数学专业英语(吴炯圻)翻译5-A

数学专业英语(5)

Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@§2.5 Basic Concepts of Cartesian GeometryTEXT A The coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily we do not talk about area by itself, instead, we talk about the area of something. This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rena Descartes (1596-1650), who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was torepresent geometric points by numbers. The procedure for points in a plane is this:Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection, denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of O and its distance form O is called the unit distance. Vertical distances along the y-axis are usually measured with the same unit distance, although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the point.Figure 2-5-1 illustrates some examples. The point with coordinates (3, 2) lies three units to the right of the y-axis and two units above the x-axis. The number 3 is called the x-coordinate of the point, 2 its y-coordinate. Points to the left of the y-axis have a negative x-coordinate; those belowthe x-axis have a negative y-coordinate. The x-coordinate of a point is sometimes called its abscissa and the y-coordinate is called its ordinate.When we write a pair of numbers such as (a, b) to represent a point, we agree that the abscissa or x-coordinate, a, is written first. For this reason, the pair (a, b) is often referred to as an ordered pair. It is clear that two ordered pairs (a, b) and (c, d) represent the same point if and only if we have a =c and b=d. Points (a, b) with both a and b positive are said to lie in the first quadrant, those with a<0 and b>0 are in the second quadrant; those with a<0 and b<0 are in the third quadrant; and those with a>0 and b<0 are in the fourth quadrant. Figure 2-5-1 shows one point in each quadrant.The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin). These lines determine three mutually perpendicular planes, and each point inspace can be completely described by specifying, with appropriate regard for signs, its distances from these planes. We shall discuss three-dimensional Cartesian geometry in more detail later on; for the present we confine our attention to plane analytic geometry.NotationsRena Descartes (1596-1650)French scientific philosopher who developed a theory known as the mechanical philosophy. This philosophy was highly influential until superseded by Newton's methodology, and maintained, for example, that the universe was a plenum in which no vacuum could exist. Descartes was the first to make a graph, allowing a geometric interpretation of a mathematical function and giving his name to Cartesian coordinates (originated from Pappus’problem).Descartes believed that a system of knowledge should start from first principles and proceed mathematically to a series of deductions, reducing physics to mathematics. In Discours de la Méthode (1637), he advocated the systematic doubting of knowledge, believing as Plato that sense perception and reason deceive us and that man cannot have real knowledge of nature. The only thing that he believed he could be certain of was that he was doubting, leading to his famous phrase "Cogito ergo sum," (I think, therefore I am). From this one phrase, he derived the rest of philosophy, including the existence of God.Pappus’ problem The full enunciation of the problem is rather involved, but the most important case is to find the locus of a point such that the product of the perpendiculars on m given straight lines shall be in a constant ratio to the product of the perpendiculars on n other given straight lines. The ancients had solved this geometrically for the case m = 1, n = 1, and the case m = 1, n = 2. Pappus had further stated that, if m = n = 2, the locus is a conic, but he gave no proof; Descartes also failed to prove this by pure geometry, but he showed that the curve is represented by an equation of the second degree, that is, a conic; subsequently Newton gave an elegant solution of the problem by pure geometry.TEXT B Geometric figuresA geometric figure, such as a curve in the plane, is a collection of points satisfying one of more special conditions. By translating these conditions into expressions, involving the coordinates x and y, we obtain one or more equations which characterize the figure in question. For example, consider a circle of radius r with its center at the origin, as shown in Figure 2-5-2. Let P be an arbitrary point on this circle, and suppose P has coordinates (x, y). Then the line segment OP is the hypotenuse of a right triangle whose legs have lengths |x| and |y| and hence, by the theorem of Pythagoras,x2+y2=r2.The equation, called a Cartesian equation of the circle, is satisfied by all points (x, y) on the circle and by no others, so the equation completely characterizes the circle. This example illustrates how analytic geometry is used to reduce geometrical statements about points to analytical statements about real numbers.Throughout their historical development, calculus and analytic geometry have been intimately intertwined. New discoveries in one subject led to improvements in the other. The development of calculus and analytic geometry in this book is similar to the historical development, in that the two subjects are treated together. However, our primary purpose is to discuss calculus, Concepts from analytic geometry that are required for this purpose will be discussed as needed. Actually, only a few very elementary concepts of plane analytic geometry are required to understand the rudiments of calculus. A deeper study of analytic geometry is needed to extend the scope and applications of calculus, and this study will be carried out in later chapters using vector methods as well as the methods of calculus. Until then, all that is required form analytic geometry is a little familiarity with drawing graph of function.TEXT C Sets of points in the planeWe have already shown that there is a one-to-one correspondence between points in a plane and pairs of numbers(x, y). Certain sets of points in the plane may be of special interest. For example, we may wish to examine the set of points comprising the circumference of a certain circle, or the set of points constituting the interior of a certain triangle. One may wonder if such sets of point may be succinctly described in a compact mathematical notation.We may write{|(x, y)|y=2x} (1)to describe the set of ordered pairs (x, y), or corresponding points, such that the ordinate is equal to twice the abscissas. In effect, then, the vertical line in (1) is read “such that “. By “the graph of the set of ordered pairs” is meant the set of all points of the plane corresponding to the setof ordered pairs. The student will readily infer that the set of points constituting the graph lies on a straight line.Consider the set{(x, y)|y=x2}.Consistent with our previous interpretation, this symbol represents the set of ordered pairs (x, y) such that the ordinate is equal to the square of the abscissa. Here, the total graph comprises a simple recognizable geometrical figure, a curve known as a parabola.On the basis of these two examples, one may be tempted to believe that any arbitrarily drawn curve, which of course determines a set of points or ordered pairs, could be described succinctly by a simple equation. Unfortunately, this is not the case. For example, the broken line in figure 2-2-3 is one of such curves.Consider now the set{(x, y)|y>2x}to describe the set of points (x, y) whose ordinate is greater than twice its abscissa. In the case, our set of points constitutes not a curve, but a region of the coordinate plane.SUPPLEMENT Conic SectionThe conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a planethat is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola. The curve produced by a plane intersecting both nappes is a hyperbola.Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under the assumption that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes that are any of the four types of conic sections are possible.A conic section may more formally be defined as the locus of a point P that moves in the plane of a fixed point F called the focus and a fixedline d called the conic section directrix (with F not on d) such that the ratio of the distance of P from F to its distance from d is a constant e called the eccentricity. If e=0, the conic is a circle, if 0<e<1, the conic is an ellipse, if e=1, the conic is a parabola, and if e>1, it is a hyperbola.A conic section with conic section directrix at x=0, focus at (p, 0), and eccentricity e>0 has Cartesian equationwhere p is called the focal parameter. Plugging in p givesfor an ellipse,for a parabola, andfor a hyperbola.The polar equation of a conic section with focal parameter p is given byFive points in a plane determine a conic, as do five tangent lines in a plane. This follows from the fact that a conic section is a quadratic curve, which has general formso dividing through by a to obtainleaves five constants. Five points, (x i, y i) for i=1, …, 5, therefore determine the constants uniquely. The geometric construction of a conic section from five points lying on it is called the Braikenridge-Maclaurin Construction. The explicit equation for this conic is given by the equationTwo conics that do not coincide or have an entire straight line in common cannot meet at more than four points. There is an infinite family of conics touching four lines. However, of the eleven regions into which plane division cuts the plane, only five can contain a conic section which is tangent to all four lines. Parabolas can occur in one region only (which also contains ellipses and one branch of hyperbolas), and the only closed region contains only ellipses.Let a polygon of 2n sides be inscribed in a given conic, with the sides of the polygon being termed alternately "odd" and "even" according to some definite convention. Then the n(n-2)points where an odd side meet a nonadjacent even side lie on a curve of order n-2. This fact includes Pascal’s Theorem as a special case.NotationsEllipse, parabola and hyperbolaAn ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1and r2 from two fixed points F1and F2(the foci) separated by a distance of 2c is a given positive constant 2a.Problem Give similar definitions of parabola and hyperbola as above.A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus); a hyperbola is a conic section defined as the locus of all points in the plane the difference of whose distances from two fixed points (the foci) separated by a given distance is a given positive constant.Pascal’s Theorem (1640)The dual of Brianchon's theorem, discovered by B. Pascal in 1640 when he was just 16 years old. It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.A very special case happens when the conic degenerates into two lines, however the theorem still holds although this particular case is usually called Pappus theorem. In 1847, Möbius published the following generalization of Pascal's theorem: if all intersectionpoints (except possibly one) of the lines prolonging two opposite sides of a (4n-2)-gon inscribed in a conic section are collinear, then the same is true for the remaining point.Brianchon's Theorem (1806)The dual of Pascal's theorem. It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.In 1847, Möbius gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a (4n-2)-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line. Brianchon's theorem and Pascal's theorem are Example of dual geometric objects of Duality Principle.Duality Principle All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line." The principle was enunciated by Gergonne (1826), A similar duality exists for reciprocation as first enunciated by Poncelet (1818).Problem 1In the following four sets, which one describes the set of points (x, y) whose ordinate is greater than twice its abscissa?A {(x, y)|y<2x}B {(x, y)|y>2x}C {(x, y)|x<2y}D {(x, y)|x<2y}Problem 2 A conic section is defined to be any of the curves of intersection of a plane with a cone, if the plane of the intersection is parallel to an element of the cone, the curve of intersection is called ____.A a circleB an ellipseC a parabolaD a hyperbola。

