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数学与应用数学专业概率论的发展大学毕业论文英文文献翻译及原文

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数学与应用数学专业论文英文文献翻译

数学与应用数学专业论文英文文献翻译

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。

数学专业毕业设计文献翻译

数学专业毕业设计文献翻译

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。

数学与应用数学英文文献及翻译

数学与应用数学英文文献及翻译

（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。

概率论与数理统计英文文献

概率论与数理统计英文文献

Introduction to probability theory andmathematical statisticsThe theory of probability and the mathematical statistic are carries on deductive and the induction science to the stochastic phenomenon statistical rule, from the quantity side research stochastic phenomenon statistical regular foundation mathematics discipline, the theory of probability and the mathematical statistic may divide into the theory of probability and the mathematical statistic two branches. The probability uses for the possible size quantity which portrays the random event to occur. Theory of probability main content including classical generally computation, random variable distribution and characteristic numeral and limit theorem and so on. The mathematical statistic is one of mathematics Zhonglian department actually most directly most widespread branches, it introduced an estimate (rectangular method estimate, enormousestimate), the parameter supposition examination, the non-parameter supposition examination, the variance analysis and the multiple regression analysis, the fail-safe analysis and so on the elementary knowledge and the principle, enable the student to have a profound understanding tostatistics principle function. Through this curriculum study, enables the student comprehensively to understand, to grasp the theory of probability and the mathematical statistic thought and the method, grasps basic and the commonly used analysis and the computational method, and can studies in the solution economy and the management practice question using the theory of probability and the mathematical statistic viewpoint and the method.Random phenomenonFrom random phenomenon, in the nature and real life, some things are interrelated and continuous development. In the relationship between each other and developing, according to whether there is a causal relationship, very different can be divided into two categories: one is deterministic phenomenon. This kind of phenomenon is under certain conditions, will lead to certain results. For example, under normal atmospheric pressure, water heated to 100 degrees Celsius, is bound to a boil. This link is belong to the inevitability between things. Usually in natural science is interdisciplinary studies and know the inevitability, seeking this kind of inevitable phenomenon.Another kind is the phenomenon of uncertainty. This kind of phenomenon is under certain conditions, the resultis uncertain. The same workers on the same machine tools, for example, processing a number of the same kind of parts, they are the size of the there will always be a little difference. As another example, under the same conditions, artificial accelerating germination test of wheat varieties, each tree seed germination is also different, there is strength and sooner or later, respectively, and so on. Why in the same situation, will appear this kind of uncertain results? This is because, we say "same conditions" refers to some of the main conditions, in addition to these main conditions, there are many minor conditions and the accidental factor is people can't in advance one by one to grasp. Because of this, in this kind of phenomenon, we can't use the inevitability of cause and effect, the results of individual phenomenon in advance to make sure of the answer. The relationship between things is belong to accidental, this phenomenon is called accidental phenomenon, or a random phenomenon.In nature, in the production, life, random phenomenon is very common, that is to say, there is a lot of random phenomenon. Issue such as: sports lottery of the winning Numbers, the same production line production, the life of the bulb, etc., is a random phenomenon. So we say: randomphenomenon is: under the same conditions, many times the same test or survey the same phenomenon, the results are not identical, and unable to accurately predict the results of the next. Random phenomena in the uncertainties of the results, it is because of some minor, caused by the accidental factors.Random phenomenon on the surface, seems to be messy, there is no regular phenomenon. But practice has proved that if the same kind of a large number of repeated random phenomenon, its overall present certain regularity. A large number of similar random phenomena of this kind of regularity, as we observed increase in the number of the number of times and more obvious. Flip a coin, for example, each throw is difficult to judge on that side, but if repeated many times of toss the coin, it will be more and more clearly find them up is approximately the same number.We call this presented by a large number of similar random phenomena of collective regularity, is called the statistical regularity. Probability theory and mathematical statistics is the study of a large number of similar random phenomena statistical regularity of the mathematical disciplines.The emergence and development of probability theoryProbability theory was created in the 17th century, it is by the development of insurance business, but from the gambler's request, is that mathematicians thought the source of problem in probability theory.As early as in 1654, there was a gambler may tired to the mathematician PASCAL proposes a question troubling him for a long time: "meet two gamblers betting on a number of bureau, who will win the first m innings wins, all bets will be who. But when one of them wins a (a < m), the other won b (b < m) bureau, gambling aborted. Q: how should bets points method is only reasonable?" Who in 1642 invented the world's first mechanical addition of computer.Three years later, in 1657, the Dutch famous astronomy, physics, and a mathematician huygens is trying to solve this problem, the results into a book concerning the calculation of a game of chance, this is the earliest probability theory works.In recent decades, with the vigorous development of science and technology, the application of probability theory to the national economy, industrial and agricultural production and interdisciplinary field. Many of applied mathematics, such as information theory, game theory, queuing theory, cybernetics, etc., are based on the theory of probability.Probability theory and mathematical statistics is a branch of mathematics, random they similar disciplines are closely linked. But should point out that the theory of probability and mathematical statistics, statistical methods are each have their own contain different content.Probability theory, is based on a large number of similar random phenomena statistical regularity, the possibility that a result of random phenomenon to make an objective and scientific judgment, the possibility of its occurrence for this size to make quantitative description; Compare the size of these possibilities, study the contact between them, thus forming a set of mathematical theories and methods.Mathematical statistics - is the application of probability theory to study the phenomenon of large number of random regularity; To through the scientific arrangement of a number of experiments, the statistical method given strict theoretical proof; And determining various methods applied conditions and reliability of the method, the formula, the conclusion and limitations. We can from a set of samples to decide whether can with quite large probability to ensure that a judgment is correct, and can control the probability of error.- is a statistical method provides methods are used in avariety of specific issues, it does not pay attention to the method according to the theory, mathematical reasoning.Should point out that the probability and statistics on the research method has its particularity, and other mathematical subject of the main differences are:First, because the random phenomena statistical regularity is a collective rule, must to present in a large number of similar random phenomena, therefore, observation, experiment, research is the cornerstone of the subject research methods of probability and statistics. But, as a branch of mathematics, it still has the definition of this discipline, axioms, theorems, the definitions and axioms, theorems are derived from the random rule of nature, but these definitions and axioms, theorems is certain, there is no randomness.Second, in the study of probability statistics, using the "by part concluded all" methods of statistical inference. This is because it the object of the research - the range of random phenomenon is very big, at the time of experiment, observation, not all may be unnecessary. But by this part of the data obtained from some conclusions, concluded that the reliability of the conclusion to all the scope.Third, the randomness of the random phenomenon, refers to the experiment, investigation before speaking. After the real results for each test, it can only get the results of the uncertainty of a certain result. When we study this phenomenon, it should be noted before the test can find itself inherent law of this phenomenon.The content of the theory of probabilityProbability theory as a branch of mathematics, it studies the content general include the probability of random events, the regularity of statistical independence and deeper administrative levels.Probability is a quantitative index of the possibility of random events. In independent random events, if an event frequency in all events, in a larger range of stable around a fixed constant. You can think the probability of the incident to the constant. For any event probability value must be between 0 and 1.There is a certain type of random events, it has two characteristics: first, only a finite number of possible results; Second, the results the possibility of the same. Have the characteristics of the two random phenomenon called"classical subscheme".In the objective world, there are a large number of random phenomena, the result of a random phenomenon poses a random event. If the variable is used to describe each random phenomenon as a result, is known as random variables.Random variable has a finite and the infinite, and according to the variable values is usually divided into discrete random variables and the discrete random variable. List all possible values can be according to certain order, such a random variable is called a discrete random variable; If possible values with an interval, unable to make the order list, the random variable is called a discrete random variable.The content of the mathematical statisticsIncluding sampling, optimum line problem of mathematical statistics, hypothesis testing, analysis of variance, correlation analysis, etc. Sampling inspection is to pair through sample investigation, to infer the overall situation. Exactly how much sampling, this is a very important problem, therefore, is produced in the sampling inspection "small sample theory", this is in the case of the sample is small, the analysis judgment theory.Also called curve fitting and optimal line problem. Some problems need to be according to the experience data to find a theoretical distribution curve, so that the whole problem get understanding. But according to what principles and theoretical curve? How to compare out of several different curve in the same issue? Selecting good curve, is how to determine their error? ...... Is belong to the scope of the optimum line issues of mathematical statistics.Hypothesis testing is only at the time of inspection products with mathematical statistical method, first make a hypothesis, according to the result of sampling in reliable to a certain extent, to judge the null hypothesis.Also called deviation analysis, variance analysis is to use the concept of variance to analyze by a handful of experiment can make the judgment.Due to the random phenomenon is abundant in human practical activities, probability and statistics with the development of modern industry and agriculture, modern science and technology and continuous development, which formed many important branch. Such as stochastic process, information theory, experimental design, limit theory, multivariate analysis, etc.译文：概率论和数理统计简介概率论与数理统计是对随机现象的统计规律进行演绎和归纳的科学，从数量侧面研究随机现象的统计规律性的基础数学学科，概率论与数理统计又可分为概率论和数理统计两个分支。

