数学专业英语[1]－The Real Number System

hehe123

Exercise ' n: X& A, O3 Q3 f3 i

Ⅰ. Answer the following questions :

1. What are the necessary conditions for operation to become an object of scientific approach?

2. Point out the main points the 2^nd and the 3^rd definitions emphasize as compared with the first definition.

Ⅱ. 1. Translate the third definitions of OR due to S. Beer.

2. Translate the following sentences into Chinese ;

ⅰ) It was G. Gantor who first introduced the concept of the set as object of mathematical study.

ⅱ) The definition of probability due to Laplace provoked a great deal of argument when it was applied;

ⅲ) Nowadays, we usually adopted measure theoretic foundations of probability initiated by A. N. Kolomogorov.

hehe123

数学专业英语－The Theory of Graphs ) H9 {7 Z4 y( G

In this chapter, we shall introduce the concept of a graph and show that graphs can be defined by square matrices and versa.

* m+ [ u: o8 N6 y/ [# S1 g3 ~' A; \

1. Introduction

Graph theory is a rapidly growing branch of mathematics. The graphs discussed 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 theory also grew out of an interesting physical problem, the so-called Konigsberg bridge problem. (this problem is discussed in Section 2) The outstanding Swiss mathematician, Leonhard Euler (1707-1783) solved the problem in 1736, thus laying the foundation for this branch of mathematics. Accordingly, Euler is called the father of graph theory.

2 x) h% F$ y2 n! i& Z0 g

Gustay Robert Kirchoff (1824-1887), a German physicist, applied graph theory in his study of electrical networks. In1847, he used graphs to solve systems of linear equations arising from electrical networks, thus developing an important class of graphs called trees.

In 1857, Arthur Caylcy discovered trees while working on differential equations. Later, he used graphs in his study of isomers of saturated hydrocarbons.

Camille Jordan (1838-1922), a French mathematician, William Rowan Hamilton, and Oystein Ore and Frank Harary, two American mathematicians, are also known for their outstanding contributions to graph theory.

$ ?; O4 w2 D3 r& N3 d

Graph theory has important applications in chemistry, genetics, management science, Markov chains, physics, psychology, and sociology.

& }: {5 [1 l4 H4 s& P( |: M

Throughout this chapter, you will find a very close relationship between graphs and matrices.

2. The Konigsberg Bridge Problem

0 h* X% Z! V6 z) ^7 p4 Z+ Y/ L

The 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 islands B and C. There are seven bridges linking the four parts of the city.

/ x* u. {7 }0 P( X! U: {

Residents of the city used to take evening walks from one section of the city 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 counting 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.

; m8 s8 f) n0 \

7 P! R o" R# N$ P1 Z! Y

9 a8 K$ S5 t( K* M% |, {0 {

7 a6 y$ e' F" e1 X) J: V0 |7 }

5 J2 d% g+ U1 x9 P2 I! o# x

Fig .1 The city of Konigsberg

9 S- l4 `# V5 n* s) A

In 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 case.

! K, W+ R5 p7 @, s4 v; k5 ]% f

$ s) p0 q. Q) J& C

) G% A& e4 @4 d; M1 Z

Fig .2 A mathematical model for the Konigsberg bridge problem

6 _' c. V) X, {

Let 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 they represent and are called vertices. The arcs or lines joining these points represent 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 trace this figure without lifting your pencil from paper or going over the same edge twice? Euler proved that a figure like this can be traced without lifting pencil 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 more than two vertices in the figure have an odd number of edges connecting them-----in fact,they all do.

hehe123

1. Graphs

Let 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 information, 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 there 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 ) connected by arcs or line segments. Such a figure is called a graph. The points are the vertices of the graph; 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 the graph. The vertices of graph are usually denoted by the letters A, B, C, and so on. An edge joining the 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２ has 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 edge 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.