数学专业英语教学大纲

数学专业英语教学大纲数学专业英语教学大纲《数学专业英语》教学大纲《一：数学专业英语》课程说明：课程代码：（一）课程代码： 08130011 ：课程代码：课程英文名称（二）课程英文名称： probability theory ：课程英文名称：：开课对象（三）开课对象：数学本科各专业：开课对象：：课程性质（四）课程性质：专业教育课程，专业数学英语是为数学专业本科大学生必须具备阅读基：课程性质：本外文数学文章能力而开设的一门课程，从专业词汇到数学知识的中文英文互相翻译到论文的协作规范等等都包容在本门课程的教学中。

：教学目的（五）教学目的：为数学专业本科大学生阅读英文数学文章和发表数学英文文章做培训，：教学目的：同时扩大学生英语知识面，为相关大学生的研究生考试做英语保温和一定强化。

：教学内容（六）教学内容：主要内容有：专业文章阅读和翻译初级阶段，专业词汇的掌握，专业文：教学内容：献的查阅，论文的翻译和写作，论文发表规范和程序等：学时数（七）学时数、学分数及学时具体分配：学时数、学时数：68-74 学时学分数：4 学分教学内容数学专业英语阅读和翻译初级精读入门专业文献英文数学论文协作基础查阅文献基本知识合计 2 讲授 4 26-28 24-26 6-8 2 2 4 68-74 2 实验/实践合计 4 28-30 24-26 8-10：教学方式（八）教学方式：课堂教学（九）考核方式和成绩记载：考核方式和成绩记载：考核方式为考试,严格考核学生的出勤情况,达到学籍管理规定的旷课量取消考试资格。

综合成绩根据平时成绩和期末成绩评定，平时成绩占 40\%，期末成绩占 60\%。

二：讲授各章的基本要求上学期部分学期部分第一章：数学专业英语阅读和翻译初级教学要点：教学要点：通过本章的教学，让学生初步接触数学专业英语文章和词汇，懂得初步英语文献翻译方法。

教学时数：教学时数：4 课时教学内容：教学内容：第一节：数学专业英语基本特点第二节：专业英语阅读和翻译 2第二章：精读入门教学要点：教学要点：通过本章的教学，让学生熟悉和掌握数学专业文章和一般英语文章写作的不同，使学生大量接触数学一般的主要不同专业文章中的基本术语和普遍常用词汇，懂得英文习作的一般习惯。

