数学专业英语论文

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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。

数学论文英语写作范文

120词英语范文-帮忙：2篇英语作文（120字以上）高二水准,2篇英第一片 Is a Training Class or Family Teacher Necessary? More and more middle school students are going to all kinds of training classes or having family teachers at the weekend。

There are two different viewpoints about it。

On the one hand, some think that studying following the family teachers is better than self study。

In my opinion, as students, we should really know whether we need a training class or family teacher。

First, make sure that you need them, and they would be helpful, then choose a reasonable one。

Just remember, once you start and never give up。

关于数学的作文关于数学的作文 200字的篇一：关于数学生活中，处处有数学。

例如：买菜啦！买文具啦！量布等等，都需要用到数学。

这个学期，老师教了一个新知识，是小数的乘法和除法。

这个知识，可帮了我大忙啊！昨天晚上，我妈妈一起去买桔子。

桔子是1.8元一斤，妈妈买了4.5斤，本应该付钱8.1元。

可是营业员粗心大意，不知道怎么算的，算成了9元钱。

还好我利用了这个学期新教的知识，在脑子里算过一便后，马上纠正了营业员的失误。

不仅营业员阿姨夸我聪明，这么小都会小数乘除法了，而且在回家的路上，妈妈还表扬我，给她省了0.9元，并且学过的知识能在生活中活用。

数学专业英语论文

四符号说明),,(h y x ：坐标的空间位置；The space coordinates;),,(h y x f ：重金属的空间分布浓度；The concentration of heavy metal spatial distribution;i z ：空间某点的属性值；A point of attribute values;i d ：点),(y x 到点),(i i y x 的水平距离；the horizontal distance of Point (x, y) to (Xi, Yi);p ：反距离加权插值函数的加权幂指数；Weighted exponential inverse distance weighted interpolation function; i ω ：反距离加权插值函数中各点浓度的权重； The weight of each point concentration of inverse distance weighted interpolation function; R ：相关系数矩阵；Correlation matrix. λ：特征根；Characteristic root;e ：特征向量；The feature vector;ij l ：主成分载荷；Principal component loadings;PC ：主成分贡献率；Principal component contribution rate;0x ：最优参考序列；The optimal reference sequence;()max ∆：两级最大差；two Level maximum difference()min ∆：两级最小差；Two level smallest difference;ij ξ：关联度；Correlation degree;i E ：综合评价系数；Comprehensive evaluation coefficient;:ε隶属度系数；Membership degree coefficient;ihC '：第i 个评价因子污染强度最大的超标值（以p 级准则值为界）； the largest pollution exceeding the standards value of the ith evaluation factor intensity (p grade criterion value for the sector);ip S ：第i 个评价因子p 级准则限值；The ith evaluation factor intensity of P grade criterion limit;r ：超标因子个数；The number of Exceed the standard factors;n ：评价因子总数，当0=r 时，取0=ε；The total number of evaluation factor, when 0=r , 0=ε;i d ：各区"i C 与"1S 之间的相对距离；The relative distance between the "i C and "1S ;j δ：各级标准隶属度向量"j S 与"1S 之间的相对距离。

数学专业英语第六讲论文写作常见错误分析

不要过于烦琐.
• 不要用On the Study of ... 或On the Existence of... 之类作为标题.
• 标题通常是一个名词性短语, 不是一个完整的句子,
• 无需主、谓、宾齐全, 其中的动词一般以分词或动名词的形式出现, 有时冠词可以省略. • 偶尔有以疑问句为标题的论文, 但不应以陈述句为标题. • 为方便二次检索, 标题尽量避免使用复杂的数学式子.
• [2] name(s) of author(s), title of the book (in italics),
publishinghouse, city of publication, year of publication.
16
Mathematical English
• [3] name(s) of author(s), title of the paper, in: title of the book (in italics), publishing house, city of publication, year of publication, numbers of the first and last pages of the paper.
improvements for you.
9
Mathematical English
数学论文对英语的要求其实并不高. 而且我们有功能强大的 LATEX(WinEdt) 编辑软件, 可以轻而易举地编排出有复
杂公式、美观漂亮的版面.
但是, 有些作者在论文写作中存在不少问题.
下面从论文的各个组成部分对一些常见的错误进行分析并举
发表论文的目的, 一是公开自己的成果并得到学术界的承认, 二是为了同行之间的交流.

数学专业英语2-11C(精选5篇)

数学专业英语2-11C（精选5篇）第一篇：数学专业英语2-11C数学专业英语论文数学专业英语论文英文原文：2-12CSome basic principles of combinatorial analysisMany problems in probability theory and in other branches of mathematics can be reduced to problems on counting the number of elements in a finite set. Systematic methods for studying such problems form part of a mathematical discipline known ascombinatorial analysis. In this section we digress briefly to discuss some basic ideas in combinatorial analysis that are useful in analyzing some of the more complicated problems of probability theory.If all the elements of a finite set are displayed before us, there is usually no difficulty in counting their total number. More often than not, however, a set is described in a way that makes it impossible or undesirable to display all its elements. For example,we might ask for the total number of distinct bridge hands that can be dealt. Each player is dealt 13 cards from a 52-card deck. The number of possible distinct hands is the same as the number of different subsets of 13 elements that can be formed from a set of 52 elements.Since this number exceeds 635 billion, a direct enumeration of all the possibilities is clearly not the best way to attack this problem; however, it can readily be solved by combinatorial analysis.This problem is a special case of the more general problem of counting the number of distinct subsets of k elements that may be formed from a set of n elements (When we say that a set has n elements,we mean that it has n distinct elements.Such a set is sometimes called an n-element set.),where n k. Let us denote this number by f(n,k).It has long been known thatn(12.1)f(n,k)k,n where, as usual k denotes the binomial coefficient,n n!k k!(n k)!52In the problem of bridge hands we havef(52,13)13635,013,559,600different hands that a player can be dealt.There are many methods known for proving (12.1). A straightforward approach is to form each subset of k elements by choosing the elements one at a time. There are n possibilities for the first choice, n 1 possibilities for the second choice, and n(k1) possibilities for the kth choice. If we make all possible choices in this1manner we obtain a total ofn(n1)(n k1)n! (n k)!subsets of k elements. Of course, these subsets are not all distinct. For example, ifk3the six subsetsa,b,c,b,c,a,c,a,b,a,c,b,c,b,a, b,a,carc all equal. In general, this method of enumeration counts each k-element subset exactly k! times. Therefore we must divide the number n!/(n k)! by k! to n obtain f(n,k). This gives us f(n,k)k, as asserted.译文：组合分析的一些基本原则许多概率论和其他一些数学分支上的问题，都可以简化成基于计算有限集合中元素数量的问题。

数学专业英语作文（学习体会）[五篇材料]

数学专业英语作文（学习体会）[五篇材料]第一篇：数学专业英语作文(学习体会)About learning the Mathematics English We major in Information and computing science , which belongs to Department of Mathematics.And also we need to learn Mathematics English in this term.As a smoothly through this term student , I benefited a lot.Firstly , I think learning English needs a lot of practice.We need to practice listening , writing , reading and thinking in English.But for the Math ematics English , it’s difficult.Because there are many different.Mathematics English always describe objective facts and truth.There are many long sentence and professional words and phrases , which is different to our Oral English.So learning Mathematics English is a challenge.Unfortunately , I don’t learn it well.Although English is the common language of communications for world diplomacy, economics and defence, then let us be as fluent and proficient in it as possible, for the sake of our own country's advancement and prestige.May be we will repent learning poorly in Mathematics English when we write graduation thesis for searching the English thesis.Secondly , I learn a little about Mathematics English.We study some texts about Mathematics, Equation ,Geometry ,Set , Function , Sequences and Their Limits , The Derivative of a Function , Elementary Mathematical Logic and so on.Also I learn some useful things , for example , some professional words and phrases , and how to describe some sample objective facts and truth , how to describe some sample formula in English.May be it helpless in my graduation thesis , but it will keeps in my mind for many years!Thirdly , the teacher’s teaching methods is very well.Hegave us a stage and let us play fully!We can learn together and there full of cordial and friendly atmosphere where we can make common progress and common development.Some students seized the opportunity but some let the chance slip away.I caught the opportunity and did my best!Thank for teacher giving the stage let me play my role!To sum up, my college life and learning life still exist many shortcomings.I will be in the next few months of this struggle, make oneself to make greater progress.第二篇：专业英语学习体会专业英语学习体会英语作为一门重要的学科，正在由于科技的飞速发展、信息全球化、社会化的日趋接近以及国际语言的特殊身份而身价倍增，成为了跨入二十一世纪所必备的三张通行证之一。

数学专业英语(Doc版).14

数学专业英语－MathematicansLeonhard Euler was born on April 15,1707,in Basel, Switzerland, the son of a mathematician and Caivinist pastor who wanted his son to become a pastor a s well. Although Euler had different ideas, he entered the University of Basel to study Hebrew and theology, thus obeying his father. His hard work at the u niversity and remarkable ability brought him to the attention of the well-known mathematician Johann Bernoulli (1667—1748). Bernoulli, realizing Euler’s tal ents, persuaded Euler’s father to change his mind, and Euler pursued his studi es in mathematics.At the age of nineteen, Euler’s first original work appeared. His paper failed to win the Paris Academy Prize in 1727; however this loss was compensated f or later as he won the prize twelve times.At the age of 28, Euler competed for the Pairs prize for a problem in astrono my which several leading mathematicians had thought would take several mont hs to solve.To their great surprise, he solved it in three days! Unfortunately, th e considerable strain that he underwent in his relentless effort caused an illness that resulted in the loss of the sight of his right eye.At the age of 62, Euler lost the sight of his left eye and thus became totally blind. However this did not end his interest and work in mathematics; instead, his mathematical productivity increased considerably.On September 18, 1783, while playing with his grandson and drinking tea, Eul er suffered a fatal stroke.Euler was the most prolific mathematician the world has ever seen. He made s ignificant contributions to every branch of mathematics. He had phenomenal m emory: He could remember every important formula of his time. A genius, he could work anywhere and under any condition.George cantor (March 3, 1845—June 1,1918),the founder of set theory, was bo rn in St. Petersburg into a Jewish merchant family that settled in Germany in 1856.He studied mathematics, physics and philosophy in Zurich and at the University of Berlin. After receiving his degree in 1867 in Berlin, he became a lecturer at the university of Halle from 1879 to 1905. In 1884,under the stra in of opposition to his ideas and his efforts to prove the continuum hypothesis, he suffered the first of many attacks of depression which continued to hospita lize him from time to time until his death.The thesis he wrote for his degree concerned the theory of numbers; however, he arrived at set theory from his research concerning the uniqueness of trigon ometric series. In 1874, he introduced for the first time the concept of cardinalnumbers, with which he proved that there were “more”transcendental numb ers than algebraic numbers. This result caused a sensation in the mathematical world and became the subject of a great deal of controversy. Cantor was troub led by the opposition of L. Kronecker, but he was supported by J.W.R. Dedek ind and G. Mittagleffer. In his note on the history of the theory of probability, he recalled the period in which the theory was not generally accepted and cri ed out “the essence of mathematics lies in its freedom!”In addition to his work on the concept of cardinal numbers, he laid the basis for the concepts of order types, transfinite ordinals, and the theory of real numbers by means of fundamental sequences. He also studied general point sets in Euclidean space a nd defined the concepts of accumulation point, closed set and open set. He wa s a pioneer in dimension theory, which led to the development of topology.Kantorovich was born on January 19, 1912, in St. Petersburg, now called Leni ngrad. He graduated from the University of Leningrad in 1930 and became a f ull professor at the early age of 22.At the age of 27, his pioneering contributi ons in linear programming appeared in a paper entitled Mathematical Methods for the Organization and planning of production. In 1949, he was awarded a S talin Prize for his contributions in a branch of mathematics called functional a nalysis and in 1958, he became a member of the Russian Academy of Science s. Interestingly enough, in 1965,kantorovich won a Lenin Prize fo r the same o utstanding work in linear programming for which he was awarded the Nobel P rize. Since 1971, he has been the director of the Institute of Economics of Ma nagement in Moscow.Paul R. Halmos is a distinguished professor of Mathematics at Indiana Univers ity, and Editor-Elect of the American Mathematical Monthly. He received his P h.D. from the University of Illinois, and has held positions at Illinois, Syracuse, Chicago, Michigan, Hawaii, and Santa Barbara. He has published numerous b ooks and nearly 100 articles, and has been the editor of many journals and se veral book series. The Mathematical Association of America has given him the Chauvenet Prize and (twice) the Lester Ford award for mathematical expositio n. His main mathematical interests are in measure and ergodic theory, algebraic, and operators on Hilbert space.Vito Volterra, born in the year 1860 in Ancona, showed in his boyhood his e xceptional gifts for mathematical and physical thinking. At the age of thirteen, after reading Verne’s novel on the voyage from earth to moon, he devised hi s own method to compute the trajectory under the gravitational field of the ear th and the moon; the method was worth later development into a general proc edure for solving differential equations. He became a pupil of Dini at the Scu ola Normale Superiore in Pisa and published many important papers while still a student. He received his degree in Physics at the age of 22 and was made full professor of Rational Mechanics at the same University only one year lat er, as a successor of Betti.Volterra had many interests outside pure mathematics, ranging from history to poetry, to music. When he was called to join in 1900 the University of Rome from Turin, he was invited to give the opening speech of the academic year. Volterra was President of the Accademia dei Lincei in the years 1923-1926. H e was also the founder of the Italian Society for the Advancement of Science and of the National Council of Research. For many years he was one of the most productive scientists and a very influential personality in public life. Whe n Fascism took power in Italy, Volterra did not accept any compromise and pr eferred to leave his public and academic activities.Vocabularypastor 牧师 hospitalize 住进医院theology 神学 thesis 论文strain 紧张、疲惫transcendental number 超越数relentless 无情的sensation 感觉，引起兴趣的事prolific 多产的controversy 争论，辩论depression 抑郁；萧条，不景气essence 本质，要素transfinite 超限的Note0. 本课文由几篇介绍数学家生平的短文组成，属传记式体裁。

