数学专业英语[1]－The Real Number System - 第4页 - 数学专业英语及翻译 - 数学建模社区-数学中国

hehe123 发表于 2004-11-27 12:50

VocabularyStatistics统计学 in tables 列成表Statistical 统计的 tabular列成表的Statistical data统计数据 sample样本Statistical method统计方法 inference推理，推断Original data原始数据 reliance信赖Qualitative定性的 forecasting预测Quantitative定量的 in common parlance按一般说法Chronological年代学的 conducive有帮助的Time series时间序列 grope摸索Cyclical循环的 akin to类似Period周期 apt to易于Periodic周期的 undue不适当的Prognosticate预测 sociology社会学Authenticity可信性，真实性 phase相位；方面Synonymously同义的 categories范畴，类型Correlative相互关系的，相依的 concurrent会合的，一致的，同时发生的

hehe123 发表于 2004-11-27 12:50

Notes
1. It is the purpose of this volume not to enter into the highly mathematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.意思是：本书的目的并不是要深入到这个论题的有关高深的数学与理论的方面，而是要讨论它的更为初等和更为常用的方面，not…but rather 意思是“不是…而是”，而rather than意思是“宁愿…而不”，两者意思相近但有差别（主要表现为强调哪方面的差别）。2. Without an adequate understanding of statistics, the investigator in the social sciences may frequently be like the blind man groping in a dark closet for a black cat that isn’t there.意思是：没有对统计学的适当的理解，社会科学的研究人员就会常常像一个盲人在一个暗室里摸索一只并不存在的黑猫一样，这里investigator 指一般研究人员，请比较与researcher，scientist等词的异同。3. It is essential that he have a working knowledge of the processes of collection and that he be able…这里essential是虚拟形容词，所以后面用he have；he be able…have和be前面省去了should。4. The distinction is one of kind rather than of amount.意思是：这种区别是类型的而不是数量的。5. …the distinction that is being made is not so much of kind as of location. 参看第二课注2关于not so much…as的解释，并比较注4和注5两句。6. （To attempt to say what will happen in the future） requires…of any mechanical forecasting device.括号里的部分为此句的主语，谓语是requires， 后面是三个并列宾语短语。

hehe123 发表于 2004-11-27 12:51

ExerciseI. Translate the following passages into Chinese:1. Statistical methods, or statistics, range from the most elementary descriptive devices which may be understood by anyone, to those extremely complicated mathematical procedures which are comprehended by only the most expert theoreticians.2. Although a person may never collect statistical data for his own use and may always use published sources, it is essential that he have a working knowledge of the processes of collection and that he be able to evaluate the reliability of the data he proposes to use. II Answer the following questions according to the content of this lesson.1. In defining statistics, what are the four procedures about the numerical data?2. How are the numerical data presented usually?3. Give examples for qualitative differentiation and quantitative classification of statistical data .

