数学专业外文文献翻译

第3章 最小均方算法3．1 引言最小均方(LMS ,least-mean-square)算法是一种搜索算法，它通过对目标函数进行适当的调整[1]—[2],简化了对梯度向量的计算。

由于其计算简单性，LMS 算法和其他与之相关的算法已经广泛应用于白适应滤波的各种应用中[3]-[7]。

为了确定保证稳定性的收敛因子范围，本章考察了LMS 算法的收敛特征。

研究表明，LMS 算法的收敛速度依赖于输入信号相关矩阵的特征值扩展[2]—[6]。

在本章中，讨论了LMS 算法的几个特性，包括在乎稳和非平稳环境下的失调[2]—[9]和跟踪性能[10]-[12]。

本章通过大量仿真举例对分析结果进行了证实。

在附录B 的B ．1节中，通过对LMS 算法中的有限字长效应进行分析，对本章内容做了补充。

LMS 算法是自适应滤波理论中应用最广泛的算法，这有多方面的原因。

LMS 算法的主要特征包括低计算复杂度、在乎稳环境中的收敛性、其均值无俯地收敛到维纳解以及利用有限精度算法实现时的稳定特性等。

3．2 LMS 算法在第2章中，我们利用线性组合器实现自适应滤波器，并导出了其参数的最优解，这对应于多个输入信号的情形。

该解导致在估计参考信号以d()k 时的最小均方误差。

最优(维纳)解由下式给出：10w R p-= (3.1)其中，R=E[()x ()]Tx k k 且p=E[d()x()] k k ，假设d()k 和x()k 联合广义平稳过程。

如果可以得到矩阵R 和向量p 的较好估计，分别记为()R k ∧和()p k ∧，则可以利用如下最陡下降算法搜索式(3．1)的维纳解：w(+1)=w()-g ()w k k k μ∧w()(()()w())k p k R k k μ∧∧=-＋２ (3.2) 其中，k ＝0，1，2，…,g ()w k ∧表示目标函数相对于滤波器系数的梯度向量估计值。

一种可能的解是通过利用R 和p 的瞬时估计值来估计梯度向量，即 ()x()x ()TR k k k ∧=()()x()p k d k k ∧= (3.3) 得到的梯度估计值为()2()x()2x()x ()()T w g k d k k k k w k ∧=-+2x()(()x ()())Tk d k k w k =-+ 2()x()e k k =- (3.4)注意，如果目标函数用瞬时平方误差2()e k 而不是MSE 代替，则上面的梯度估计值代表了真实梯度向量，因为2010()()()()2()2()2()()()()Te k e k e k e k e k e k e k w w k w k w k ⎡⎤∂∂∂∂=⎢⎥∂∂∂∂⎣⎦2()x()e k k =-()w g k ∧= (3.5)由于得到的梯度算法使平方误差的均值最小化．因此它被称为LMS 算法，其更新方程为 (1)()2()x()w k w k e k k μ+=+ (3.6) 其中，收敛因子μ应该在一个范围内取值，以保证收敛性。

Some Properties of Solutions of Periodic Second OrderLinear Differential Equations1. Introduction and main resultsIn this paper ， we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna’s value distribution theory of meromorphic functions [12, 14, 16］。

In addition ， we will use the notation )(f σ，)(f μand )(f λto denote respectively the order of growth ， the lower order of growth and the exponent of convergence of the zeros of a meromorphic function f ，)(f e σ(［see 8］）,the e —type order of f(z), is defined to ber f r T f r e ),(log lim)(+∞→=σ Similarly ， )(f e λ,the e —type exponent of convergence of the zeros of meromorphic functionf , is defined to be rf r N f r e )/1,(log lim )(++∞→=λ We say that )(z f has regular order of growth if a meromorphic function )(z f satisfiesrf r T f r log ),(log lim )(+∞→=σ We consider the second order linear differential equation0=+''Af fWhere )()(z e B z A α=is a periodic entire function with period απω/2i =。

附件1：外文资料翻译译文第1章预备知识双曲守恒律系统是应用在出现在交通流，弹性理论，气体动力学，流体动力学等等的各种各样的物理现象的非常重要的数学模型。

一般来说，古典解非线性双曲方程柯西问题解的守恒定律仅仅适时局部存在于初始数据是微小和平滑的.这意味着震波在解决方案里相配的大量时间里出现。

既然解是间断的而且不满足给定的传统偏微分方程式，我们不得不去研究广义的解决方法，或者是满足分布意义的方程式的函数.我们考虑到如下形式的拟线性系统, (1.0.1)这里是代表物理量密度的未知矢量向量，是给定表示保守项的适量函数，这些方程式通常被叫做守恒律.让我们假设一下，是(1.0.1)在初始数据. (1.0.2)下的传统解。

使成为消失在紧凑子集外的函数的一类。

我们用乘以(1.0.1)并且使的部分，得到. (1.0.3)定义1.0.1 有，,有界函数叫做在以原始数据为边界条件下，(1.0.1)初值问题的一个弱解，在(1.0.3)适用于所有.非线性系统守恒理论的一个重要方面是这些方程解的存在疑问性.它正确的帮助解答在手边的已经建立的自然现象的模型的问题，而且如果在问题是适定的.为了得到一个总体的弱解或者一个考虑到双曲守恒律的普遍的解，一个为了在(1.0.1)右手边增加一个微小抛物摄动限：(1.0.4)在这是恒定的.我们首先应该得到一个关于柯西问题(1.0.4),(1.0.2)对于任何一个依据下列抛物方程的一般理论存在的的解的序列：定理1.0.2 (1)对于任意存在的, (1.0.4)的柯西问题在有界可测原始数据(1.0.2)对于无限小的总有一个局部光滑解，仅依赖于以原始数据的.(2)如果解有一个推理的估量对于任意的,于是解在上存在.(3)解满足：如果.( 4)特别的，如果在(1.0.4)系统中的一个解以(1.0.5)形式存在，这里是在上连续函数，,如果(1.0.6) 这里是一个正的恒量，而且当变量趋向无穷大或者趋向于0时，趋向于0.证明.在(1)中的局部存在的结果能简单的通过把收缩映射原则应用到解的积分表现得到，根据半线性抛物系统标准理论.每当我们有一个先验的局部解的评估，明显的本地变量一步一步扩展到，因为逐步变量依据基准.取得局部解的过程清晰地表现在(3)中的解的行为.定理1.0.2的(1)-(3)证明的细节在[LSU,Sm]看到.接下来是Bereux和Sainsaulieu未发表的证明(cf. [Lu9, Pe])我们改写方程式(1.0.5)如下：(1.0.7)当.然后. (1.0.8) 以初值(1.0.8)的解能被格林函数描写：. (1.0.9)由于，,(1.0.9)转化为.(1.0.10)因此对于任意一个，有一个正的下界.在定理1.0.2中获得的解叫做粘性解.然后我们有了粘性解的序列,,如果我们再假如是在关于参数的空间上一致连续，即存在子序列（仍被标记）如下, 在上弱对应(1.0.11) 而且有子序列如下，弱对应(1.0.12) 在习惯于成长适当成长性.如果，a.e.，(1.0.13)然后明显的是(1.01)使在(1.0.4)的趋近于0的一个初始值(1.0.2)的一个弱解.我们如何得到弱连续(1.0.13)的关于粘度解的序列的非线性通量函数？补偿密实度原理就回答了这个问题.为什么这个理论叫补偿密实度？粗略的讲，这个术语源自于下列结果：如果一个函数序列满足(1.0.14)与下列之一或者(1.0.15) 当趋近于0时弱相关，总之，不紧密.然而，明显的，任何一个在(1.0.15)中的弱紧密度能补偿使其成为的紧密度.事实上，如果我们将其相加，得到(1.0.16)当趋近于0时弱相关,与(1.0.14)结合意味着的紧密度.在这本书里，我们的目标是介绍一些补偿紧密度方法对标量守恒律的应用，和一些特殊的两到三个方程式系统.此外，一些具有松弛扰动参量的物理系统也被考虑进来。

数学专业英语第三版课文翻译章本文将根据数学专业英语第三版课文《Step by Step Thinking》进行翻译。

"Step by Step Thinking"is an article that introduces the concept of step-by-step thinking in mathematics.It highlights the importance of breaking down complex problems into smaller,more manageable steps in order to solve them effectively.The article begins by stating that step-by-step thinking is a fundamental skill in mathematics.It emphasizes the need to approach problems by breaking them downinto smaller components,as this helps to clarify the problem and identify potential solutions.The author argues that this approach is not only applicable tomathematics but also to various other fields,as it promotes clearer thinking and problem-solving abilities.The article then discusses the step-by-step thinking process in more detail.It suggests that the first step is tocarefully read and understand the problem, ensuring that all relevant information is identified.This is followed by breaking the problem down into smaller sub-problems or steps,each of which can be solved individually.The author emphasizes the need to be systematic and organized during this process,as it helps to prevent mistakes and confusion.Furthermore,the article highlights the importance of logical reasoning in step-by-step thinking.It states that each step should be justified with logical reasoning,ensuring that the solution is based on sound mathematical principles.The author advises against skipping steps or making assumptions without proper justification,as this can lead to erroneous results.The article also provides examples to illustrate the step-by-step thinking approach.It presents a complex problem and demonstrates how breaking it down into smaller steps can simplify the solution process.By solving each step individually and logically connecting them,the problem can be solved effectively.In conclusion,"Step by Step Thinking" emphasizes the significance of step-by-step thinking in mathematics and problem-solving. It encourages readers to approach problems systematically,breaking them down into smaller components,and justifying eachstep with logical reasoning.This approach promotes clearer thinking and enhances problem-solving abilities,not only in mathematics but also in other disciplines.。