数学建模论文英文

数学建模论文英文Abstract:Mathematical modeling is an essential tool in various scientific and engineering disciplines, facilitating the understanding and prediction of complex systems. This paper explores the fundamental principles of mathematical modeling, its applications, and the methodologies employed in constructing and analyzing models. Through case studies, we demonstrate the power of mathematical models in solving real-world problems.Introduction:The introduction of mathematical modeling serves as a foundation for the entire paper. It provides an overview of the significance of mathematical modeling in modern problem-solving and sets the stage for the subsequent sections. It also outlines the objectives and scope of the paper.Literature Review:This section reviews existing literature on mathematical modeling, highlighting the evolution of the field, key concepts, and the diverse range of applications. It also identifies gaps in current knowledge that the present study aims to address.Methodology:The methodology section describes the approach taken to construct and analyze mathematical models. It includes theselection of appropriate mathematical tools, the formulation of the model, and the validation process. This section is crucial for ensuring the scientific rigor of the study.Model Development:In this section, we delve into the process of model development, including the identification of variables, the establishment of relationships, and the formulation of equations. The development of the model is presented in a step-by-step manner to ensure clarity and reproducibility.Case Studies:Case studies are presented to demonstrate the practical application of mathematical models. Each case study is carefully selected to illustrate the versatility and effectiveness of mathematical modeling in addressing specific problems.Results and Discussion:This section presents the results obtained from the application of the mathematical models to the case studies. The results are analyzed to draw insights and conclusions about the effectiveness of the models. The discussion also includes an evaluation of the model's limitations and potential areas for improvement.Conclusion:The conclusion summarizes the key findings of the paper and reflects on the implications of the study. It also suggests directions for future research in the field of mathematical modeling.References:A comprehensive list of references is provided to acknowledge the sources of information and ideas presented in the paper. The references are formatted according to a recognizedcitation style.Appendices:The appendices contain any additional information that supports the paper, such as detailed mathematical derivations, supplementary data, or extended tables and figures.Acknowledgments:The acknowledgments section, if present, expresses gratitudeto individuals or organizations that contributed to the research but are not authors of the paper.This structure ensures that the mathematical modeling paperis comprehensive, logically organized, and adheres to academic standards.。

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)

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。

数学专业英语第七讲论文写作及投稿

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【写作环节】 17、图表是放在后面，还是插入在文章中。(看投稿须知要求)。 18、字数是否符合杂志要求（有的杂志对字数也有要求。比如最多8个版面。他一般会告诉你怎么推测自己的文章占几个版面，比如有的杂志大约是 8000 个字符数 (characters)（包括空格）为一个版面）。 19、全文的字号是否符合要求。（一般是12号字(12-point)，双倍行距（Double space））。 20、是否进行了致谢（国外的文章一般都会致谢。一般要求写受什么基金资助，谁对文章改了，谁进行了技术帮助，谁提供了一些实验材料等；其中很多杂志基金资助一般写在Footnote中）。
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论文的结构
Title(标题) Abstract(摘要) Keywords(关键词) Table of contents(目录) Nomenclature(术语表) Introduction(引言) Method(方法) 正 Results(结果) 文 Discussion(讨论) Conclusion(结论) Acknowledgement(致谢) Notes(注释) References(参考文献) Appendix(附录)