hehe123 发表于 2004-11-27 12:51

数学专业英语[9]－Linear Programming

Linear Programming is a relatively new branch of mathematics.The cornerstone of this exciting field was laid independently bu Leonid V. Kantorovich,a Russian mathematician,and by Tjalling C,Koopmans, a Yale economist,and George D. Dantzig,a Stanford mathematician. Kantorovich’s pioneering work was motivated by a production-scheduling problem suggested by the Central Laboratory of the Leningrad Plywood Trust in the late 1930’s. The development in the United States was influenced by the scientific need in World War II to solve logistic military problems, such as deploying aircraft and submarines at strategic positions and airlifting supplies and personnel.
The following is a typical linear programming problem:
A manufacturing company makes two types of television sets: one is black and white and the other is color. The company has resources to make at most 300 sets a week. It takes $180 to make a black and white set and $270 to make a color set. The company does not want to spend more than $64,800 a week to make television sets. If they make a profit of $170 per black and white set and $225 per color set, how many sets of each type should the company make to have a maximum profit?
This problem is discussed in detail in Supplementary Reading Material Lesson 14.
Since mathematical models in linear programming problems consist of linear inequalities, the next section is devoted to such inequalities.
Recall that the linear equation lx+my+n=0 represents a straight line in a plane. Every solution (x,y) of the equation lx+my+n=0 is a point on this line, and vice versa.
An inequality that is obtained from the linear equation lx+my+n=0 by replacing the equality sign “=” by an inequality sign < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to) is called a linear inequality in two variables x and y. Thus lx+my+n≤0, lx+my+n≥0 are all linear liequalities. A solution of a linear inequality is an ordered pair (x,y) of numbers x and y for which the inequality is true.
EXAMPLE 1 Graph the solution set of the pair of inequalities
<v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path extrusionok="f" connecttype="rect" gradientshapeok="t"></v:path><LOCK aspectratio="t" v:ext="edit"></LOCK></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape>
<v:shape><v:imagedata></v:imagedata><v:textbox style="mso-next-textbox: #_x0000_s1026"></v:textbox><w:wrap type="tight"></w:wrap></v:shape>SOLUTION Let A be the solution set of the inequality x+y-7≤0 and B be that of the inequality x-3y +6 ≥0 .Then A∩B is the solution set of the given pair of inequalities. Set A is represented by the region shaded with horizontal lines and set B by the region shaded with vertical lines in Fig.1. Therefore the crossed-hatched region represents the solution set of the given pair of inequalities. Observe that the point of intersection (3.4) of the two lines is in the solution set.
Generally speaking, linear programming problems consist of finding the maximum value or minimum value of a linear function, called the objective function, subject to some linear conditions, called constraints. For example, we may want to maximize the production or profit of a company or to maximize the number of airplanes that can land at or take off from an airport during peak hours; or we may want to minimize the cost of production or of transportation or to minimize grocery expenses while still meeting the recommended nutritional requirements, all subject to certain restrictions. Linear programming is a very useful tool that can effectively be applied to solve problems of this kind, as illustrated by the following example.
 EXAMPLE 2 Maximize the function f(x,y)=5x+7y subject to the constraints
x≥0 y≥0
x+y-7≤0
2x-3y+6≥0
 SOLUTION First we find the set of all possible pairs(x,y) of numbers that satisfy all four inequalities. Such a solution is called a feasible sulution of the problem. For example, (0,0) is a feasible solution since (0,0) satisfies the given conditions; so are (1,2) and (4,3).



 Secondly, we want to pick the feasible solution for which the given function f (x,y) is a maximum or minimum (maximum in this case). Such a feasible solution is called an optimal solution.


 Since the constraints x ≥0 and y ≥0 restrict us to the first quadrant, it follows from example 1 that the given constraints define the polygonal region bounded by the lines x=0, y=0,x+y-7=0, and 2x-3y+6=0, as shown in Fig.2.
<v:shape><v:imagedata></v:imagedata></v:shape>
Fig.2.
 Observe that if there are no conditions on the values of x and y, then the function f can take on any desired value. But recall that our goal is to determine the largest value of f (x,y)=5x+7y where the values of x and y are restricted by the given constraints: that is, we must locate that point (x,y) in the polygonal region OABC at which the expression 5x+7y has the maximum possible value.
 With this in mind, let us consider the equation 5x+7y=C, where C is any number. This equation represents a family of parallel lines. Several members of this family, corresponding to different values of C, are exhibited in Fig.3. Notice that as the line 5x+7y=C moves up through the polygonal region OABC, the value of C increases steadily. It follows from the figure that the line 5x+7y=43 has a singular position in the family of lines 5x+7y=C. It is the line farthest from the origin that still passes through the set of feasible solutions. It yields the largest value of C: 43.(Remember, we are not interested in what happens outside the region OABC) Thus the largest value of the function f(x,y)=5x+7y subject to the condition that the point (x,y) must belong to the region OABC is 43; clearly this maximum value occurs at the point B(3,4).
<v:shape><v:imagedata></v:imagedata></v:shape>
Fig.3.
 Consider the polygonal region OABC in Fig.3. This shaded region has the property that the line segment PQ joining any two points P and Q in the region lies entirely within the region. Such a set of points in a plane is called a convex set. An interesting observation about example 2 is that the maximum value of the objective function f occurs at a corner point of the polygonal convex set OABC, the point B(3,4).
 The following celebrated theorem indicates that it was not accidental.
 THEOREM (Fundamental theorem of linear programming) A linear objective function f defined over a polygonal convex set attains a maximum (or minimum) value at a corner point of the set.
 We now summarize the procedure for solving a linear programming problem:
1. Graph the polygonal region determined by the constraints.
2. Find the coordinates of the corner points of the polygon.
3. Evaluate the objective function at the corner points.
4. Identify the corner point at which the function has an optimal value.

