向,取得重大突破.     

一个不等式的推广与证明

推广是发现问题的基本途径,是提出猜想的基本方式,是迁移数学知识的基本手段,是创造数学知识的基本策略,也是数学发现的基本策略和重要手段.数学推广,可能产生新问题、新方法和新理论.数学推广,可以促进知识的条理化、一般化和系统化.学会数学推广,可以促进知识理解,激活数学思维,催生创新灵感.     

利用几何画板制作动态的空间直角坐标系工具

如果有一个制作简单且使用方便的动态的空间直角坐标系工具,那么可以快捷制作一个空间图形.例如,制作立体几何中的动态空间几何体、动态空间曲线、动态空间曲面等教学课件.有了这个坐标系可以从不同的角度观察图形的真实形状,可以很方便地观察几何体(或曲线或曲面)的正视图、俯视图、侧视图.有助于学生建立空间概念,提高学生的空间想象能力和思维能力,激发学生的学习兴趣,提高课堂效率.     

星系螺旋结构的气体激波理论

本文用星际气体自引力星系激波来解释星系的螺旋结构、恒星的扰动引力场并非必要条件.我们首先证明,即使扰动引力场为零,也可以存在局部的星系激波解.这种解要求|ω_η0|>α,而且只要气体的密度反差比较大,就只能用激波解来解释螺旋结构.用叠代的方法求出了星际气体的自引力激波宏图.对一种特定的扰动引力场模拟气体自引力,可以在速度平面上定性分析激波解的特性.初始原星系盘中的物质分布不均匀性,通过缠卷过程、不稳定性增长和波动叠加.可以发展成星系激波宏图.这样,对星系激波的起源,演化和维持给出一个完整的图象.利用这个图象,可以解释星系螺旋结构的大量观测结果和分类特性.     

带有限L^p函数的一个刻画及其应用——献给陆善镇教授75华诞

利用Shannon-Nyquist采样定理的思想,本文给出带有限L^p函数类的一个刻画,并且利用这个刻画得到Schrodinger方程的若干估计,如Strichartz型和倒时空Strichartz估计等,同时也给出若干已知估计的简单证明.     

A simple classroom model for the spread of a disease

In this paper, we give an account of the mathematics that took place in a high-school classroom in connection with two extremely simple models for the spread of an infectious disease within a population. We also provide some ideas as to how the model might be made somewhat more realistic, and briefly discuss the possibility for developing more sophisticated models.     

A general recurrence interpolation formula and its applications to multivariate interpolation

Following the ideas of several papers by G. Mühlbach, a general recurrence interpolation formula is obtained that contains as particular cases some extended Newton and Aitken-Neville interpolation formulas. The exposition of the problem allows us to show the applications of this formula to multivariate interpolation. This is the principal aim of this work. Some simple examples are given to show the variety of applications.     

Dynamics of periodically kicked oscillators

We review some recent results surrounding a general mechanism for producing chaotic behavior in periodically kicked oscillators. The key geometric ideas are illustrated via a simple linear shear model.     

Inverse programming: Theory and examples

In this paper some simple ideas about ordered sets and inverse functions are developed into a theory of inverse pairs of optimization problems. This relationship is shown to permit the use of root-finding methods to find a solution for one problem through a sequence of solutions to the other. A variety of problems showing this relationship are presented as examples.This research was partially supported by the National Research Council of Canada under grants A7675 and A7751.     

Interpolation of nonlinear partial differential operators and generation of differentiale evolutions

By interpolating between Sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain. Hence some techniques from the theory of ordinary differential equations can be applied to some p.d.e.'s. Using these ideas, we study a class of nonlinear evolutions in a Banach space. We obtain some very simple existence and continuous dependence results. The theory is applicable to reaction-diffusion equations, dispersion equations, and hyperbolic equations before shocks develop.     

The general analytical and numerical solution for the modified KdV equation with convergence analysis

We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd.     

Three new algorithms based on the sequential secant method

We introduce three new implementations of the sequential secant method for solving nonlinear simultaneous equations. Following the ideas of Gragg and Stewart, we store orthogonal factorizations of some of the matrices involved. Degeneracy in the increments of the independent variable is corrected according to simple and theoretically justified procedures. Some numerical experiences are also given.     

Generic injectivity and stability of inverse problems for connections

We consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a compact Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral geometry question of some attenuated geodesic ray transform through a pseudolinearization argument. We show injectivity (up to natural obstructions) and stability estimates for both the linear and nonlinear problems for generic simple metrics and generic connections and Higgs fields, including the real-analytic ones. We consider the problems on simple manifolds to make the exposition of the main ideas clear and concise, many results of this paper still hold under some assumptions weaker than simplicity.     

Superreals and superfunctions

The system of superreal numbers introduced by D. O. Tall shows that it is possible to develop a limited theory of infinitesimals (and infinite numbers) in an elementary context, using only classical mathematical ideas and methods. It is used here to present a simple and intuitively straightforward introduction to the delta function and its derivatives. The treatment has the conceptual advantage of exhibiting the delta function as a genuine function in some sense, while at the same time making it clear how that sense differs from that appropriate to ‘ordinary’ functions.