5 t; g- K- w7 ?, a& d9 f' p

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 matrix 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 is

Clearly 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 matrix representation of some graph. For example, consider the matrix

A B C D

Clearly, this is the matrix representation of the graph in 图5.

hehe123

Vocabulary

Network 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

hehe123

Notes

1. 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 mathematician 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。

２．After all, by the fundamental counting 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.

意思是：毕竟，由基本的计算原理知，可能的路径的总数，不会超过5040个。然而逐一地去考察这些路径是否有一条路适合题意，那是太耗费时间了，that works 意思是：“有效”，这里可理解为：“适合题意”。

3.It is possible to trace the figure without lifting your pencil from paper or going the same edge twice?

意思是：是否能够跟踪图形而使你的铅笔不离开纸且不走过同一条边两次呢？这一句在英语上等同于without lifting your pencil from paper and without going over the same edge twice.

1. . . .in fact, they all do.

这里they代表顶点vertices; do 代表have an odd number of edges connecting them.

2. A is called the valence or degree of the vertex, denoted by v(A).

注意denoted 前面的逗号，可使读者不至于误会v(A)是用来记vertex的。这里v(A)是用来记Ａ的Ｖalence.

６. the entry that belongs to the row headed by X and column headed by Y gives the number of edges from vertex X to vertex Y.

意思是：属于Ｘ行，Ｙ列这一项的数字给出了从顶点Ｘ到顶点Ｙ的边数。这里the row headed by X意是冠以Ｘ的行，可简称Ｘ行或等Ｘ行。

hehe123

Exercise + c8 S5 g% b0 ~ {" c5 I

Ⅰ.answer the following questions:

1. How is the Konigsberg Bridge problem stated?

2. According to Euler’s theorem, why is the answer of the Konigsberh Bridge Problem negative?

Ⅱ.Translate the following passages into Chinese:

When a number of electrical components are connected together, we are said to have an electrical network. The junction between two or more components in a network are called nodes of the network, Each path joining a pair of nodes and through interconnections is best described by a diagram which eliminates all the electrical properties of the components. This graph is obtained by redrawing the circuit of the network with lines replacing the electrical components.

The graph makes clear the existence of a number of closed paths which may be traced along the branches. Such closed paths are called loops. Of the total number of loops of a network, a certain number of independent loops may be chosen. One way of choosing a set of independent loops is as follows:form, from the network, a sub-network by removing branches until no loops remain, although each node is still connected by a single path to another node. Such a structure is called a tree of the network.

Ⅲ.Translate the following sentences into English (in each sentence, make use of the phrase given in bracket):

下面简写The Konigsberg Bridge problem 为Ｋ.B.问题

１．Ｋ.B.问题只不过是尤拉所证明的定理的一个特例。(a special case)

２．从尤拉关于图论的一个定理，即可得Ｋ.B.问题的答案。(follows immediately from.)

３．Ｋ.B.问题的不可能性是尤拉定理的一个直接结果。(a direct consequence of)

hehe123

The mathematics to which our youngsters are exposed at school is. With rare exceptions, based on the classical yes-or-no, right-or-wrong type of logic. It normally doesn’t include one word about probability as a mode of reasoning or as a basis for comparing several alternative conclusions. Geometry, for instance, is strictly devoted to the “if-then” type of reasoning and so to the notion (idea) that any statement is either correct or incorrect.

However, it has been remarked that life is an almost continuous experience of having to draw conclusions from insufficient evidence, and this is what we have to do when we make the trivial decision as to whether or not to carry an umbrella when we leave home for work. This is what a great industry has to do when it decides whether or not to put $50000000 into a new plant abroad. In none of these case and indeed, in practically no other case that you can suggest, can one proceed by saying:” I know that A, B, C, etc. are completely and reliably true, and therefore the inevitable conclusion is~~” For there is another mode of reasoning, which does not say: This statement is correct, and its opposite is completely false.” But which say: There are various alternative possibilities. No one of these is certainly correct and true, and no one certainly incorrect and false. There are varying degrees of plausibility—of probability—for all these alternatives. I can help you understand how these plausibility’s compare; I can also tell you reliable my advice is.”