hehe123 发表于 2004-11-27 12:52

Vocabularylinear programming 线形规划 quadrant 象限objective function 目标函数 convex 凸的constraints 限制条件，约束条件 convex set 凸集feaseble solution 容许解，可行解 corner point 偶角点optimal solution 最优解 simplex method 单纯形法

hehe123 发表于 2004-11-27 12:52

Notes1. A Yale economist, a Stanford mathematician 这里Yale Stanford 是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)2. subject to some lincar conditions 解作“在某些线形条件的限制下”。试比较下面各词组在用法上的异同： subject to; under the condition(s) of; satisfying the condition(s).3. celebrated theorem 意思为“著名定理”。4. Since…it follows…与Notice…it follow…都是数学中常用的句型，以表达“根据什么，可得什么”这一意思，请参看附录III。5. 本课多次用到recall, observe, notice, remember等词，用以提醒读者一些已知的事实或定理，读者可从这些例句中体会这些词的用法。请参看附录III。

hehe123 发表于 2004-11-27 12:53

Exercise
I.Translate the following passage into Chinesre:
 Although satisfactory for two-variable linear programs, the corner-point method is not computationally effective when extended beyond three-variable situations. Its usefulness is restricted to introducing the general idea of linear programming, and the need for more poweful solution techniques is obvious. The simplex method, devised by George Dantzig in the late 1940’s, was, central to the explosion in linear programming applications that occurred in the 1950’s and 1960’s. Although more powerful techniques exist for solving special problems, the simplex method is still the most computationally effective general method available for solving the widest variety of linear programming problems.




II.Read the content of this lesson carefully and complete the following sentences:
1. Linear programming problems consist of


2. The set of feasible solutions of a linear programming problem is 
3. A polygonal region is a region bounded by 
4. A convex set in a plane is





III.Translate the following sentences into English:
1. 点(3,4)是两条直线x+y-7=0和2x-3y+6=0的交点。
2. (3,4)是例2的最优解。
3. 计算目标函数在偶角点处的值，然后进行比较，求出目标函数的最大值或最小值。

hehe123 发表于 2004-11-27 12:54

数学专业英语[10]－Groups and Rings

<TABLE width="90%" border=0>

<TR>
<TD width="100%"><IMG src="http://www.shumo.com/bbs/Skins/Default/topicface/face1.gif" align=absMiddle border=0> 数学专业英语－Groups and Rings

During the present century modern abstract algebra has become more and more important as a tool for research not only in other branches of mathematics but even in other sciences .Many discoveries in abstract algebra itself have been made during the past years and the spirit of algebraic research has definitely tended toward more abstraction and rigor so as to obtain a theory of greatest possible generality. In particular, the concepts of group ,ring,integral domain and field have been emphasized.
The notion of an abstract group is fundamental in all sciences ,and it is certainly proper to begin our subject with this concept. Commutative additive groups are made into rings by assuming closure with respect to a second operation having some of the properties of ordinary multiplication. Integral domains and fields are rings restricted in special ways and may be fundamental concepts and their more elementary properties are the basis for modern algebra.