数学与应用数学专业论文英文文献翻译

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 cu bic 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 a uniquepolynomial 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 exactlyre- 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 the interpolationconditions 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 the exercisesasks 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 ofpoints where the function is to be evaluated. The output, v, is the same length as u and has elements ))xterpkvyin(k,(,()uOur 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 can begin 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 usedfor 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。

数学与应用数学英文文献及翻译

数学与应用数学英文文献及翻译

（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。

数学与应用数学专业论文英文文献翻译

数学与应用数学专业论文英文文献翻译

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。

数学专业英语论文(含中文版)

数学专业英语论文(含中文版)

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 berf r T f r e ),(log lim)(+∞→=σSimilarly, )(f e λ，the e-type exponent of convergence of the zeros of meromorphic function f , is defined to berf r N f r e )/1,(loglim)(++∞→=λWe say that )(z f has regular order of growth if a meromorphic function )(z f satisfiesrf r T f r log ),(log lim)(+∞→=σWe consider the second order linear differential equation0=+''Af fWhere )()(z e B z A α=is a periodic entire function with period απω/2i =. The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19].When )(z A is rational in ze α，Bank and Laine [6] proved the following theoremTheorem A Let )()(z e B z A α=be a periodic entire function with period απω/2i = and rational in zeα.If )(ζB has poles of odd order at both ∞=ζ and 0=ζ, then for everysolution )0)((≠z f of (1.1), +∞=)(f λBank [5] generalized this result: The above conclusion still holds if we just suppose that both ∞=ζ and 0=ζare poles of )(ζB , and at least one is of odd order. In addition, the stronger conclusion)()/1,(l o g r o f r N ≠+ (1.2) holds. When )(z A is transcendental in ze α, Gao [10] proved the following theoremTheorem B Let ∑=+=p j jj b g B 1)/1()(ζζζ,where )(t g is a transcendental entire functionwith 1)(<g σ, p is an odd positive integer and 0≠p b ，Let )()(ze B z A =.Then anynon-trivia solution f of (1.1) must have +∞=)(f λ. In fact, the stronger conclusion (1.2) holds.An example was given in [10] showing that Theorem B does not hold when )(g σis any positive integer. If the order 1)(>g σ , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theoremsTheorem 1 Let )()(ze B z A α=,where )()/1()(21ζζζg g B +=,1g and 2g are entire functions with 2g transcendental and )(2g μnot equal to a positive integer or infinity, and 1g arbitrary. If Some properties of solutions of periodic second order linear differential equations )(z f and )2(i z f π+are two linearly independent solutions of (1.1), then+∞=)(f e λOr2)()(121≤+--g f e μλWe remark that the conclusion of Theorem 1 remains valid if we assume )(1g μ isnotequaltoapositiveintegerorinfinity,and2g arbitraryand stillassume )()/1()(21ζζζg g B +=,In the case when 1g is transcendental with its lower order not equal to an integer or infinity and 2g is arbitrary, we need only to consider )/1()()/1()(*21ηηηηg g B B +==in +∞<<η0,ζη/1<.Corollary 1 Let )()(z e B z A α=,where )()/1()(21ζζζg g B +=,1g and 2g are entire functions with 2g transcendental and )(2g μno more than 1/2, and 1g arbitrary.(a) If f is a non-trivial solution of (1.1) with +∞<)(f e λ,then )(z f and )2(i z f π+are linearly dependent.(b)If 1f and 2f are any two linearly independent solutions of (1.1), then +∞=)(21f f e λ.Theorem 2 Let )(ζg be a transcendental entire function and its lower order be no more than 1/2. Let )()(z e B z A =,where ∑=+=p j jj b g B 1)/1()(ζζζand p is an odd positive integer,then +∞=)(f λ for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.We remark that the above conclusion remains valid if∑=--+=pj jjbg B 1)()(ζζζWe note that Theorem 2 generalizes Theorem D when )(g σis a positive integer or infinity but2/1)(≤g μ. Combining Theorem D with Theorem 2, we haveCorollary 2 Let )(ζg be a transcendental entire function. Let )()(z e B z A = where ∑=+=p j jj b g B 1)/1()(ζζζand p is an odd positive integer. Suppose that either (i) or (ii)below holds:(i) )(g σ is not a positive integer or infinity; (ii) 2/1)(≤g μ;then +∞=)(f λfor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.2. Lemmas for the proofs of TheoremsLemma 1 ([7]) Suppose that 2≥k and that 20,.....-k A A are entire functions of period i π2,and that f is a non-trivial solution of0)()()(2)(=+∑-=k i j j z yz A k ySuppose further that f satisfies )()/1,(logr o f r N =+; that 0A is non-constant and rationalin ze ，and that if 3≥k ，then 21,.....-k A A are constants. Then there exists an integer qwith k q ≤≤1 such that )(z f and )2(i q z f π+are linearly dependent. The same conclusionholds if 0A is transcendental in ze ，andf satisfies )()/1,(logr o f r N =+，and if 3≥k ，thenas ∞→r through a set1L of infinite measure, wehave )),((),(j j A r T o A r T =for 2,.....1-=k j .Lemma 2 ([10]) Let )()(z e B z A α=be a periodic entire function with period 12-=απωi and betranscendental in z e α, )(ζB is transcendental and analytic on +∞<<ζ0.If )(ζB has a pole of odd order at ∞=ζ or 0=ζ(including those which can be changed into this case by varying the period of )(z A and Eq . (1.1) has a solution 0)(≠z f which satisfies )()/1,(logr o f r N =+,then )(z f and )(ω+z f are linearly independent. 3. Proofs of main resultsThe proof of main results are based on [8] and [15].Proof of Theorem 1 Let us assume +∞<)(f e λ.Since )(z f and )2(i z f π+are linearly independent, Lemma 1 implies that )(z f and )4(i z f π+must be linearly dependent. Let )2()()(i z f z f z E π+=,Then )(z E satisfies the differential equation222)()()(2))()(()(4z E cz E z E z E z E z A -''-'=, (2.1)Where 0≠c is the Wronskian of 1f and 2f (see [12, p. 5] or [1, p. 354]), and )()2(1z E c i z E =+πor some non-zero constant 1c .Clearly, E E /'and E E /''are both periodic functions with period i π2,while )(z A is periodic by definition. Hence (2.1) shows that 2)(z E is also periodic with period i π2.Thus we can find an analytic function )(ζΦin +∞<<ζ0,so that )()(2z e z E Φ=Substituting this expression into (2.1) yieldsΦΦ''+ΦΦ'-ΦΦ'+Φ=-2222)(43)(4ζζζζcB (2.2)Since both )(ζB and )(ζΦare analytic in }{+∞<<=ζζ1:*C ,the V aliron theory [21, p. 15] gives their representations as)()()(ζζζζb R B n =，)()()(11ζφζζζR n =Φ， （2.3）where n ，1n are some integers, )(ζR and )(1ζR are functions that are analytic and non-vanishing on }{*∞⋃C ，)(ζb and )(ζφ are entire functions. Following the same arguments as used in [8], we have),(),()/1,(),(φρρφρφρS b T N T ++=， （2.4） where )),((),(φρφρT o S =.Furthermore, the following properties hold [8])}(),(max{)()()(222E E E E f eL eR e e e λλλλλ===,)()()(12φλλλ=Φ=E eR ,Where )(2E eR λ(resp, )(2E eL λ) is defined to berE r N R r )/1,(loglim2++∞→(resp, rE r N R r )/1,(loglim2++∞→),Some properties of solutions of periodic second order linear differential equationswhere )/1,(2E r N R (resp. )/1,(2E r N L denotes a counting function that only counts the zeros of 2)(z E in the right-half plane (resp. in the left-half plane), )(1Φλis the exponent of convergence of the zeros of Φ in *C , which is defined to beρρλρlog )/1,(loglim)(1Φ=Φ++∞→NRecall the condition +∞<)(f e λ,we obtain +∞<)(φλ.Now substituting (2.3) into (2.2) yields+'+'+-'+'++=-21112111112)(43)()()()()(4φφζζφφζζζφζζζζζR R n R R n R cb R n n)222)1((1111111112112φφφφζφφζφφζζζ''+''+'''+''+'+'+-R R R R R n R R n n n （2.5）Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .Proof of Corollary 1 (b). Suppose 1f and 2f are linearlyindependentand +∞<)(21f f e λ,then +∞<)(1f e λ,and +∞<)(2f e λ.We deduce from the conclusion of Corollary 1 (a) that )(z f j and )2(i z f j π+are linearly dependent, j = 1; 2.Let)()()(21z f z f z E =.Then we can find a non-zero constant2c suchthat )()2(2z E c i z E =+π.Repeating the same arguments as used in Theorem 1 by using the fact that 2)(z E is also periodic, we obtain2)()(121≤+--g E e μλ,a contradiction since 2/1)(2≤g μ.Hence +∞=)(21f f e λ.Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies)()/1,(logr o f r N =+. We deduce 0)(=f e λ, so )(z f and )2(i z f π+ are linearlydependent by Corollary 1 (a). However, Lemma 2 implies that )(z f and )2(i z f π+are linearly independent. This is a contradiction. Hence )()/1,(log r o f r N ≠+holds for each non-trivial solution f of (1.1). This completes the proof of Theorem 2.Acknowledgments The authors would like to thank the referees for helpful suggestions to improve this paper. References[1] ARSCOTT F M. Periodic Di®erential Equations [M]. The Macmillan Co., New Y ork, 1964. [2] BAESCH A. On the explicit determination of certain solutions of periodic differentialequations of higher order [J]. Results Math., 1996, 29(1-2): 42{55.[3] BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differentialequations [J].Complex V ariables Theory Appl., 1997, 34(1-2): 7{17.[4] BANK S B. On the explicit determination of certain solutions of periodic differential equations[J]. Complex V ariables Theory Appl., 1993, 23(1-2): 101{121.[5] BANK S B. Three results in the value-distribution theory of solutions of linear differentialequations [J].Kodai Math. J., 1986, 9(2): 225{240.[6] BANK S B, LAINE I. Representations of solutions of periodic second order linear differentialequations [J]. J. Reine Angew. Math., 1983, 344: 1{21.[7] BANK S B, LANGLEY J K. Oscillation theorems for higher order linear differential equationswith entire periodic coe±cients [J]. Comment. Math. Univ. St. Paul., 1992, 41(1): 65{85.[8] CHIANG Y M, GAO Shi'an. On a problem in complex oscillation theory of periodic secondorder lineardifferential equations and some related perturbation results [J]. Ann. Acad. Sci. Fenn. Math., 2002, 27(2):273{290.一些周期性的二阶线性微分方程解的方法1． 简介和主要成果在本文中，我们假设读者熟悉的函数的数值分布理论[12，14，16]的基本成果和数学符号。