（外文翻译从原文第一段开始翻译，翻译了约2000字）勾股定理是已知最早的古代文明定理之一。

这个著名的定理被命名为希腊的数学家和哲学家毕达哥拉斯。

毕达哥拉斯在意大利南部的科托纳创立了毕达哥拉斯学派。

他在数学上有许多贡献，虽然其中一些可能实际上一直是他学生的工作。

毕达哥拉斯定理是毕达哥拉斯最著名的数学贡献。

据传说，毕达哥拉斯在得出此定理很高兴，曾宰杀了牛来祭神，以酬谢神灵的启示。

后来又发现2的平方根是不合理的，因为它不能表示为两个整数比，极大地困扰毕达哥拉斯和他的追随者。

他们在自己的认知中，二是一些单位长度整数倍的长度。

因此2的平方根被认为是不合理的，他们就尝试了知识压制。

它甚至说，谁泄露了这个秘密在海上被淹死。

毕达哥拉斯定理是关于包含一个直角三角形的发言。

毕达哥拉斯定理指出，对一个直角三角形斜边为边长的正方形面积，等于剩余两直角为边长正方形面积的总和图1根据勾股定理，在两个红色正方形的面积之和A和B，等于蓝色的正方形面积，正方形三区因此，毕达哥拉斯定理指出的代数式是：对于一个直角三角形的边长a，b和c，其中c是斜边长度。

虽然记入史册的是著名的毕达哥拉斯定理，但是巴比伦人知道某些特定三角形的结果比毕达哥拉斯早一千年。

现在还不知道希腊人最初如何体现了勾股定理的证明。

如果用欧几里德的算法使用，很可能这是一个证明解剖类型类似于以下内容：六^维-论~文.网“一个大广场边a+ b是分成两个较小的正方形的边a和b分别与两个矩形A和B，这两个矩形各可分为两个相等的直角三角形，有相同的矩形对角线c。

四个三角形可安排在另一侧广场a+b中的数字显示。

在广场的地方就可以表现在两个不同的方式：1。

由于两个长方形和正方形面积的总和：2。

作为一个正方形的面积之和四个三角形：现在，建立上面2个方程，求解得因此，对c的平方等于a和b的平方和（伯顿1991）有许多的勾股定理其他证明方法。

一位来自当代中国人在中国现存最古老的含正式数学理论能找到对Gnoman和天坛圆路径算法的经典文本。

Assume that you have a guess U(n) of the solution. If U(n) is close enough to the exact solution, an improved approximation U(n + 1) is obtained by solving the linearized problemwhere have asolution.has. In this case, the Gauss-Newton iteration tends to be the minimizer of the residual, i.e., the solution of minUIt is well known that for sufficiently smallAndis called a descent direction for , where | is the l2-norm. The iteration iswhere is chosen as large as possible such that the step has a reasonable descent.The Gauss-Newton method is local, and convergence is assured only when U(0)is close enough to the solution. In general, the first guess may be outside thergion of convergence. To improve convergence from bad initial guesses, a damping strategy is implemented for choosing , the Armijo-Goldstein line search. It chooses the largestinequality holds:|which guarantees a reduction of the residual norm by at least Note that each step of the line-search algorithm requires an evaluation of the residualAn important point of this strategy is that when U(n) approaches the solution, then and thus the convergence rate increases. If there is a solution to the scheme ultimately recovers the quadratic convergence rate of the standard Newton iteration. Closely related to the above problem is the choice of the initial guess U(0). By default, the solver sets U(0) and then assembles the FEM matrices K and F and computesThe damped Gauss-Newton iteration is then started with U(1), which should be a better guess than U(0). If the boundary conditions do not depend on the solution u, then U(1) satisfies them even if U(0) does not. Furthermore, if the equation is linear, then U(1) is the exact FEM solution and the solver does not enter the Gauss-Newton loop.There are situations where U(0) = 0 makes no sense or convergence is impossible.In some situations you may already have a good approximation and the nonlinear solver can be started with it, avoiding the slow convergence regime.This idea is used in the adaptive mesh generator. It computes a solution on a mesh, evaluates the error, and may refine certain triangles. The interpolant of is a very good starting guess for the solution on the refined mesh.In general the exact Jacobianis not available. Approximation of Jn by finite differences in the following way is expensive but feasible. The ith column of Jn can be approximated bywhich implies the assembling of the FEM matrices for the triangles containing grid point i. A very simple approximation to Jn, which gives a fixed point iteration, is also possible as follows. Essentially, for a given U(n), compute the FEM matrices K and F and setNonlinear EquationsThis is equivalent to approximating the Jacobian with the stiffness matrix. Indeed, since putting Jn = K yields In many cases the convergence rate is slow, but the cost of each iteration is cheap.The nonlinear solver implemented in the PDE Toolbox also provides for a compromise between the two extremes. To compute the derivative of the mapping , proceed as follows. The a term has been omitted for clarity, but appears again in the final result below.The first integral term is nothing more than Ki,j.The second term is “lumped,” i.e., replaced by a diagonal matrix that contains the row j j = 1, the second term is approximated bywhich is the ith component of K(c')U, where K(c') is the stiffness matrixassociated with the coefficient rather than c. The same reasoning can beapplied to the derivative of the mapping . Finally note that thederivative of the mapping is exactlywhich is the mass matrix associated with the coefficient . Thus the Jacobian ofU) is approximated bywhere the differentiation is with respect to u. K and M designate stiffness and mass matrices and their indices designate the coefficients with respect to which they are assembled. At each Gauss-Newton iteration, the nonlinear solver assembles the matrices corresponding to the equationsand then produces the approximate Jacobian. The differentiations of the coefficients are done numerically.In the general setting of elliptic systems, the boundary conditions are appended to the stiffness matrix to form the full linear system: where the coefficients of and may depend on the solution . The “lumped”approach approximates the derivative mapping of the residual by The nonlinearities of the boundary conditions and the dependencies of the coefficients on the derivatives of are not properly linearized by this scheme. When such nonlinearities are strong, the scheme reduces to the fix-pointiter ation and may converge slowly or not at all. When the boundary condition sare linear, they do not affect the convergence properties of the iteration schemes. In the Neumann case they are invisible (H is an empty matrix) and in the Dirichlet case they merely state that the residual is zero on the corresponding boundary points.Adaptive Mesh RefinementThe toolbox has a function for global, uniform mesh refinement. It divides each triangle into four similar triangles by creating new corners at the midsides, adjusting for curved boundaries. You can assess the accuracy of the numerical solution by comparing results from a sequence of successively refined meshes.If the solution is smooth enough, more accurate results may be obtained by extra polation. The solutions of the toolbox equation often have geometric features like localized strong gradients. An example of engineering importance in elasticity is the stress concentration occurring at reentrant corners such as the MATLAB favorite, the L-shaped membrane. Then it is more economical to refine the mesh selectively, i.e., only where it is needed. When the selection is based ones timates of errors in the computed solutions, a posteriori estimates, we speak of adaptive mesh refinement. Seeadapt mesh for an example of the computational savings where global refinement needs more than 6000elements to compete with an adaptively refined mesh of 500 elements.The adaptive refinement generates a sequence of solutions on successively finer meshes, at each stage selecting and refining those elements that are judged to contribute most to the error. The process is terminated when the maximum number of elements is exceeded or when each triangle contributes less than a preset tolerance. You need to provide an initial mesh, and choose selection and termination criteria parameters. The initial mesh can be produced by the init mesh function. The three components of the algorithm are the error indicator function, which computes an estimate of the element error contribution, the mesh refiner, which selects and subdivides elements, and the termination criteria.The Error Indicator FunctionThe adaption is a feedback process. As such, it is easily applied to a lar gerrange of problems than those for which its design was tailored. You wantes timates, selection criteria, etc., to be optimal in the sense of giving the mostaccurate solution at fixed cost or lowest computational effort for a given accuracy. Such results have been proved only for model problems, butgenerally, the equid is tribution heuristic has been found near optimal. Element sizes should be chosen such that each element contributes the same to the error. The theory of adaptive schemes makes use of a priori bounds forsolutions in terms of the source function f. For none lli ptic problems such abound may not exist, while the refinement scheme is still well defined and has been found to work well.The error indicator function used in the toolbox is an element-wise estimate of the contribution, based on the work of C. Johnson et al. For Poisson'sequation –f -solution uh holds in the L2-normwhere h = h(x) is the local mesh size, andThe braced quantity is the jump in normal derivative of v hr is theEi, the set of all interior edges of thetrain gulation. This bound is turned into an element-wise error indicator function E(K) for element K by summing the contributions from its edges. The final form for the toolbox equation Becomeswhere n is the unit normal of edge and the braced term is the jump in flux across the element edge. The L2 norm is computed over the element K. This error indicator is computed by the pdejmps function.The Mesh RefinerThe PDE Toolbox is geared to elliptic problems. For reasons of accuracy and ill-conditioning, they require the elements not to deviate too much from beingequilateral. Thus, even at essentially one-dimensional solution features, such as boundary layers, the refinement technique must guarantee reasonably shaped triangles.When an element is refined, new nodes appear on its mid sides, and if the neighbor triangle is not refined in a similar way, it is said to have hanging nodes. The final triangulation must have no hanging nodes, and they are removed by splitting neighbor triangles. To avoid further deterioration oftriangle quality in successive generations, the “longest edge bisection” scheme Rosenberg-Stenger [8] is used, in which the longest side of a triangle is always split, whenever any of the sides have hanging nodes. This guarantees that no angle is ever smaller than half the smallest angle of the original triangulation. Two selection criteria can be used. One, pdead worst, refines all elements with value of the error indicator larger than half the worst of any element. The other, pdeadgsc, refines all elements with an indicator value exceeding a user-defined dimensionless tolerance. The comparison with the tolerance is properly scaled with respect to domain and solution size, etc.The Termination CriteriaFor smooth solutions, error equi distribution can be achieved by the pde adgsc selection if the maximum number of elements is large enough. The pdead worst adaption only terminates when the maximum number of elements has been exceeded. This mode is natural when the solution exhibits singularities. The error indicator of the elements next to the singularity may never vanish, regardless of element size.外文翻译假定估计值，如果是最接近的准确的求解,通过解决线性问题得到更精确的值当为正数时,( 有一个解，即使也有一个解都是不需要的。