Groups





DEFINITION A non-empty set G of elements a,b,…is said to form a group with respect to 0 if:
I. G is closed with respect to 0
II. The associative law holds in G, that is 
aо(bоc)=(aоb)оc
for every a, b, c of G
Ⅲ. For every a and b of G there exist solutions χ and У in G of the equations
 aοχ=b yοa=b
A group is thus a system consisting of a set of elements and operation ο with respect to which G forms a group. We shall generally designate the entire system by the set G of its elements and shall call G a group. The notation used for the operation is generally unimportant and may be taken in as convenient a way as possible. 
DEFINITION A group G is called commutative or abelian if
aοb=bοa


For every a and b of G.


An elementary physical example of an abelian group is a certain rotation group. We let G consist of the rotations of the spoke of a wheel through multiples of 90º and aοb be the result of the rotation a followed by the rotation b. The reader will easily verify that G forms a group with respect to ο and that aοb=bοa. There is no loss of generality when restrict our attention to multiplicative groups, that is, write ab in stead of aοb.



EQUIVALENCE





In any study of mathematical systems the concept of equivalence of systems of the same kind always arises. Equivalent systems are logically distinct but we usually can replace any one by any other in a mathematical discussion with no loss of generality. For groups this notion is given by the definition: let G and G´ be groups with respective operations o and o´,and let there be a1-1 correspondence
S : a<v:shapetype> <v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path extrusionok="f" connecttype="rect" gradientshapeok="t"></v:path><LOCK aspectratio="t" v:ext="edit"></LOCK></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>a´ (a in G and a´ in G´)


between G and G´ such that
 (aοb)´=a´ο b´ 
for all a, b of G. then we call G and G´equivalent(or simply, isomorphic)groups.
The relation of equivalence is an equivalence relation in the technical sense in the set of all groups. We again emphasize that while equivalent groups may be logically distinct they have identical properties.
The groups G and G´ of the above definition need not be distinct of course and o´ may be o. when this is the case the self-equivalence S of G is called an automorphism. 
 I: a <v:shape><v:imagedata></v:imagedata></v:shape>a


Of G, but other automorphisms may also exist.
 
Rings 



A ring is an additive abelian group B such that 
I. the set B is closed with respect to a second operation designated by multiplication; that is , every a and b of B define a unique element ab of B.
II. multiplication is associative; that is 
a (bc) = (ab)c
for every a, b, c of B.
Ⅲ. The distributive laws 
 a (b+c) = ab +ac (b+c) a=ba +ca


hold for every a, b, c of B.
The concept of equivalence again arises. We shall write
 B ≌ B′


to mean that B and B′are equivalent.






</TD></TR></TABLE>

hehe123 发表于 2004-11-27 12:54

Vocabulary
Group 群 rigor 严格ring 环 generalization 推广 integral domain 整环 Abelian group 阿贝尔群commutative additive group 可交换加法群 rotation 旋转automorphism 自同构

hehe123 发表于 2004-11-27 12:55

数学专业英语[11]－Historical introduction of Calculus

The Two Basic Concepts of Calculus



The remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in physics,engineering,chemistry,geology,biology, and other fields including,rather recently,some of the social sciences.
To give the reader an idea of the many different types of problems that can be treatedby the methods of calculus,we list here a few sample questions.
With what speed should a rocket be fired upward so that it never returns to earth? What is the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter Ｌ? What volume of material is removed from a solid sphere of radius 2 r if a hole of redius r is drilled through the center? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour,by how much will it increase at the end of two hours? If a ten-pound force stretches an elastic spring one inch,how much work is required to stretch the spring one foot?
These examples,chosen from various fields,illustrate some of the technical questions that can be answered by more or less routine applications of calculus.
Calculus is more than a technical tool－it is a collection of fascinating and exeiting idea that have interested thinking men for centuries.These ideas have to do with speed,area,volume,rate of growth,continuity,tangent line,and otherconcepts from a varicty of fields.Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is its unifying power.Most of these ideas can be formulated so that they revolve around two rather specialized problems of a geometric nature.We turn now to a brief description of these problems.
Consider a cruve C which lies above a horizontal base line such as that shown in Fig.1. We assume this curve has the property that every vertical line intersects it once at most.The shaded portion of the figure consists of those pointe which lie below the curve C , above the horizontal base,and between two parallel vertical segments joining C to the base.The first fundamental problem of calculus is this: To assign a number which measures the area of this shaded region.
Consider next a line drawn tangent to the curve,as shown in Fig.1. The second fundamental problem may be stated as follows:To assign a number which measures the steepness of this line.
<v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path extrusionok="f" connecttype="rect" gradientshapeok="t"></v:path><LOCK aspectratio="t" v:ext="edit"></LOCK></v:shapetype><v:shape><v:imagedata></v:imagedata><v:textbox style="mso-next-textbox: #_x0000_s1026"></v:textbox><w:wrap type="tight"></w:wrap></v:shape>
 





