数学与应用数学论文中英文资料外文翻译文献

数学与应用数学论文中英文资料外文翻译文献

数学与应用数学论文中英文资料外文翻译文献UNITS OF M EASUR EM ENT AND FUNC TIONAL FOR M ( V o t i n g O u t c o m e s a n d C a m p a i g n E x p e n d i t u r e s )In the voting outcome equation in (2.28), R = 0.505. Thus, the share of campaign expenditures explains just over 50 percent of the variation in the election outcomes for this sample. This is a fairly sizable portionTwo important issues in applied economics are (1) understanding how changing theunits of measurement of the dependent and/or independent variables affects OLS estimates and (2) knowing how to incorporate popular functional forms used in e conomi c s i nt o regres s i o n analysis. The mathemati c s ne e ded for a ful l un de rs t anding of functional form issues is reviewed in Appendix A.The Effects of Changing Units of Measurement on OLSStatisticsIn Example 2.3, we chose to measure annual salary in thousands of dollars, and t he return on e quit y was mea s ured as a perc e n t (ra t her than a s a dec i ma l). I t is c ruci a l to know how salary and roe are measured in this example in order to make sense of the estimates in equation (2.39). We must also know that OLS estimates change in entirely expected ways when the units of measurement of the dependent and independent variables change. In Example2.3, suppose that, rather than measuring s a l ary in thousands of do l la rs, we m ea s u re it i n doll a rs. Let sal a rdol be sal a ry i n dollars (salardol =845,761 would be interpreted as $845,761.). Of course, salardol has a simple relationship to the salary measured in thousands of dollars: salardol ? 1,000? salary. We do not need to actually run the regression of salardol on roe to know that the estimated equation is: salaˆrdol = 963,191 +18,501 roe.We obtain the intercept and slope in (2.40) simply by multiplying the intercept and theslope in (2.39) by 1,000. This gives equations (2.39) and (2.40) the same interpretation.Looking at (2.40), if roe = 0, then salaˆrdol = 963,191, so the predicted salary is $963,191 [the same value we obtained from equation (2.39)]. Furthermore, if roe increases by one, then the predicted salary increases by $18,501; again, this isw hat w econcluded from our earlier analysis of equation (2.39).Generally, it is easy to figure out what happens to the intercept and slope estimates when the dependent variable changes units of measurement. If the dependent variable is multiplied by the constant c—which means each value in the s a m ple is multi pl i ed b yc—t h en t he OLS in t ercept a nd s lope esti m at es are als o multiplied by c. (This assumes nothing has changed about the independent variable.) In the CEO salary example, c ?1,000 in moving from salary to salardol.Chapter 2T he Sim pl e Re g re s sion ModelWe can also use the CEO salary example to see what happens when we change the units of measurement of the independent variable. Define roedec =roe/100 to be t he d e cimal equiva l ent of ro e; t hus, roedec =0.23 means a return o n equi ty of23 percent. To focus on changing the unitsof measurement of the independent variable, we return to our original dependent variable, salary, which is measured in thousands of dollars. When we regress salary onroedec, we obtain salˆary =963.191 + 1850.1 roedec.T he coef fi ci e nt on roedec is 100 times t he coe ffi cient on roe i n (2.39). This i s as it should be. Changing roe by one percentage point is equivalent to Δroedec = 0.01. From (2.41), if Δ roedec = 0.01, then Δ salˆary = 1850.1(0.01) = 18.501, which is what is obtained by using (2.39). Note that, in moving from (2.39) to (2.41), the independentv ariable was divided b y 100, and so t h e OLS slope estim a te was multiplied by 100, preserving the interpretation of the equation. Generally, if the independent variable is divided or multiplied by some nonzero constant, c, then the OLS slope coefficient is also multiplied or divided by c respectively.The intercept has not changed in (2.41) because roedec =0 still corresponds to a z ero retur n on equity. In ge n eral, changin g t he uni t s of m easurem e nt of only the independent variable does not affect the intercept.In the previous section, we defined R-squared as a goodness-of-fit measure for OLS regression. We can also ask what happens to R2 when the unit of measurement of either the independent or the dependent variable changes. Without doing any algebra, we should know the result: the goodness-of-fit of the model should not depend on the units of measurement of our variables. For example, the amount of variation in salary, explained by the return on equity, should not depend on whether salary is measured in dollars or in thousands of dollars or on whether return on equity is a percent or a decimal. This intuition can be verified mathematically: using the definition of R2, it can be shown that R2 is, in fact, invariant to changes in the units of y or x.Incor por a ting Nonlinear ities in Simple R egressionSo far we have focused on linear relationships between the dependent and independent variables. As we mentioned in Chapter 1, linear relationships are notn early gen er a l enou g h for a ll e co nomi c a pplications. F ortuna t ely, it is rathe r e a s y to incorporate many nonlinearities into simple regression analysis by appropriately defining the dependent and independent variables. Here we will cover two possibilities that often appear in applied work.In reading applied work in the social sciences, you will often encounter re gr es sion equati o ns w he re the de pende nt varia bl e a pp ears in l og arit hm i c f orm. W h y is this done? Recall the wage-education example, where we regressed hourly wage on years of education. We obtained a slope estimate of 0.54 [ see equation (2.27)], which means that each additional year of education is predicted to increase hourly wage by54 cents.B ecaus e of t he l i near n at ur e of (2.27), 54 c ents is the i ncrea s e f or e i ther the fi rst year of education or the twentieth year; this may not be reasonable.Suppose, instead, that the percentage increase in wage is the same given one m ore yea r of e ducation. Model (2.27) does no t im ply a c onst a nt per c entag e i nc re ase: the percentage increases depends on the initial wage. A model that gives (approximately) a constant percentage effect is log(wage) =β 0 +β 1educ + u,(2.42) where log(.) denotes the natural logarithm. (See Appendix A for a review of logarithms.) In particular, if Δu =0, then % Δwage = (100* β 1) Δ educ.(2.43) N otice ho w we mult i ply β 1 b y 100 t o g et the perc e ntage change in w a ge give n one additional year of education. Since the percentage change in wage is the same for each additional year of education, the change in wage for an extra year of education increases aseducation increases; in other words, (2.42) implies an increasing return to education.B y e x ponenttiat i ng (2.42), we c an w ri t e wage =ex p(β 0+β 1educ + u). T his equationis graphed in Figure 2.6, with u = 0.Estimating a model such as (2.42) is straightforward when using simple regression. Just define the dependent variable, y, to be y = log(wage). The i ndependent v ar i able is represented b y x = e duc. The mechanics of O L S are the sa m e as before: the intercept and slope estimates are given by the formulas (2.17) and (2.19). In other words, we obtain β ˆ0 andβ ˆ1 from the OLS regression of log(wage) on educ.E X A M P L E 2 . 1 0( A L o g W a g e E q u a t i o n )Using the same data as in Example 2.4, but using log(wage) as the dependent variable, we obtain the following relationship: log(ˆwage) =0.584 +0.083 educ(2.44) n = 526, R =0.186.The coefficient on educ has a percentage interpretation when it is multiplied by 100: wage increases by 8.3 percent for every additional year of education. This is what economists mean when they refer to the “return to another year of education.”It is important to remember that the main reason for using the log of wage in (2.42) is to impose a constant percentage effect of education on wage. Once equation (2.42) is obtained, the natural log of wage is rarely mentioned. In particular, it is not correct to say that another year of education increases log(wage) by 8.3%.The intercept in (2.42) is not very meaningful, as it gives the predicted log(wage), when educ =0. The R-squared shows that educ explains about 18.6 percent of the variation in log(wage) (not wage). Finally, equation (2.44) might not capture all of the non-linearity in the relationship between wage and schooling.If there are“diplomae ffects,”t hen t he twelft h ye a r of e ducat i on—gradu a ti on from hi gh s c hool—c ould be worth much more than the eleventh year. We will learn how to allow for this kind of nonlinearity in Chapter 7. Another important use of the natural log is in obtaining a constant elasticity model.E X A M P L E 2 . 1 1( C E O S a l a r y a n d F i r m S a l e s )We can estimate a constant elasticity model relating CEO salary to firm sales. The data set is the same one used in Example 2.3, except we now relate salary to sales. Let sales be annual firm sales, measured in millions of dollars. A constant elasticity model is log(salary =β 0 +β 1log(sales) +u, (2.45) where β 1 is the elasticity of s a l ary w ith respe c t to sal es. T h is model fa ll s under t he simple regressio n model by defining the dependent variable to be y = log(salary) and the independent variable to be x = log(sales). Estimating this equation by OLS givesPart 1Regression Analysis with Cross-Sectional Data l og(salˆary)= 4.822 ?+0.257 log(sa le s)(2.46)n =209, R = 0.211.The coefficient of log(sales) is the estimated elasticity of salary with respect to sales. It implies that a 1 percent increase in firm sales increases CEO salary by about 0.257 pe r cent—t he usual int e rp re tation of an e la s ti c ity.The two functional forms covered in this section will often arise in the remainder of this text. We have covered models containing natural logarithms here because they a ppear so freque nt ly in a ppl ied wo r k. The i nterpr e tat i on of such m odel s w i l l n ot be much different in the multiple regression case.It is also useful to note what happens to the intercept and slope estimates if we change the units of measurement of the dependent variable when it appears in logarithmic form.B ecaus e t he ch a nge t o log ar i thm i c form approx i mates a proportionate change, i t makes sense that nothing happens to the slope. We can see this by writing the rescaled variable as c1yi for each observation i. The original equation is log(yi) =β 0 +β 1xi +ui. If we add log(c1) to both sides, we get log(c1) + log(yi) + [log(c1) β 0] +β 1xi + ui, orlog(c1yi) ? [log(c1) +β 0] +β 1xi +ui.(Remember that the sum of the logs is equal to the log of their product as shown in Appendix A.) Therefore, the slope is still ? 1, but the intercept is now log(c1) ? ? 0. Similarly, if the independent variable is log(x), and we change the units of measurement of x before taking the log, the slope remains the same but the intercept does not change. You will be asked to verify these claims in Problem 2.9.We end this subsection by summarizing four combinations of functional forms available from using either the original variable or its natural log. In Table 2.3, x and y stand for the variables in their original form. The model with y as the dependent variable and x as the independent variable is called the level-level model, because each variable appears in its level form. The model with log(y) as the dependentv ariable a nd x as t he independent va r i able i s called t he l og-level m od el. We w i ll no t explicitly discuss the level-log model here, because it arises less often in practice. In any case, we will see examples of this model in later chapters.Chapter 2Th e Simpl e R e gr e s s i on M od e lTable 2.3The last column in Table 2.3 gives the interpretation of β 1. In the log-level model, 100* β 1 i s so m e t imes called the s emi-elasti c ity of y wit h re s pe ct to x. As we mentioned in Example 2.11, in the log-log model, β1 is the elasticity of y with respect to x. Table 2.3 warrants careful study, as we will refer to it often in the r em aind er of the text.The Meaning of“Linear”RegressionThe simple regression model that we have studied in this chapter is also called the simple linear regression model. Yet, as we have just seen, the general model also allows for certain nonlinear relationships. So what does “linear”mean here? You can se e b y looking a t equ a ti o n (2.1) tha t y =β 0 +β 1x + u. The key i s t hat t his equati on i s linear in the parameters, β 0 and β 1. There are no restrictions on how y and x relate to the original explained and explanatory variables of interest. As we saw in Examples 2.7 and 2.8, y and x can be natural logs of variables, and this is quite common in applications. But we need not stop there. For example, nothing prevents us from using simple regression to estimate a model such as cons =β 0 +β 1√inc+u, where cons is annual consumption and inc is annual income.While the mechanics of simple regression do not depend on how y and x are defined, the interpretation of the coefficients does depend on their definitions. For successful empirical work, it is much more important to become proficient at interpreting coefficients than to become efficient at computing formulas such as (2.19). We will get much more practice with interpreting the estimates in OLS regression lines when we study multiple regression.There are plenty of models that cannot be cast as a linear regression model because they are not linear in their parameters; an example is cons = 1/(β 0 +β 1inc) + u.E s t im a ti on of such mode l s ta ke s us into t he real m of t he nonli ne ar regressi on model, which is beyond the scope of this text. For most applications, choosing a model that can be put into the linear regression framework is sufficient.EXPECTED VAL UES AND VAR IANCES OF THE OLSESTIM ATOR SI n Sec t ion 2.1, we defined the popula t ion m ode l y =β 0 +β 1x +u, a nd w e claimed that the key assumption for simple regression analysis to be useful is that the expected value of u given any value of x is zero. In Sections 2.2, 2.3, and 2.4, we discussed the algebraic properties of OLS estimation. We now return to the population model and study the statistical properties of OLS. In other words, we now view β ˆ0 a nd β ˆ1 as e s timat ors for th e pa rameters ? 0 and ? 1 t ha t appear in t he popula t ion model. This means that we will study properties of the distributions of ? ˆ0 and ? ˆ1 over different random samples from the population. (Appendix C contains definitions of estimators and reviews some of their important properties.)Unbiasedness of OLSW e be g i n by establishing the unbi a s e dne s s of OLS unde r a simple set of assumptions.For future reference, it is useful to number these assumptions using the prefix“S LR”for simple linear regression. The first assumption defines the population model.测量单位和函数形式在投票结果方程（2.28）中，R²=0.505。