英文数学论文

This paper is concerned with the Cauchy problem of nonlinear wave equations with potential, strong, and nonlinear damping terms. Firstly, by using variational calculus and compactness lemma, the existence of standing waves of the ground states is obtained. Then the instability of the standing wave is shown by applyingpotential-well arguments and concavity methods. Finally, we show how small the initial data are for the global solutions to exist.Keywords:wave equations; nonlinear damping terms; strong damping terms; global existence; blow-upIntroductionConsider the Cauchy problem for nonlinear wave equations with potential, strong, andnonlinear damping terms,{utt−Δu−ωΔut+V(x)u+|ut|m−2ut=|u|p−2u,u(0,x)=u0,ut(0,x)=u1,in [0,T)×Rn,in Rn,(1)where p>2, m≥2, T>0, ω>0,u0∈H1(Rn),u1∈L2(Rn),(2)and2<p≤2nn−2,for n≥3;p>2,for n≤2.(3)With the absence of the strong damping term Δut, and the damping term ut (see [1]), (1.1) can be viewed as an interaction between one ormore discrete oscillators and a field or continuous medium [2].For the case of linear damping ( ω=0, m=2) and nonlinear sources, Levine [3] showed that the solutions to (1.1) with negative initial energy blow-up for the abstract version. For the nonlinear damping and source terms ( ω=0, m>2, p>2, V(x)=0), the abstract version has been considered by many researchers [4]–[12]. For instance, Georgiev and Todorova [4] prove that if m≥p, a global weak solution exists for any initial data, while if 2<m<p the solution blows up in finite time when the initial energy is sufficiently negative. In [5], Todorova considers the additional restriction on m. Ikehata[6] considers the solutions of (1.1) with small positive initial energy, using the so-called ‘potential-well’ theory. The case of strong damping (ω>0, m=2, V(x)=0) and nonlinear source terms ( p>2) has been studied by Gazzola and Squassina in [1]. They prove the global existence of solutions with initial data in the potential well and show that every global solution is uniformly bounded in the natural phase space. Moreover, they prove finite time blow-up for solutions with high energy initial data.However, they do not consider the case of a nonlinear damping term( ω>0, m>2, p>2).To the best of our knowledge, little work has been carried out on the existence and instability of the standing wave for (1.1). In this paper, we study the existence of a standing wave with ground state (ω=1), which is the minimal action solution of thefollowing elliptic equation:−Δϕ+V(x)ϕ=|ϕ|p−2ϕ.(4)Based on the characterization of the ground state and the localwell-posedness theory[7], we investigate the instability of the standing wave for the Cauchy problem (1.1). Finally, we derive a sufficient condition of global existence of solutions to the Cauchy problem (1.1) by using the relation between initial data and the ground state solution of (1.4). It should be pointed out that these results in the present paper are unknown to (1.1) before.For simplicity, throughout this paper we denote ∫Rn⋅dx by ∫⋅dx and arbitrary positive constants by C.Preliminaries and statement of main resultsWe define the energy space H in the course of nature asH:={φ∈H1(Rn),∫V(x)|φ|dx<∞}.(5)By its definition, H is a Hilbert space, continuously embeddedin H1(Rn), when endowed with the inner product as follows:⟨φ,ϕ⟩H:=∫(∇φ∇ϕ¯+V(x)φϕ¯)dx,(6)whose associated norm is denoted by ∥⋅∥H. If φ∈H, then∥φ∥H=(∫|∇φ|2dx+∫V(x)|φ|2dx)12.(7)Throughout this paper, we make the following assumptions on V(x):⎧⎩⎨⎪⎪⎪⎪inf x∈RnV(x)=V¯(x)>0,V(x) is a C1 bounded measurable function on Rn,lim x→∞V(x)=∞.(8)According to [1] and [7], we have the following local well-posedness for the Cauchy problem (1.1).Proposition 2.1If (1.2) and (1.3) hold, then there exists a unique solution u(t,x)of the Cauchy problem (1.1) on a maximal time interval[0,T), for some T∈(0,∞) (maximalexistence time) such thatu(t,x)∈C([0,T);H1(Rn))∩C1([0,T);L2(Rn))∩C2([0,T);H−1(Rn)) ,ut(t,x)∈C([0,T);H1(Rn))∩Lm([0,T)×Rn),(9)and either T=∞or T<∞and lim t→T−∥u∥H1=∞.Remark 2.2From Proposition 2.1, it follows that m=p is the critical case, namely for p≤m, a weak solution exists globally in time for any compactly supported initial data; whilefor m<p, blow-up of the solution to the Cauchy problem (1.1) occurs.We define the functionalsS(ϕ):=12∫|∇ϕ|2dx+12∫V(x)|ϕ|2dx−1p∫|ϕ|pdx,(10)R(ϕ):=∫|∇ϕ|2dx+∫V(x)|ϕ|2dx−∫|ϕ|pdx,(11)for ϕ∈H1(Rn), and we define the setM:={ϕ∈H1∖{0};R(ϕ)=0}.(12)We consider the constrained variational problemdM:=inf{supλ≥0S(λϕ):R(ϕ)<0,ϕ∈H1∖{0}}.(13)For the Cauchy problem (1.1), we define unstable and stable sets, K1and K2, asfollows:K1≡{ϕ∈H1(Rn)∣R(ϕ)<0,S(ϕ)<dM},K2≡{ϕ∈H1(Rn)∣R(ϕ)>0,S(ϕ)<dM}∪{0}.(14)The main results of this paper are the following.Theorem 2.3There exists Q∈M such that(a1) S(Q)=inf MS(ϕ)=dM;(a2) Q is a ground state solution of (1.4).From Theorem 2.3, we have the following.Lemma 2.4Let Q(x)be the ground state of (1.4). If (1.3) holds, thenS(Q)=min MS(ϕ).(15)Theorem 2.5Assume that (1.2)-(1.3) hold and the initial energy E(0)satisfiesE(0)=<12∫|u1|2dx+12(∫|∇u0|2dx+∫V(x)|u0|2dx)−1p∫|u0|pdxp−2 2p(∫|∇Q|2dx+∫V(x)|Q|2dx).(16)(b1) If2<m<p, and there exists t0∈[0,T)such that u(t0)∈K1, then the solution u(x,t)of the Cauchy problem (1.1) blows up in a finite time.(b2) If2<p≤m, there exists t0∈[0,T)such that u(t0)∈K2, then the solution u(x,t)of the Cauchy problem (1.1)globally existson[0,∞). Moreover, for t∈[0,∞), u(x,t)satisfies∥ut∥22+p−2p(∫|∇u0|2dx+∫V(x)|u0|2dx)<p−2p(∫|∇Q|2dx+∫V(x)|Q |2dx).(17)Variational characterization of the ground stateIn this section, we prove Theorem 2.3.Lemma 3.1The constrained variational problemdM:=inf{supλ≥0S(λϕ):R(ϕ)<0,ϕ∈H1∖{0}},(18)is equivalent tod1:=inf{supλ≥0S(λϕ):R(ϕ)=0,ϕ∈H1∖{0}}=infϕ∈MS(ϕ),(19)and dM provided2<p≤2nn−2as well as2≤m<p.ProofLet ϕ∈H1. SinceS(λϕ)=λ22(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λpp∫|ϕ|pdx,(20)it follows thatddλS(λϕ)=λ(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λp−1∫|ϕ|pdx.(21)Thus by 2≤m<p, one sees that there exists some λ1≥0 such thatsupλ≥0S(λϕ)=S(λ1ϕ)=λ21(12∫|∇ϕ|2dx+12∫V(x)|ϕ|2dx−λp−21p∫|ϕ|pdx),(22)where λ1 uniquely depends on ϕ and satisfies∫|∇ϕ|2dx+∫V(x)|ϕ|2dx−λp−21∫|ϕ|pdx=0.(23)Sinced2dλ2S(λϕ)=∫|∇ϕ|2dx+∫V(x)|ϕ|2dx−pλp−2∫|ϕ|pdx,(24)which together with p>2 and (3.6) implies that d2dλ2S(λϕ)|λ=λ1<0, wehavesupλ≥0S(λϕ)=p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)pp−2(∫|ϕ|pdx)2p−2. (25)Therefore, the above estimates lead todM==inf{supλ≥0S(λϕ):R(ϕ)<0,ϕ∈H1∖{0}}inf{p−22p(∫|∇ϕ|2dx+∫V (x)|ϕ|2dx)pp−2(∫|ϕ|pdx)2p−2:R(ϕ)<0}.(26)It is easy to see thatd1==inf{supλ≥0S(λϕ):R(ϕ)=0,ϕ∈H1∖{0}}infϕ∈M{p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)}.(27)From (2.3)-(2.5), on M one hasS(ϕ)=p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx).(28)It follows that d1=infϕ∈MS(ϕ) and S(ϕ)>0 on M.Next we establish the equivalence of the two minimization problems (3.1) and (3.2).For any ϕ0∈H1and R(ϕ0)<0, let ϕβ(x)=βϕ0. There exists a β0∈(0,1)suchthat R(ϕβ0)=0, and from (3.8) we getsupλ≥0S(λϕβ0)===p−22p(∫|∇ϕβ0|2dx+∫V(x)|ϕβ0|2dx)pp−2(∫|ϕβ0|pdx)2p−2p−22pβ2pp−20(∫|∇ϕ0|2dx+∫V(x)|ϕ0|2dx)pp−2β2pp−20(∫|ϕ0|pdx)2p−2p−22p(∫|∇ϕ0|2dx+∫V(x)|ϕ0|2dx)pp−2(∫|ϕ0|pdx)2p−2.(29)Consequently the two minimization problems (3.8) and (3.9) are equivalent, that is, (3.1) and (3.2) are equivalent.Finally, we prove dM>0by showing d1>0in terms of the above equivalence.Since 2<p≤2nn−2, the Sobolev embedding inequality yields∫|ϕ|pdx≤C(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)p2.(30)From R(ϕ)=0, it follows that∫|∇ϕ|2dx+∫V(x)|ϕ|2dx=∫|ϕ|pdx≤C(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)p2,(31 )which together with p>2 implies∫|∇ϕ|2dx+∫V(x)|ϕ|2dx≥C>0.(32)Therefore, from (3.10), we getS(ϕ)≥C>0,ϕ∈M.(33)Thus from the equivalence of the two minimization problems (3.1) and (3.2) one concludes that dM>0 for 2<p≤2nn−2.This completes the proof of Lemma 3.1. □Proposition 3.2S is bounded below on M and dM>0.ProofFrom (2.3)-(2.6), on M one hasS(ϕ)=p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx).(34)It follows that S(ϕ)>0 on M. So S is bounded below on M. From (2.6) we have dM>0. □Proposition 3.3Letϕλ(x)=λϕ(x), forϕ∈H1∖{0}andλ>0. Then there exists auniqueμ>0 (depending onϕ) such that R(ϕμ)=0. Moreover,R(ϕλ)>0,for λ∈(0,μ);R(ϕλ)<0,for λ∈(μ,∞);(35)andS(ϕμ)≥S(ϕλ),∀λ>0.