“张冠李戴”又如何——点差法失效的成因与结果分析

1前言点差法在解析几何中的重要地位与"神奇"效果,是每一位高中数学教师所熟知的.所以在圆锥曲线的教学过程中,教师对点差法的高度重视是绝对有理由,也是符合教学实际的.但由于大部分教师(包括笔者)对点差法的理解又是"局限"的,或者说有盲点所以在实际教学过程中还是多少存在着一     

也谈"点差法"的改进

点差法,又叫代点相减法,是解决圆锥曲线中的点弦问题非常重要,也非常简便的方法之一.利用这个设而不求的方法能快速、准确地得到弦的中点坐标与弦的斜率之间的关系式.同时我们也知道,点差法本身存在一个较大的缺陷.     

由引例探究圆锥曲线的“中点弦”

直线与圆锥曲线相交所得“中点弦”问题,是解析几何中的重要内容之一,也是高考的一个热点问题。解决此类问题,常规思路主要有两种：一是利用代数法结合根与系数的关系求解;二是利用点差法处理。本文以教材中一道双曲线“中点弦”问题为引例,展开探讨。     

Black-Scholes期权定价模型的五点式混合差分方法

研究了欧式看涨期权定价问题的差分方法,将Black-Scholes方程等价代换为标准抛物型偏微分方程,在时间方向上采用前、后差商,空间方向上采用五点差分格式,再引入参数θ建立一个稳定的混合差分格式.根据Von Neumann条件证明了该格式的稳定性及收敛性,并通过数值计算的实际应用,结果表明该算法适用于到期日较长的期权...     

一维不定常流方程的Harten解

为了利用一个二阶五点差分格式,对一维不定常流方程,计算强平面爆炸波遇不动固壁正反射的解,本文采用一定的技巧构造了边界和次边界的差分格式,并推算了初值的自模拟解析解.同时对初值的奇性提出了处理方法,从而满意地得到了这个较为困难问题的计算结果.本文提出和所利用的差分格式在处理激波间断问题方面具有实用意义.     

用点差法解决直线被二次曲线所截弦问题的研究

圆锥曲线问题是历届高考的重头戏．其中，设点作差法(简称为“点差法”)在解决直线被二次曲线所截弦的问题中有着广泛运用．在初学点差法时，我由于没有吃透它的实质，做起题来思路很乱．经过反复思考，我终于对点差法有了比较清晰的认识，并与另一重要方法——利用韦达定理求解作了一番比较，得出一些规律，在此想与大家交流一下．     

小麦茎秆与单穗重的相关性分析

主要是分析茎秆与单穗重和产量的相关性.以6个小麦品种为材料,分别测取了穗下茎长,茎粗,穗长,株高,节壁厚,节重及单穗重,对茎质与穗重进行了相关分析,结果表明:茎秆性状与穗重存在不同程度的相关性.其中,茎粗与单穗重呈正相关;节壁厚与单穗重呈正相关;节壁重与单穗重呈正相关;而株高和茎长与单穗重的关系不明显且较为复杂,应分情况讨论.     

点差法与向量联手简解中点弦问题

中点弦问题是直线与圆锥曲线的重要题型,也是高考的热点问题.在解答中点弦问题中的一个比较理想的方法是,点差法与直线斜率联合解题.它比用根与系数的关系和直线斜率联合解题,具有"设而不求"减少运算量的功效,但美中不足的是,有时需要对斜率的存在性进行分类讨论,甚至在运算变形过程中还要进行第二次分类,很容易造成逻辑上的混乱和表达上的困难,常给人"会而不对,对而不全,全而不美"的解题感受.向量是解决直线问题的一把利剑,若将点差法与向量联手,则可达到一种新的解题效果和解题体验.     

点差法及其应用

在研究直线被圆锥曲线截得中点弦问题时,常设出弦端点坐标,并代入圆锥曲线方程得两式,将两式相减.这种解题方法,不妨叫设点求差法,简称点差法,其解题的主要步骤有:1.设弦的端点坐标;2.代入方程两式相减;3.建立端点与中点的坐标关系;4.求弦所在直线斜率.点差法解题过程规律化,运算简单化,适     

强群胚及其性质

首先提出了强群胚,强子群胚,陪集和商强群胚的定义,接着讨论了k重强群胚,然后给出了它们的性质和同态定理并予以严格证明.此外提出的k重相容的强群胚将在流形变换,李群胚和系统结构的兼容性等方面有重要的应用.     

New representation of the remainder in the Bernstein approximation

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.     

一类广义Gauss型求积公式

基于被积函数在n次第一类和第二类Chebyshev多项式的零点处的差商,该本构造了两种Gauss型求积公式. 这些求积公式包含了某些已知结果作为特例.更重要的是这些新结果与Gauss-Turan求积公式有密切的联系.     