数学专业外文翻译---幂级数的展开及其应用In the us n。

we XXX its convergence n。

a power series always converges to a n。

We can use simple power series。

as well as XXX quadrature methods。

to find this n。

However。

this n will address another issue: can an arbitrary n f(x) be expanded into a power series?XXX n will address this XXX power series can be seen as an n of reality。

so we can start to solve the problem of expanding a n f(x) into a power series by considering f(x) and polynomials。

To do this。

we will introduce the following formula without proof:Taylor'XXX that if a n f(x) has derivatives of order n+1 in a neighborhood of x=x0.then we can use the following XXX:f(x)=f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0)^2+。

+f^(n)(x0)(x-x0)^n+r_n(x)Here。

r_n(x) represents the remainder term.XXX (x) is given by (x-x)n+1.This formula is of the (9-5-1) type for the Taylor series。

重庆理工大学数学专业英语学院学号姓名年月 2012年12月17日CONTROLLABILITY OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELA Y可控的无穷时滞中立型泛函微分方程In this article, we establish a result about controllability to the following class of partial neutral functional di fferential equations with infinite delay:0,),()(0≥⎪⎩⎪⎨⎧∈=++=∂∂t x xt t F t Cu ADxt Dxt t βφ (1)在这篇文章中，我们建立一个关于可控的结果偏中性与无限时滞泛函微分方程的下面的类： 0,),()(0≥⎪⎩⎪⎨⎧∈=++=∂∂t x xt t F t Cu ADxt Dxt tβφ (1) where the state variable (.)x takes values in a Banach space ).,(E and the control (.)u is given in[]0),,,0(2>T U T L ,the Banach space of admissible control functions with U a Banach space. C is abounded linear operator from U into E, A : D(A) ⊆ E → E is a linear operator on E, B is the phase space of functions mapping (−∞, 0] into E, which will be specified later, D is a bounded linear operator from B into E defined byB D D ∈-=ϕϕϕϕ,)0(0状态变量(.)x 在).,(E 空间值和控制用(.)u 受理控制范围[]0),,,0(2>T U T L 的Banach 空间，Banach 空间。