(36)ProofBy (2.3) and (2.4), we haveS(ϕλ)=λ22(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λpp∫|ϕ|pdx,R(ϕλ)=λ2(∫|∇ϕ| 2dx+∫V(x)|ϕ|2dx)−λp∫|ϕ|pdx.(37)From the definition of M, there exists a unique μ>0 such that R(ϕμ)=0. Moreover,R(ϕλ)>0,for λ∈(0,μ);R(ϕλ)<0,for λ∈(μ,∞).(38)SinceddλS(ϕλ)=λ−1R(ϕλ),(39)and R(ϕμ)=0, we have S(ϕμ)≥S(ϕλ), ∀λ>0. □Next, we solve the variational problem (2.6).We first give a compactness lemma in [8].Lemma 3.4Let1≤p<N+2N−2when N≥3and1≤p<∞when N=1,2. Then theembedding H↪Lp+1is compact.In the following, we prove Theorem 2.3.Proof of Theorem 2.3According to Proposition 3.2, we let {ϕn,n∈N}⊂M be a minimizing sequence for(2.6), that is,R(ϕn)=0,S(ϕn)→dM.(40)From (3.15) and (3.16), we know ∥∇ϕn∥22 is bounded for all n∈N. Then there exists a subsequence {ϕnk,k∈N}⊂{ϕn,n∈N}, such that{ϕnk}⇀ϕ∞weakly in H1.(41)For simplicity, we still denote {ϕnk,k∈N} by {ϕn,n∈N}. So we have ϕn⇀ϕ∞weakly in H1.(42)By Lemma 3.4, we haveϕn→ϕ∞strongly in L2(Rn),(43)ϕn→ϕ∞strongly in Lp(Rn).(44)Next, we prove that ϕ∞≠0 by contradiction. If ϕ∞≡0, from (3.18) and (3.19), we haveϕn→0strongly in L2(Rn),(45)ϕn→0strongly in Lp(Rn).(46)Since ϕn∈M, R(ϕn)=0, we obtain(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)→0,n→∞.(47)On the other hand, from (2.3) we have(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−2p∫|ϕn|pdx→dM.(48)From (3.20)-(3.21) we obtain (∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)→dM. According to Proposition 3.2, dM≥c>0. This is in contradictory to (3.22). Thus ϕ∞≠0.According to Proposition 3.3, we take Q=(ϕ∞)μwith μ>0uniquely determined by the condition R(Q)=R[(ϕ∞)μ]=0. From (3.17)-(3.19), we have(ϕ∞)μ→Q strongly in L2(Rn),(49)(ϕ∞)μ→Q strongly in Lp(Rn),(50)(ϕ∞)μ⇀Q weakly in H1(Rn).(51)Since R(ϕn)=0 and by Proposition 3.3, we getS[(ϕ∞)μ]≤S(ϕn).(52)From (3.23)-(3.26), we haveS(Q)≤lim−−−n→∞S[(ϕ∞)μ]≤lim n→∞S(ϕn)=inf MS.(53)Since Q≠0 and R(Q)=0, then Q∈M. Therefore from (3.27), Q solves the minimization problemS(Q)=minϕ∈MS(ϕ).(54)Thus we complete the proof of (a1) of Theorem 2.3.Next we prove (a2) of Theorem 2.3.Since Q is a solution of the problem (3.28), there exists a Lagrange multiplier ΛsuchthatδQ[S+ΛR]=0,(55)where δϕG denotes the variation of G(ϕ) about ϕ. By the formulaδϕG(ϕ)=∂∂ηG(ϕ+ηδϕ)∣∣η=0,(56)we haveδϕ[S+ΛR]=(1+2Λ)∫(−Δϕ⋅δϕ+V(x)ϕ⋅δϕ)dx−(1+pΛ)∫ϕ|ϕ|p−2δϕdx,(57 )where δϕ denotes the variation of ϕ. From (3.29), we get(1+2Λ)∫(|∇Q|2+V(x)|Q|2)dx=(1+pΛ)∫|Q|pdx.(58)From R(Q)=0, we have∫(|∇Q|2+V(x)|Q|2)dx=∫|Q|pdx.(59)From (3.31) and (3.32), we obtain Λ=0. Thus from (3.29) and (3.30), we have−ΔQ+V(x)Q=|Q|p−2Q.(60)This indicates that Q is a solution of (1.4) as (1.4) is the Euler Lagrange equation of the functional S. Thus we prove (a2) of Theorem 2.3.So far, we have completed the proof of Theorem 2.3. □Blow-up and global existenceIn this section, we prove Theorem 2.5.Lemma 4.1Let E(0)<dM, then K1and K2are invariant under the flow generated by (1.1).ProofSuppose that u0∈K1and u(t)be the solution of (1.1). From Pucci and Serrin [10]and (2.4), we haveS(u(t))≤<E(t)=E(0)−∫t0∥∥ut(s,⋅)∥∥mmds−∫t0∥∥∇ut(s,⋅)∥∥22ds S(Q)=dM,t∈[0,T).(61)To prove u(t)∈K1, we need to proveR(u(t))<0,t∈[0,T).(62)If (4.1) is not true, by continuity, there would exist a t¯>0 such that R(u(t¯))=0 because R(u0)<0. It follows that u(t¯)∈M. This isimpossible because S(u(t¯))<S(Q) and S(Q)=minϕ∈MS(ϕ). Thus (4.1)is true. So K1 is invariant under the flow generated by (1.1). Similarly, we can show that K2 is also invariant under the flow generated by (1.1). This completes the proof of Lemma 4.1. □Lemma 4.2Let the initial data u0, u1satisfy (1.2) and let u(t,x)be a local solution of the Cauchy problem (1.1)on[0,T). If there exists a u0∈K1and E(0)<dM, then theinequality∫(|∇u|2+V(x)|u|2)dx>2pp−2dM(63)is fulfilled for t∈[0,T).ProofAccording to Lemma 4.1, wehave ∫|u|pdx>∫(|∇u|2+V(x)|u|2)dx for t∈[0,T). Using the aboveinequality and the identity (2.3), we getdM≤p−22p∫(|∇u|2+V(x)|u|2)dx,(64)which completes the proof of Lemma 4.2. □Next, we prove Theorem 2.5.Proof of Theorem 2.5By (2.7), we have S(u0)≤E(0)<S(Q).Firstly, we prove (b1) of Theorem 2.5. From the energy identity we have∫t0∥∥ut(s,⋅)∥∥mmds+∫t0∥∥∇ut(s,⋅)∥∥22ds=E(0)−E(t)≤dM,(65)for all t≥0.Denoting J(t)=∥u(t,⋅)∥22, we haveJ′′(t)=2∥∥ut(t,⋅)∥∥22−2R(u(t))−2∫∇ut(t,x)∇u(t,x)dx−2∫u(t, x)ut(t,x)∣∣ut(t,x)∣∣m−2dx.(66)Using the Hölder inequality and the interpolation ine quality, we obtain ∣∣∣∫u(t,x)ut(t,x)∣∣ut(t,x)∣∣m−2dx∣∣∣≤≤∥∥u(t,⋅)∥∥m∥∥ut(t,⋅)∥∥m−1m∥∥u(t,⋅)∥∥δ2∥∥u(t,⋅)∥∥1−δp∥∥ut(t,⋅)∥∥m−1m,(67)with δ=1m−1p12−1p. From R(u)<0, we have∫V(x)|u|2dx<∥u∥pp,(68)which together with Lemma 4.1 give∥∥u(t,⋅)∥∥δ2∥∥u(t,⋅)∥∥1−δp∥∥ut(t,⋅)∥∥m−1m≤C∥∥ut(t,⋅)∥∥m−1m∥∥u(t,⋅)∥∥1−pm−δ+pδ2p∥∥u(t,⋅)∥∥pmp.(69)Using the Young inequality and 1−pm−δ+pδ2=0, we have∥∥u(t,⋅)∥∥δ2∥∥u(t,⋅)∥∥1−δp∥∥ut(t,⋅)∥∥m−1m≤C(ε)∥∥ut(t,⋅)∥∥mm+ε∥∥u(t,⋅)∥∥pp,∫∇ut(t,x)∇u(t,x)dx≤∥∥∇u(t,⋅)∥∥2∥∥∇ut(t,⋅)∥∥2≤C(ε)∥∥∇ut(t,⋅)∥∥22+ε∥∥∇u(t,⋅)∥∥22.(70)Since−R(u(t))≥≥−R(u(t))+δ(E(t)−E(0))(1−δp)∥∥u(t,⋅)∥∥pp+δ2∥∥ut(t,⋅)∥∥22+(δ2−1)(∥∥∇u(t,⋅)∥∥22+∫V(x)∣∣u(t,⋅)∣∣2dx)−δE(0),(71)we have12J′′(t)+C(ε)∥∥ut(t,⋅)∥∥mm+C(ε)∥∥∇ut(t,⋅)∥∥22+ε∥∥∇u(t,⋅)∥∥22≥(1+δ2)∥∥ut(t,⋅)∥∥22+(1−δp−ε)∥∥u(t,⋅)∥∥pp+(δ2−1−ε)(∥∥∇u(t,⋅)∥∥22+∫V(x)∣∣u(t,⋅)∣∣2dx)−δE(0),(72)where the constant δ>2 is chosen as follows.Since E(0)<dM, we choose the constant δso that2pdMpdM−(p−2)E(0)<δ<p.(73)This guarantees that δ>2. Then, using this choice and Lemma 4.2, we get (δ2−1)(∥∥∇u(t,⋅)∥∥22+∫V(x)∣∣u(t,⋅)∣∣2dx)−δE(0)≥(δ2−1)2pp−2dM−δE(0)≥0.(74)If the constant δ is fixed, we choose the constant ε such thatC1=1−δp−ε>0.(75)Finally, using the inequality (4.6) and Lemmas 4.1 and 4.2 we have J′′(t)+C(ε)∥∥ut(t,⋅)∥∥mm+C(ε)∥∥∇ut(t,⋅)∥∥22+ε∥∥∇u(t,⋅)∥∥2 2≥C1∥∥u(t,⋅)∥∥pp≥C1(∥∥∇u(t,⋅)∥∥22+∫V(x)∣∣u(t,⋅)∣∣2dx)≥C12pp−2dM,(76)where C1>0. Since (4.3), integrating (4.8) over [0,t] we haveJ′(t)≥C1pdMp−2t−C(ε)d+J′(0),(77)from which one concludes that there exists a t1 such that J′(t)|t=t1>0. Hence, J(t) is increasing for t>t1 (which is the interval of existence). Since R(u)<0, there exists a t2such that ∥u(t,x)∥pp is increasing for t>t2. When t is large enough, thequantities∥ut(t,x)∥mm and ∥∇u(t,x)∥22 are small enough. Otherwise, assume that there is t∗ such that∥ut(t,x)∥mm>∥ut(t∗,x)∥mm forall t>t∗. By integrating the inequality, we obtain a contradiction with (4.3) and E(t)≥0.Thus in these cases, the quantity(1−δp−ε)∥∥u(t,⋅)∥∥pp+(δ2−1)(∥∥∇u(t,⋅)∥∥22+∫V(x)∣∣u(t,⋅)∣∣2dx)−δE(0)−C(ε)∥∥ut(t,⋅)∥∥mm−C(ε)∥∥∇ut(t,⋅)∥∥22−ε∥∥∇u(t,⋅)∥∥22(78)will eventually become positive. Therefore for t large enough, from (4.6)and (4.7) we haveJ′′(t)≥(1+δ2)∥∥ut(t,⋅)∥∥22.(79)Using the Hölder inequality, we getJ(t)J′′(t)≥2+δ8[J′(t)]2.(80)Since[J−δ−68(t)]′′=−δ−68J−δ+108(t)[J(t)J′′(t)−2+δ8[J′(t)]2],(81 )from (4.10) we have [J−δ−68(t)]′′≤0. Therefore J−δ−68(t) is concave for sufficiently large t, and there exists a finite time T∗ such thatlim t→T∗J−δ−68(t)=0.(82)From assumption on V(x), we obtain∫V(x)|u|2dx≥V¯∫|u|2dx.(83)Thus one get T<∞ andlim t→T−∥u∥H=∞.(84)Now we complete the proof of (b1) of Theorem 2.5.Next, we prove (b2) of Theorem 2.5.From (2.7) and Lemma 4.1, we obtain E(0)<S(Q). It followsthat u0(x,t)satisfiesR(u0)=∫|∇u0|2dx+∫V(x)|u0|2dx−∫|u0|pdx>0,(85)which will be proved by contradiction. If (4.11) is not true, then we have R(u0)≤0. Thus there exists 0<μ≤1 such that u0≠0 andR(μu0)=μ2(∫|∇u0|2dx+∫V(x)|u0|2dx)−μp∫|u0|pdx=0,(86)which implies μu0∈M.On the other hand, for 0<μ≤1, u0∈K2and (2.3) yieldS(μu0)<μ2(∫|∇u0|2dx+∫V(x)|u0|2dx)≤p−22p∫|Q|pdx=S(Q),(87)which is contradictory to Lemma 3.1.Therefore, by R(u0)>0and Lemma 4.1, we have R(u)>0and E(t)≤E(0)<S(Q).ThusE(t)−1pS(u)≤E(0),(88)namely12∫|ut|2dx+p−22p(∫|∇u|2dx+∫V(x)|u|2dx)≤E(0).(89)Therefore we have established the boundof u(x,t) in H1 for t∈[0,T) and thus the solution u(x,t) of (1.1) exists globally on t∈[0,∞).From (4.11), E(0)<S(Q) and Lemma 3.1, we have the estimate (2.8).Thus, we complete the proof of Theorem 2.5. □Competing interestsThe author declares that they have no competing interests.ReferencesJ. Differ. Equ.. 150, 203–214。