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。

数学专业英语词汇（A）

数学专业英语词汇（A）a priori bound 先验界限a priori distribution 先验分布a priori probability 先验概率a summable a可和的abacus 算盘abbreviate 略abbreviation 简化abel equation 阿贝耳方程abel identity 阿贝耳恒等式abel inequality 阿贝耳不等式abel summation method 阿贝耳求和法abelian algebra 阿贝耳代数abelian differential 阿贝耳微分abelian equation 阿贝耳方程abelian extension 阿贝耳扩张abelian function 阿贝耳函数abelian function field 阿贝耳函数域abelian functor 阿贝耳函子abelian group 交换群abelian groupoid 阿贝耳广群abelian integral 阿贝耳积分abelian summation 阿贝耳求和法thermocouple pyrometer 热电偶高温计abelian theorem 阿贝耳定理abelian variety 阿贝耳簇abridge 略abridged notation 简算记号abscissa 横坐标abscissa of absolute convergence 绝对收敛坐标abscissa of summability 可和性坐标abscissa of uniform convergence 一致收敛横坐标absolute 绝对形absolute address 绝对地址absolute class field 绝对类域absolute coding 绝对编码absolute cohomology 绝对上同调absolute conic 绝对二次曲线absolute convergence 绝对收敛absolute curvature vector 绝对曲率向量absolute deviation 绝对偏差absolute differential calculus 绝对微分学absolute error 绝对误差absolute extremes 绝对极值absolute extremum 绝对极值absolute frequency 绝对频率absolute geometry 绝对几何absolute homology group 绝对同岛absolute homotopy group 绝对同伦群absolute inequality 绝对不等式absolute instability 绝对不稳定性absolute maximum 绝对极大值absolute minimum 绝对极小值absolute moment 绝对矩absolute neighborhood 绝对邻域absolute neighborhood retract 绝对邻域收缩核absolute norm 绝对范数absolute number 不名数absolute parallelism 绝对平行性absolute quadric 绝对二次曲面absolute ramification index 绝对分歧指数absolute rotation 绝对旋转absolute singular homology group 绝对奇异同岛absolute space 绝对空间absolute space time 绝对时空absolute stability 绝对稳定性absolute term 常数项absolute unit 绝对单位absolute value 绝对值absolute value sign 绝对值符号absolute velocity 绝对速度absolutely additive measure 完全加性测度absolutely compact set 绝对紧集absolutely complete system 绝对完备系absolutely continuous 绝对连续的absolutely continuous distribution 绝对连续分布absolutely continuous function 绝对连续函数absolutely continuous measure 绝对连续测度absolutely continuous part 绝对连续部分absolutely continuous transformation 绝对连续变换absolutely convergent 绝对收敛的absolutely convergent integral 绝对收敛积分absolutely convergent series 绝对收敛级数absolutely convex hull 绝对凸包absolutely discontinuous function 绝对不连续函数absolutely integrable 绝对可积的absolutely irreducible character 绝对不可约特征absolutely irreducible representation 绝对不可约表示absolutely irreducible variety 绝对不可约簇absolutely normal number 绝对范数absolutely prime ideal 绝对素理想absolutely semisimple algebra 绝对半单代数absolutely simple group 绝对单群absolutely summable sequence 绝对可和序列absolutely unbiased estimator 绝对无偏估计量absolutely unramified extension 绝对非分歧扩张absorbing barrier 吸收障碍absorbing medium 吸收媒体absorbing set 吸收集absorbing state 吸收态absorption 吸收absorption coefficient 吸收系数absorption curve 吸收曲线absorption factor 吸收因数absorption index 吸收指数absorption law 吸收律absorption probability 吸收概率abstract 抽象的abstract algebra 抽象代数abstract algebraic geometry 抽象代数几何abstract automaton 抽象自动机abstract category 抽象范畴abstract code 理想码abstract complex 抽象复形abstract group 抽象群abstract interval function 抽象区间函数abstract mathematics 抽象数学abstract number 不名数abstract ordered simplicial complex 抽象有序单纯复形abstract simplex 抽象单形abstract simplicial subcomplex 抽象单子复形abstract space 抽象空间abstraction 抽象abstraction operator 抽象算子absurd 谬论的absurdity 谬论abundance ratio 丰度比abundant number 过剩数accelerated motion 加速运动acceleration 加速度acceleration of convergence 收敛性的加速acceleration of gravity 重力加速度acceptable quality level 合格质量水平acceptance 肯定acceptance inspection 接受检查acceptance limit 接受界限acceptance line 接受线acceptance number 接受数acceptance probability 接受概率acceptance region 接受区域acceptance zone 接受带access 存取access speed 存取速度access time 存取时间accessibility 可达性accessible boundary point 可达边界点accessible ordinal number 可达序数accessible point 可达点accessible set 可达集accessible vertex 可达顶点accessory extremal 配连极值accidental 偶然的accidental coincidence 偶然符合accidental error 随机误差accommodation 第accumulated error 累积误差accumulating point 聚点accumulation 累积accumulation point 聚点accumulator 累加器存储器accuracy 准确性accuracy grade 准确度accuracy of measurement 测量精确度accuracy rating 准确度acnod 孤点acount 计算action integral 酌积分action variable 酌变量active restriction 有效限制actual 真实的actual infinity 实无穷acute 尖锐的acute angle 锐角acute angled triangle 锐角三角形acute triangle 锐角三角形acuteness 锐度acyclic 非循环的acyclic complex 非循环复形acyclic graph 环道自由图acyclic model theorem 非循环模型定理ad infinitum 无穷地adams circle 阿达姆斯圆adams extrapolation method 阿达姆斯外插法adaptability 适应性adaptation 适应adapted basis 适应基add 加added circuit 加法电路addend 加数adder 加法器addition 加法addition formulas 加法公式addition sign 加号addition system 加法系addition table 加法表addition theorem 加法定理addition theorem of probability 概率的加法定理additional 加法的additional code 附加代码additional condition 附加条件additional error 附加误差additive 加法的additive category 加性范畴additive class 加性类additive functional 加性泛函数additive functional transformation 加性泛函变换additive group 加法群additive interval function 加性区间函数additive inverse element 加性逆元素additive number theory 堆垒数论additive operator 加性算子additive process 加性过程additive relation 加性关系additive separable 加法可分的additive theory of numbers 堆垒数论additive valuation 加法赋值additively commutative ordinal numbers 加性交换序数additivity 加法性address 地址address part 地址部分address register 地址寄存器addressing 指定箱位adele 阿代尔adele group 阿代尔群adequate 适合的adherent point 触点adhesion 附着adjacency 邻接adjacency matrix 邻接矩阵adjacent angles 邻角adjacent edge 邻棱adjacent side 邻边adjacent supplementary angles 邻角adjacent vertex 邻顶adjoint boundary value problem 伴随边值问题adjoint determinant 伴随行列式adjoint difference equation 伴随差分方程adjoint differential equation 伴随微分方程adjoint differential expression 伴随微分式adjoint form 伴随形式adjoint function 伴随函数adjoint functor 伴随函子adjoint graph 导出图adjoint group 伴随群adjoint hilbert problem 伴随希耳伯特问题adjoint integral equation 伴随积分方程adjoint kernel 伴随核adjoint linear map 伴随线性映射adjoint matrix 伴随阵adjoint operator 伴随算子adjoint process 伴随过程adjoint representation 伴随表示adjoint space 伴随空间adjoint surface 伴随曲面adjoint system 伴随系adjoint system of differential equations 微分方程的伴随系adjoint transformation 伴随算子adjoint vector 伴随向量adjunct 代数余子式adjunction 附加adjunction of an identity element 单位元的附加adjustable point 可胆点adjustment 蝶admissibility limit 容许界限admissible 容许的admissible category 容许范畴admissible chart 容许图admissible control 可行控制admissible decision function 容许判决函数admissible decision rule 容许判决函数admissible deformation 容许形变admissible domain 容许区域admissible function 容许函数admissible homomorphism 容许同态admissible hypothesis 容许假设admissible lifting 容许提升admissible map 容许映射admissible sequence 容许序列admissible space 容许空间admissible strategy 容许策略admissible subgroup 容许子群admissible test 容许检定admissible value 容许值affine algebraic set 仿射代数集affine collineation 仿射直射变换affine connection 仿射联络affine coordinates 平行坐标affine curvature 仿射曲率affine differential geometry 仿射微分几何学affine distance 仿射距离affine figure 仿射图形affine function 仿射函数affine geometry 仿射几何学affine group 仿射群affine group scheme 仿射群概型affine isothermal net 仿射等温网affine length 仿射长度affine line 仿射直线affine normal 仿射法线affine parameter 仿射参数affine principal curvature 仿射助率affine rational transformation 仿射有理变换affine space 仿射空间affine sphere 仿射球面affine surface 仿射曲面affine transformation 仿射变换affine variety 仿射簇affinely connected manifold 仿射连通廖affinely connected space 仿射连通空间affinity 仿射变换affirmation 肯定affirmative proposition 肯定命题affix 附标after effect 后效酌aggregate 集aggregation 聚合agreement 一致air coordinates 空间坐标airy function 亚里函数airy integral 亚里积分aitken interpolation 艾特肯插值aitken interpolation formula 艾特肯插值公式albanese variety 阿尔巴内斯簇aleph 阿列夫aleph zero 