Math problems about BaseballSummaryBaseball is a popular bat-and-ball game involving both athletics and wisdom. There are strict restrictions on the material, size and manufacture of the bat.It is vital important to transfer the maximum energy to the ball in order to give it the fastest batted speed during the hitting process.Firstly, this paper locates the center-of-percussion (COP) and the viberational node based on the single pendulum theory and the analysis of bat vibration.With the help of the synthesizing optimization approach, a mathematical model is developed to execute the optimized positioning for the “sweet spot”, and the best hitting spot turns out not to be at the end of the bat. Secondly, based on the basic model hypothesis, taking the physical and material attributes of the bat as parameters, the moment of inertia and the highest batted ball speed (BBS) of the “sweet spot” are evaluated using different parameter values, which enables a quantified comparison to be made on the performance of different bats.Thus finally explained why Major League Baseball prohibits “corking” and metal bats.In problem I, taking the COP and the viberational node as two decisive factors of the “sweet zone”, models are developed respectively to study the hitting effect from the angle of energy conversion.Because the different “sweet spots” decided by COP and the viberational node reflect different form of energy conversion, the “space-distance” concept is introduced and the “Technique for Order Preferenceby Similarity to Ideal Solution (TOPSIS) is used to locate the “sweet zone” step by step. And thus, it is proved that t he “sweet spot” is not at the end of the bat from the two angles of specific quantitative relationship of the hitting effects and the inference of energy conversion.In problem II, applying new physical parameters of a corked bat into the model developed in Problem I, the moment of inertia and the BBS of the corked bat and the original wood bat under the same conditions are calculated. The result shows that the corking bat reduces the BBS and the collision performance rather than enhancing the “sweet spot” effect. On the other hand, the corking bat reduces the moment of inertia of the bat, which makes the bat can be controlled easier. By comparing the two Team # 8038 Page 2 of 20 conflicting impacts comprehensively, the conclusion is drawn that the corked bat will be advantageous to the same player in the game, for which Major League Baseball prohibits “corking”.In problem III, adopting the similar method used in Problem II, that is, applying different physical parameters into the model developed in Problem I, calculate the moment of inertia and the BBS of the bats constructed by different material to analyze the impact of the bat material on the hitting effect. The data simulation of metal bats performance and wood bats performance shows that the performance of the metal bat is improved for the moment of inertia is reduced and the BBS is increased. Our model and method successfully explain why Major League Baseball, for the sake of fair competition, prohibits metal bats.In the end, an evaluation of the model developed in this paper is given, listing itsadvantages s and limitations, and providing suggestions on measuring the performance of a bat.Restatement of the ProblemExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit.Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?2.1 Analysis of Problem IFirst explain the “sweet spot” on a baseball bat, and then develop a model that helps ex plain why this spot isn’t at the end of the bat.[1]There are a multitude of definitions of the sweet spot:1) the location which produces least vibrational sensation (sting) in the batter's hands2) the location which produces maximum batted ball speed3) the location where maximum energy is transferred to the ball4) the location where coefficient of restitution is maximum5) the center of percussionFor most bats all of these "sweet spots" are at different locations on the bat, so one is often forced to define the sweet spot as a region.If explained based on torque, this “sweet spot” might be at the end of the bat, which is known to be empirically incorrect.This paper is going to explain this empirical paradox by exploring the location of the sweet spot from a reasonable angle.Based on necessary analysis, it can be known that the sweet zone, which is decided by the center-of-percussion (COP) and the vibrational node, produces the hitting effect abiding by the law of energy conversion.The two different sweet spots respectively decided by the COP and the viberational node reflect different energy conversions, which forms a two-factor influence.2.2 Analysis of Problem IIProblem II is to explain whether “corking” a bat enhances the “sweet spot” effect and why Major League Baseball prohibits “corking”.[4]In order to find out what changes will occur after corking the bat, the changes of the bat’s parameters should be analyzed first:1) The mass of the corked bat reduces slightly than before;2) Less mass (lower moment of inertia) means faster swing speed;3) The mass center of the bat moves towards the handle;4) The coefficient of restitution of the bat becomes smaller than before;5) Less mass means a less effective collision；6) The moment of inertia becomes smaller.[5][6]By analyzing the changes of the above parameters of a corked bat, whether the hitting effect of the sweet spot has been changed could be identified and then the reason for prohibiting “corking” might be cle ar.2.3 Analysis of Problem IIIFirst, explain whether the bat material imposes impacts on the hitting effect; then, develop a model to predict different behavior for wood or metal bats to find out the reason why Major League Baseball prohibits metal bats?The mass (M) and the center of mass (CM) of the bat are different because of the material out of which the bat is constructed. The changes of the location of COP and moment of inertia ( I bat ) could be inferred.[2][3]Above physical attributes influence not only the swing speed of the player (the less the moment of inertia-- I bat is, the faster the swing speed is) but also the sweet spot effect of the ball which can be reflected by the maximum batted ball speed (BBS). The BBS of different material can be got by analyzing the material parameters that affect the moment of inertia.Then, it can be proved that the hitting effects of different bat material are different.3.Model Assumptions and Symbols3.1 Model Assumptions1) The collision discussed in th is paper refers to the vertical collision on the “sweet spot”;2) The process discussed refers to the whole continuous momentary process starting from the moment the bat contacts the ball until the moment the ball departs from the bat;3) Both the bat and the ball discussed are under common conditions.3.2 Instructions Symbolsa kinematic factor kthe rotational inertia of the object about its pivot point I0the mass of the physical pendulum Mthe location of the center-of-mass relative to the pivot point dthe distance between the undetermined COP and the pivot Lthe gravitational field strength gthe moment-of-inertia of the bat as measured about the pivot point on the handleI the swing period of the bat on its axis round the pivotTthe length of the bat sthe distance from the pivot point where the ball hits the bat zvibration frequency fthe mass of the ball m4.Modeling and Solution4.1 Modeling and Solution to Problem I 4.1.1 Model Preparation 1) Analysis of the pushing force or pressure exerted on hands[1] Team # 8038 Page 7 of 20 Fig. 4-1 As showed in Fig. 4-1:If an impact force F were to strike the bat at the center-of-mass (CM) then point P would experience a translational acceleration - the entire bat would attempt to accelerate to the left in the same direction as the applied force, without rotating about the pivot point.If a player was holding the bat in his/her hands, this would result in an impulsive force felt inthe hands.If the impact force F strikes the bat below the center-of-mass, but above the center-of-percussion, point P would experience both a translational acceleration in the direction of the force and a rotational acceleration in the opposite direction as the bat attempts to rotate about its center-of-mass. The translational acceleration to the left would be greater than the rotational acceleration to the right and a player would still feel an impulsive force in the hands.If the impact force strikes the bat below the center-of-percussion, then point P would still experience oppositely directed translational and rotational accelerations, but now the rotational acceleration would be greater.If the impact force strikes the bat precisely at the center-of-percussion, then the translational acceleration and the rotational acceleration in the opposite direction exactly cancel each other.method: Instead of being distributed throughout the entire object, let the mass of the physical pendulum M be concentrated at a single point located at a distance L from the pivot point.This point mass swinging from the end of a string is now a "simple" pendulum, and its period would be the same as that of the original physical pendulum if the distance L wasThis location L is known as the "center-of-oscillation". A solid object which oscillates about a fixed pivot point is called a physical pendulum.When displaced from its equilibrium position the force of gravity will attempt to return the object to its equilibrium position, while its inertia will cause it to overshoot.As a result of this interplay between restoring force and inertia the object will swing back and forth, repeating its cyclic motion in a constant amount of time. This time, called the period, depends on the mass of the object M , the location of the center-of-mass relative to the pivot point d , the rotational inertia of the object about its pivot point I 0 and the gravitational field strength g according to4.1.2 Solutions to the two “sweet spot” regions1) Locating the COP[1][4]Determining the parameters：a. mass of the bat M ;b. length of the bat S (the distance between Block 1 and Block 5 in Fig 4-3);c. distance between the pivot and the center-of-mass d ( the distance between Block2 and Block3 in Fig. 4-3);d .swing period of the bat on its axis round the pivot T (take an adult male as an example: the distance between the pivot and the knob of the bat is 16.8cm (the distance between Block 1 and Block 2 in Fig. 4-3);e.distance between the undetermined COP and the pivot L (the distance between Block 2 and Block 4 in Fig. 4-3, that is the turning radius) .Fig.4-3 Table 4-1Block1 knobBlock2 pivotBlock 3 the center-of-mass（CM）Block 4 t he center of percussion (COP)Block 5the end of the batCalculation method of COP[1][4]:distance between the undetermined COP and the pivot:T 2g L= 4π 2 ( g is the gravity acceleration）（4-3）moment of inertia:I0 = T 2 MgL 4π 2 ( L is the turning radius, M is the mass) （4-4）Results:The reaction force on the pivot is less than 10% of the bat-and-ball collision force. When the ball falls on any point in the “sweet spot” region, the area where the collision force reduction is less than 10% is (0.9 L ,1.1L) cm, which is called “Sweet Zone 1”.2) Determining the vibrational nodeThe contact between bat and ball, we consider it a process of wave ransmission.When the bat excited by a baseball of rapid flight, all of these modes, (as well as some additional higher frequency modes) are excited and the bat vibrates .We depend on the frequency modes ,list the following two modes:The fundamental bending mode has two nodes, or positions of zero displacement). One is about 6-1/2 inches from the barrel end close to the sweet spot of the bat. The other at about 24 inches from the barrel end (6 inches from the handle) at approximately the location of a right-handed hitter's right hand.Fundamental bending mode 1 (215 Hz) The second bending mode has three nodes, about 4.5 inches from the barrel end, a second near the middle of the bat, and the third at about the location of a right-handed hitter's left hand.Second bending mode 2 (670 Hz) The figures show the two bending modes of a freely supported baseball bat.The handle end of the bat is at the right, and the barrel end is at the left. The numbers on the axis represent inches (this data is for a 30 inch Little League wood baseball bat). These figures were obtained from a modal analysis experiment. In this opinion we prefer to follow the convention used by Rod Cross[2] who defines the sweet zone as Team # 8038 Page 11 of 20 the region located between the nodes of the first and second modes of vibration (between about 4-7 inches from the barrel end of a 30-inch Little League bat).The solving time in accordance with the searching times and backtrack times. It isobjective to consider the two indices together.4.1.3 Optimization Modelwood bat (ash)swing period T 0.12sbat mass M 876.0 15gbat length S 86.4 cmCM position d 41.62cmcoefficient of restitution BBCOR 0.4892initialvelocity vin 7.7m /sswing speed vbat 15.3 m/sball mass mball850.5gAdopting the parameters in the above table and based on the quantitative regions in sweet zone 1 and 2 in 4.1.2, the following can be drawn:[2] Sweet zone 1 is (0.9 L ,1.1L) = (50cm , 57.8358cm)Sweet zone 2 is ( L* , L* ) = (48.41cm,55.23cm)define the position of Block 2 which is the pivot as the origin of the number axis, and x as a random point on the number axis.Optimization modeling[2]The TOPSIS method is a technique for order preference by similarity to ideal solution whose basic idea is to transform the integrated optimal region problem into seeking the difference among evaluation objects—“distance”. That is, to determine the most ideal position and the acceptable most unsatisfactory position according to certain principals, and then calculate the distance between each evaluation object and Team # 8038 Page 12 of 20 the most ideal position and the distance between each evaluation object and the acceptable most unsatisfactory position.Finally, the “sweet zone” can be drawn by an integrated c omparison.Step 1 : Standardization of the extent value Standardization is performed via range transformation,x * = a dimensionless quantity，and x * ∈[0,1]Step 2:x min = min{0.9 L, L* }x max = max{1.1L, L* }x ∈( x min , x max ) ;Step 3: Calculating the distance The Euclidean distance of the positive ideal position is:The Euclidean distance of the negative ideal position is:Step 4: Seeking the integrated optimal region The integrated evaluation index of theevaluation object is:……………………………（4-5）Optimization positioningConsidering bat material physical attributes of normal wood, when the period is T = 0.12s and the vibration frequency is f = 520 HZ, the ideal “sweet zone” extent can be drawn as [51.32cm , 55.046cm] .As this consequence showed, the “sweet spot” cannot be at the end the bat. This conclusion can also be verified by the model for problem II.4.1.4 Verifying the “sweet spot” is not at the end of the bat1) Analyzed from the hitting effect According to Formula 4-11 and Table 4-2, the maximum batted-ball-speed of Team # 8038 Page 13 of 20 the “sweet spot” can be calculated as BBS sweet = 27.4 m / s , and the maximum batted-ball-speed of the bat end can be calculated as BBS end = 22.64 m / s . It is obvious that the “sweet spot” is not at the end of the bat.2) Analyzed from the energy According to the definition of “sweet spot” and the method of locating the “sweet spot”, energy loss should be minimized in order to transfer the maximum energy to the ball.When considering the “sweet spot” region from angle of torque, the position for maximum torque is no doubt at the end of the bat. But this position is also the maximum rebounded point according to the theory of force interaction. Rebound wastes the energy which originally could send the ball further.To sum up the above points: it can be proved that the “sweet spot” is not at the end of the bat by studying the quantitative relationship of the hitting effect and the inference of the energy transformation.4.2 Modeling and Solution to Problem II4.2.1 Model Preparation 1) Introduction to corked bat[5][6]: Fig 4-7As shown in Fig 4-7, Corking a bat the traditional way is a relatively easy thing to do. You just drill a hole in the end of the bat, about 1-inch in diameter, and about 10-inches deep. You fill the hole with cork, super balls, or styrofoam - if you leave the hole empty the bat sounds quite different, enough to give you away. Then you glue a wooden plug, like a 1-inch dowel, in to the end. Finally you sand the end to cover the evidence.Some sources suggest smearing a bit of glue on the end of the bat and sprinkling sawdust over it so help camouflage the work you have done.2) Situation studied：Situation of the best hitting effect: vertical collision occurs between the bat and the ball, and the energy loss of the collision is less than 10% and more than 90% of the momentum transfers from the bat to the ball (the hitting point is the “sweet spot”). Team # 8038 Page 14 of 203) Analysis of COR After the collision the ball rebounded backwards and the bat rotated about its pivot. The ratio of ball speeds (outgoing / incoming) is termed the collision efficiency, e A . A kinematic factor k , which is essentially the effective massof the bat, is defined as…………………………………………………………（4-6）I bat where I nat is the moment-of-inertia of the bat as measured about the pivot point on the handle, and z is the distance from the pivot point where the ball hits the bat. Once the kinematic factor k has been determined and the collision efficiency e A has been measured, the BBCOR is calculated from…………………………………………（4-7）Physical parameters vary with the material:The hitting effect of the “sweet spot” varies with the d ifferent bat material.It is related with the mass of the ball M , the center-of-mass ( CM ), the location of the center-of-mass d , the location of COP L , the coefficient of restitution BBCOR and the moment-of-inertia of the bat I bat .4.2.2 Controlling variable method analysisM is the mass of the object；is the location of the center-of-mass relative to the d pivot point；is the gravitational field strength；bat is the moment-of-inertia of the bat g I as measured about the pivot point on the handle; z is the distance from the pivot point where the ball hits the bat；vinl speed just before collision. The following formulas are got by sorting the above variables[1]:…………………………………………… （4-8 ）is the incoming ball speed；vbat is the bat swing………………………………………………（4-9）…………………………………………（4-10）Associating the above three formulas with formula (4-6) and (4-7), the formulas among BBS , the mass M , the center-of-mass ( CM ), the location of COP, the coefficient of restitution BBCOR and the moment-of-inertia of the bat I bat are:………………………（4-11）……………………………………………………………（4-12）………………………………………………………………（4-13）It can be known form formula (4-11), (4-12) and (4-13):1) When the coefficient of restitution BBCOR and mass M of the material changes, BBS will change;2) When mass M and the location of center-of mass CM changes, I bat changes, which is the dominant factor deciding the swing speed.4.2.3 Analysis of corked bat and wood bat [5][6]It makes the game unfair to increase the hitting accuracy by corking the bat.4.2.4 Reason for prohibiting corking[4]If the swing speed is unchanged, the corked bat cannot hit the ball as far as the wood bat, but it grants the player more reaction time and increases the accuracy. Influenced by a multitude of random factors, vertical collision cannot be assured in each hitting.The following figure shows the situation of vertical collision between the bat and the ball:In order to realize the best hitting effect, all of the BBS drawn from the above calculating results are assumed to be vertical collision.But in a professional baseball game, because the hitting accuracy is also one of the decisive factors, increasing the hitting accuracy equals to enhance the hitting effect. After cording the bat, the moment-of-inertia of the bat reduces, which improves the player’s capability of controlling the bat.Thus, the hitting is more accurate, which makes the game unfair.To sum up, in order to avoiding the unfairness of a game, Major League Baseball prohibits “corking”4.3 Modeling and Solution to Problem IIIAccording to the model developed in Problem I, the hitting effect of the “sweet spot” depends on the mass of the ball M , the center-of-mass ( CM ), the location of CM d , the location of COP L , the coefficient of restitution BBCOR and the moment-of-inertia of the bat I bat . An analysis of metal bat and wood bat is made.4.3.1 Analysis of metal bat and wood bat [8][9]I bat (1) is the moment-of-inertia of the metal bat, and I bat (2) is moment-of-inertia of the wood bat.Conclusion: Because the hitting part is hollow for the metal bat, the CM is closer tothe handle of bat for an aluminum bat than a wood bat. I bat of metal bat is less than I bat of the wood bat, which increases the swing speed.It means the professional players are able to watch the ball travel an additional 5-6 feet before having to commit to a swing, which makes the hitting more accurate to damage the fairness of the game.4.3.2 Reason for prohibiting the metal bat [4]Through the studies on the above models: 【4.3.1-(1)】proves the best hitting effect of a metal bat is better than a wood bat.【4.3.1-(2）】proves the hitting accuracy of a metal bat is better than a wood bat.To sum up，the metal bat is better than the wood bat in both the two factors, which makes the game unfair.And that’s why Major League Baseball prohibits metal bat.5.Strengths and Weaknesses of the Model5.1.Strengths1) The model, with the help of the single-pendulum theory and the analysis of the vibration of the bat and the ball, locates the COP and vibrational node of bats respectively, and locates the “sweet spot” influenced by multitudes of factors utilizing the integrated optimization method.The overall optimized solution makes the “sweet spot” more persuasive.2) The Model analyzes the integrated performance of different material bats from the aspects of the maximum initial velocity (BBS) and the hitting accuracy, and explains why corked bats are prohibited successfully.3) Deriving results from the controlling variable method analysis and taking the Law of Energy Conservation and the theories of structural dynamics as foundation enable to avoid the complicated mechanical analysis and derivation.5.2 WeaknessesThe model fails to evaluate the performance of bats exactly from the angle of the game relationship between the maximum initial velocity (BBS) and the accuracy when evaluating the hitting effect.6.References [1]/~drussell/bats-new/sweetspot.html [2] Mathematical Modeling Contest: Selection and Comment on Award-winning Papers [3].au/~cross/baseball.htm[4]Adair, R.K.1994.The Physics of Baseball. New York: Harper Perennial.[5]D.A.Russell,"Hoop frequency as a predictor of performance for softball bats," Engineering of Sport 5 Vol.2, pp.641-647 (International Sports Engineering Association, 2004).Proceedings of the 5th International Conference on the Engineering of Sport, UC Davis, September 11-15, 2004.[6] A.M.Nathan, "Some Remarks on Corked Bats" (June 10, 2003)[7] ESPN Baseball Tonight, on June 3, 2003 aired a nice segment in which Buck Showalter showed how to cork a bat, drilling the hole, filling it with cork, and plugging the end.[8] R.M. Greenwald R.M., L.H.Penna , and J.J.Crisco,"Differences in Batted Ball Speed with Wood and Aluminum Baseball Bats: A Batting Cage Study," J. Appl.Biomech., 17, 241-252 (2001).[9] J.J.Crisco, R.M.Greenwald, J.D.Blume, and L.H.Penna, "Batting performance ofwood and metal baseball bats," Med. Sci. Sports Exerc., 34(10), 1675-1684 (2002) [10] Robert K. Adair, The Physics of Baseball, 3rd Ed., (Harper Collins, 2002)[11] R. Cross, "The sweet spot of a baseball bat," Am. J. Phys., 66(9), 771-779 (1998)[12] A. M. Nathan, "The dynamics of the baseball-bat collision," Am. J. Phys., 68(11), 979-990 (2000)[13] K.Koenig, J.S.Dillard, D.K.Nance, and D.B.Shafer, "The effects of support conditions on baseball bat testing," Engineering of Sport 5 Vol.2, pp.87-93 (International Sports Engineering Association, 2004).Proceedings of the 5th International Conference on the Engineering of Sport, UC Davis, September 11-15, 2004.棒球的数学问题简介：棒球是一个受欢迎的bat-and-ball游戏,既包括体育和智慧。