基于连分式与Newton-Padé逼近的数值积分

首先利用Newton-Pade表中部分序列推导出连分式,提出逆差商算法,算出关于高阶导数与高阶差商的连分式插值余项.接着,构造基于此类连分式的有理求积公式与相应的复化求积公式,算出相应的求积余项,研究表明,在一定条件下,求积公式序列一致收敛于积分真值.然后,为保证连分式计算顺利进行,研究连分式分母非0的充分条件.最后,若干数值算例表明,对某些函数采用新提出的复化有理求积公式计算数值积分,所得结果优于采用Simpson公式.     

Absolute stability analysis of Lurie control system with multiple delays: An integral-equality approach

This paper investigates the stability problem for the nominal and uncertain Lurie control system with multiple delays, respectively. With the delay divided into two segments, a novel Lyapunov functional is introduced and some new delay-dependent stability criteria are derived by employing integral-equality technique. It is theoretically proved that the obtained criteria are less conservative than some existing ones. Two examples are given to illustrate the effectiveness of the proposed results.     

A family of Steffensen type methods with seventh-order convergence

In this paper, based on some known fourth-order Steffensen type methods, we present a family of three-step seventh-order Steffensen type iterative methods for solving nonlinear equations and nonlinear systems. For nonlinear systems, a development of the inverse first-order divided difference operator for multivariable function is applied to prove the order of convergence of the new methods. Numerical experiments with comparison to some existing methods are provided to support the underlying theory.     

Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.     

基于新集对分析的粗糙集模型

首先,在这篇文章里两个集合里的元素被分为三个相对独立的部分来研究。接下来,一种定量信息——元素数目被用来描述这两个集合间的关系,然后提出了基于元素数目的新集对分析思想;另一方面,引用含糊集中真隶属度和假隶属度的概念,提出了一种模糊集对联系度。然后,这两种集对分析思想分别被用到粗糙集理论中作为粗糙集模型的一种推广,此外还讨论了一些相关的性质。     

Reciprocal polynomial extrapolation vs Richardson extrapolation for singular perturbed boundary problems

The reciprocal polynomial extrapolation was introduced in Amat et al. (J Comput Math 22(1):1?C10, 2004), where its accuracy and stability were studied and a linear scalar test problem was analyzed numerically. In the present work, a new step in the implementation of the reciprocal polynomial extrapolation, ensuring at least the same behavior as the Richardson extrapolation, is proposed. Looking at the reciprocal extrapolation as a Richardson extrapolation where the original data is nonlinearly modified, the improvements that we will obtain should be justified. Several theoretical analysis of the new extrapolation, including local error estimates and stability properties, are presented. A comparison between the two extrapolation techniques is performed for solving some boundary problems with perturbation controlled by a small parameter ?. Using two specific boundary problems, the error and the robustness of the new technique using centered divided differences in a uniform mesh are investigated numerically. They turn out to be better than those presented by the Richardson extrapolation. Finally, investigations on the accuracy when using a special non-uniform discretization mesh are presented. A numerical comparison with the Richardson extrapolation for this particular case, where we present some improvements, is also performed.     

Numerical simulations and modeling for stochastic biological systems with jumps

This paper gives a numerical method to simulate sample paths for stochastic differential equations (SDEs) driven by Poisson random measures. It provides us a new approach to simulate systems with jumps from a different angle. The driving Poisson random measures are assumed to be generated by stationary Poisson point processes instead of Lévy processes. Methods provided in this paper can be used to simulate SDEs with Lévy noise approximately. The simulation is divided into two parts: the part of jumping integration is based on definition without approximation while the continuous part is based on some classical approaches. Biological explanations for stochastic integrations with jumps are motivated by several numerical simulations. How to model biological systems with jumps is showed in this paper. Moreover, method of choosing integrands and stationary Poisson point processes in jumping integrations for biological models are obtained. In addition, results are illustrated through some examples and numerical simulations. For some examples, earthquake is chose as a jumping source which causes jumps on the size of biological population.     

Discrete time dynamic multi-leader–follower games with stage-depending leaders under feedback information

Discrete time dynamic Stackelberg games with alternating leaders are recently brought forward by and dynamic programming algorithms are extended to discrete time dynamic Stackelberg games of two players with alternating leaders under feedback information structure. In many practical problems, there are multiple leaders and multiple followers. Moreover, the positions of leaders and followers may change at some stages. We aim to extend the results of discrete time dynamic Stackelberg games with alternating leaders to multi-leader–follower games and to obtain a new type of game, dynamic multi-leader–follower game with stage-depending leaders. To simplify the problem, all players in the model are divided into two groups. At each stage, players in one group act as leaders and the players in other group are followers. In the subsequent stage, the positions may go to the contrary. Actually, whether the players in some group act as leaders or not at some stage, depends on the information at the previous stage(s). Furthermore, dynamic programming algorithms are extended to dynamic multi-leader–follower game with stage-depending leaders under feedback information structure in this paper.