阿列夫零alexander cohomology module 亚历山大上同担alexander cohomology theory 亚历山大上同帝alexander matrix 亚历山大阵alexander polynomial 亚历山大多项式algebra 代数学algebra of events 事件场algebra of logic 逻辑代数algebra of tensors 张量代数algebra over k 环k上的代数algebraic 代数的algebraic adjunction 代数的附加algebraic affine variety 仿射代数集algebraic algebra 代数的代数algebraic branch point 代数分歧点algebraic calculus 代数计算algebraic closure 代数闭包algebraic closure operator 代数闭包算子algebraic complement 代数余子式algebraic cone 代数锥algebraic correspondence 代数对应algebraic curve 代数曲线algebraic equation 代数方程algebraic expression 代数式algebraic extension 代数扩张algebraic form 代数形式algebraic fraction 代数分式algebraic function 代数函数algebraic function field 代数函数域algebraic geometry 代数几何学algebraic group 代数群algebraic hull 代数包algebraic hypersurface of the seconed order 二阶代数超曲面algebraic integer 代数整数algebraic irrational number 代数无理数algebraic lie algebra 代数的李代数algebraic logic of pocket calculator 袖珍计算机的代数逻辑algebraic multiplicity 代数重度algebraic number 代数数algebraic number field 代数数域algebraic number theory 代数数论algebraic operation 代数运算algebraic polynomial 代数多项式algebraic singularity 代数奇点algebraic space 代数空间algebraic spiral 代数螺线algebraic structure 代数结构algebraic sum 代数和algebraic surface 代数曲面algebraic system 代数系algebraic variety 代数簇algebraically closed field 代数闭域algebraically dependent elements 代数相关元algebraically equivalent 代数等价的algebraically independent elements 代数无关元algebraization 代数化algebro geometric 代数几何的algebroid function 代数体函数algebroidal function 代数体函数algorithm 算法algorithm of division 辗转相除法algorithm of euclid 欧几里得算法algorithm theory 算法论algorithmic language 算法语言algorithmization 算法化aligned systematic sampling 列系统抽样alignment chart 列线图aliquot part 整除部分alligation 混合法allocation problem 配置问题allowable 容许的allowable defects 容许靠allowable error 容许误差allowance 允许almost all 几乎处处almost bounded function 殆有界函数almost certain convergence 几乎必然收敛almost complex manifold 殆复廖almost convergent sequence 殆收敛序列almost equivalent 殆等价almost everywhere 几乎处处almost impossible event 殆不可能事件almost invariant set 殆不变集almost periodic function 殆周期函数almost periodicity 殆周期性almost significant 殆显著的alpha capacity 容量alpha limit set 极限集alphabetical 字母的alphanumeric 字母数字式alphanumeric representation of information 信息的字母数字表示alternate angles 错角alternating chain 交错链alternating differential form 外微分形式alternating differential of differential form 微分形式的交错微分alternating direction method 交替方向法alternating form 交错形式alternating function 反对称函数alternating group 交错群alternating harmonic series 莱布尼兹级数alternating knot 交错纽结alternating matrix 交错矩阵alternating method 交错法alternating product 外积alternating sequence 交错序列alternating series 交错级数alternating series test 交错级数检验alternating sum 交错和alternating tensor 交错张量alternating tensor density 交错张量密度alternating tree 交错树alternation 交错alternative 交错;择一alternative algebra 交错代数alternative field 交错域alternative hypothesis 择一假设alternative normal form 析取范式alternative proposition 选言命题altitude 高度altitude theorem 高度定理amalgamated product 融合积amalgamated subcategory 融和子范畴amalgamation 合并ambient space 环绕空间ambiguous point 歧点amicable numbers 亲和数amount 量amphicheiral knot 双向纽结ample divisor 丰富除子amplitude 振幅;角amplitude of a complex number 复数角analog computer 模拟计算机analogous 类似的analogue display 相似表示analogue method 相似法analogy 类似analysis 数学分析analysis of time series 时间序列分析analysis of variance 方差分析analytic 分析的analytic arc 解析弧analytic completion 解析开拓analytic continuation 解析开拓analytic curve 解析曲线thermocouple pyrometer 热电偶高温计analytic dynamics 分析动力学analytic expression 解析式analytic function 分析函数analytic function of several variables 多元解析函数analytic geometry 分析几何学analytic index 解析指数analytic manifold 解析廖analytic method 解析法analytic prime number theory 解析素数论analytic proof 解析证明analytic proposition 解析命题analytic set 解析集analytic space 解析空间analytic transformation 解析变换analytical differential 解析微分analytical geometry 分析几何学analytical hierarchy 解析分层analytical mapping 全纯映射analytical transformation 全纯映射analytically continuable 可解析开拓的analytically independent 解析无关的analytically irreducible variety 解析不可约簇analytically representable function 解析可表示的函数anastigmatic 去象散的anchor ring 环面ancillary statistic 辅助统计量and circuit 与电路andre permutation 安得列置换andre polynomial 安得列多项式anger function 安格尔函数angle 角angle at center 圆心角angle between chord and tangent 弦和切线的角angle function 角函数angle of 落后角angle of advance 超前角angle of attack 迎角angle of contact 接触角angle of contingence 切线角angle of declination 俯角angle of diffraction 衍射角angle of incidence 入射角angle of inclination 斜角angle of intersection 相交角angle of lead 超前角angle of reflection 反射角angle of refraction 折射角angle of rotation 旋转角angle of torsion 挠率角angle preserving 保角的angle preserving map 保角映象angular 角的angular acceleration 角加速度angular coefficient 角系数angular coordinates 角坐标angular correlation 角相关angular derivative 角微商angular dispersion 角色散angular displacement 角位移angular distance 角距angular distribution 角分布angular domain 角域angular excess 角盈angular frequency 角频率angular magnification 角放大率angular measure 角测度angular metric 角度量angular momentum 角动量angular momentum conservation law 角动量守恒律angular motion 角运动angular neighborhood 角邻域angular transformation 角变换angular unit 角的单位angular velocity 角速度anharmonic oscillation 非低振动anharmonic ratio 交比anisotropic 蛤异性的anisotropic body 蛤异性体anisotropy 蛤异性annihilator 零化子annular 环annulator 零化子annulus 圆环anomalous magnetic moment 反常磁矩anomalous propagation 反常传播anomalous scattering 反常散射anomaly 近点角antecedent 前项anti automorphism 反自同构anti hermitian form 反埃尔米特形式anti isomorphic lattice 反同构格anti isomorphism 反同构anti position 反位置anti reflexiveness 反自反性anti semiinvariant 反半不变量antianalytic function 反解析函数antichain 反链anticlockwise 逆时针的anticlockwise revolution 逆时针回转anticlockwise rotation 逆时针回转anticoincidence 反重合anticoincidence method 反重法anticommutation 反交换anticommutative 反交换的anticommutativity 反交换性anticommutator 反换位子antiderivative 不定积分的antiholomorphic 反全纯的antihomomorphism 反同态antiisomorphy 反同构antilinear 反线性的antilinear mapping 反线性映射antilinear transformation 反线性变换antilogarithm 反对数antimode 反方式antimodule 反模antinode 反结点antinomy 二律背反antiorder homomorphism operator 反序同态算子antiordered set 反有序集合antiparallel 逆平行的antiplane 反平面antipodal map 对映映射antipodal point 对映点antipodal set 对映集antipode 对映点antipoints 反点antiradical 反根式antiresonance 反共振antisymmetric 反对称的antisymmetric function 反对称函数antisymmetric matrix 反对称矩阵antisymmetric relation 反对称关系antisymmetric tensor 反对称张量antisymmetrical state 反对称态antithesis 反题antitone mapping 反序映射antitone sequence 反序列antitonic function 反序函数antitonicity 反序性antitony 反序性antitrigonometric function 反三角函数antiunitary 反酉的aperiodic damping 非周期衰减aperiodic motion 非周期运动aperiodicity 非周期性aperture ratio 口径比apex 顶点apex angle 顶角apical angle 顶角apolar 非极性的apolarity 从配极性apothem 边心距apparent error 貌似误差apparent force 表观力apparent motion 表观运动apparent orbit 表观轨迹apparent singularity 貌似奇性application 应用applied mathematics 应用数学applied mechanics 应用力学approach 接近approach infinity 接近无穷大approximability 可逼近性approximable 可逼近的approximate 近似的;使近似approximate calculation 近似计算approximate construction 近似准approximate continuity 近似连续性approximate derivative 近似导数approximate differentiability 近似可微性approximate differential 近似微分approximate formula 近似公式approximate integration 近似积分approximate limit 近似极限approximate lower semi continuity 近似下半连续性approximate method 近似法approximate number 近似数approximate partial derivative 近似偏导函数approximate partial derived function 近似偏导函数approximate partial differential 近似偏微分approximate solution 近似解approximate total differentiability 