数学与应用数学专业论文英文文献翻译Chapter 3InterpolationInterpolation is the process of defining a function that takes on specified values at specified points. This chapter concentrates on two closely related interpolants, the piecewise cubic spline and the shape-preserving piecewise cubic named “pchip”.3.1 The Interpolating PolynomialWe all know that two points determine a straight line. More precisely, any two points in the plane, ),(11y x and ),(11y x , with )(21x x ≠ , determine a unique first degree polynomial in x whose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same straight line graph.This generalizes to more than two points. Given n points in the plane, ),(k k y x ，n k ,,2,1 =, with distinct k x ’s, there is aunique polynomial in x of degree less than n whose graph passes through the points. It is easiest to remember that n , the number of data points, is also the number of coefficients, although some of the leading coefficients might be zero, so the degree might actually be less than 1-n . Again, there are many different formulas for the polynomial, but they all define the same function.This polynomial is called the interpolating polynomial because it exactly re- produces the given data.n k y x P k k ,,2,1,)( ==，Later, we examine other polynomials, of lower degree, that only approximate the data. They are not interpolating polynomials.The most compact representation of the interpolating polynomial is the La- grange form.∑∏⎪⎪⎭⎫ ⎝⎛--=≠k k k j j k j y x x x x x P )( There are n terms in the sum and 1-n terms in each product, so this expression defines a polynomial of degree at most 1-n . If )(x P is evaluated at k x x = , all the products except the k th are zero. Furthermore, the k th product is equal to one, so the sum is equal to k y and theinterpolation conditions are satisfied.For example, consider the following data set:x=0:3;y=[-5 -6 -1 16];The commanddisp([x;y])displays0 1 2 3-5 -6 -1 16 The Lagrangian form of the polynomial interpolating this data is)16()6()2)(1()1()2()3)(1()6()2()3)(2()5()6()3)(2)(1()(--+----+---+-----=x x x x x x x x x x x x x P We can see that each term is of degree three, so the entire sum has degree at most three. Because the leading term does not vanish, the degree is actually three. Moreover, if we plug in 2,1,0=x or 3, three of the terms vanish and the fourth produces the corresponding value from the data set.Polynomials are usually not represented in their Lagrangian form. More fre- quently, they are written as something like523--x xThe simple powers of x are called monomials and this form of a polynomial is said to be using the power form.The coefficients of an interpolating polynomial using its power form,n n n n c x c x c x c x P ++++=---12211)(can, in principle, be computed by solving a system of simultaneous linear equations⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡------n n n n n n n n n n n y y y c c c x x x x x x x x x 21212122212121111111 The matrix V of this linear system is known as a Vandermonde matrix. Its elements arej n kj k x v -=, The columns of a Vandermonde matrix are sometimes written in the opposite order, but polynomial coefficient vectors in Matlab always have the highest power first.The Matlab function vander generates Vandermonde matrices. For our ex- ample data set,V = vander(x)generatesV =0 0 0 11 1 1 18 4 2 127 9 3 1Thenc = V\y’computes the coefficientsc =1.00000.0000-2.0000-5.0000In fact, the example data was generated from the polynomial 523--x x .One of the exercises asks you to show that Vandermonde matrices are nonsin- gular if the points k x are distinct. But another one of theexercises asks you to show that a Vandermonde matrix can be very badly conditioned. Consequently, using the power form and the Vandermonde matrix is a satisfactory technique for problems involving a few well-spaced and well-scaled data points. But as a general-purpose approach, it is dangerous.In this chapter, we describe several Matlab functions that implement various interpolation algorithms. All of them have the calling sequencev = interp(x,y,u)The first two input arguments, x and y, are vectors of the same length that define the interpolating points. The third input argument, u, is a vector of points where the function is to be evaluated. The output, v, is the same length as u and has elements ))xterpyvuink(k(,,)(Our first such interpolation function, polyinterp, is based on the Lagrange form. The code uses Matlab array operations to evaluate the polynomial at all the components of u simultaneously.function v = polyinterp(x,y,u)n = length(x);v = zeros(size(u));for k = 1:nw = ones(size(u));for j = [1:k-1 k+1:n]w = (u-x(j))./(x(k)-x(j)).*w;endendv = v + w*y(k);To illustrate polyinterp, create a vector of densely spaced evaluation points.u = -.25:.01:3.25;Thenv = polyinterp(x,y,u);plot(x,y,’o’,u,v,’-’)creates figure 3.1.Figure 3.1. polyinterpThe polyinterp function also works correctly with symbolic variables. For example, createsymx = sym(’x’)Then evaluate and display the symbolic form of the interpolating polynomial withP = polyinterp(x,y,symx)pretty(P)produces-5 (-1/3 x + 1)(-1/2 x + 1)(-x + 1) - 6 (-1/2 x + 3/2)(-x + 2)x-1/2 (-x + 3)(x - 1)x + 16/3 (x - 2)(1/2 x - 1/2)xThis expression is a rearrangement of the Lagrange form of the interpolating poly- nomial. Simplifying the Lagrange form withP = simplify(P)changes P to the power formP =x^3-2*x-5Here is another example, with a data set that is used by the other methods in this chapter.x = 1:6;y = [16 18 21 17 15 12];disp([x; y])u = .75:.05:6.25;v = polyinterp(x,y,u);plot(x,y,’o’,u,v,’-’);produces1 2 3 4 5 616 18 21 17 15 12creates figure 3.2.Figure 3.2. Full degree polynomial interpolation Already in this example, with only six nicely spaced points, we canbegin to see the primary difficulty with full-degree polynomial interpolation. In between the data points, especially in the first and last subintervals, the function shows excessive variation. It overshoots the changes in the data values. As a result, full- degree polynomial interpolation is hardly ever used for data and curve fitting. Its primary application is in the derivation of other numerical methods.第三章 插值多项式插值就是定义一个在特定点取给定值得函数的过程。

第一段翻译(2)：what is the exact value of the number pai?a mathematician made an experiment in order to find his own estimation of the number pai.in his experiment,he used an old bicycle wheel of diameter 63.7cm.he marked the point on the tire where the wheel was touching the ground and he rolled the wheel straight ahead by turning it 20 times.next,he measured the distance traveled by the wheel,which was 39.69 meters.he divided the number 3969 by 20*63.7 and obtained 3.115384615 as an approximation of the number pai.of course,this was just his estimate of the number pai and he was aware that it was not very accurate.数π的精确值是什么？一位数学家做了实验以便找到他自己对数π的估计。

在试验中，他用了一直径63.1厘米的旧自行车轮。

他在车轮接触地面的轮胎上做了标记，而且将车轮向前转动20次。

接下来，他测量了车轮经过的距离，是39.69米。

他用3969除20*63.7得到了数π的近似值3.115384615。

当然，这只是对数π的估计值，并且他也意识到不是很准确。

第二段翻译(5)：one of the first articles which we included in the "History Topics" section archive was on the history of pai.it is a very popular article and has prompted many to ask for a similar article about the number e.there is a great contrast between the historical developments of these two numbers and in many ways writing a history of e is a much harder task than writing one of pai.the number e is,compared to pai,a relative newcomer on the mathematical scene.我们包括在“历史专题”部分档案中的第一篇文章就是历史上的π，这是一篇很流行的文章，也促使许多人想了解下一些有关数e的类似文章。