数学专业英语论文(含中文版)

数学专业英语论文(含中文版)

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 successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.In this article, we give su fficient conditions for controllability of some partial neutral functional di fferential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies 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; infinity delay1 IntroductionIn 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 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. Cis 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 specified later, D is a bounded linear operator from B into E defined 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 defined by]0,(),()(-∞∈+=θθθt x xtF is an E-valued nonlinear continuous mapping on B ⨯ℜ+.The problem of controllability of linear and nonlinear systems repr esented by ODE in finit dimensional space was extensively studied. Many authors extended the controllability concept to infinite 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 infinite delay to study [23]. In recent years, the theory of neutral functional di fferential equations with infinite delay in infinitedimension was deve loped 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 defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We shall assume conditions that assure global existence and give the su fficient conditions for controllability of some partial neutral functional di fferential 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 infinite 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 of functions mapping (−∞, 0] into E, and satisfies the following fundamental axioms that were first introduced in [13] and widely discussedin [16].(A)There exist a positive constant H and functions K(.), M(.):++ℜ→ℜ,with K continuous and M locally bounded, such that, for any ℜ∈σand 0>a ,if x : (−∞, σ + a] → E, B x ∈σ and (.)x is continuous on [σ, σ+a], then, for every t in [σ, σ+a], the following conditions hold:(i) B xt ∈, (ii) Bt x H t x ≤)(,which is equivalent toB H ϕϕ≤)0(or every B ∈ϕ(iii) Bσσσσx t M s x t K xtts 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 satisfies 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 defined 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 continuoussemigroup ))((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∈ andx 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 satisfies(i) S(0) = 0, (ii) for any y ∈ E, t → S(t)y is strongly 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 aconstant 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 constantsM and ω,such that t e M t S ω≤')( for all t ≥ 0.Notice that the controllability of a class of non-de nsely defined functional di fferential equations was studied in [12] in the finite delay case.、2 Main ResultsWe start with introducing the following definition.Definition 1 Let T > 0 and ϕ ∈ B. We consider the following definition.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) ⎰∈tA D Dxsds 0)( for t ∈ [0, T ) ,(iii) ⎰⎰+++=ts tt ds x s F s Cu Dxsds A D Dx 0),()(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 ts 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), defined on (−∞,+∞) .Definition 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) satisfies 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 that (see [20]) the linear operator W from U into D(A) defined 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 constants1N and 2N satisfying 1N C ≤and 21~N W ≤-，then, Eq. (1) is controllable on J providedthat1))(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 tt s t ⎰⎰-+-+'+=000)()(),()()()(ϕOrdsx s B s t S D t S x D t x tst ⎰-'+'+=+∞→00),()(lim)()(λλϕds s Cu B s t S t⎰-'++∞→0)()(limλλThen, an arbitrary integral solution x(.) of Eq. (1) on (−∞, δ) , δ > 0, satisfies x(δ) = e1 if and only ifdss Cu B s t S ds x s F s S d d D S x D e ts ⎰⎰-'+-+'+=+∞→001)()(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 )(~011t ds x s B s t S D S x D e W ts⎰-'-'--=+∞→-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⎰-+'+=ts t ds z s F s t S dtd 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 )}(),()(limτττδτλδλ⎰-'-+∞→Without loss of generality, suppose thatω ≥ 0. Using similar arguments as in [1], we can seehat, for every1z ,)(2ϕδZ z ∈and t ∈ [0, δ] ,∞-+≤-210021)())(())((z z K e M D t Pz t Pz δδωβAs K is continuous and1)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 verifies x(δ) = e1.Remark 1 Suppose that all linear operators W from U into D(A) defined byds s Cu B s b S Wu )()(limλδλ⎰-'=+∞→0 ≤ a < b ≤ T, T > 0, induce invertible operators W ~ on KerW U b a L /)],,([2,such that thereexist positive constants N1 and N2 satisfying 1N C ≤ and21~N W ≤-,taking NT =δ,N large enough and following [1]. A similar argument as 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 neutral functional di fferenti al equations with infinite delay. J Math Anal Appl, 2004, 294: 438–461[2] Adimy M, Ezzinbi K. A class of linear partial neutral functional differential equations withnondense domain. J Dif Eq, 1998, 147: 285–332[3] Arendt W. Resolvent positive operators and integrated semigroups. Proc London Math Soc,1987, 54(3):321–349[4] Atmania R, Mazouzi S. Controllability of semilinear integrodifferential equations withnonlocal conditions. Electronic J of Diff Eq, 2005, 2005: 1–9[5] Balachandran K, Anandhi E R. Controllability of neutral integrodifferential infinite delaysystems in Banach spaces. Taiwanese J Math, 2004, 8: 689–702[6] Balasubramaniam P, Ntouyas S K. Controllability for neutral stochastic functional differentialinclusionswith infinite delay in abst ract space. J Math Anal Appl, 2006, 324(1): 161–176、[7] Balachandran K, Balasubramaniam P, Dauer J P. Local null controllability of nonlinearfunctional differ-ential systems in Banach space. J Optim Theory Appl, 1996, 88: 61–75 [8] Balasubramaniam P, Loganathan C. Controllability of functional differential equations withunboundeddelay in Banach space. J Indian Math Soc, 2001, 68: 191–203[9] Bouzahir H. On neutral functional differential equations. Fixed Point Theory, 2005, 5: 11–21 The 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。

国外数学期刊英汉对照

国外数学期刊英汉对照