数学专业英语(只是部分,不是很完整)

Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition “+”, subtraction “-” , multiplication “×”, division “÷” and equality “=”. 数学家研究的是概念和命题，公理，公设，定义和定理都是命题。符号是数学中一个特殊而有用的工具，常用于表达概念和命题。公式，图形和图表都是不同的符号……..
1－B Equation
An equation is a statement of the equality between two equal numbers or number symbols. Equations are of two kinds---- identities and equations of condition. An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation. 等式是关于两个数或者数的符号相等的一种描述。等式有两种－恒等式和条件等式。算术或者代数恒等式都是等式。这种等式的两端要么一样，要么经过执行指定的运算后变成一样。

数学英文论文

070451 Controlling chaos based on an adaptive nonlinear compensatingmechanism*Corresponding author,Xu Shu ,email:123456789@Abstract The control problems of chaotic systems are investigated in the presence of parametric u ncertainty and persistent external distu rbances based on nonlinear control theory. B y designing a nonlinear compensating mechanism, the system deterministic nonlinearity, parametric uncertainty and disturbance effect can be compensated effectively. The renowned chaotic Lorenz system subject to parametric variations and external disturbances is studied as an illustrative example. From Lyapu nov stability theory, sufficient conditions for the choice of control parameters are derived to guarantee chaos control. Several groups of experiments are carried out, including parameter change experiments, set-point change experiments and disturbance experiments. Simulation results indicate that the chaotic motion can be regulated not only to stead y states but also to any desired periodic orbits with great immunity to parametric variations and external distu rbances.Keywords: chaotic system, nonlinear compensating mechanism, Lorenz chaotic systemPACC: 05451. IntroductionChaotic motion, as the peculiar behavior in deterministic systems, may be undesirable in many cases, so suppressing such a phenomenon has been intensively studied in recent years. Generally speaking chaos suppression and chaos synchronization[1-4 ]are two active research fields in chaos control and are both crucial in application of chaos. In the following letters we only deal with the problem of chaos suppression and will not discuss the chaos synchronization problem.Since the early 1990s, the small time-dependent parameter perturbation was introduced by Ott,Grebogi, and Y orke to eliminate chaos,[5]many effective control methods have been reported in various scientific literatures.[1-4,6-36,38-44,46] There are two lines in these methods. One is to introduce parameter perturbations to an accessible system parameter, [5-6,8-13] the other is to introduce an additive external force to the original uncontrolled chaotic system. [14-37,39-43,47] Along the first line, when system parameters are not accessible or can not be changed easily, or the environment perturbations are not avoided, these methods fail. Recently, using additive external force to achieve chaos suppression purpose is in the ascendant. Referring to the second line of the approaches, various techniques and methods have been proposed to achieve chaos elimination, to mention only a few:(ⅰ) linear state feedback controlIn Ref.[14] a conventional feedback controller was designed to drive the chaotic Duffing equation to one of its inherent multiperiodic orbits.Recently a linear feedback control law based upon the Lyapunov–Krasovskii (LK) method was developed for the suppression of chaotic oscillations.[15]A linear state feedback controller was designed to solve the chaos control problem of a class of new chaotic system in Ref.[16].(ⅱ) structure variation control [12-16]Since Y u X proposed structure variation method for controlling chaos of Lorenz system,[17]some improved sliding-mode control strategies were*Project supported by the National Natural Science Foundation of C hina (Grant No 50376029). †Corresponding au thor. E-mail:zibotll@introduced in chaos control. In Ref.[18] the author used a newly developed sliding mode controller with a time-varying manifold dynamic to compensate the external excitation in chaotic systems. In Ref.[19] the design schemes of integration fuzzy sliding-mode control were addressed, in which the reaching law was proposed by a set of linguistic rules. A radial basis function sliding mode controller was introduced in Ref.[20] for chaos control.(ⅲ) nonlinear geometric controlNonlinear geometric control theory was introduced for chaos control in Ref.[22], in which a Lorenz system model slightly different from the original Lorenz system was studied considering only the Prandtl number variation and process noise. In Ref.[23] the state space exact linearization method was also used to stabilize the equilibrium of the Lorenz system with a controllable Rayleigh number. (ⅳ)intelligence control[24-27 ]An intelligent control method based on RBF neural network was proposed for chaos control in Ref.[24]. Liu H, Liu D and Ren H P suggested in Ref.[25] to use Least-Square Support V ector Machines to drive the chaotic system to desirable points. A switching static output-feedback fuzzy-model-based controller was studied in Ref.[27], which was capable of handling chaos.Other methods are also attentively studied such as entrainment and migration control, impulsive control method, optimal control method, stochastic control method, robust control method, adaptive control method, backstepping design method and so on. A detailed survey of recent publications on control of chaos can be referenced in Refs.[28-34] and the references therein.Among most of the existing control strategies, it is considered essentially to know the model parameters for the derivation of a controller and the control goal is often to stabilize the embedded unstable period orbits of chaotic systems or to control the system to its equilibrium points. In case of controlling the system to its equilibrium point, one general approach is to linearize the system in the given equilibrium point, then design a controller with local stability, which limits the use of the control scheme. Based on Machine Learning methods, such as neural network method[24]or support vector machine method,[25]the control performance often depends largely on the training samples, and sometimes better generalization capability can not be guaranteed.Chaos, as the special phenomenon of deterministic nonlinear system, nonlinearity is the essence. So if a nonlinear real-time compensator can eliminate the effect of the system nonlinearities, chaotic motion is expected to be suppressed. Consequently the chaotic system can be controlled to a desired state. Under the guidance of nonlinear control theory, the objective of this paper is to design a control system to drive the chaotic systems not only to steady states but also to periodic trajectories. In the next section the controller architecture is introduced. In section 3, a Lorenz system considering parametric uncertainties and external disturbances is studied as an illustrative example. Two control schemes are designed for the studied chaotic system. By constructing appropriate L yapunov functions, after rigorous analysis from L yapunov stability theory sufficient conditions for the choice of control parameters are deduced for each scheme. Then in section 4 we present the numerical simulation results to illustrate the effectiveness of the design techniques. Finally some conclusions are provided to close the text.2. Controller architectureSystem differential equation is only an approximate description of the actual plant due to various uncertainties and disturbances. Without loss of generality let us consider a nonlinear continuous dynamic system, which appears strange attractors under certain parameter conditions. With the relative degree r n(n is the dimension of the system), it can be directly described or transformed to the following normal form:121(,,)((,,)1)(,,,)(,,)r r r z z z z za z v wb z v u u d z v u u vc z v θθθθθθθθ-=⎧⎪⎪⎪=⎪=+∆+⎨⎪ ++∆-+⎪⎪ =+∆+⎪=+∆⎩ (1) 1y z =where θ is the parameter vector, θ∆ denotes parameter uncertainty, and w stands for the external disturbance, such that w M ≤with Mbeingpositive.In Eq.(1)1(,,)T r z z z = can be called external state variable vector,1(,,)T r n v v v += called internal state variable vector. As we can see from Eq.(1)(,,,,)(,,)((,,)1)d z v w u a z v w b z v uθθθθθθ+∆=+∆+ ++∆- (2)includes system nonlinearities, uncertainties, external disturbances and so on.According to the chaotic system (1), the following assumptions are introduced in order to establish the results concerned to the controller design (see more details in Ref.[38]).Assumption 1 The relative degree r of the chaotic system is finite and known.Assumption 2 The output variable y and its time derivatives i y up to order 1r -are measurable. Assumption 3 The zero dynamics of the systemis asymptotically stable, i.e.,(0,,)v c v θθ=+∆ is asymptotically stable.Assumption 4 The sign of function(,,)b z v θθ+∆is known such that it is always positive or negative.Since maybe not all the state vector is measurable, also (,,)a z v θθ+∆and (,,)b z v θθ+∆are not known, a controller with integral action is introduced to compensate theinfluenceof (,,,,)d z v w u θθ+∆. Namely,01121ˆr r u h z h z h z d------ (3) where110121112100ˆr i i i r r r r i i ii r i i d k z k k k z kz k uξξξ-+=----++-==⎧=+⎪⎪⎨⎪=----⎪⎩∑∑∑ (4)ˆdis the estimation to (,,,,)d z v w u θθ+∆. The controller parameters include ,0,,1i h i r =- and ,0,,1i k i r =- . Here011[,,,]Tr H h h h -= is Hurwitz vector, such that alleigenvalues of the polynomial121210()rr r P s s h sh s h s h --=+++++ (5)have negative real parts. The suitable positive constants ,0,,1i h i r =- can be chosen according to the expected dynamic characteristic. In most cases they are determined according to different designed requirements.Define 1((,,))r k sign b z v θμ-=, here μstands for a suitable positive constant, and the other parameters ,0,,2i k i r =- can be selected arbitrarily. After011[,,,]Tr H h h h -= is decided, we can tune ,0,,1i k i r =- toachievesatisfyingstaticperformances.Remark 1 In this section, we consider a n-dimensional nonlinear continuous dynamic system with strange attractors. By proper coordinate transformation, it can be represented to a normal form. Then a control system with a nonlinear compensator can be designed easily. In particular, the control parameters can be divided into two parts, which correspond to the dynamic characteristic and the static performance respectively (The theoretic analysis and more details about the controller can be referenced to Ref.[38]).3. Illustrative example-the Lorenz systemThe Lorenz system captures many of the features of chaotic dynamics, and many control methods have been tested on it.[17,20,22-23,27,30,32-35,42] However most of the existing methods is model-based and has not considered the influence ofpersistent external disturbances.The uncontrolled original Lorenz system can be described by112121132231233()()()()x P P x P P x w x R R x x x x w xx x b b x w =-+∆++∆+⎧⎪=+∆--+⎨⎪=-+∆+⎩ (6) where P and R are related to the Prendtl number and Rayleigh number respectively, and b is a geometric factor. P ∆, R ∆and b ∆denote the parametric variations respectively. The state variables, 1x ,2x and 3x represent measures of fluid velocity and the spatial temperature distribution in the fluid layer under gravity ， and ,1,2,3i w i =represent external disturbance. In Lorenz system the desired response state variable is 1x . It is desired that 1x is regulated to 1r x , where 1r x is a given constant. In this section we consider two control schemes for system (6).3.1 Control schemes for Lorenz chaotic system3.1.1 Control scheme 1The control is acting at the right-side of the firstequation (1x), thus the controlled Lorenz system without disturbance can be depicted as1122113231231x Px Px u xRx x x x x x x bx y x =-++⎧⎪=--⎨⎪=-⎩= (7) By simple computation we know system (7) has relative degree 1 (i.e., the lowest ordertime-derivative of the output y which is directly related to the control u is 1), and can be rewritten as1122113231231z Pz Pv u vRz z v v v z v bv y z =-++⎧⎪=--⎨⎪=-⎩= (8) According to section 2, the following control strategy is introduced:01ˆu h z d=-- (9) 0120010ˆ-d k z k k z k uξξξ⎧=+⎪⎨=--⎪⎩ (10) Theorem 1 Under Assumptions 1 toAssumptions 4 there exists a constant value *0μ>, such that if *μμ>, then the closed-loop system (8), (9) and (10) is asymptotically stable.Proof Define 12d Pz Pv =-+, Eq.(8) can be easily rewritten as1211323123z d u v Rz z v v vz v bv =+⎧⎪=--⎨⎪=-⎩ (11) Substituting Eq.(9) into Eq.(11) yields101211323123ˆz h z d dv R z z v v v z v bv ⎧=-+-⎪=--⎨⎪=-⎩ (12) Computing the time derivative of d and ˆdand considering Eq.(12) yields12011132ˆ()()dPz Pv P h z d d P Rz z v v =-+ =--+- +-- (13) 0120010000100ˆ-()()ˆ=()d k z k k z k u k d u k d k z k d d k dξξξ=+ =--++ =-- - = (14)Defining ˆdd d =- , we have 011320ˆ()()dd d P h P R z P z v P v P k d=- =+- --+ (15) Then, we can obtain the following closed-loop system101211323123011320()()z h z dvRz z v v v z v bv d Ph PR z Pz v Pv P k d⎧=-+⎪=--⎪⎨=-⎪⎪=+---+⎩ (16) To stabilize the closed-loop system (16), a L yapunovfunction is defined by21()2V ςς=(17)where, ςdenotes state vector ()123,,,Tz v v d, isthe Euclidean norm. i.e.,22221231()()2V z v v dς=+++ (18) We define the following compact domain, which is constituted by all the points internal to the superball with radius .(){}2222123123,,,2U z v v d zv v dM +++≤(19)By taking the time derivative of ()V ςand replacing the system expressions, we have11223322*********01213()()(1)V z z v v v v dd h z v bv k P d R z v P R P h z d P v d P z v d ς=+++ =----++ +++-- (20) For any ()123,,,z v v d U ∈, we have: 222201230120123()()(1)V h z v b v k P dR z v PR Ph z d P v d d ς≤----+ ++++ ++ (21)Namely,12300()(1)22020V z v v dPR Ph R h R P ς⎡⎤≤- ⎣⎦++ - 0 - - 1 - 2⨯00123(1)()2Tb PR Ph P k P z v v d ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥0 ⎢⎥2⎢⎥++⎢⎥- - - +⎢⎥⎣22⎦⎡⎤⨯ ⎣⎦(22) So if the above symmetrical parameter matrix in Eq.