SIAM J. DISCRETE MATH.V ol. 26, No. 1, pp. 193–205ROMAN DOMINATION ON 2-CONNECTED GRAPHS∗CHUN-HUNG LIU†AND GERARD J. CHANG‡Abstract. A Roman dominating function of a graph G is a function f: V (G) →{0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f is w(f) = . The Roman domination number of G is the minimum weight of a Roman dominating function of G Chambers,Kinnersley, Prince, and West [SIAM J. Discrete Math.,23 (2009), pp. 1575–1586] conjectured that ≤[2n/3] for any 2-connected graph G of n vertices.This paper gives counterexamples to the conjecture and proves that≤max{[2n/3], 23n/34}for any 2-connected graph G of n vertices. We also characterize 2-connected graphs G for which = 23n/34 when 23n/34 > [2n/3].Key words. domination, Roman domination, 2-connected graphAMS. subject classifications. 05C69, 05C35D O I. 10.1137/0807330851. Introduction. Articles by ReVelle [14, 15] in the Johns Hopkins Magazine suggested a new variation of domination called Roman domination; see also [16] for an integer programming formulation of the problem. Since then, there have been several articles on Roman domination and its variations [1, 2, 3, 4, 5, 7, 8, 9, 10,11, 13, 17, 18, 19]. Emperor Constantine imposed the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armies, stationary and traveling. Each vertex (city) that has no army must have a neighboring vertex with a traveling army. Stationary armies then dominate their own vertices; a vertex with two armies is dominated by its stationary army, and its open neighborhood is dominated by the traveling army.In this paper, we consider (simple) graphs and loopless multigraphs G with vert ex set V (G) and edge set E(G). The degree of a vertex v∈V (G) is the number of edges incident to v. Note that the number of neighbors of v may be less than degGv in a loopless multigraph. A Roman dominating function of a graph G is a function f:V(G) →{0, 1, 2} such that whenever f(v) = 0, there exists a vertex u adjacent to v such that f(u) = 2. The weight of f, denoted by w(f), is defined as.For any subgraph H of G, let w(f,H) =. The Roman dominationnumber of G is the minimum weight of a Roman dominating function.Among the papers mentioned above, we are most interested in the one by Chambers et al. [2] in which extremal problems of Roman domination are discussed.In particular, they gave sharp bounds for graphs with minimum degree 1 or 2 and boundsof + and . After settling some special cases, they gave the following conjecture in an earlier version of the paper [2].Conjecture (Chambers et al. [2]). For any 2-connected graph G of n vertices, ≤[2n/3]。

Concrete MathematicsR. L. Graham, D. E. Knuth, O. Patashnik《Concrete Mathematics》，1.3 THE JOSEPHUS PROBLEM R. L. Graham, D. E. Knuth, O. Patashnik Sixth printing, Printed in the United States of America 1989 by Addison-Wesley Publishing Company，Reference 1-4pages具体数学R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克《具体数学》，1.3，约瑟夫环问题R.L.格雷厄姆，D.E.克努特，O.帕塔希尼克第一版第六次印刷于美国，韦斯利出版公司，1989年，引用8-16页1.递归问题本章研究三个样本问题。

这三个样本问题给出了递归问题的感性知识。

它们有两个共同的特点：它们都是数学家们一直反复地研究的问题；它们的解都用了递归的概念，按递归概念，每个问题的解都依赖于相同问题的若干较小场合的解。

2.约瑟夫环问题我们最后一个例子是一个以Flavius Josephus命名的古老的问题的变形，他是第一世纪一个著名的历史学家。

据传说，如果没有Josephus的数学天赋，他就不可能活下来而成为著名的学者。

在犹太|罗马战争中，他是被罗马人困在一个山洞中的41个犹太叛军之一，这些叛军宁死不屈，决定在罗马人俘虏他们之前自杀，他们站成一个圈，从一开始，依次杀掉编号是三的倍数的人，直到一个人也不剩。