国外数学期刊英汉对照作者:admin 日期:2009-06-17字体大小: 小中大Advances in Applied Mathematics 应用数学Advances in Applied Probability 应用概率论进展Advances in Computational Mathematics 计算数学进展Advances in Mathematics 数学进展Algebra Colloquium 代数学讨论会Algebras and Representation Theory 代数和表示理论American Mathem atical Monthly 美国数学月刊American Statistician 美国统计员Annals of Applied Probability 应用概率论年报Annals of Global Analysis and Geometry 整体分析与几何学年报Annals of Mathematics and Artificial Intelligence 人工智能论题年报Annals of Operations Research 运筹学研究年报Annals of Probability 概率论年报Annals of Pure and Applied Logic 抽象和应用逻辑年报Annals of Statistics 统计学年报Annals of The Institute of Statistical Mathematics 统计数学学会年报Applicable Algebra in Engineering Communication and Computing 代数在工程通信与计算中的应用Applied and Computational Harmonic Analysis 调和分析应用和计算Applied Categorical Structures 应用范畴结构Applied Mathematics and Computation 应用数学与计算Applied Mathematics and Optimization 应用数学与最优化Applied Mathematics Letters 应用数学快报Archive for History of Exact Sciences 科学史档案Archive for Mathem atical Logic 数理逻辑档案Archive for Rational Mechanics and Analysis 理性力学和分析档案Archives of Computational Methods in Engineering 工程计算方法档案Asymptotic Analysis 渐近线分析Autonomous Robots 机器人British Journal of Mathematical & Statistical Psychology 英国数学与统计心理学杂志Bulletin of The American Mathematical Society 美国数学会快报Bulletin of the London Mathematical Society 伦敦数学会快报Calculus of Variations and Partial Differential Equations 变分法与偏微分方程Combinatorics Probability & Computing 组合概率与计算Combustion Theory and Modeling 燃烧理论建模Communications in Algebra 代数通讯Communications in Contemporary Mathematics 当代数学通讯Communications in Mathematical Physics 数学物理学通讯Communications in Partial Differential Equations 偏微分方程通讯Communications in Statistics-Simulation and Computation 统计通讯–模拟与计算Communications on Pure and Applied Mathematics 纯数学与应用数学通讯Computational Geometry-Theory and Applications 计算几何- 理论与应用Computational Optimization and Applications 优化计算与应用Computational Statistics & Data Analysis 统计计算与数据分析Computer Aided Geometric Design 计算机辅助几何设计Computer Physics Communications 计算机物理通讯Computers & Mathematics with Applications 计算机与数学应用Computers & Operations Research 计算机与运筹学研究Concurrent Engineering-Research and Applications 共点工程- 研究与应用Conformal Geometry and Dynamics 投影几何与力学Decision Support Systems 决策支持系统Designs Codes and Cryptography 编码设计与密码系统Differential Geom etry and Its Applications 微分几何及其应用Discrete and Continuous Dynamical Systems 离散与连续动力系统Discrete Applied Mathematics 应用离散数学Discrete Computational Geometry 离散计算几何学Discrete Event Dynamic Systems-Theory and Applications 离散事件动态系统- 理论和应用Discrete Mathematics 离散数学Educational and Psychological Measurement 教育与心理测量方法Engineering Analysis with Boundary Elem ents 工程边界元素分析Ergodic Theory and Dynamical System s 遍历理论和动力系统European Journal of Applied Mathematics 欧洲应用数学杂志European Journal of Combinatorics 欧洲组合数学杂志European Journal of Operational Research 欧洲运筹学杂志Experimental Mathematics 实验数学Expert Systems with Applications 专家系统应用Finite Fields and Their Applications 有限域及其应用Foundations of 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国际系统科学杂志International Statistical Review 国际统计评论Inverse Problems 反比问题Journal of Algebra 代数学报Journal of Algebraic Combinatorics 代数组合数学学报Journal of Algebraic Geometry 代数几何学报Journal of Algorithm s 算法学报Journal of Applied Mathematics and Mechanics 数学应用与力学学报Journal of Applied Probability 应用概率杂志Journal of Approximation Theory 近似值理论杂志Journal of Combinatorial Optimization 组合最优化学报Journal of Combinatorial Theory Series B 组合理论学报B辑Journal of Combinatorial Theory Series A 组合理论学报A辑Journal of Computational Acoustics 声学计算杂志Journal of Computational Analysis and Applications 计算分析与应用学报Journal of Computational and Applied Mathematics 计算与应用数学杂志Journal of Computational Biology 生物计算杂志Journal of Computational Mathematics 计算数学学报Journal of Computational Neuroscience 神经系统计算杂志Journal of Computational Neuroscience 神经系统计算杂志Journal of Differential Equations 微分方程组学报Journal of Econom etrics 计量经济学会会刊Journal of Engineering Mathematics 工程数学学报Journal of Functional Analysis 泛函分析学报Journal of Geometry and Physics 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Theoretical Probability 概率理论杂志K-Theory K-理论Lecture Notes in Economics and Mathematical Systems 经济学和数学体系讲座Lecture Notes on Mathematics 数学讲座Letters in Mathematical Physics 数学物理学通讯Linear Algebra and Its Applications 线性代数及其应用Mathematical Biosciences 数理生物学Mathematical Finance 数理财金学Mathematical Geology 数理地质学Mathematical Intelligencer 数学情报Mathematical Logic Quarterly 数理逻辑学报Mathematical Methods in the Applied Sciences 数学应用学科Mathematical Methods of Operations Research 运筹学研究Mathematical Models & Methods in Applied Sciences 数学模拟与应用方法Mathematical Physics Electronic J 数学物理电子杂志Mathematical Proceedings of the Society 数学学会学报Mathematical Programming 数学规划Mathematical Social Sciences 数学社会科学Mathematics and Computers in Simulation 数学与电脑模拟Mathematics and Mechanics of Solids 数学与固体力学Mathematics Archives 数学档案库Mathematics Magazine 数学杂志Mathematics of Computation 计算数学Mathematics of Control Signals and Systems 控制信号系统数学Memoirs of the American Mathematical Society 美国数学会备忘录Modem Logic 现代逻辑学Nonlinear Analysis-Theory Methods & Applications 非线性分析- 理论与应用Nonlinearity 非线性特性Notices 短评Numerical Algorithms 数字算法Numerical Methods for Partial Differential Equations 偏微分方程式数值方法Oxford Bulletin of Economics and Statistics 牛津大学经济与统计快报Potential Analysis 位势分析Proceedings of the American Mathem atical Society 美国数学会会议录Proceedings of the London Mathematical Society 伦敦数学会会议录Proceedings of the Royal Society of Edinburgh Section A-Mathem atics 英国爱丁堡皇家学会会议录A分册数学Quarterly Journal of Mathematics 数学季刊Quarterly Journal of Mechanics and Applied Mathematics 数学与应用数学季刊Quarterly of Applied Mathematics 应用数学季刊Queueing Systems 排列系统Random Structures and Algorithms 随机结构与算法Reports on Mathematical Physics 数学物理学报告Representation Theory 表示法理论Robotics and Autonomous Systems 机器人技术和自动系统Rocky Mountain Journal of Mathematics 数学难题学报Set-Valued Analysis 精点分析Statistical Papers 统计学论文Statistics & Probability Letters 统计和概率通讯Statistics and Computing 统计和计算Stochastic Analysis and Applications 随机分析和应用Stochastic Environm ental Research and Risk Assessm ent 随机环境论研究和风险评估Stochastic Processes and Their Applications 随机过程及其应用Studies in Applied Mathematics 应用数学研究The College Mathematics Journal 大学数学杂志The Electronic Journal of Combinatorics 组合数学电子期刊Theory of Computing Systems 计算方法理论Theory of Probability and Its Applications 概率理论及其应用程序理论Topology 拓扑学Topology and Its Applications 拓扑学及其应用Transactions of the American Mathematical Society 美国数学会学报Applied Numerical Mathematics《应用数值数学》荷兰Annales Scientifiques de l'École Normale Supérieure《高等师范学校科学纪事》法国Applied and Computational Harmonic Analysis《应用和计算谐波分析》美国Applied Stochastic Models in Business and Industry《商业与工业应用随机模型》英国Acta Applicandae Mathematicae 《应用数学学报》荷兰Advances in Computational Mathematics《计算数学进展》荷兰Annals of Mathematics and Artificial Intelligence《数学与人工智能纪事》荷兰Annals of Operations Research《运筹学纪事》荷兰Annals of the Institute of Statistical Mathematics《统计数理研究所纪事》日本。