近似全可微性approximate total differential 近似全微分approximate upper semi continuity 近似上半连续性approximate value 近似值approximate value in excess 过剩近似值approximately equal 近似等于approximately semicontinuous 近似半连续的approximation 逼近approximation by excess 过剩逼近approximation calculus 近似计算approximation error 近似误差approximation function 逼近函数approximation in excess 过剩逼近approximation method 近似法approximation methods in physics 物理学中的逼近法approximation theorem 逼近定理approximation theory 逼近理论arbitrarily small 任意小arbitrary 任意的arbitrary constant 任意常数arbitrary element 任意元素arbitrary parameter 任意参数arbitrary small number 任意小数arc 弧arc component 弧分量arc cosecant 反余割arc cosine 逆余弦arc cotangent 逆余切arc hyperbolic function 反双曲函数arc length 弧长arc of a circle 圆弧arc secant 反正割arc set 弧集arc sine 逆正弦arc sine distribution 反正弦分布arc sine law 反正弦定律arc sine transformation 反正弦变换arc tangent 反正切arch 拱形archaeometry 考古测量学archimedean 阿基米德性的archimedean group 阿基米德群archimedean semigroup 阿基米德半群archimedean space 阿基米德空间archimedean total order 阿基米德全序archimedean valuation 阿基米德赋值archimedes axiom 阿基米德公理archimedes spiral 阿基米德螺线archimedically ordered field 阿基米德有序域archimedically ordered number field 阿基米德有序数域arcwise connected set 弧连通集arcwise connected space 弧连通空间arcwise connectedness 弧连通性are 公亩area 面积area function 面积函数area of a circle 圆面积area preserving mapping 保面积映射areal coordinates 重心坐标areal derivative 面积导数areal element 面积元素areal integral 面积分areal velocity 面积速度argand plane 复数平面argument 自变数;辐角argument function 辐角函数argument of a function 函数的自变数argument principle 辐角原理argumentation 论证aristotelian logic 亚里斯多德逻辑学arithmetic 算术arithmetic al function 数论函数arithmetic difference 算术差arithmetic division 算术除法arithmetic element 运算元素arithmetic expression 算术表达式arithmetic function 数论函数arithmetic genus 算术狂arithmetic geometric mean 算术几何平均arithmetic geometric series 算术几何级数arithmetic logic of pocket calculator 袖珍计算机的算术逻辑arithmetic mean 算术平均arithmetic number 正实数arithmetic of algebraic number fields 代数数域的数论arithmetic of algebras 代数的数论arithmetic of local fields 局部域的数沦arithmetic operation 算术操作算术运算arithmetic progression 算术级数arithmetic subgroup 算术子群arithmetic unit 运算元素arithmetical hierarchy 算术谱系arithmetical predicate 算术谓词arithmetical triangle 帕斯卡三角形arithmetics 算术arithmetization 算术化arithmometer 四则计算机arrangement 排列array 排列arrow 射artificial variable 人工变量artificial variable method 人工变量法artin conductor 阿廷前导子artin conjecture 阿廷猜想artin reciprocity law 阿廷互反禄artinian module 阿丁模artinian ring 阿丁环ascending 上升的ascending chain condition 升链条件ascending difference 后向差分ascending power series 升幂级数ascending powers 升幂ascending sequence 递升序列aspherical space 非球面空间asphericity 非球面牲assemblage 集assembler 汇编assertion sign 断定号assignable cause 可指定的原因assignment problem 配置问题associate equation 相伴方程associated equation 相伴方程associated fiber bundle 相伴的纤维丛associated form 连带形式associated function 连带函数associated graded module 相伴分次模associated graded ring 形式环associated homogeneous equation 相伴齐次方程associated homogeneous system 相伴齐次组associated laguerre polynomial 连带的拉盖尔多项式associated legendre function 相伴勒让德函数associated legendre polynomial 连带的勒让德多项式associated minimal surface 相伴极小曲面associated power series 相伴幂级数associated prime ideal 相伴素理想associated radius of convergence 相伴收敛半径associated space 相伴空间associated spherical harmonic 相伴球面低associated surface 连带曲面associated system 相伴系associated undirected graph 相伴无向图association 结合associative algebra 结合代数associative law 结合律associative law for series 级数的结合律associativity 结合性assume 假定assumption 假定assumption formula 假定公式asterisk 星号asteroid 星形线asymmetric 非对称的asymmetric relation 非对称关系asymmetric variety 非对称簇asymmetrical 非对称的asymmetrical graph 恒等图asymmetry 非对称性asymptote 渐近线asymptote of curve 曲线的渐近线asymptotic 浙近的asymptotic behavior 渐近状态asymptotic circle 渐近圆asymptotic cone 渐近锥面asymptotic convergence 渐近收敛asymptotic curvature 渐近曲率asymptotic curve 渐近曲线asymptotic density 渐近密度asymptotic direction 渐近方向asymptotic efficiency 渐近效率asymptotic expansion 渐近展开asymptotic formula 渐近公式asymptotic line 渐近线asymptotic mean value 渐近平均值asymptotic minimal basis 渐近极小基asymptotic order 渐近阶asymptotic path 渐近路线asymptotic plane 渐近平面asymptotic point 渐近点asymptotic rate of convergence 渐近收敛速度asymptotic series 渐近级数asymptotic solution 渐近解asymptotic stability 渐近稳定性asymptotic surface 渐近曲面asymptotic unbiased estimator 渐近无偏估计量asymptotic value 渐近值asymptotically efficient estimator 渐近有效估计量asymptotically equal 渐近相等asymptotically equal sequence 渐近相等序列asymptotically equivalent function 渐近等价函数asymptotically normal distribution 渐近正态分布asymptotically normally distributed 渐近正规分布的asymptotically stable 渐近稳定的asymptotically stable solution 渐近稳定解asynchronous computer 异步计算机atiyah singer index theorem 阿蒂亚辛格指数定理atlas 坐标邻域系atom 原子atomic element 原子元素atomic formula 原子公式atomic lattice 原子格atomic proposition 原子命题atomicity 原子性attaching map 接着映射attenuation 衰减attenuation constant 衰减常数attenuation factor 衰减因数attenuator 衰减器attraction 引力attractive force 引力attractor 吸引区attribute 属性augend 被加数augmentation 扩张augmentation preserving map 增广保存映射augmented complex 扩张复形augmented matrix 增广矩阵austausch 交换autocorrelation 自相关autocorrelation coefficient 自相关系数autocorrelation function 自相关函数autocorrelogram 自相关图autocovariance 自协方差autocovariance function 自协方差函数autodistributivity 自分配性automata 自动机automatic check 自动检验automatic coding 自动编码automatic computation 自动计算automatic computer 自动计算机automatic control 自动控制automatic control system 自动控制系统automatic control theory 自动控制理论automatic programming 自动程序设计automatic testing 自动检验automation 自动化automaton 自动机automaton graph 自动机图automorphic form 自守形式automorphic function 自守函数automorphism 自同构automorphism group 自同构群autonomous system 自治系统autoparallel curve 自平行曲线autoregression 自回归autoregression equation 自回归方程autoregressive process 自回归过程autoregressive transformation 自回归变换auxiliary 辅助的auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 相伴齐次方程auxiliary function 辅助函数auxiliary line 辅助线auxiliary variable 辅助变数average 平均值average deviation 平均偏差average error 平均误差average life 平均寿命average outgoing quality 平均出厂质量average outgoing quality limit 平均出厂质量极限average quality protection 平均品质保护average sample number 平均样本数average speed 平均速度average term 普通项average time 平均时间average value 平均值averaging 取平均数averaging method 平均法averaging operator 平均算子axes of coordinates 座标轴axial 轴的axial symmetry 轴对称axial vector 轴向量axially symmetric flow 轴对称流axialsymmetric vector field 轴对称向量场axiom 公理axiom of accessibility 可达性公理axiom of choice 选择公理axiom of completeness 完备性公理axiom of comprehension 概括公理axiom of constructibility 可构成性公理axiom of constructivity 可构成性公理axiom of continuity 连续公理axiom of extensionality 外延性公理thermocouple pyrometer 热电偶高温计axiom of infinity 无穷性公理axiom of power set 幂集公理axiom of reducibility 可化归性公理axiom of regularity 正则性公理axiom of subsets 子集公理axiom of substitution 替换公理axiom of sum set 并集公理axiom of the empty set 空集公理axiom of union 并集公理axiom scheme 公理格式axiomatic 公理的axiomatic method 公理法axiomatic set theory 公理论的集论;公理集合论axiomatic system 公理系统axiomatics 公理学axiomatization 公理化axiomatize 公理化axioms of congruence 叠合公理axioms of continuity 连续公理axioms of denumerability 可数公理axioms of incidence 关联公理axis 轴axis of a cone 锥轴axis of abscissas 横坐标轴axis of absolute convergence 绝对收敛轴axis of affinity 仿射轴axis of convergence 收敛轴axis of coordinate 坐标轴axis of curvature 曲率轴axis of ordinates 纵坐标轴axis of projection 射影轴axis of reals 实轴axis of revolution 回转轴axis of rotation 回转轴axis of symmetry 对称轴axisymmetric 轴对称的axonometric perspective 轴测投影法axonometric projection 轴测投影法axonometry 轴测法azimuth 方位角azimuthal 方位角的。