但是在这些叛军中的Josephus和他没有被告发的同伴觉得这么做毫无意义，所以他快速的计算出他和他的朋友应该站在这个恶毒的圆圈的哪个位置。

在我们的变形了的问题中，我们以n个人开始，从1到n编号围成一个圈，我们每次消灭第二个人直到只剩下一个人。

例如，这里我们以设n= 10做开始。

Differential CalculusNewton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successfulaccomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch ofcalculus,differential calculus.In this article, we give su fficient conditions for controllability of some partial neutral functional di fferential equations with in finite delay. We suppose that the linear part is not necessarily densely de fined but satis fies the resolvent estimates of the Hille-Yosida theorem. The results are obtained using the integrated semigroups theory. An application is given to illustrate our abstract result. Key words Controllability; integrated semigroup; integral solution; in finity delay1 IntroductionIn this article, we establish a result about controllability to the following class of partial neutral functional di fferential equations with in finite delay:0,),()(0≥⎪⎩⎪⎨⎧∈=++=∂∂t x xt t F t Cu ADxt Dxt t βφ (1) where the state variable (.)x takes values in a Banach space ).,(E and the control(.)u isgiven in []0),,,0(2>T U T L ,the Banach space of admissible control functions with U a Banach space. C is a bounded linear operator from U into E, A : D(A) ⊆ E → E is a linear operator on E, B is the phase space of functions mapping (−∞, 0] into E, which will be speci fied later, D is a bounded linear operator from B into E de fined byB D D ∈-=ϕϕϕϕ,)0(00D is a bounded linear operator from B into E and for each x : (−∞, T ] → E, T > 0, and t ∈ [0, T ], xt represents, as usual, the mapping from (−∞, 0] into E de fined by]0,(),()(-∞∈+=θθθt x xtF is an E-valued nonlinear continuous mapping on B ⨯ℜ+.The problem of controllability of linear and nonlinear systems represented by ODE in finit dimensional space was extensively studied. Many authors extended the controllability concept to in finite dimensional systems in Banach space with unbounded operators. Up to now, there are a lot of works on this topic, see, for example, [4, 7, 10, 21]. There are many systems that can be written as abstract neutral evolution equations with in finite delay to study [23]. In recent years, the theory of neutral functional di fferential equations with in finite delay in in finite dimension was developed and it is still a field of research (see, for instance, [2, 9, 14, 15] and the references therein). Meanwhile, the controllability problem of such systems was also discussed by many mathematicians, see, for example, [5, 8]. The objective of this article is to discuss the controllability for Eq. (1), where the linear part is supposed to be non-densely de fined but satis fies the resolventestimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the sufficient conditions for controllability of some partial neutral functional differential equations with infinite delay. The results are obtained using the integrated semigroups theory and Banach fixed point theorem. Besides, we make use of the notion of integral solution and we do not use the analytic semigroups theory.Treating equations with in finite delay such as Eq. (1), we need to introduce the phase space B. To avoid repetitions and understand the interesting properties of the phase space, suppose that ).,(B B is a (semi)normed abstract linear space offunctions mapping (−∞, 0] into E, and satis fies the following fundamental axioms that were first introduced in [13] and widely discussed in [16].(A) There exist a positive constant H and functions K(.),M(.):++ℜ→ℜ,with K continuous and M locally bounded, such that, forany ℜ∈σand 0>a ,if x : (−∞, σ + a] → E, B x ∈σ and (.)x iscontinuous on [σ, σ+a], then, for every t in [σ, σ+a], the followingconditions hold:(i) B xt ∈, (ii) B tx H t x ≤)(,which is equivalent to B H ϕϕ≤)0(or every B ∈ϕ (iii) B σσσσx t M s x t K xt t s B )()(sup )(-+-≤≤≤(A) For the function (.)x in (A), t → xt is a B -valued continuous function for t in [σ, σ + a].(B) The space B is complete.Throughout this article, we also assume that the operator A satis fies the Hille-Yosida condition : (H1) There exist and ℜ∈ω，such that )(),(A ρω⊂+∞ and {}M N n A I n n ≤≥∈---ωλλωλ,:)()(sup （2） Let A0 be the part of operator A in )(A D de fined by {}⎩⎨⎧∈=∈∈=)(,,)(:)()(000A D x for Ax x A A D Ax A D x A D It is well known that )()(0A D A D =and the operator 0A generates a strongly continuous semigroup ))((00≥t t T on )(A D . Recall that [19] for all )(A D x ∈ and 0≥t ,one has )()(000A D xds s T f t ∈ and x t T x sds s T A t )(0)(00=+⎪⎭⎫ ⎝⎛⎰. We also recall that 00))((≥t t T coincides on )(A D with the derivative of the locally Lipschitz integrated semigroup 0))((≥t t S generated by A on E, which is, according to [3, 17, 18], a family of bounded linear operators on E, that satis fies(i) S(0) = 0,(ii) for any y ∈ E, t → S(t)y is strong ly continuous with values in E, (iii) ⎰-+=sdr r s r t S t S s S 0))()(()()(for all t, s ≥ 0, and for any τ > 0 there exists a constant l(τ) > 0, such thats t l s S t S -≤-)()()(τ or all t, s ∈ [0, τ] .The C0-semigroup 0))((≥'t t S is exponentially bounded, that is, there exist two constants M and ω,such that t e M t S ω≤')( for all t ≥ 0.Notice that the controllability of a class of non-densely de fined functional di fferential equations was studied in [12] in the finite delay case.、2 Main ResultsWe start with introducing the following de finition.De finition 1 Let T > 0 and ϕ ∈ B. We consider the following de finition.We say that a function x := x(., ϕ) : (−∞, T ) → E, 0 < T ≤ +∞, is an integral solution of Eq. (1) if(i) x is continuous on [0, T ) ,(ii)⎰∈t A D Dxsds 0)( for t ∈ [0, T ) , (iii)⎰⎰+++=t s t t ds x s F s Cu Dxsds A D Dx 00),()(ϕfor t ∈ [0, T ) ,(iv) )()(t t x ϕ= for all t ∈ (−∞, 0].We deduce from [1] and [22] that integral solutions of Eq. (1) are given for ϕ ∈ B, such that )(A D D ∈ϕ by the following system⎪⎩⎪⎨⎧-∞∈=∈+-'+'=⎰+∞→],0,(),()(),,0[,)),()(()(lim )(0t t t x t t ds x s F s Cu B s t S D t S Dxt t s ϕλϕλ 、 (3)Where 1)(--=A I B λλλ.To obtain global existence and uniqueness, we supposed as in [1] that (H2) 1)0(0<D K .(H3) E F →B ⨯+∞],0[:is continuous and there exists 0β> 0, such that B -≤-21021),(),(ϕϕβϕϕt F t F for ϕ1, ϕ2 ∈ B and t ≥ 0. (4) Using Theorem 7 in [1], we obtain the following result.Theorem 1 Assume that (H1), (H2), and (H3) hold. Let ϕ ∈ B such that D ϕ ∈ D(A). Then, there exists a unique integral solution x(., ϕ) of Eq. (1), de fined on (−∞,+∞) .De finition 2 Under the above conditions, Eq. (1) is said to be controllable on the interval J = [0, δ], δ > 0, if for every initial function ϕ ∈ B with D ϕ ∈ D(A) and for any e1 ∈ D(A), there exists a control u ∈ L2(J,U), such that the solution x(.) of Eq. (1) satis fies 1)(e x =δ.Theorem 2 Suppose that(H1), (H2), and (H3) hold. Let x(.) be the integral solution of Eq. (1) on (−∞, δ) , δ > 0, and assume tha t (see [20]) the linear operator W from U into D(A) de fined byds s Cu B s S Wu )()(limλδλδ⎰-'=+∞→， （5）nduces an invertible operator W ~on KerW U J L /),(2，such that there exist positive constants 1N and 2N satisfying 1N C ≤and 21~N W≤-，then, Eq. (1) is controllable on J provided that 1))(2221000<++δδωδωδβδβK e M N N e M D ， （6） Where )(max :0t K K t δδ≤≤=.Proof Following [1], when the integral solution x(.) of Eq. (1) exists on (−∞, δ) , δ > 0, it is given for all t ∈ [0, δ] byds s Cu s t S dt d ds x s F s t S dt d D t S x D t x t t s t ⎰⎰-+-+'+=000)()(),()()()(ϕ Ords x s B s t S D t S x D t x t s t ⎰-'+'+=+∞→00),()(lim)()(λλϕ ds s Cu B s t S t⎰-'++∞→0)()(lim λλ Then, an arbitrary integral solution x(.) of Eq. (1) on (−∞, δ) , δ > 0, satis fies x(δ) = e1 if and only ifds s Cu B s t S ds x s F s S d d D S x D e t s ⎰⎰-'+-+'+=+∞→0001)()(lim ),()()(λλδδδδϕδThis implies that, by use of (5), it su ffices to take, for all t ∈ J, {})()()(lim~)(01t ds s Cu B s t S W t u t ⎰-'=+∞→-λλ {})(),()(lim )(~0011t ds x s B s t S D S x D e W t s ⎰-'-'--=+∞→-λλϕδδin order to have x(δ) = e1. Hence, we must take the control as above, and consequently, the proof is reduced to the existence of the integral solution given for all t ∈[0, δ] by⎰-+'+=t s t ds z s F s t S dt d D t S z D t Pz 00),()()(:))((ϕ {ϕδδδD S z D z W C s t S dt d t )()(~)(001'---=⎰- ds s d z F B S )}(),()(lim 0τττδτλδλ⎰-'-+∞→ Without loss of generality, suppose that ω ≥ 0. Using simila r arguments as in [1], we can see hat, for every 1z ,)(2ϕδZ z ∈and t ∈ [0, δ] ,∞-+≤-210021)())(())((z z K e M D t Pz t Pz δδωβAs K is continuous and 1)0(0<K D ,we can choose δ > 0 small enough, such that1)2221000<++δδωδωδββK e M N N e M D .Then, P is a strict contraction in )(ϕδZ ,and the fixed point of P gives the unique integral olution x(., ϕ) on (−∞, δ] that veri fies x(δ) = e1.Remark 1 Suppose that all linear operators W from U into D(A) de fined byds s Cu B s b S Wu )()(lim 0λδλ⎰-'=+∞→0 ≤ a < b ≤ T, T > 0, induce invertible operators W ~ on KerW U b a L /)],,([2,such that there exist positive constants N1 and N2 satisfying 1N C ≤ and21~N W ≤-,taking N T =δ,N large enough and following [1]. A similar argumentas the above proof can be used inductively in 11],)1(,[-≤≤+N n n n δδ,to see that Eq. (1) is controllable on [0, T ] for all T > 0.Acknowledgements The authors would like to thank Prof. Khalil Ezzinbi and Prof. Pierre Magal for the fruitful discussions.References[1] Adimy M, Bouzahir H, Ezzinbi K. Existence and stability for some partial neutralfunctional di fferential equations with in finite delay. J Math Anal Appl, 2004, 294: 438–461[2] Adimy M, Ezzinbi K. A class of linear partial neutral functional di fferentialequations with nondense domain. J Dif Eq, 1998, 147: 285–332[3] Arendt W. Resolvent positive operators and integrated semigroups. Proc LondonMath Soc, 1987, 54(3):321–349[4] Atmania R, Mazouzi S. Controllability of semilinear integrodi fferentialequations with nonlocal conditions. Electronic J of Di ff Eq, 2005, 2005: 1–9[5] Balachandran K, Anandhi E R. Controllability of neutral integrodi fferentialin finite delay systems in Banach spaces. Taiwanese J Math, 2004, 8: 689–702[6] Balasubramaniam P, Ntouyas S K. Controllability for neutral stochasticfunctional di fferential inclusionswith in finite delay in abstract space. J Math Anal Appl, 2006, 324(1): 161–176、[7] Balachandran K, Balasubramaniam P, Dauer J P. Local null controllability ofnonlinear functional di ffer-ential systems in Banach space. J Optim Theory Appl, 1996, 88: 61–75[8] Balasubramaniam P, Loganathan C. Controllability of functional di fferentialequations with unboundeddelay in Banach space. J Indian Math Soc, 2001, 68: 191–203[9] Bouzahir H. On neutral functional differential equations. Fixed Point Theory, 2005, 5: 11–21The study of differential equations is one part of mathematics that, perhaps more than any other, has been directly inspired by mechanics, astronomy, and mathematical physics. Its history began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equation arising from problems in geometry and mechanics. There early discoveries, beginning about 1690, gradually led to the development of a lot of “special tricks” for solving certain special kinds of differential equations. Although these special tricks are applicable in mechanics and geometry, so their study is of practical importance.微分方程牛顿和莱布尼茨，完全相互独立，主要负责开发积分学思想的地步，迄今无法解决的问题可以解决更多或更少的常规方法。