数学与应用数学专业数学教学中的师生互动研究大学毕业论文外文文献翻译及原文

数学与应用数学专业数学教学中的师生互动研究大学毕业论文外文文献翻译及原文

毕业设计（论文）外文文献翻译文献、资料中文题目：数学教学中的师生互动研究文献、资料英文题目：文献、资料来源：文献、资料发表（出版）日期：院（部）：专业：数学与应用数学专业班级：姓名：学号：指导教师：翻译日期： 2017.02.14毕业设计（论文）外文文献翻译译文题目：数学教学中的师生互动研究专业：数学与应用数学专业外文资料翻译译文数学教学中的师生互动研究[摘要] 基于心理学、教育学及数学学科特点的分析可知, 数学教学中的师生互动有着深厚的理论底蕴.师生互动的内涵体现在主体平等、认知协调、情感共鸣及教学互补四个方面.实施师生合作的教学模式是师生互动的必然要求, 为此, 在数学教学过程中, 教师应建立平等民主的师生关系;加强师生间的交流与合作;创设探索性的问题情境;实施计划导控.[ 关键词] 数学教学;师生互动;教学模式长期以来, 数学教学中教师一言堂、满堂灌现象严重, 学生所领会的通常只是教师或课本编写者的观点 , 他们很少有表达自己思想观点的机会.这样 , 充当被动听讲角色的学生就不能发挥自己的主动性,知识掌握的效果会大打折扣, 并导致能力培养的速度减缓 , 严重阻碍学生创新精神和实践能力的培养.大学数学教育界近年来逐渐重视教学改革, 问题解决、启发式、研究式已渐入人心, 但这些改革方案往往停留在理论探讨上, 实际教学中仍旧是注入式盛行 .究其原因, 主要在于教师的教学观没有根本转变, 教学改革仅重视“教”的活动而忽视“学”的活动 , 因而不能准确地了解学生的真实思维活动 , 不能科学地把握教学的进程和节奏 , 极大地影响了教学效果 .为此, 就有必要对数学教学中的师生互动问题做出探讨.一数学教学中师生互动的理论基础从发展心理学的角度看, 实践活动是人的心理、认识、意识产生和发展的基础.人的心理是在活动中形成和发展的, 人的个性的丰富程度取决于他的活动内容 .如果剥夺人的活动自由, 他的心理发展就会受到阻碍 .在学生如何获得知识和经验的问题上, 美国教育学家杜威的观点值得我们借鉴 .他认为 , 知识和经验靠灌输是不行的, 要靠学生在各种实际活动中通过不断地发现和解决问题来丰富和改造自己的经验 , 获得真知 ;他主张发掘学生的智慧潜力 , 让学生在教学中扮演一种充分积极的角色 , 使学生的兴趣和才能得到自由的充分的发展 .现代教学论认为, 在教学过程中应当使学生系统地掌握学科的基础知识和技能, 以满足其认知的需要 , 而且还应促进学生的发展, 满足学生增长智力、培养能力和兴趣特长、形成良好的个性心理品质和发挥创造力的需要等等 .要做到既保证学生的认识任务又保证学生发展任务的全面完成, 需要以师生双方的积极性和主动性作为基础, 否则认识和发展双重任务的圆满完成是很难想象的.按照教学过程中的矛盾分析, 教学过程中教师和学生共同构成了教学的主体, 师生双方扮演着不同的角色, 各自从事着具有相互联系、相互制约的活动.钟启泉先生认为 ,“教学的本质在于沟通和合作, 教与学的关系是沟通中的相互作用关系, 教育者与受教育者的关系是交互主体性的伙伴关系”[1] .在教学过程的某个时刻可以由教师对教学起支配作用, 教学活动的中心在教师, 然而在另一个时刻则由学生起支配作用 .这个时刻 , 教学活动的内容、组织安排和速度可能主要由学生的学习活动来决定, 并要求教师的活动围绕着学生的学习活动来进行.比如在学生独立探索、实验研究时, 学生需要的仅是教师的引导、点拨和鼓励, 知识结论再也不是一味由教师告知.因此 , 只有把教师和学生都看成是教学活动中能动的角色和要素, 把师生双方的关系看成互为主体、互相依存、互相配合的关系 , 才能真正反映教学活动的本质, 使师生的生命活力在课堂上得到充分发挥 , 使教学过程本身具有生成新因素的能力, 具有自身的、由师生共同创造的活力 .从数学学习特点的角度看 , 数学的高度抽象性极易造成教学内容与学生的现实生活的分离, 相应的数学教学活动常常受到歪曲、变异、形式化、表面化 .在数学教学中, 由于数学教材是按照数学理论的逻辑体系编写的, 而这种逻辑体系中的知识呈现顺序与数学理论的真实发现过程往往是相反的(真实发现过程常用“分析法”, 而逻辑体系则采用“演绎法”), 从而 , 根据教材所进行的学习往往是“反思维过程”的活动 :把数学当成纯粹的数学推理 , 当成“逻辑推理”的一种形式来学习, 数学本来具有的丰富多彩性、变化性都被深深地掩盖起来 .显然,。