(22) is positive definite, then V is negative and definite, which implies that system (16) is asymptotically stable based on L yapunov stability theory.By defining the principal minor determinants of symmetrical matrix in Eq.(22) as ,1,2,3,4i D i =, from the well-known Sylvester theorem it is straightforward to get the following inequations:100D h => (23)22004RD h =-> (24)23004R b D bh =-> (25)240302001()(1)(2)821[2(1)]08P M D k P D b PR Ph PR D Pb Ph R PR Ph =+-+++--+++>(26)After 0h is determined by solving Inequalities (23) to (25), undoubtedly, the Inequalities (26) can serve effectively as the constraints for the choice of 0k , i.e.20200031(1)(2)821[2(1)]8P M b PR Ph PR D Pb Ph R PR Ph k P D ++++ ++++>- (27)Here,20200*31(1)(2)821[2(1)]8P M b PR Ph PR D Pb Ph R PR Ph P D μ++++ ++++=-.Then the proof of the theorem 1 is completed. 3.1.2 Control scheme 2Adding the control signal on the secondequation (2x ), the system under control can be derived as112211323123x P x P x x R x x x x u xx x bx =-+⎧⎪=--+⎨⎪=-⎩ (28) From Eq.(28), for a target constant 11()r x t x =,then 1()0xt = , by solving the above differential equation, we get 21r r x x =. Moreover whent →∞,3r x converges to 12r x b . Since 1x and 2x havethe same equilibrium, then the measured state can also be chosen as 2x .To determine u , consider the coordinate transform:122133z x v x v x=⎧⎪=⎨⎪=⎩ and reformulate Eq.(28) into the following normal form:1223121231231zRv v v z u vPz Pv v z v bv y z =--+⎧⎪=-⎨⎪=-⎩= (29) thus the controller can be derived, which has the same expression as scheme 1.Theorem 2 Under Assumptions 1, 2, 3 and 4, there exists a constant value *0μ>, such that if *μμ>, then the closed-loop system (9), (10) and (29) is asymptotically stable.Proof In order to get compact analysis, Eq.(29) can be rewritten as12123123z d u v P z P v vz v bv =+⎧⎪=-⎨⎪=-⎩ (30) where 2231d Rv v v z =--Substituting Eq.(9) into Eq.(30),we obtain:1012123123ˆz h z d dv P z P v v z v bv ⎧=-+-⎪=-⎨⎪=-⎩ (31) Giving the following definition:ˆdd d =- (32) then we can get22323112123212301()()()()dRv v v v v z R Pz Pv Pz Pv v v z v bv h z d =--- =--- ----+ (33) 012001000ˆ-()d k z k k z k u k d u k dξξ=+ =--++ = (34) 121232123010ˆ()()()(1)dd d R Pz Pv Pz Pv v v z v bv h z k d=- =--- --+-+ (35)Thus the closed-loop system can be represented as the following compact form:1012123123121232123010()()()(1)zh z d v Pz Pv v z v bv d R Pz Pv Pz Pv v v z v bv h z k d⎧=-+⎪⎪=-⎪=-⎨⎪=---⎪⎪ --+-+⎩(36) The following quadratic L yapunov function is chosen:21()2V ςς=(37)where, ςdenotes state vector ()123,,,Tz v v d , is the Euclidean norm. i.e.,22221231()()2V z v v dς=+++ (38) We can also define the following compact domain, which is constituted by all the points internalto the super ball with radius .(){}2222123123,,,2U z v v d zv v dM =+++≤ (39)Differentiating V with respect to t and using Eq.(36) yields112233222201230011212322321312()(1)(1)()V z z v v v v dd h z P v bv k dP R h z d P z v z v v P b v v d P v d P z v d z v d ς=+++ =----+ +++++ ++--- (40)Similarly, for any ()123,,,z v v d U ∈, we have: 2222012300112133231()(1)(1)(2V h z P v b v k dPR h z d P z v v P b d P v d d M z dς≤----+ +++++ ++++ + (41)i.e.,12300()(12)22V z v v dPR M h P h P Pς⎡⎤≤- ⎣⎦+++ - -2 - 0 ⨯ 001230(12)(1)2TP b PR M h P k z v v d ⎡⎤⎢⎥⎢⎥⎢⎥ - ⎢⎥⎢⎥⎢⎥ ⎢⎥22⎢⎥⎢⎥ +++ - - -+⎢⎥⎣22⎦⎡⎤⨯ ⎣⎦(42) For brevity, Let1001(12)[(222)82(23)]P PR M h b PR P h M P b α=++++++ ++(43) 2201[(231)(13)]8P M P b b PR h α=+-+++ (44)230201(2)[2(12)8(2)(4)]PM P b P P PR M h P b Ph P α=++ +++ ++- (45)Based on Sylvester theorem the following inequations are obtained:100D h => (46)22004PD h P =-> (47)3202PMD bD =-> (48)403123(1)0D k D ααα=+---> (49)where,1,2,3,4i D i =are the principal minordeterminants of the symmetrical matrix in Eq.(42).*0k μ>*12331D αααμ++=- (50)The theorem 2 is then proved.Remark 2 In this section we give two control schemes for controlling chaos in Lorenz system. For each scheme the control depends on the observed variable only, and two control parameters are neededto be tuned, viz. 0h and 0k . According to L yapunov stability theory, after 0h is fixed, the sufficient condition for the choice of parameter 0k is also obtained.4. Simulation resultsChoosing 10P =,28R =, and 8/3b =, the uncontrolled Lorenz system exhibits chaotic behavior, as plotted in Fig.1. In simulation let the initial values of the state of thesystembe 123(0)10,(0)10,(0)10x x x ===.x1x 2x1x 3Fig.1. C haotic trajectories of Lorenz system (a) projected on12x x -plane, (b) projected on 13x x -plane4.1 Simulation results of control the trajectory to steady stateIn this section only the simulation results of control scheme 2 are depicted. The simulation results of control scheme 1 will be given in Appendix. For the first five seconds the control input is not active, at5t s =, control signal is input and the systemtrajectory is steered to set point2121(,,)(8.5,8.5,27.1)T Tr r r x x x b =, as can be seen inFig.2(a). The time history of the L yapunov function is illustrated in Fig.2(b).t/sx 1,x 2,x 3t/sL y a p u n o v f u n c t i o n VFig.2. (a) State responses under control, (b) Time history of the Lyapunov functionA. Simulation results in the presence ofparameters ’ changeAt 9t s =, system parameters are abruptly changed to 15P =,35R =, and 12/3b =. Accordingly the new equilibrium is changedto 2121(,,)(8.5,8.5,18.1)T Tr r r x x x b =. Obviously, aftervery short transient duration, system state converges to the new point, as shown in Fig.3(a). Fig.4(a) represents the evolution in time of the L yapunov function.B. Simulation results in the presence of set pointchangeAt 9t s =, the target is abruptly changedto 2121(,,)(12,12,54)T Tr r r x x x b =, then the responsesof the system state are shown in Fig.3(b). In Fig.4(b) the time history of the L yapunov function is expressed.t/sx 1,x 2,x 3t/sx 1,x 2,x 3Fig.3. State responses (a) in the presence of parameter variations, (b) in the presence of set point changet/sL y a p u n o v f u n c t i o n Vt/sL y a p u n o v f u n c t i o n VFig.4. Time history of the Lyapunov fu nction (a) in the presence of parameter variations, (b) in the presence of set point changeC. Simulation results in the presence ofdisturbanceIn Eq.(5)external periodic disturbance3cos(5),1,2,3i w t i π==is considered. The time responses of the system states are given in Fig.5. After control the steady-state phase plane trajectory describes a limit cycle, as shown in Fig.6.t/sx 1,x 2,x 3Fig.5. State responses in the presence of periodic disturbancex1x 3Fig.6. The state space trajectory at [10,12]t ∈in the presence ofperiodic disturbanceD. Simulation results in the presence of randomnoiseUnder the influence of random noise,112121132231233xPx Px x Rx x x x u xx x bx εδεδεδ=-++⎧⎪=--++⎨⎪=-+⎩ (51) where ,1,2,3i i δ= are normally distributed withmean value 0 and variance 0.5, and 5ε=. The results of the numerical simulation are depicted in Fig.7,which show that the steady responses are hardly affected by the perturbations.t/sx 1,x 2,x 3t/se 1,e 2,e 3Fig.7. Time responses in the presence of random noise (a) state responses, (b) state tracking error responses4.2 Simulation results of control the trajectory to periodic orbitIf the reference signal is periodic, then the system output will also track this signal. Figs.8(a) to (d) show time responses of 1()x t and the tracking trajectories for 3-Period and 4-period respectively.t/sx 1x1x 2t/sx 1x1x 2Fig.8. State responses and the tracking periodic orbits (a)&( b)3-period, (c)&(d) 4-periodRemark 3 The two controllers designed above solved the chaos control problems of Lorenz chaoticsystem, and the controller design method can also beextended to solve the chaos suppression problems of the whole Lorenz system family, namely the unified chaotic system.[44-46] The detail design process and close-loop system analysis can reference to the author ’s another paper.[47] In Ref.[47] according to different positions the scalar control input added,three controllers are designed to reject the chaotic behaviors of the unified chaotic system. Taking the first state 1x as the system output, by transforming system equation into the normal form firstly, the relative degree r (3r ≤) of the controlled systems i s known. Then we can design the controller with the expression as Eq.(3) and Eq.(4). Three effective adaptive nonlinear compensating mechanisms are derived to compensate the chaotic system nonlinearities and external disturbances. According toL yapunov stability theory sufficient conditions for the choice of control parameters are deduced so that designers can tune the design parameters in an explicit way to obtain the required closed loop behavior. By numeric simulation, it has been shown that the designed three controllers can successfully regulate the chaotic motion of the whole family of the system to a given point or make the output state to track a given bounded signal with great robustness.5. ConclusionsIn this letter we introduce a promising tool to design control system for chaotic system subject to persistent disturbances, whose entire dynamics is assumed unknown and the state variables are not completely measurable. By integral action the nonlinearities, including system structure nonlinearity, various disturbances, are compensated successfully. It can handle, therefore, a large class of chaotic systems, which satisfy four assumptions. Taking chaotic Lorenz system as an example, it has been shown that the designed control scheme is robust in the sense that the unmeasured states, parameter uncertainties and external disturbance effects are all compensated and chaos suppression is achieved. Some advantages of this control strategy can be summarized as follows: (1) It is not limited to stabilizing the embeddedperiodic orbits and can be any desired set points and multiperiodic orbits even when the desired trajectories are not located on the embedded orbits of the chaotic system.(2) The existence of parameter uncertainty andexternal disturbance are allowed. The controller can be designed according to the nominal system.(3) The dynamic characteristics of the controlledsystems are approximately linear and the transient responses can be regulated by the designer through controllerparameters ,0,,1i h i r =- .(4) From L yapunov stability theory sufficientconditions for the choice of control parameters can be derived easily.(5) The error converging speed is very fast evenwhen the initial state is far from the target one without waiting for the actual state to reach the neighborhood of the target state.AppendixSimulation results of control scheme 1.t/sx 1,x 2,x 3t/sL y a p u n o v f u n c t i o n VFig.A1. (a) State responses u nder control, (b) Time history of the Lyapunov functiont/sx 1,x 2,x 3t/sx 1,x 2,x 3Fig.A2. State responses (a) in the presence of parameter variations, (b) in the presence of set point changet/sL y a p u n o v f u n c t i o n Vt/sL y a p u n o v f u n c t i o n VFig.A3. Time history of the L yapu nov fu nction (a) in the presence of parameter variations, (b) in the presence of set point changet/sx 1,x 2,x 3Fig.A4. State responses in the presence of periodic disturbanceresponsest/sx 1,x 2,x 3Fig.A5. State responses in the presence of rand om noiset/sx 1x1x 2Fig.A6. State response and the tracking periodic orbits (4-period)References[1] Lü J H, Zhou T S, Zhang S C 2002 C haos Solitons Fractals 14 529[2] Yoshihiko Nagai, Hua X D, Lai Y C 2002 C haos Solitons Fractals 14 643[3] Li R H, Xu W , Li S 2007 C hin.phys.16 1591 [4]Xiao Y Z, Xu W 2007 C hin.phys.16 1597[5] Ott E ，Greb ogi C and Yorke J A 1990 Phys.Rev .Lett. 64 1196 [6]Yoshihiko Nagai, Hua X D, Lai Y C 1996 Phys.Rev.E 54 1190 [7] K.Pyragas, 1992 Phys. Lett. A 170 421 [8] Lima,R and Pettini,M 1990 Phys.Rev.A 41 726[9] Zhou Y F, Tse C K, Qiu S S and Chen J N 2005 C hin.phys. 14 0061[10] G .Cicog na, L.Fronzoni 1993 Phys.Rew .E 30 709 [11] Rakasekar,S. 1993 Pramana-J.Phys.41 295 [12] Gong L H 2005 Acta Phys.Sin.54 3502 (in C hinese) [13] Chen L,Wang D S 2007 Acta Phys.Sin.56 0091 (in C hinese) [14] C hen G R and Dong X N 1993 IEEE Trans.on Circuits andSystem-Ⅰ:Fundamental Theory and Applications 40 9 [15] J.L. Kuang, P.A. Meehan, A.Y.T. Leung 2006 C haos SolitonsFractals 27 1408[16] Li R H, Xu W, Li S 2006 Acta Phys.Sin.55 0598 (in C hinese) [17] Yu X 1996 Int.J.of Systems Science 27 355[18] Hsun-Heng Tsai, C hyu n-C hau Fuh and Chiang-Nan Chang2002 C haos,Solitons Fractals 14 627[19] Her-Terng Yau and C hieh-Li C hen 2006 C hao ,SolitonsFractal 30 709[20] Guo H J, Liu J H, 2004 Acta Phys.Sin.53 4080 (in C hinese) [21] Yu D C, Wu A G , Yang C P 2005 Chin.phys.14 0914 [22] C hyu n-C hau Fuh and Pi-Cheng Tu ng 1995 Phys.Rev .Lett.752952[23] Chen L Q, Liu Y Z 1998 Applied Math.Mech. 19 63[24] Liu D, R en H P, Kong Z Q 2003 Acta Phys.Sin.52 0531 (inChinese)[25] Liu H, Liu D and Ren H P 2005 Acta Phys.Sin.54 4019 (inChinese)[26] C hang W , Park JB, Joo YH, C hen GR 2002 Inform Sci 151227[27] Gao X, Liu X W 2007 Acta Phys.Sin. 56 0084 (in C hinese) [28] Chen S H, Liu J, Lu J 2002 C hin.phys.10 233 [29] Lu J H, Zhang S. 2001 Phys. Lett. A 286 145[30] Liu J, Chen S H, Xie J. 2003 C haos Solitons Fractals 15 643 [31] Wang J, Wang J, Li H Y 2005 C haos Solitons Fractals 251057[32] Wu X Q, Lu JA, C hi K. Tse, Wang J J, Liu J 2007 ChaoSolitons Fractals 31 631[33] A.L.Fradkov , R .J.Evans, 2002 Preprints of 15th IF AC W orldCongress on Automatic Control 143[34] Zhang H G 2003 C ontrol theory of chaotic systems (Shenyang:Northeastern University) P38 (in C hinese)[35] Yu-Chu Tian, Moses O.Tadé, David Levy 2002Phys.Lett.A.296 87[36] Jose A R , Gilberto E P, Hector P, 2003 Phys. Lett. A 316 196 [37] Liao X X, Yu P 2006 Chaos Solitons Fractals 29 91[38] Tornambe A, V aligi P.A 1994 Measurement, and C ontrol 116293[39] Andrew Y.T.Leung, Liu Z R 2004 Int.J.Bifurc.C haos 14 2955 [40] Qu Z L, Hu,G .,Yang,G J, Qin,G R 1995 Phys.Rev .Lett.74 1736 [41] Y ang J Z, Qu Z L, Hu G 1996 Phys.Rev.E.53 4402[42] Shyi-Kae Yang, C hieh-Li Chen, Her-Terng Yau 2002 C haosSolitons Fractals 13 767。