数学专业英语(Doc版).25

数学专业英语－The Theory of GraphsIn this chapter, we shall introduce the concept of a graph and show that graph s can be defined by square matrices and versa.1.IntroductionGraph theory is a rapidly growing branch of mathematics. The graphs discusse d in this chapter are not the same as the graphs of functions that we studied previously, but a totally different kind.Like many of the important discoveries and new areas of learning, graph the ory also grew out of an interesting physical problem, the so-called Konigsberg bridge problem. (this problem is discussed in Section 2) The outstanding Swis s mathematician, Leonhard Euler (1707-1783) solved the problem in 1736, thus laying the foundation for this branch of mathematics. Accordingly, Euler is ca lled the father of graph theory.Gustay Robert Kirchoff (1824-1887), a German physicist, applied graph theor y in his study of electrical networks. In1847, he used graphs to solve systems of linear equations arising from electrical networks, thus developing an importa nt class of graphs called trees.In 1857, Arthur Caylcy discovered trees while working on differential equati ons. Later, he used graphs in his study of isomers of saturated hydrocarbons.Camille Jordan (1838-1922), a French mathematician, William Rowan Hamilt on, and Oystein Ore and Frank Harary, two American mathematicians, are also known for their outstanding contributions to graph theory.Graph theory has important applications in chemistry, genetics, management s cience, Markov chains, physics, psychology, and sociology.Throughout this chapter, you will find a very close relationship between gra phs and matrices.2.The Konigsberg Bridge ProblemThe Russian city of Konigsberg (now Kaliningrad, Russia) lies on the Pregel River.(See Fig.1) It consists of banks A and D of the river and the two island s B and C. There are seven bridges linking the four parts of the city. Residents of the city used to take evening walks from one section of the cit y to another and go over some of these bridges. This, naturally, suggested the following interesting problem: can one walk through the city crossing each bridge exactly once? The problem sounds simple, doesn’t it?You might want to try a few paths before going any further. After all, by the fundamental countin g principle, the number of possible paths cannot exceed 7!=5040. Nonetheless, it would be time consuming to look at each of them to find one that works.Fig .1 The city of KonigsbergIn 1736, Euler proved that no such walk is possible. In fact, he proved a far more general theorem, of which the Konigsberg bridge problem is a special ca se.Fig .2 A mathematical model for the Konigsberg bridge problemLet us construct a mathematical model for this problem.rcplace each area of the city by a point in a plane. The points A, B, C,and D denote the areas th ey represent and are called vertices. The arcs or lines joining these points repr esent the represent the respective bridges. (See图２)They are called edges. The Konigsberg bridge problem can now be stated as follows: Is it possible to tra ce this figure without lifting your pencil from paper or going over the same e dge twice? Euler proved that a figure like this can be traced without lifting pe ncil and without traversing the same edge twice if and only if it has no more than weo vertices with an odd number of edges joining them. Observe that m ore than two vertices in the figure have an odd number of edges connecting t hem-----in fact,they all do.1. GraphsLet us return to the example Friendly Airlines flies to the five cities, Boston (B), Chicago (C), Detroit (D), Eden (E), and Fairyland (F) as follows: it has direct daily flights from city B to cities C, D, and F, from C to B, D, and E; from D to B, C, and F, from E to C, and from F to B and D. This informat ion, though it sounds complicated, can be conveniently represented geometrically, as in 图3. Each city is represented by a heavy dot in the figure; an arc or a line segment between two dots indicates that ther e is a direct flight between these cities.What does this figure have in common with 图２? Both consist of points (denoted by thick dots ) co nnected by arcs or line segments. Such a figure is called a graph. The points are the vertices of the gra ph; the arcs and line segments are its edges. More generally, we make the following definition:A graph consists of a finite set of points, together with arcs or line segments connecting some of them. These points are called the vertices of the graph; the arcs and line segments are called the edges og thegraph. The vertices of graph are usually denoted by the letters A, B, C, and so on. An edge joining th e vertices A and B is denoted by AB or A-B.Fig .3图２and 图3 are graphs. Other graphs are shown in 图4. The graph in图２has four vertices A, B, C, and D, and seven edges AB, AB, AC, BC, BD, CD, and BD. For the graph in图4b, there are four vertices, A, B, C, and D, but only two edges AD and CD. Consider the graph in图4c, it contains an ed ge emanating from and terminating at the same vertex A. Such an edge is called a loop. The graph in 图4d contains two edges between the vertices A and C and a loop at the vertex C.The number of edges meeting at a vertex A is called the valence or degree of the vertex, denoted by v (A). For the graph in图4b, we have v(A)=1, v(B)=0, v(C)=1, and v(D)=2. In图4b, we have v(A)=3, v(B) =2, and v(C)=4.A graph can conveniently be described by using a square matrix in which the entry that belong to the row headed by X and the column by Y gives the number of edges from vertex X to vertex Y. This m atrix is called the matrix representation of the graph; it is usually denoted by the letter M.The matrix representation of the graph for the Konigsberg problem isClearly the sum of the entries in each row gives the valence of the corresponding vertex. We have v(A) =3, v(B)=5, v(C)=3, as we would expect.Conversely, every symmetric square matrix with nonnegative integral entries can be considered the ma trix representation of some graph. For example, consider the matrixA B C DClearly, this is the matrix representation of the graph in 图5.VocabularyNetwork 网络Electrical network 电网络Isomer 异构体emanate 出发，引出Saturated hydrocarbon 饱和炭氢化合物terminate 终止，终结Genetics 遗传学valence 度Management sciences 管理科学node 结点Markov chain 马尔可夫链interconnection 相互连接Psychology 心理学 Konigsberg bridge problem 康尼格斯堡桥问题Sociology 社会学Line-segment 线段Notes1. Camille Jordan, a French mathematician, William Rowan Hamilton and . . .注意：a French mathematician 是Camille Jordan 的同位语不要误为W.R.Hamilton 是a French mathe matician 同位语这里关于W.R.Hamilton 因在本文前几节已作介绍，所以这里没加说明。