有关概率论的外文文献

有关概率论的外文文献

文档下载亿万文档免费下载毕业论文外语考试资格考试IT计算机高等教育教学研究经管营销总结/汇报工作范文高中教育初中教育小学教育当前位置：文档下载 > 所有分类 > 人文社科 > 教育学/心理学 > 有关概率论的外文文献免费下载此文档有关概率论的外文文献SAS Statistical Analysis Software And Logistic RegressionI. Overview:SAS is called the Statistics Analysis System, the first from the University of North Carolina's two post-graduate preparation of biostatistics, and in 1976 the Institute of SAS software is established e, the formal SAS software launched. SAS is a large-scale decision support for integrated information systems, but the software system functions limited to the first statistical analysis, since the statistical analysis is still an important part of its core functionality. the current SAS version is 9.0 version, the size is about 1G. After years of development, SAS has been around more than 120 countries and regions, nearly 30,000 institutions that have a direct users over three million people, across the financial, medical and health, production, transport, communications, government and education and scientific research. In Britain and the United States and other countries, skilled using SAS for statistical analysis isthe conditions for many companies and research institutions selection. In data processing and statistical analysis, SAS system known as the international standard software systems, and in 96 ~ 97 years hasbeen selected as the first choice for the establishment of a database product. SAS is called the Big Mac statistical software sector. The other example of this is as follows: in a harsh strict world-famous U.S. FDA drug approval process, the statistical analysis of the drug test results is carried out SAS and other software will be voided! Even a simple and standard deviation are void! This shows theauthority of the SAS.SAS is a combination of SAS software system, which is a combination of multiple functionalmodules, the basic part of BASE SAS module. BASE SAS module is the core of the SAS system,which assume the main task of data management and user management environment for the conduct of the user of language processing, call the other SAS modules and products. In other words, SAS systems, we start the BASE SAS module, which in addition has its own data management, programming andcomputing descriptive statistics, the SAS system or the central dispatching room. It can stand alone, but also with other products or modules together form a complete system. Each module can be installed and updated through the installation process very easy. SAS system has a flexible interface and powerful extension of the functional modules in the basis of BASE SAS, you can add the followingdifferent modules and a variety of new features: SAS / STAT (statistical analysis module), SAS / GRAPH (graphics module) , SAS / QC (quality control module), SAS / ETS (Econometric and time series analysis module), SAS / OR (operations research module), SAS / IML (interactive matrix programming language module), SAS / FSP ( fast data-processing module of the interactive menu system), SAS / AF (interactive full-screen application system software modules) and so on. SAS has a intelligent drawing system, it not only painted a variety of charts, but also draw the map. SAS provides a wide range of statistical process, each process contains a great deal of any option. Users can set a series of data processing to realize more complex statistical analysis. In addition, SAS also offers a variety of probability analysis function, quantile function, the sample statistics functions and random number generator function, so that users can request easily special statistics.2. operationSAS was developed from the mainframe system, the core operation is the process-driven, after many years of development, SAS has now become a complete set of computer language, and its user interface is also fully embodied the characteristics: It uses MDI (Multiple Document interface), the user input program in the PGM window, the results of the analysis in the form of text output in the OUTPUT window. using the program, users can complete all the work, including statistical analysis, forecasting, modeling and simulation, sampling and so on. However, this makes the beginners to learn SASlanguage, entry is more difficult. The Windows SAS version accord to different user groups to develop a number of graphical user interface, graphical user interface of these different characteristics, use very convenient. However, due to limit, and not to promote the focus of SAS, so the vast majority of people do not understand.3.the basic operation and basic concepts of SAS3.1 Dataset (dataset) and the databaseStatistics are for the operation of the data, files which is filled with SAS data is named dataset. in the capacity as the data sets,data sets also included in different library (for the time being itunderstood as a database). SAS in the library is divided into two types of permanent and temporary. As the name suggests, the existence of a permanent library in the data set is permanent (as long as youdo not delete it), temporary library in the data sets from the SASyou automatically be deleted. As for the concept of SAS in the database, the simplest to understand is a directory, a directory of stored data sets.第1页下一页文档免费下载：有关概率论的外文文献（下载1-4页，共4页）我要评论TOP相关主题∙有关单片机的外文文献∙有关旅游的外文文献∙有关jsp的外文文献∙下载外文文献的网站∙关于单片机的外文文献∙关于plc的外文文献∙论文的外文文献怎么找∙查找外文文献的网站相关文档概率论与数理统计英文文献概率论与数理统计英文文献_数学_自然科学_专业资料。

应用数学专业外文翻译

应用数学专业外文翻译

本科毕业论文外文翻译外文译文题目（中文）：具体数学：汉诺塔问题学院:专业:学号:学生姓名:指导教师:日期: 二○一二年六月1 Recurrent ProblemsTHIS CHAPTER EXPLORES three sample problems that give a feel for what’s to c ome. They have two traits in common: They’ve all been investigated repeatedly by mathe maticians; and their solutions all use the idea of recurrence, in which the solution to eac h problem depends on the solutions to smaller instances of the same problem.1.1 THE TOWER OF HANOILet’s look first at a neat little puzzle called the Tower of Hanoi,invented by the Fr ench mathematician Edouard Lucas in 1883. We are given a tower of eight disks, initiall y stacked in decreasing size on one of three pegs:The objective is to transfer the entire tower to one of the other pegs, movingonly one disk at a time and never moving a larger one onto a smaller.Lucas furnished his toy with a romantic legend about a much larger Tower of Brah ma, which supposedly has 64 disks of pure gold resting on three diamond needles. At th e beginning of time, he said, God placed these golden disks on the first needle and orda ined that a group of priests should transfer them to the third, according to the rules abov e. The priests reportedly work day and night at their task. When they finish, the Tower will crumble and the world will end.It's not immediately obvious that the puzzle has a solution, but a little thought (or h aving seen the problem before) convinces us that it does. Now the question arises：What's the best we can do？That is，how many moves are necessary and suff i cient to perfor m the task?The best way to tackle a question like this is to generalize it a bit. The Tower of Brahma has 64 disks and the Tower of Hanoi has 8；let's consider what happens if ther e are TL disks.One advantage of this generalization is that we can scale the problem down even m ore. In fact, we'll see repeatedly in this book that it's advantageous to LOOK AT SMAL L CASES first. It's easy to see how to transfer a tower that contains only one or two di sks. And a small amount of experimentation shows how to transfer a tower of three.The next step in solving the problem is to introduce appropriate notation：NAME ANO CONQUER. Let's say that T n is the minimum number of moves that will t ransfer n disks from one peg to another under Lucas's rules. Then T1is obviously 1 , an d T2= 3.We can also get another piece of data for free, by considering the smallest case of all：Clearly T0= 0，because no moves at all are needed to transfer a tower of n = 0 disks! Smart mathematicians are not ashamed to think small,because general patterns are easier to perceive when the extreme cases are well understood(even when they are trivial).But now let's change our perspective and try to think big；how can we transfer a la rge tower? Experiments with three disks show that the winning idea is to transfer the top two disks to the middle peg, then move the third, then bring the other two onto it. Thi s gives us a clue for transferring n disks in general：We first transfer the n−1 smallest t o a different peg (requiring T n-1moves), then move the largest (requiring one move), and finally transfer the n−1 smallest back onto the largest (req uiring another T n-1moves). Th us we can transfer n disks (for n > 0)in at most 2T n-1+1 moves：T n≤2T n—1+1，for n > 0.This formula uses '≤' instead of '=' because our construction proves only that 2T n—1+1 mo ves suffice; we haven't shown that 2T n—1+1 moves are necessary. A clever person might be able to think of a shortcut.But is there a better way? Actually no. At some point we must move the largest d isk. When we do, the n−1 smallest must be on a single peg, and it has taken at least Tmoves to put them there. We might move the largest disk more than once, if we're n n−1ot too alert. But after moving the largest disk for the last time, we must trans fr the n−1 smallest disks (which must again be on a single peg)back onto the largest；this too re quires T n−1moves. HenceT n≥ 2T n—1+1，for n > 0.These two inequalities, together with the trivial solution for n = 0, yieldT0=0；T n=2T n—1+1 , for n > 0. (1.1)(Notice that these formulas are consistent with the known values T1= 1 and T2= 3. Our experience with small cases has not only helped us to discover a general formula, it has also provided a convenient way to check that we haven't made a foolish error. Such che cks will be especially valuable when we get into more complicated maneuvers in later ch apters.)A set of equalities like (1.1) is called a recurrence (a. k. a. recurrence relation or r ecursion relation). It gives a boundary value and an equation for the general value in ter ms of earlier ones. Sometimes we refer to the general equation alone as a recurrence, alt hough technically it needs a boundary value to be complete.The recurrence allows us to compute T n for any n we like. But nobody really like to co m pute fro m a recurrence，when n is large；it takes too long. The recurrence only gives indirect, "local" information. A solution to the recurrence would make us much h appier. That is, we'd like a nice, neat, "closed form" for Tn that lets us compute it quic kly，even for large n. With a closed form, we can understand what T n really is.So how do we solve a recurrence? One way is to guess the correct solution,then to prove that our guess is correct. And our best hope for guessing the solution is t o look (again) at small cases. So we compute, successively,T3= 2×3+1= 7; T4= 2×7+1= 15; T5= 2×15+1= 31; T6= 2×31+1= 63.Aha! It certainly looks as ifTn = 2n−1，for n≥0. (1.2)At least this works for n≤6.Mathematical induction is a general way to prove that some statement aboutthe integer n is true for all n≥n0. First we prove the statement when n has its smallest v alue,no; this is called the basis. Then we prove the statement for n > n0,assuming that it has already been proved for all values between n0and n−1, inclusive; this is called th e induction. Such a proof gives infinitely many results with only a finite amount of wo rk.Recurrences are ideally set up for mathematical induction. In our case, for exampl e，(1.2) follows easily from (1.1)：The basis is trivial，since T0 = 20−1= 0.And the indu ction follows for n > 0 if we assume that (1.2) holds when n is replaced by n−1：T n= 2T n+1= 2(2n−1−1)+1=2n−1.Hence (1.2) holds for n as well. Good! Our quest for T n has ended successfully.Of course the priests' task hasn't ended；they're still dutifully moving disks,and wil l be for a while, because for n = 64 there are 264−1 moves (about 18 quintillion). Even at the impossible rate of one move per microsecond, they will need more than 5000 cent uries to transfer the Tower of Brahma. Lucas's original puzzle is a bit more practical, It requires 28−1 = 255 moves, which takes about four minutes for the quick of hand.The Tower of Hanoi recurrence is typical of many that arise in applications of all kinds. In finding a closed-form expression for some quantity of interest like T n we go t hrough three stages：1 Look at small cases. This gives us insight into the problem and helps us in stages2 and 3.2 Find and prove a mathematical expression for the quantity of interest.For the Tower of Hanoi, this is the recurrence (1.1) that allows us, given the inc lination，to compute T n for any n.3 Find and prove a closed form for our mathematical expression.For the Tower of Hanoi, this is the recurrence solution (1.2).The third stage is the one we will concentrate on throughout this book. In fact, we'll fre quently skip stages I and 2 entirely, because a mathematical expression will be given to us as a starting point. But even then, we'll be getting into subproblems whose solutions will take us through all three stages.Our analysis of the Tower of Hanoi led to the correct answer, but it r equired an“i nductive leap”；we relied on a lucky guess about the answer. One of the main objectives of this book is to explain how a person can solve recurrences without being clairvoyant. For example, we'll see that recurrence (1.1) can be simplified by adding 1 to both sides of the equations：T0+ 1= 1;T n + 1= 2T n-1+ 2, for n >0.Now if we let U n= T n+1，we haveU0 =1；U n= 2U n-1，for n > 0. (1.3)It doesn't take genius to discover that the solution to this recurrence is just U n= 2n；he nce T n= 2n −1. Even a computer could discover this.Concrete MathematicsR. L. Graham, D. E. Knuth, O. Patashnik《Concrete Mathematics》，1.1 ，The Tower Of HanoiR. L. Graham, D. E. Knuth, O. PatashnikSixth printing, Printed in the United States of America1989 by Addison-Wesley Publishing Company，Reference 1-4 pages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》，1.1，汉诺塔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1-4页1 递归问题本章将通过对三个样本问题的分析来探讨递归的思想。