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)

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。

“数学史与数学教育”英语论文

Teacher education through the history of mathematicsFulvia FuringhettiPublished online: 16 February 2007# Springer Science + Business Media B.V. 2007Abstract In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers‟ belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the …laboratory of mathematics education‟carried out in a 2 year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.Key words history of mathematics . mathematics teacher education . cognitive root . evolutionary mode . situated modeThe difference between us and the pupils entrusted to our care lies only in this, that we have traveled a longer tract of the parabola of life. If the pupils do not understand, the fault is with the teacher who does not know how to explain. Nor does it do any goodto load the responsibility onto the lower schools. We must take the pupils as they are,and recall what they have forgotten, or studied under a different nomenclature. If the teacher torments his students, and instead of winning their love, he arouses hatred against himself and the science he teaches, not only will his teaching be negative, butthe necessity of living with so many little enemies will be a continual torment for him. Each makes his own fortune, good or bad. Who causes his own troubles, cries alone.So said Jove, as Homer reported (Odyssey, I, 34). With these principles, dear readerand colleague, may you live happily.(the mathematician Peano, 1924)Educ Stud Math (2007) 66:131–143DOI 10.1007/s10649-006-9070-0F. Furinghetti (*)Dipartimento Di Matematica, Università Di Genova, Via Dodecaneso 35, Genova 16146, Italye-mail: furinghe@dima.unige.itDewey said somewhere that subject matter is a prime source of pedagogical insights. Almost no educators really believe this, I think, except in the trivial sense of hopingthat teachers, textbooks writers, and curriculum designers …know their mathematics‟. Even many mathematicians, who ought to know better, have no interest in lookingbelow the instrumental or formal surface of mathematics in order to get clues abouthow to present it more effectively.(the mathematics educator Wheeler, 1989, pp. 282–283)Any embryo of sciences presents this double aspect: monster as a fetus; marvel as a germ. (the historian Gino Loria, unknown source)1 IntroductionThis paper deals with the problem of designing strategies for teacher education programs. The three quotations at the beginning of the paper should direct the reader‟s attention to the three main issues at stake: knowledge for teaching, view of mathematics and its teaching, history of mathematics in mathematics teaching.Borko et al. (1992) have pointed out that …knowledge for teaching‟ is comprised of …subject matter knowledge‟ and …pedagogical matter knowledge‟. There is a third emerging component, that is teachers‟ beliefs about mathematics and its teaching, whose importancein shaping instructional practice is widely recognized, (see Leder, Pehkonen & Törner, 2002, Thompson, 1992). Cooney (1999, p. 168) has noted that “Preservice teachers come to teacher education programs with notions about what constitutes teaching. Not surprisingly, their view of the teaching of mathematics is, more or less, consistent with the way they experienced learning mathematics.” The consequence of this is that “their learning experiences often counter current reform efforts in mathematics education”. Analogously Skott (2001) found that the novice teacher he studied specifically referred to experiences from his own upper secondary education. To combat this situation teacher education programs need to offer challenging situations that help to elaborate personal and meditated beliefs about mathematics and its teaching. One of the suggestions (Thompson, 1992) is the introduction of courses in the history of mathematics.The history of mathematics is not a new issue in teacher education programs, see (Schubring et al., 2000). The way history has been employed for this purpose has varied and has led to different outputs. In some cases, as discussed in (Heiede, 1996), the presence of history consisted simply in delivering a course in the history of mathematics with the aim of giving a historical background to teachers‟ mathematics knowledge, as advocated by many researchers, Freudenthal (1981) for one. In other cases, (see Arcavi, Bruckheimer, & Ben-Zvi, 1982, 1987; Swetz, 1995), packages of historical materials focused on mathematical concepts difficult to teach were used to deepen teachers‟ pedagogical reflection on these concepts. Some studies, (see Hsieh, C.-J. & Hsieh, F.-J., 2000; Hsieh, 2000; Philippou & Christou, 1998a, b), paid particular attention to the link between history and mathematics teachers‟beliefs and attitudes about mathematics. The overall conclusion drawn by the designers of the courses in which history was used is that teachers‟ preparation may have benefits from historical knowledge. A few studies, (see Fraser & Koop, 1978; Stander, 1991), report no significant difference in teacher preparation after the use of historical materials.In my opinion, the different conclusions drawn by the various authors are not to be ascribed to the role of the history of mathematics in teacher education, but to the way of132 F. Furinghettidealing with history in teacher education programs. The outputs of these programs are not …method-free‟, that is to say the history of mathematics presented in these courses may havea different influence in teacher education according to the method that has been used to insert it into the program.In this paper I refer to an education program for prospective teachers in which thehistory of mathematics was introduced not per se, but as a mediator of knowledge for teaching. The aim was to make the participants reflect on the meaning of mathematical objects through experiencing historical moments of their construction. It was intended thatthis reflection would promote an appropriation of meaning for teaching mathematicalobjects that counteracts the passive reproduction of the style of mathematics teaching the prospective teachers have experienced as students.2 To experience the construction of mathematical objects through historyThe arguments that support the use of history in the classroom apply –with some adaptations to the different context –also to the case of teacher education. They may be summarized by the sentence of Thomas (2002, p. 46), according to which “Mathematical facts, without some understanding of why they are the way they are, are almost impossibleto learn.” Another reason quoted in the literature for using history in mathematics educationis the fact that history promotes cultural understanding. As Jahnke et al. (2000, p. 292) putit: “Integrating history in mathematics invites us to place the development of mathematicsin the scientific and technological context of a particular time and in the history of ideas and society, and also to consider the history of teaching mathematics from perspectives that lie outside the established disciplinary subject boundaries.” They also mention replacement,since “Integrating history in mathematics replaces the usual with something different: itallows mathematics to be seen as an intellectual activity, rather than as just a corpus of knowledge or a set of techniques.” The cultural understanding and replacement promoted by history have some link with the need to humanize mathematics education that is often advocated in the literature, (see Bidwell, 1993; Brown, 1996). This link is outlined by Tymoczko (1994, p. 335) as follows: “To introduce students to humanistic mathematics is to introduce them to a human adventure, an adventure that humans have actually partaken of in history.”I ascribe great importance to the enrichment of the view of mathematics provided bycultural understanding and replacement; nevertheless in this paper I am more concernedwith a third aspect –reorientation –which is strictly linked to the aim of making the prospective teachers experience again the construction of mathematical objects. Accordingto Jahnke et al. (2000, p. 292) “Integrating history in mathematics challenges one‟s perceptions through making the familiar unfamiliar. [...] The history of mathematics has the virtue of …astonishing with what comes of itself‟ [...].” To walk in the foreign and unknown landscape provided by history forces us to look around in a different manner and brings tolight elements which otherwise would escape. In such a context one better grasps the roots around which the mathematical concepts were built over the centuries. Through reorientation the learners involved in the process (in our case prospective teachers) areforced to find their own path towards the appropriation of meaning of mathematical objects.I point out that here …meaning‟ has two dimensions: meaning of mathematical objects and meaning of objects to be taught.I see a strict relation between the roots of concepts identified by history and thecognitive roots of concepts described by Tall (2003) as concepts which are (potentially) meaningful to the student at the time, yet contain the seeds of cognitive expansion to formal Teacher education through the history of mathematics 133definitions and later theoretical development. Tall (2003) mentions three representational worlds: embodied, symbolic-proceptual, formal-axiomatic. These three worlds encompass different modes of operating, which Tall put in relation with Bruner‟s modes of mental representation. Though the term …embodied‟ is used in a number of different ways within contemporary cognitive science, it may be said that all researchers “share a focus on theintimate relation between cognition, mind, and living bodily experience in the world, that is, on the ways in which complex adaptive behavior emerges from physical experience in biologically-constrained systems”. (Núñez, Edwards, & Matos, 1999, p. 49) The term …embodied‟ is used by Tall (2003, p. 4) “to refer to thought built fundamentally on sensory perception as opposed to symbolic operation and logical deduction. This gives to the term …embodied‟ a more focused meaning in mathematical thinking”. The symbolic-proceptual world refers to the triad …concept, process acting on it, symbol as a pivot between the two‟. In school the mathematical objects are often approached in the second, or, even worst, in the third world. About this fact Skemp (1969) claims that a purely logical approach provides only the final product of the mathematical discovery and does not generate in the learner the processes by which mathematical discoveries are made, so that such an approach teaches the mathematical thought, not the mathematical thinking. When mathematics is presented in a polished and finished way the learners (both in school and in teacher education courses) have difficulties in identifying the cognitive roots of concepts in the magma of processes, concepts, and rules they have at their disposal. The embodied world is a suitable place to start the construction of mathematical objects, it is the world wherethe “somatic” understanding mentioned in Tang (2004) may happen. History providesthe means to work in this world, as shown in examples such as the introduction of the derivative in Gravemeijer and Doorman (1999).3 The experimentOur experiment is set in a prospective teacher education program aimed at specializing in teaching mathematics, physics or both. An official diploma certifies the specialization. The full program lasts two academic years; it is centered on courses in pedagogy, psychology, information and communication technology, mathematics education, laboratory of mathematics education, physics education, laboratory of physics education. In the second year there are courses on foundations of mathematics, on foundations of physics, on history of mathematics, on history of physics. In the program the prospective teachers spend some weeks in practical training in school assisted by an experienced teacher: at the beginning simply as observers and afterwards also acting as teachers (they assign tasks, deliver lectures, assess students‟ performances).Most participants have no teaching experience. They are graduates in scientific or technological disciplines (most in mathematics); thus it may be assumed that they have a strong preparation in the content of the disciplines they will teach, in particular, in mathematics. Preliminary interviews carried out at the beginning of the training program have confirmed what was noted in Section 1: they already have some beliefs about mathematics and its teaching and, in particular, they have difficulties in considering ways of teaching a given mathematical topic other than the way that they have seen in their school days. Usually they had formal teaching in school and the university courses have stressed this side of their subject matter preparation. When they attend the theoretical lectures on mathematics education they have difficulties in appreciating the usefulness and applicability 134 F. Furinghettiof the educational theories. A good environment for overcoming this scant interest at the beginning of the program and for linking theory and practice is the seven credits (= 42 h) course carried out in the first semester of the first year called “Laboratory of mathematicseducation”. In this course the theoretical notions of mathematics education and pedagogy have to be applied to practice: the participants are requested to produce teaching sequences suitable for the classroom under the guidance of an instructor. Physically the laboratory of mathematics education is carried out in a computer laboratory so that the participants may use information technology including the Internet. Moreover the disposition of desks and chairs allows peer discussions among participants and with the instructor.In the case here presented there were 15 participants. They prepared a teaching sequence on algebra for 9th–10th grade students. They worked in groups of three assisted and prompted by their instructor. The instructor was an experienced teacher with a good background in research in mathematics education and in history of mathematics. These qualities were fundamental for carrying out the experiment successfully.The activity with history on which I focus in this paper concerns the work preparatory to the design of teaching sequences; the aim of this preparatory work was to acquaint prospective teachers with the cognitive roots of the concepts and processes that their potential students will meet in algebra. This integration of history in the laboratory of mathematics education does not require that the teaching sequences produced by the participants encompass historical parts, rather it requires that history is an inspirer of strategies of teaching.The phases of the experiment were:(1) Analysis of the national mathematical programs from elementary school up to the last year of high school. The aim of this activity was not only to familiarize the participants with the content to be taught and with the methodologies suggested by the guidelinesof the Ministry of Education, but also to identify the continuity through the different school levels. The participants chose a theme (in our case it was algebra) and followedits evolution in the successive grades of the programs. They also looked at some textbooks commonly used in Italian schools to confront the official guidelines of the Ministry of Education with school practice. The instructor helped them, since they were able to identify the mathematical topics, but not the skills required to develop them nor the presence of these skills across the different themes of the programs. In this phasethe orientation towards the passive reproduction of what they had seen in their practice emerged. They tended to work without attending to the context in which they will actas teachers. For this reason it was necessary to recall the educational literature on algebra discussed in the theoretical courses of mathematics education, see (Bednarz, Kieran, & Lee, 1996; Puig, 2004; Radford, 2000; Schoenfeld & Arcavi, 1988).(2) Exploiting the history of mathematics for analysing mathematical concepts. After the analysis of the programs, the prospective teachers started to work in the history of mathematics. The aim of this phase was to foster the identification of the cognitive roots of algebra through the lens of history. The main steps in the history of algebra had been previously outlined by the instructor and afterwards the prospective teachers had at their disposal manuals on the history of mathematics, readers with original sources translated into their first language, and internet resources. The participants had to work hard, firstly to identify the periods and the authors in the history of algebra which were relevant to their purpose, and afterwards to analyse the selected materials to identify the cognitive roots and their development. This was the phase where the prospective teachers couldlive again the experience of constructing mathematical objects.Teacher education through the history of mathematics 135(3) Design of the teaching sequence. The sequences designed by the prospective teachers must fill the programs‟ requirements and be suitable to the relevant school level.Within each group the prospective teachers discussed the historical materials theyfound and their relationship to the mathematical topic they intended to teach in their sequence. The instructor stimulated them to work in strict contact with the classroom context in which their sequence was to be set and intervened with advice and the confrontation of their opinions. The prospective teachers were encouraged to apply the notions taught in the theoretical courses of mathematics education and pedagogy. Atthe end of the laboratory they were given the opportunity to propose segments of the sequences they had produced to volunteer pupils and to test their reactions.(4) Discussion among groups and confrontation of the produced sequences. In this phasethe focus went back from history to the classroom in order to discuss how thecognitive roots may be developed in the teaching sequence. There was an exchange of ideas and new questions were raised. New ideas were generated spontaneously by the discussion, as advocated by Simon (1994).In describing our experiment we cannot ignore the problem of our prospective teachers‟preparation in the history of mathematics. History is introduced in our course as an artifact. According to Verillon and Rabardel (1995) an artifact becomes a tool when the users are able to use it for their own aims. It is obvious that in the case of the artifact …history‟ teachers need to know something about the history of mathematics. The extent and the deepness of knowledge have consequences in the use of history for teaching. Those who know history through secondary sources such as treatises on the history of mathematics are likely to reproduce the view of the author of treatises instead of developing their own view and, in addition, they make their choice under the influence of the choices made by the experts. Many researchers (see Arcavi & Bruckheimer, 2000; Jahnke et al., 2000) recommend the use of original sources for better grasping the roots of concepts. There are even more integralist positions: Jahnke (1994) advocates that teachers need to know how research in the history of mathematics is performed.In practice, there is the need for mediating different positions. In our course the historical preparation of the participants was not homogeneous: only a few of them had attended university courses on history of mathematics. The course in the history of mathematics contemplated in the education program is short (21 h) and is delivered in the second year, after the laboratory discussed here. To homogenize the audience the instructor provided the prospective teachers with different kinds of sources (for example, treatises and readers of history, web addresses, and handouts) to be explored under her guidance for producing the required teaching sequence. The participants had to look for information, to choose among different sources, to interpret original historical passages, to evaluate many elements, and to make their own choices. One may say that they were forced to reproduce in a minor waythe work of historians. This activity introduced a sense of community of practice among the prospective teachers and made them accept the collaboration inside the groups, between groups, and with the instructor.4 CommentsAs a result of the activities in the laboratory of mathematics education our prospective teachers produced different kinds of materials: plans for teaching sequences, exercises and problems, reports on the experimentation with segments of the teaching sequences they136 F. Furinghettiplanned for the classroom. History affected their products in two modes. One mode, that we may term …evolutionary‟, consisted in looking at the evolution of a given concept through the centuries. Reading some treatises on the history of mathematics and some histories of algebra found on the Internet fostered this mode. This mode offered to our prospective teachers the opportunity of identifying the elements that made a certain stream of thinking dominant. The historian Parshall (1988) uses the metaphor taken from Darwin‟s theory to explain whyViète‟s algebra, based on the generality of the method of analysis, supplanted its predecessors (Diophantus‟ and geometrical algebra). The reflection on the different streams in the development of algebra brings to the fore the power of the winner, that is the main ideas of algebraic method that shape modern presentation.An example of the result of this mode of using history in the laboratory of mathematics education is the plan for working in the classroom on second degree equations produced by a group of three teachers, see Fig. 1. This schematization was accompanied by an explanation. The ideas our prospective teachers took from history for designing their plan may be summarized as follows:& History provides meaningful examples of algorithms and methods that allow exploitation of the operational nature of mathematical objects& History suggests the development of the concepts in a visual/perceptual environment such as that provided by geometry.The other mode with which history affected prospective teachers‟ products was mainly raised by reading original passages of selected historical authors. I use the term …situated‟ to describe this mode because the cognition was situated in the historical context so that it was possible to understand the reasoning of the mathematicians who contributed to the development of concepts. With the situated mode one recovers the cognitive roots through the historical roots and avoids sticking only to the polished theory that comes at the end of the evolutionary process.Second degree equations(1) Second degree equationsas a historical process(2) Graphicalrepresentation in theCartesian plane(3) Algebraicproperties(1) Second degree equazions as a historical processSecond degreeequations inMesopotamiaGeometricalgebra inGreeceDiophantus Al-Khwarizmi’ssolving formulaViète andHornermethod(2) Graphical representation in the Cartesian planeSecond degreeequations andgraphics of parabolasSolutions inthe CartesianplaneAlgebraicmanipulations forfinding thesolving formulaAncient problemstranslated intomodernformalism(3) Algebraic propertiesShort notes on complex solutions and the fundamentaltheorem of algebraFactorizationFig. 1 Plan for working in the classroom on second degree equationsTeacher education through the history of mathematics 137In the following I give two examples of the activities promoted by reading original sources to approach the passage from arithmetic to algebra. Problems taken from medieval treatises of arithmetic were presented to teachers. The first set of problems was taken from the collection of 53 problems titled Propositiones ad acuendos juvenes (Problems to sharpen the young) written by Alcuin of York during the time of Charlemagne. A reliable translation of the original Latin text into Italian was available (Franci, 2005), so that the difficulty of interpreting an unknown language was avoided. Note that, here and in the other medieval problems considered the authors give empirical (numerical) solutions, sometimes without explaining the way they arrived to their solution. One of the problems discussed by the participants was the following (Hadley & Singmaster, 1992, p. 120):39. De quodam emptore in oriente –An oriental merchantA man in the east wanted to buy 100 assorted animals for 100 shillings. He orderedhis servant to pay 5 shillings for a camel, one shilling for an ass and one shilling for 20 sheep. How many camels, asses, and sheep did he buy?Solution. Five nineteens are 95, that is, 19 camels are bought for 95 shillings, beingfive nineteens. Add another one, that is buy one ass for one shilling, making 96 [shillings]. Then four twenties are 80, so for four shillings buy 80 sheep. Then add 19and 1 and 80 making 100. So there are 100 animals. And add 95 and 1 and 4, making100 shillings. So there are both 100 animals and 100 shillings.Our prospective teachers quickly produced the following equations in the unknowns c (camels), a (asses), s (sheep): c t a t s ? 100 and 5c t a t s=20 ? 100. I feel that they could have considered this problem a variation of the so-called realistic mathematics problems if the solution provided by Alcuin had not puzzled them. They were asked to discuss how students might react to the translation of this problem from the verbal form to the algebraic form of equation. The struggle of the author around his numerical solution shed light on one of the main differences between arithmetic and algebra. Arithmetic goes from known (here numbers and numerical attempts) to unknown (the sought solutions), while algebra goes from unknown to known. Thus algebra has to be seen not just as a generalization of arithmetic, but rather as a change in the method of work, which is applied when one starts the solving process by naming c, a, s the unknowns. Since this method is the core of algebra, our prospective teachers came back to work on this method: they read some of Al-Khwarizmi‟s passages about the solution of second degree equations. This author uses a geometrical method of …cutting and pasting‟ squares and rectangles. It is notable that he begins his explanation with this sentence (Al-Khwarizmi, 1831, p. 13):The figure to explain this, the sides of which are unknown. It represents the square,the which, or the root of which, you wish to know. This is the figure AB, each side ofwhich may be considered as one of its roots.Al-Khwarizmi‟s method is close to the analytical method, because this initial assumptionthat the equation he wishes to solve is satisfied.Another aspect of Alcuin‟s problem that puzzled our prospective teachers was the fact thatthe role of variables and unknowns is shuffled. Initially c, a, s are unknown, but at the end c and s play the role of variables in the relation 80c=19s. This brought our prospective teachers to reflect on issues that they have underestimated, for example the ambiguity of the two roles and the fact that understanding the role of variables “provides the basis for the transition from the arithmetic to algebra and is necessary for the meaningful use of all advanced mathematics”(see Schoenfeld & Arcavi, 1988, p. 420). They conjectured that the solution given in the old 138 F. Furinghettitext comes from a table of values made by Alcuin and easily linked the solutions of this problem with graphical representations in the Cartesian plane. The teaching sequence generated by this exercise was enhanced by a set of word problems similar to the problem of the oriental merchant conceived with the aim of providing pupils with situations that link the translation into algebraic language with the discussion of the graphical representations. Another activity concerned Problem 47 from the medieval treatise of arithmetic byDell‟Abbaco (1964) [here presented in my translation]:A gentleman asked his servant to bring him seven apples from the garden. He said:“You will meet three doorkeepers and each of them will ask you for half of all applesplus two taken from the remaining apples.” How many apples must the servant pluckif he wishes to have seven apples left?In the original version the problem is written in an old fashioned Italian language withwords and syntax which are no longer used in the modern language. The solution is givenin the style used in the previous problem by Alcuin, through numerical calculations without。