数学专业英语(Doc版).15

数学专业英语－Notations and Abbreviations (I) Learn to understand N set of natural numbersZ set of integersR set of real numbersC set of complex numbers+ plus; positive－minus; negative×multiplied by; times÷divided by＝equals; is equal to≡identically equal to≈,≌approximately equal to＞greater than≥greater than or equal to＜less than≤less than or equal to》much greater than《much less thansquare rootcube rootnth root│a│ absolute value of an! n factoriala to the power n ; the nth power of a[a] the greatest integer≤athe reciprocal of aLet A, B be sets∈ belongs to ; be a member ofnot belongs tox∈A x os amember of A∪ unionA∪B A union B∩ intersectionA∩B A intersection BA B A is a subset of B;A is contained in B A B A contains Bcomplement of Athe closure of Aempty set( ) i=1,2,…,r j=1,2,…,s r-by-s（r×s）matrix││I,j=1,2,…,n determinant of order ndet( ) the determinant of the matrix ( )vector Fx=( , ,…, ) x is an n-tuple of‖‖the norm of …‖ parallel to┴ perpendicular tothe exponential function of xlin x the logarithmic function of xsie sinecos cosinetan tangentsinh hyperbolic sinecosh hyperbolic cosinethe inverse of ff is the composite or the composition of u and vthe limit of …as n approaches ∞(as x approaches )x a x approaches a, the differential coefficient of y; the 1st derivative of y , the nth derivative of ythe partial derivative of f with respect to xthe partial derivative of f with respect to ythe indefinite integral of fthe definite integral of f between a and b (from a to b) the increment of xdifferential xsummation of …the sum of the terms indicated∏the product of the terms indicated=> impliesis equivalent to（）round brackets; parantheses[ ] square brackets{ } braces。

关于数学专业英语课程教学的一些探讨与体会

关于数学专业英语课程教学的一些探讨与体会【摘要】数学专业英语课程是面向数学系学生所开始的一门专业任选课，它具有英语、数学、英语与数学相结合等特点，要求学生具备一定的数学知识和英语知识。

本文针对数学专业英语教学的现状，结合此门课程所具有的特点，来探讨如何提高学生学习数学专业英语的乐趣，提高学生阅读数学专业英语文献的能力，同时介绍一些数学的前沿研究热点。

【关键词】数学专业英语；教学探讨；教学方法如今越来越多的本科生毕业后到国外进行研究生学习，首要面对的一个紧要问题就是要听懂国外教师上课。

虽然在出国之前经历过各种各样的英语考试，口语和听力有了一定的提高，但专业英语词汇量比较缺乏。

数学专业英语课是针对数学系本科生所开设的一门专业任选课，课程开设的目的是帮助本科生能掌握一定的数学专业词汇，提高阅读数学专业文献的能力，促进我国的数学教育与国际上的交流，紧跟国际数学的研究步伐。

此课程的开设有助于扩大本科生的视野，知道一些数学专业词汇的英文表达，进一步会撰写一篇英语科技论文。

我们学院通常在第四学期开设专业英语选修课，从前两年的教学效果来看，绝大部分学生对此门课程兴趣不大，抱着拿两学分的心态，因此在课堂不能和教师进行良好的互动，课堂浑然无趣。

本文从以下四个方面指出在教学中所存在的一些问题，并探讨数学专业英语课程的教学改革方法与策略。

一、教学课时量较少目前学院采用吴炯圻老师的数学英语课本-第二版，此门课程性质是专业任选课，安排总课时为32。

课本内容总共6章，分别介绍了数学专业的阅读和翻译初阶；精读课文；专业文选；数学英语专业论文写作基础；查阅英语数学文献的基本知识和数学文献常用英语词汇。

从课本内容来看，涉及内容较多，如第二章内容有12小节，从课时安排来看，要想讲完全部内容有难度，因此在课堂教学中，我们选取了与数学分析、高等代数和概率论与数理统计等学科相关的内容来讲解，如函数的思想；线性空间中的相关与无关集；概率论与数理统计中的基本术语，通过讲解这些内容，让同学们能够联想到之前学过的数学内容，从而会主动用专业英语来表述。

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

数学专业英语数学专业英语课后答案2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place 汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。

数学专业英语(5)

Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@§2.5 Basic Concepts of Cartesian GeometryTEXT A The coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily we do not talk about area by itself, instead, we talk about the area of something. This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rena Descartes (1596-1650), who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was torepresent geometric points by numbers. The procedure for points in a plane is this:Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection, denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of O and its distance form O is called the unit distance. Vertical distances along the y-axis are usually measured with the same unit distance, although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the point.Figure 2-5-1 illustrates some examples. The point with coordinates (3, 2) lies three units to the right of the y-axis and two units above the x-axis. The number 3 is called the x-coordinate of the point, 2 its y-coordinate. Points to the left of the y-axis have a negative x-coordinate; those belowthe x-axis have a negative y-coordinate. The x-coordinate of a point is sometimes called its abscissa and the y-coordinate is called its ordinate.When we write a pair of numbers such as (a, b) to represent a point, we agree that the abscissa or x-coordinate, a, is written first. For this reason, the pair (a, b) is often referred to as an ordered pair. It is clear that two ordered pairs (a, b) and (c, d) represent the same point if and only if we have a =c and b=d. Points (a, b) with both a and b positive are said to lie in the first quadrant, those with a<0 and b>0 are in the second quadrant; those with a<0 and b<0 are in the third quadrant; and those with a>0 and b<0 are in the fourth quadrant. Figure 2-5-1 shows one point in each quadrant.The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin). These lines determine three mutually perpendicular planes, and each point inspace can be completely described by specifying, with appropriate regard for signs, its distances from these planes. We shall discuss three-dimensional Cartesian geometry in more detail later on; for the present we confine our attention to plane analytic geometry.NotationsRena Descartes (1596-1650)French scientific philosopher who developed a theory known as the mechanical philosophy. This philosophy was highly influential until superseded by Newton's methodology, and maintained, for example, that the universe was a plenum in which no vacuum could exist. Descartes was the first to make a graph, allowing a geometric interpretation of a mathematical function and giving his name to Cartesian coordinates (originated from Pappus’problem).Descartes believed that a system of knowledge should start from first principles and proceed mathematically to a series of deductions, reducing physics to mathematics. In Discours de la Méthode (1637), he advocated the systematic doubting of knowledge, believing as Plato that sense perception and reason deceive us and that man cannot have real knowledge of nature. The only thing that he believed he could be certain of was that he was doubting, leading to his famous phrase "Cogito ergo sum," (I think, therefore I am). From this one phrase, he derived the rest of philosophy, including the existence of God.Pappus’ problem The full enunciation of the problem is rather involved, but the most important case is to find the locus of a point such that the product of the perpendiculars on m given straight lines shall be in a constant ratio to the product of the perpendiculars on n other given straight lines. The ancients had solved this geometrically for the case m = 1, n = 1, and the case m = 1, n = 2. Pappus had further stated that, if m = n = 2, the locus is a conic, but he gave no proof; Descartes also failed to prove this by pure geometry, but he showed that the curve is represented by an equation of the second degree, that is, a conic; subsequently Newton gave an elegant solution of the problem by pure geometry.TEXT B Geometric figuresA geometric figure, such as a curve in the plane, is a collection of points satisfying one of more special conditions. By translating these conditions into expressions, involving the coordinates x and y, we obtain one or more equations which characterize the figure in question. For example, consider a circle of radius r with its center at the origin, as shown in Figure 2-5-2. Let P be an arbitrary point on this circle, and suppose P has coordinates (x, y). Then the line segment OP is the hypotenuse of a right triangle whose legs have lengths |x| and |y| and hence, by the theorem of Pythagoras,x2+y2=r2.The equation, called a Cartesian equation of the circle, is satisfied by all points (x, y) on the circle and by no others, so the equation completely characterizes the circle. This example illustrates how analytic geometry is used to reduce geometrical statements about points to analytical statements about real numbers.Throughout their historical development, calculus and analytic geometry have been intimately intertwined. New discoveries in one subject led to improvements in the other. The development of calculus and analytic geometry in this book is similar to the historical development, in that the two subjects are treated together. However, our primary purpose is to discuss calculus, Concepts from analytic geometry that are required for this purpose will be discussed as needed. Actually, only a few very elementary concepts of plane analytic geometry are required to understand the rudiments of calculus. A deeper study of analytic geometry is needed to extend the scope and applications of calculus, and this study will be carried out in later chapters using vector methods as well as the methods of calculus. Until then, all that is required form analytic geometry is a little familiarity with drawing graph of function.TEXT C Sets of points in the planeWe have already shown that there is a one-to-one correspondence between points in a plane and pairs of numbers(x, y). Certain sets of points in the plane may be of special interest. For example, we may wish to examine the set of points comprising the circumference of a certain circle, or the set of points constituting the interior of a certain triangle. One may wonder if such sets of point may be succinctly described in a compact mathematical notation.We may write{|(x, y)|y=2x} (1)to describe the set of ordered pairs (x, y), or corresponding points, such that the ordinate is equal to twice the abscissas. In effect, then, the vertical line in (1) is read “such that “. By “the graph of the set of ordered pairs” is meant the set of all points of the plane corresponding to the setof ordered pairs. The student will readily infer that the set of points constituting the graph lies on a straight line.Consider the set{(x, y)|y=x2}.Consistent with our previous interpretation, this symbol represents the set of ordered pairs (x, y) such that the ordinate is equal to the square of the abscissa. Here, the total graph comprises a simple recognizable geometrical figure, a curve known as a parabola.On the basis of these two examples, one may be tempted to believe that any arbitrarily drawn curve, which of course determines a set of points or ordered pairs, could be described succinctly by a simple equation. Unfortunately, this is not the case. For example, the broken line in figure 2-2-3 is one of such curves.Consider now the set{(x, y)|y>2x}to describe the set of points (x, y) whose ordinate is greater than twice its abscissa. In the case, our set of points constitutes not a curve, but a region of the coordinate plane.SUPPLEMENT Conic SectionThe conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a planethat is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola. The curve produced by a plane intersecting both nappes is a hyperbola.Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newton was then able to derive the shape of orbits mathematically using calculus, under the assumption that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes that are any of the four types of conic sections are possible.A conic section may more formally be defined as the locus of a point P that moves in the plane of a fixed point F called the focus and a fixedline d called the conic section directrix (with F not on d) such that the ratio of the distance of P from F to its distance from d is a constant e called the eccentricity. If e=0, the conic is a circle, if 0<e<1, the conic is an ellipse, if e=1, the conic is a parabola, and if e>1, it is a hyperbola.A conic section with conic section directrix at x=0, focus at (p, 0), and eccentricity e>0 has Cartesian equationwhere p is called the focal parameter. Plugging in p givesfor an ellipse,for a parabola, andfor a hyperbola.The polar equation of a conic section with focal parameter p is given byFive points in a plane determine a conic, as do five tangent lines in a plane. This follows from the fact that a conic section is a quadratic curve, which has general formso dividing through by a to obtainleaves five constants. Five points, (x i, y i) for i=1, …, 5, therefore determine the constants uniquely. The geometric construction of a conic section from five points lying on it is called the Braikenridge-Maclaurin Construction. The explicit equation for this conic is given by the equationTwo conics that do not coincide or have an entire straight line in common cannot meet at more than four points. There is an infinite family of conics touching four lines. However, of the eleven regions into which plane division cuts the plane, only five can contain a conic section which is tangent to all four lines. Parabolas can occur in one region only (which also contains ellipses and one branch of hyperbolas), and the only closed region contains only ellipses.Let a polygon of 2n sides be inscribed in a given conic, with the sides of the polygon being termed alternately "odd" and "even" according to some definite convention. Then the n(n-2)points where an odd side meet a nonadjacent even side lie on a curve of order n-2. This fact includes Pascal’s Theorem as a special case.NotationsEllipse, parabola and hyperbolaAn ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1and r2 from two fixed points F1and F2(the foci) separated by a distance of 2c is a given positive constant 2a.Problem Give similar definitions of parabola and hyperbola as above.A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus); a hyperbola is a conic section defined as the locus of all points in the plane the difference of whose distances from two fixed points (the foci) separated by a given distance is a given positive constant.Pascal’s Theorem (1640)The dual of Brianchon's theorem, discovered by B. Pascal in 1640 when he was just 16 years old. It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.A very special case happens when the conic degenerates into two lines, however the theorem still holds although this particular case is usually called Pappus theorem. In 1847, Möbius published the following generalization of Pascal's theorem: if all intersectionpoints (except possibly one) of the lines prolonging two opposite sides of a (4n-2)-gon inscribed in a conic section are collinear, then the same is true for the remaining point.Brianchon's Theorem (1806)The dual of Pascal's theorem. It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.In 1847, Möbius gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a (4n-2)-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line. Brianchon's theorem and Pascal's theorem are Example of dual geometric objects of Duality Principle.Duality Principle All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line." The principle was enunciated by Gergonne (1826), A similar duality exists for reciprocation as first enunciated by Poncelet (1818).Problem 1In the following four sets, which one describes the set of points (x, y) whose ordinate is greater than twice its abscissa?A {(x, y)|y<2x}B {(x, y)|y>2x}C {(x, y)|x<2y}D {(x, y)|x<2y}Problem 2 A conic section is defined to be any of the curves of intersection of a plane with a cone, if the plane of the intersection is parallel to an element of the cone, the curve of intersection is called ____.A a circleB an ellipseC a parabolaD a hyperbola。