Power Series Expansion and Its ApplicationsIn the previous section, we discuss the convergence of power series, in its convergence region, the power series always converges to a function. For the simple power series, but also with itemized derivative, or quadrature methods, find this and function. This section will discuss another issue, for an arbitrary function ()f x , can be expanded in a power series, and launched into.Whether the power series ()f x as and function The following discussion will address this issue.1 Maclaurin (Maclaurin) formulaPolynomial power series can be seen as an extension of reality, so consider the function ()f x can expand into power series, you can from the function ()f x and polynomials start to solve this problem. To this end, to give here without proof the following formula.Taylor (Taylor) formula, if the function ()f x at 0x x = in a neighborhood that until the derivative of order 1n +, then in the neighborhood of the following formula :20000()()()()()()n n f x f x x x x x x x r x =+-+-++-+… (9-5-1) Among10()()n n r x x x +=-That ()n r x for the Lagrangian remainder. That (9-5-1)-type formula for the Taylor.If so 00x =, get2()(0)()n n f x f x x x r x =+++++…, (9-5-2) At this point,(1)(1)111()()()(1)!(1)!n n n n n f f x r x x x n n ξθ+++++==++ (01θ<<).That (9-5-2) type formula for the Maclaurin.Formula shows that any function ()f x as long as until the1n +derivative, n can be equal to a polynomial and a remainder.We call the following power series()2(0)(0)()(0)(0)2!!n nf f f x f f x x x n '''=+++++…… (9-5-3) For the Maclaurin series.So, is it to ()f x for the Sum functions If the order Maclaurin series (9-5-3) the first 1n + items and for 1()n S x +, which()21(0)(0)()(0)(0)2!!n nn f f S x f f x x x n +'''=++++…Then, the series (9-5-3) converges to the function ()f x the conditions1lim ()()n n s x f x +→∞=.Noting Maclaurin formula (9-5-2) and the Maclaurin series (9-5-3) the relationship between the known1()()()n n f x S x r x +=+Thus, when()0n r x =There,1()()n f x S x +=Vice versa. That if1lim ()()n n s x f x +→∞=,Units must()0n r x =.This indicates that the Maclaurin series (9-5-3) to ()f x and function as the Maclaurin formula (9-5-2) of the remainder term ()0n r x → (when n →∞).In this way, we get a function ()f x the power series expansion:()()0(0)(0)()(0)(0)!!n n n nn f f f x x f f x x n n ∞='==++++∑……. (9-5-4) It is the function ()f x the power series expression, if, the function of the power series expansion is unique. In fact, assuming the functionf (x ) can be expressed as power series20120()n n n n n f x a x a a x a x a x ∞===+++++∑……, (9-5-5)Well, according to the convergence of power series can be itemized within the nature of derivation, and then make 0x = (power series apparently converges in the 0x = point), it is easy to get()2012(0)(0)(0),(0),,,,,2!!n nn f f a f a f x a x a x n '''====…….Substituting them into (9-5-5) type, income and ()f x the Maclaurin expansion of (9-5-4) identical.In summary, if the function f (x ) contains zero in a range of arbitraryorder derivative, and in this range of Maclaurin formula in the remainder to zero as the limit (when n → ∞,), th en , the function f (x ) can start forming as (9-5-4) type of power series.Power Series()20000000()()()()()()()()1!2!!n n f x f x f x f x f x x x x x x x n '''=+-+-++-……,Known as the Taylor series.Second, primary function of power series expansionMaclaurin formula using the function ()f x expanded in power series method, called the direct expansion method.Example 1Test the function ()x f x e =expanded in power series of x . Solution because()()n x f x e =,(1,2,3,)n =…Therefore()(0)(0)(0)(0)1n f f f f '''====…,So we get the power series21112!!n x x x n +++++……, (9-5-6) Obviously, (9-5-6)type convergence interval (,)-∞+∞, As (9-5-6)whether type ()x f x e = is Sum function, that is, whether it converges to ()x f x e = , but also examine remainder ()n r x . Because1e ()(1)!xn n r x x n θ+=+ (01θ<<)，且x x x θθ≤≤,Therefore11e e ()(1)!(1)!xx n n n r x x x n n θ++=<++,Noting the value of any set x ,x e is a fixed constant, while the series (9-5-6) is absolutely convergent, so the general when the item when n →∞,10(1)!n xn +→+ , so when n → ∞,there10(1)!n xx e n +→+, From thislim ()0n n r x →∞=This indicates that the series (9-5-6) does converge to ()x f x e =, therefore21112!!x n e x x x n =+++++…… (x -∞<<+∞). Such use of Maclaurin formula are expanded in power series method, although the procedure is clear, but operators are often too Cumbersome, so it is generally more convenient to use the following power series expansion method.Prior to this, we have been a functionx-11, x e and sin x power series expansion, the use of these known expansion by power series of operations, we can achieve many functions of power series expansion. This demandfunction of power series expansion method is called indirect expansion .Example 2Find the function ()cos f x x =,0x =,Department in the power series expansion.Solution because(sin )cos x x '=,And3521111sin (1)3!5!(21)!n n x x x x x n +=-+-+-++……，（x -∞<<+∞） Therefore, the power series can be itemized according to the rules of derivation can be342111cos 1(1)2!4!(2)!n n x x x x n =-+-+-+……,（x -∞<<+∞） Third, the function power series expansion of the application example The application of power series expansion is extensive, for example, can use it to set some numerical or other approximate calculation of integral value.Example 3 Using the expansion to estimate arctan x the value of π. Solution because πarctan14= Because of357arctan 357x x x x x =-+-+…, (11x -≤≤),So there1114arctan14(1)357π==-+-+…Available right end of the first n items of the series and as an approximation of π. However, the convergence is very slow progression to get enough items to get more accurate estimates of πvalue.此外文文献选自于：Walter.Rudin.数学分析原理(英文版)[M].北京：机械工业出版社.幂级数的展开与其应用在上一节中，我们讨论了幂级数的收敛性，在其收敛域内，幂级数总是收敛于一个和函数．对于一些简单的幂级数，还可以借助逐项求导或求积分的方法，求出这个和函数．本节将要讨论另外一个问题，对于任意一个函数()f x ，能否将其展开成一个幂级数，以与展开成的幂级数是否以()f x 为和函数下面的讨论将解决这一问题．一、 马克劳林(Maclaurin)公式幂级数实际上可以视为多项式的延伸，因此在考虑函数()f x 能否展开成幂级数时，可以从函数()f x 与多项式的关系入手来解决这个问题．为此，这里不加证明地给出如下的公式．泰勒(Taylor)公式 如果函数()f x 在0x x =的某一邻域内，有直到1n +阶的导数，则在这个邻域内有如下公式：()20000000()()()()()()()()()2!!n n n f x f x f x f x f x x x x x x x r x n '''=+-+-++-+…,(951) 其中(1)10()()()(1)!n n n f r x x x n ξ++=-+．称()n r x 为拉格朗日型余项．称(951)式为泰勒公式．如果令00x =，就得到2()(0)()n n f x f x x x r x =+++++…， (952)此时，(1)(1)111()()()(1)!(1)!n n n n n f f x r x x x n n ξθ+++++==++, (01θ<<)．称(952)式为马克劳林公式．公式说明，任一函数()f x 只要有直到1n +阶导数，就可等于某个n 次多项式与一个余项的和．我们称下列幂级数()2(0)(0)()(0)(0)2!!n nf f f x f f x x x n '''=+++++…… (953) 为马克劳林级数．则，它是否以()f x 为和函数呢若令马克劳林级数(953)的前1n +项和为1()n S x +，即()21(0)(0)()(0)(0)2!!n nn f f S x f f x x x n +'''=++++…，则，级数(953)收敛于函数()f x 的条件为1lim ()()n n s x f x +→∞=．注意到马克劳林公式(952)与马克劳林级数(953)的关系，可知1()()()n n f x S x r x +=+．于是，当()0n r x =时，有1()()n f x S x +=．反之亦然．即若1lim ()()n n s x f x +→∞=则必有()0n r x =．这表明，马克劳林级数(953)以()f x 为和函数⇔马克劳林公式(952)中的余项()0n r x → (当n →∞时)．这样，我们就得到了函数()f x 的幂级数展开式：()()20(0)(0)(0)()(0)(0)!2!!n n n nn f f f f x x f f x x x n n ∞='''==+++++∑……(954)它就是函数()f x 的幂级数表达式，也就是说，函数的幂级数展开式是唯一的．事实上，假设函数()f x 可以表示为幂级数20120()n n n n n f x a x a a x a x a x ∞===+++++∑……，(955)则，根据幂级数在收敛域内可逐项求导的性质，再令0x =(幂级数显然在0x =点收敛)，就容易得到()2012(0)(0)(0),(0),,,,,2!!n nn f f a f a f x a x a x n '''====……．将它们代入(955)式，所得与()f x 的马克劳林展开式(954)完全相同．综上所述，如果函数()f x 在包含零的某区间内有任意阶导数，且在此区间内的马克劳林公式中的余项以零为极限(当n →∞时)，则，函数()f x 就可展开成形如(954)式的幂级数．幂级数()00000()()()()()()1!!n n f x f x f x f x x x x x n '=+-++-……,称为泰勒级数．二、 初等函数的幂级数展开式利用马克劳林公式将函数()f x 展开成幂级数的方法，称为直接展开法． 例1 试将函数()x f x e =展开成x 的幂级数． 解 因为()()n x f x e =,(1,2,3,)n =…所以()(0)(0)(0)(0)1n f f f f '''====…，于是我们得到幂级数21112!!n x x x n +++++……， (956)显然，(956)式的收敛区间为(,)-∞+∞，至于(956)式是否以()xf x e =为和函数，即它是否收敛于()x f x e =，还要考察余项()n r x ．因为1e ()(1)!xn n r x x n θ+=+ (01θ<<)， 且x x x θθ≤≤，所以11e e ()(1)!(1)!x x n n n r x x x n n θ++=<++． 注意到对任一确定的x 值，x e 是一个确定的常数，而级数(956)是绝对收敛的，因此其一般项当n →∞时，10(1)!n x n +→+，所以当n →∞时，有 10(1)!n x x e n +→+， 由此可知lim ()0n n r x →∞=． 这表明级数(956)确实收敛于()x f x e =，因此有21112!!x n e x x x n =+++++…… (x -∞<<+∞)． 这种运用马克劳林公式将函数展开成幂级数的方法，虽然程序明确，但是运算往往过于繁琐，因此人们普遍采用下面的比较简便的幂级数展开法．在此之前，我们已经得到了函数x-11，x e 与sin x 的幂级数展开式，运用这几个已知的展开式，通过幂级数的运算，可以求得许多函数的幂级数展开式．这种求函数的幂级数展开式的方法称为间接展开法．例2 试求函数()cos f x x =在0x =处的幂级数展开式．解 因为(sin )cos x x '=，而3521111sin (1)3!5!(21)!n n x x x x x n +=-+-+-++……,（x -∞<<+∞）， 所以根据幂级数可逐项求导的法则，可得342111cos 1(1)2!4!(2)!n n x x x x n =-+-+-+……,（x -∞<<+∞）． 三、 函数幂级数展开的应用举例幂级数展开式的应用很广泛，例如可利用它来对某些数值或定积分值等进行近似计算．例3 利用arctan x 的展开式估计π的值．解 由于πarctan14=， 又因357arctan 357x x x x x =-+-+…， (11x -≤≤)， 所以有1114arctan14(1)357π==-+-+…． 可用右端级数的前n 项之和作为π的近似值．但由于级数收敛的速度非常慢，要取足够多的项才能得到π的较精确的估计值．此外文文献选自于：Walter.Rudin.数学分析原理(英文版)[M].北京：机械工业出版社.。

2.5笛卡尔几何学的基本概念（basic concepts of Cartesian geometry）课文5-A the coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily , we do not talk about area by itself ,instead, we talk about the area of something . This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was to represent geometric points by numbers. The procedure for points in a plane is this ：Two perpendicular reference lines (called coordinate axes) are chosen, one horizont al (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection denoted by O, is called theorigin. On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance ,although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the points.Figure 2-5-1 illustrates some examples.The point with coordinates (3,2) lies three units to the right of the y-axis and two units above the x-axis.The number 3 is called the x-coordinate of the point,2 its y-coordinate. Points to the left of the y-axis have a negative x-coordinate; those below the x-axis have a negtive y-coordinate. The x-coordinateof a point is sometimes called its abscissa and the y-coordinateis called its ordinate.When we write a pair of numberssuch as (a,b) to represent a point, we agree that the abscissa or x-coordinate,a is written first. For this reason, the pair(a,b) is often referred to as an ordered pair. It is clear that two ordered pairs (a,b) and (c,d) represent the same point if and only if we have a=c and b=d. Points (a,b) with both a and b positiveare said to lie in the first quadrant ,those with a<0 and b>0 are in the second quadrant ; and those with a<0 and b<0 are in the third quadrant ; and those with a>0 and b<0 are in the fourthquadrant. Figure 2-5-1 shows one point in each quadrant.The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin) . These lines determine three mutually perpendicular planes ,and each point in space can be completely described by specifying , with appropriate regard for signs ,its distances from these planes. We shall discuee three-dimensional Cartesian geometry in more detail later on ; for the present we confine our attention to plane analytic geometry.课文5-A：笛卡尔几何坐标系正如前面所提到的，积分应用的一种是计算面积。