Basically,calculus has to do with the precise formulation and solution of these two special problems.It enables us to define the concepts of area and tangent line and to calculate the area of a given region or the steepness of a given angent line. Integral calculus deals with the problem of area while differential calculus deals with the problem of tangents.



Historical Background


The birth of integral calculus occurred more than 2000 years ago when the Greeks attempted to determine areas by a procees which they called the method of exhaustion.The essential ideas of this ,method are very simple and can be described briefly as follows:Given a region whose area is to be determined,we inscribe in it a polygonal region which approximates the given region and whose area we can easily compute.Then we choose another polygonal region which gives a better approximation,and we continue the process,taking polygons with more and more sides in an attempt to exhaust the given region.The method is illustrated for a scmicircular region in Fig.2. It was used successfully by Archimedes(287－212 B.C.) to find exact formulas for the area of a circle and a few other special figures.
 The development of the method of exhaustion beyond the point to which Archimcdcs carried it had to wait nearly eighteen centuries until the use of algebraic symbols and techniques became a standard part of mathematics. The elementary algebra that is familiar to most high-school students today was completely unknown in Archimedes’ time,and it would have been next to impossible to extend his method to any general class of regions without some convenient way of expressing rather lengthy calculations in a compact and simpolified form.
A slow but revolutionary change in the development of mathematical notations began in the 16 th century A.D. The cumbersome system of Roman numerals was gradually displaced by the Hindu-Arabic characters used today,the symbols “+”and “-”were introduced for the forst time,and the advantages of the decimal notation began to be recognized.During this same period,the brilliant successe of the Italian mathematicians Tartaglia,Cardano and Ferrari in finding algebraic solutions of cubic and quadratic equations stimulated a great deal of activity in mathematics and encouraged the growth and acceptance of a new and superior algebraic language. With the wide spread introduction of well-chosen algebraic symbols,interest was revived in the ancient method of exhaustion and a large number of fragmentary results were discovered in the 16 th century by such pioneers as Cavalieri, Toricelli, Roberval, Fermat, Pascal, and Wallis.



<v:shape><v:imagedata></v:imagedata></v:shape>
Fig.2. The method of exhaustion applied to a semicircular region.



Gradually the method of exhaustion was transformed into the subject now called integral calculus,a new and powerful discipline with a large variety of applications, not only to geometrical problems concerned with areas and volumes but also to jproblems in other sciences. This branch of mathematics, which retained some of the original features of the method of exhaustion,received its biggest impetus in the 17 th century, largely due to the efforts of Isaac Newion (1642—1727) and Gottfried Leibniz (1646—1716), and its development continued well into the 19 th century before the subject was put on a firm mathematical basis by such men as Augustin-Louis Cauchy (1789-1857) and Bernhard Riemann (1826-1866).Further refinements and extensions of the theory are still being carried out in contemporary mathematics.

页: 1 2 3 [4] 5 6 7 8 9 10 11 12 13

数学建模社区-数学中国's Archiver

数学专业英语[9]－Linear Programming

数学专业英语[10]－Groups and Rings

数学专业英语[11]－Historical introduction of Calculus