数学专业英语微分学论文翻译

数学专业英语微分学论文翻译

数学专业英语微分学论文翻译信息0801 姬彩云 200801010129Differential Calculus 微分学Historical Introduction历史介绍Newton 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 successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.The central idea of differential calculus is the notion of derivative.Like the integral,the derivative originated from a problem in geometry—the problem of finding the tangent line at a point of a curve.Unlile the integral,however,the derivative evolved very latein the history ofmathematics.The concept was not formulated until early in the 17th century when the Frenchmathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special functions.Fermat’s idea,basicall y very simple,can be understood if we referto a curve and assume that at each of its points this curve has a definite direction that can be described by a tangent line.Fermatnoticed that at certain points where the curve has a maximum or minimum,the tangent line must be horizontal.Thus the problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.This raises the more general question of determining the directionof the tangent line at an arbitrary point of the curve.It was the attempt to solve this general problem that led Fermat to discover some of the rudimentary ideas underlying the notion of derivative.At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of finding the tangent line at a point of a curve.The first person to realize that these two seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they exploited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.Although the derivative was originally formulated to study the problem of tangents,it was soon found that it also provides a way to calculate velocity and,more generally,the rate of change of afunction.In the next section we shall consider a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and mayhelp to motivate the general definition of derivative which is given below.A Problem Involving Velocity 一个设计速度的问题Suppose a projectile is fired straight up from the ground withinitial velocity of 144 feet persecond.Neglect friction,and assume the projectile is influenced only by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at time t we woule have f(t)=144 t.In actual practice,gravity causes the projectile to slow down until its velocity decreases to zero and then it drops back to earth.Physical experiments suggest that as the projectile is aloft,its height f(t) is given by the formula.2The term –16t is due to the influence of gravity.Note that f(t)=0 when t=0 and when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.The problem we wish to consider is this:To determine the velocity of the projectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we introduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient. Change in distance during time interval =f(t+h)-f(t)/h.ength of time intervalThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positive or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute value.The limit process by which v(t) is obtained from the difference quotient is written symbolically as follows: The equation is used to define velocity not only for this particular example but,more generally,for any particle moving along a straightline,provided the position function f is such that the differercequotient tends to a definite limit as h approaches zero.The example describe in the foregoing section points the way to the introduction of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choosea fixed point in this interval and introduce the differencequotient[f(x+h)-f(x)]/h.where the number h,which may be positive or negative(but notzero),is such that x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to as the average rate of change of f in the interval joining x to x+h.Now we let h approach zero and see what happens to this quotient.Ifthe quotient.If the quotient approaches some definite values as alimit(which implies that the limit is the same whether h approaches zerothrough positive values or through negative values),then this limit is called thederivative of f at x and is denoted by the symbol f’(x) (read as―f prime of x‖).Thus the formaldefinition of f’(x) may be stated as follows Definition of derivative.The derivative f’(x)is definedby the equation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。

数学专业英语论文(英文翻译中文)

数学专业英语论文(英文翻译中文)

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。

数学专业外文文献翻译

数学专业外文文献翻译

0 000 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 ()]T x 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0 000 2()x ()x (T R 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 wg 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。

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毕业设计（论文）外文文献翻译文献、资料中文题目：概率论的发展文献、资料英文题目：The development of probabilitytheory文献、资料来源：文献、资料发表（出版）日期：院（部）：专业：数学与应用数学班级：姓名：学号：指导教师：翻译日期： 2017.02.14毕业论文（设计）英文文献翻译外文文献The development of probability theorySummaryThis paper consist therefore of two parts: The first is concerned with the development of the calyculus of chance before Bernoulli in order to provide a background for the achievement of Ja kob Bernoulli and will emphasize especially the role of Leibniz. The second part deals with the relationship between Leibniz add Bernoulli an d with Bernoulli himself, particularly with the question how it came about that he introduced probability into mathematics.First some preliminary remarks:Ja kob Bernoulli is of special interest to me, because he is the founder of a mathematical theory of probability. That is to say that it is mainly due to him that a concept of probability was introduced into a field of mathematics.TextMathematics could call the calculus of games of chance before Bernoulli. This has another consequence that makes up for a whole programme: The mathematical tools of this calculus should be applied in the whole realm of areas which used a concept of probability. In other words，the Bernoullian probability theory should be applied not only togames of chance and mortality questions but also to fields like jurisprudence, medicine, etc.My paper consists therefore of two parts: The first is concerned with the development of the calculus of chance before Bernoulli in order to provide a background or the achievements of Ja kob Bernoulli and will emphasize especially the role of Leibniz. The second part deals with the relationship between Leibniz and Bernoulli and Bernoulli himself, particularly with the question how it came about that he introduced probability into mathematics.Whenever one asks why something like a calculus of probabilities arose in the 17th century, one already assumes several things: for instance that before the 17th century it did not exist, and that only then and not later did such a calculus emerge. If one examines the quite impressive literature on the history of probability, one finds that it is by no means a foregone conclusion that there was no calculus of probabilities before the 17th century. Even if one disregards numerous references to qualitative and quantitative inquiries in antiquity and among the Arabs and the Jews, which, rather freely interpreted, seem to suggest the application of a kind of probability-concept or the use of statistical methods, it is nevertheless certain that by the end of the 15th century an attempt was being interpreted.People made in some arithmetic works to solve problems of games of chance by computation. But since similar problems form the major part of the early writings on probability in the 17th century, one may be induced to ask why then a calculus of probabilities did not emerge in the late 15th century. One could say many things: For example, that these early game calculations in fact represent one branch of a development which ultimately resulted in a calculus of probabilities. Then why shouldn't one place the origin of the calculus of probabilities before the 17th after all? Quite simply because a suitable concept of probability was missing from the earlier computations. Once the calculus of probabilities had beendeveloped, it became obvious that the older studies of games of chance formed a part of the new discipline.We need not consider the argument that practically all the solutions of problems of games of chance proposed in the 15th and 16th centuries could have been viewed as inexact, and thus at best as approximate, by Fermat in the middle of the 17th century, that is, before the emergence of a calculus of probabilities.The assertion that no concept of probability was applied to games of chance up to the middle of the 17th century can mean either that there existed no concept of probability (or none suitable), or that though such a concept existed it was not applied to games of chance. I consider the latter to be correct, and in this I differ from Hacking, who argues that an appropriate concept of probability was first devised in the 17th century.I should like to mention that Hacking(Mathematician)and I agree ona number of points. For instance, on the significance of the legal tradition and of the practical ("-low") sciences: Hacking makes such factors responsible for the emergence of a new concept of probability, suited to a game calculus, while perceive them as bringing about the transfer and quantification of a pre-existent probability-concept.译文概率论的发展作者；龙腾施耐德摘要本文由两部分构成：首先是提供了一个为有关与发展雅各布 - 前伯努利相关背景，雅各布对数学做出了不可磨灭的贡献。

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