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课文9-B Terminology and notationwhen we work with a differential equation such as(9.1),it is customary to write y in place of f（x) and y' in place of f'(x)，the higher derivatives being denoted by y"，y'''，etc.Of course ，other letters such as u，v，z，etc.are also used instead of y. By the order of an equation is meant the order of the highest derivatives which appears.For example ,(9.1)is first-order equation which may be written as y'=y.The differential equation)sin(xy"yxy'3+=is one of second order.In this chapter we shall begin our study with firs-order equations which can be solved for y' and written as follows:(9.2) y'=f(x,y),Where the expression f(x,y) on the right has various special forms. A defferentiable function y=Y(x) will be called a solution of (9.2) on an interval I if the function Y and and its derivative Y' satisfy the relationY'=f[x,Y(x)]For every x in I. The simplest case occurs when f(x,y)is independent of y.In this case , (9.2) becomes(9.3) y'=Q(x),Say, where Q is assumed to be a liven function defined on some interval I. To solve the differential equation(9. 3) means to find a primitive of Q.The Second fundamental theorem of calculus tells us how to do it when Q is continuous on an open interval I. We simply integrate Q and add any constant.Thus，every solution of (9.3) is included in the formula（9.4）y=∫Q(x)dx + C,where C is any constant ( usually called an arbitrary constant of integration). The differential equation(9.3) has infinitely many 课文9—B 术语和符号当我们在求解像(9.1)式的微分方程时，习惯用y代替f(x)，用y’代替f'(x)，用高阶导数y''和y'''等表示。

当然，其他的字母如u,v,z等等，同样可以用来代替y。

微分方程和阶数指的是现在其中的高阶导数的阶。

例如，（9.1）式是一个一次方程可以写成y'=y。

微分方程)s i n(x y"yxy'3+=是一个二阶的。

在这章我们将会学习到可以求解y'的一阶微分方程。

一阶方程可以被写成这样：（9.2）y'=f(x,y),其中，右边有各个特殊形式表示。

如果对于区间I中的每一个x函数y和他的倒数满足Y'=f[x,Y(x)]那么可微函数就为（9.2）在区间I中的一个解，最简单的形式是f（x，y）与y无关。

在这种情况下，（9.2）式变成了（9.3）y'=Q(x),表明，其中Q是假定在区间中的一个给定函数，对于一个给定的函数定义在各个区间I.求解微分方程（9.3）就意味着找到原始的区间Q。

第二基本积分定理告诉我们，当Q位于一个连续的开放的区间I 时该怎么做。

我们直接对Q积分并加上任意常数。

因此，y=∫Q(x)dx + C包含了（9.3）式的所有解（9.4）y=∫Q(x)dx + C,其中C为任意常数（通常被称为积分下限的任意常数），微分方程（9.3）有无穷多个解，每个解对应一个C。

solutions, one for each value of C.If it is not possible to evaluate the integral in(9. 4)in terms of familiar function,such as polynomials，rational functions，trigonometric and inverse trigonometric functions，logarithms，and exponentials，still we consider the differential equation as having been solved if the solution can be expressed in terms of integrals of known functions. In actual practice，there are various methods for obtaining approximate evaluations of integrals which lead to useful information about the solution. Automatic high-speed computing machines are often designed with this kind of problem in mind .Example. Linear motion determined from the velocity. Suppose a particle moves along a straight line in such a way that its velocity at time t is 2sin t.Determine its position at time t..Solution. if Y(t) denotes the position at time t measured from some staring point,then the derivative Y'(t) represents the velocity at time t. We are given thatY'(t)=2sin t. Integrating，we find thatY(t)=2∫sin t dt+C=2cos t +C.This is all we can deduce about Y(t) from a knowledge of the velocity alone;some other piece of information is needed to fix the position function. We can determine C if we know the value of Y at some particular instant. For example，if Y( 0)=0，then C=2 and the position function is Y( t )=2-2cos t. But if Y( 0)=2，then C=4 and the position function is Y( t)=4-2cos t.In some respects the example just solved is typical of what happens in general.Some-where in the process of solving a first-order differential equation，an integration is required to remove the derivative y' and in this step an arbitrary如果这不是在一些常见条件下的函数对（9.4）求积分，如多项式，有理方程，三角函数和反三角函数，对数和指数函数，我们仍然要把微分方程看作是一个被解答的函数，如果这个函数可以表示成已知的函数的话。

在实际的练习中，有很多方法可以获得近似值，这些值可以给正确答案提供很多有用的信息。

自动高速运行的计算机在设计时就常常考虑到对这类问题的处理。

例如：直线运动取决于速度。

假设一个质点按照这种方法沿着一条直线运动,它的速度在t时刻是 2sin t。

在t时刻确定该质点位置。

结论：如果Y(t)表示被测点在t时刻的位置，那么，Y'(t)表示在t时刻的速率。

我们便可得出以下结论：Y'(t)=2sin t.综上所述：Y(t)=2∫sin t dt+C=2cos t +C.这就是所有我们能从速度推断出的关于Y(t)的知识；其他的一些信息需要从位置函数中整理得到。

在一些特殊的时刻，如果我们知道Y值，便可推断出C。

例如，如果Y(0)=2，那么C=4，并且位置函数是：Y( t)=4-2cos t.在某些方面，刚才例题所解决的问题是一般情况下都可行。

在求解一阶微分方程的过程中，为了消除导数y'，需要进一步进行积分，这时候就出现了一个任意常数C。

将常数Cconstant C appears. The way in which the arbitrary constant C enters into the solution will depend on the nature of the given differential equation. It may appear as an additive constant as in Equation(9. 4)，but it is more likely to appear in some other way .For example,when we solve the equation y'=y in Section 9. 3,we shall find that every solution has the form y=Ce^x.In many problems it is necessary to select from the collection of all solutions one having a prescribed value at some point. The prescribed value is called an initial condition,and the problem of determining such a solution is called an initial-value problem.This terminology originated in mechanics where,as in the above example,the prescribed value represents the displacement at some initial time.熊进学号：20083729 代入方程的解决方法依赖于所给常微分方程。

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