9 g5 p4 ^( q1 g N2 k# M

This is the kind of logic, which is developed in the theory of probability. This theory deals with not two truth-values—correct or false—but with all the in intermediate truth values: almost certainly true, very probably true, possibly true, unlikely, very unlikely, etc. Being a precise quantities theory, it does not use phrases such as those just given, but calculates for any question under study the numerical probability that it is true. If the probability has the value of 1, the answer is an unqualified “yes” or certainty. If it is zero (0), the answer is an unqualified “no” i.e. it is false or impossible. If the probability is a half (0.5), then the chances are even that the question has an affirmative answer. If the probability is tenth (0.1), then the chances are only 1 in 10 that the answer is “yes.”

8 A7 q) j4 N0 B: P4 K/ w/ b6 N

It is a remarkable fact that one’s intuition is often not very good at csunating answers to probability problems. For ex ample, how many persons must there are at least two persons in the room with the same birthday (born on the same day of the month)? Remembering that there are 356 separate birthdays possible, some persons estimate that there would have to be 50, or even 100, persons in the room to make the odds better than even. The answer, in fact, is that the odds are better than eight to one that at least two will have the same birthday. Let us consider one more example: Everyone is interested in polls, which involve estimating the opinions of a large group (say all those who vote) by determining the opinions of a sample. In statistics the whole group in question is called the “universe” or “population”. Now suppose you want to consult a large enough sample to reflect the whole population with at least 98% precision (accuracy) in 99out of a hundred instances: how large does this very reliable sample have to be? If the population numbers 200 persons, then the sample must include 105 persons, or more than half the whole population. But suppose the population consists of 10,000 persons, or 100,000 persons? In the case of 10,000 persons, or 1000,000 person? In the case of 10,000 persons, a sample, to have the stated reliability, would have to consist of 213 persons: the sample increases by only 108 when the population increases by 9800. And if you add 90000 more to the population, so that it now numbers 100000, you have to add only 4 to the sample. The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of probability rather than on intuition.

Although the subject started out (began) in the seventeenth century with games of chance such as dice and cards, it soon became clear that it had important applications to other fields of activity. In the eighteenth century Laplace laid the foundations for a theory of errors, and Gauss later develop this into a real working tool for all experimenters and observers. Any measurement or set of measurement is necessarily is necessarily inexact; and it is a matter of the highest importance to know how to take a lot of necessarily discordant data, combine them in the best possible way, and produce in addition some useful estimate of the dependability of the results. Other more modern fields of application are: in life insurance; telephone traffic problems; information and communication theory; game theory, with applications to all forms of competition, including business international politics and war; modern statistical theories, both for the efficient design of experiments and for the interpretation of the results of experiments; decision theories, which aid us in making judgments; probability theories for the process by which we learn, and many more.

----Weaver, W.

hehe123

Vocabulary 4 Q Q0 `; k! g, J8 f8 b5 w' s3 L: o, ]$ ]

Probability 概率论 permutation 置换

Plausibility 似乎合理 binomial coefficient 二次式系数

Affirmative 肯定的 generating function 母函数

Estimate 估计 even 事件

Discordant 不一致的 information and communication theory

Communication theory 通讯理论 信息与通讯论

Decision theory 决策论 game theory 对策论，博弈论

hehe123

Notes

1. Geometry, for example, is strictly devoted to the “if—then” type of reasoning and so to the notion (idea) that any statement is either correct or incorrect.

意思是：例如几何学就是严格地属于那种“如果，则”的推理类型，所以它也就属于那种对任何陈述要么是对的要么是不对的概念范围。Is devoted to 意思是：“奉献于”，这里可作：“属于”解，注意在and so to the notion~~中，在前面省去is devoted.

2. However, it has been remarked that life is an almost continuous experience when we leave home for work.

意思是：然而，人们已经注意到，生活就是这样一种几乎不断地需要我们从不充分的证据中去做出结论的经历，这就是对诸如我们离家上班时是否要带雨伞做出定时，我们所需要做的。

3. If the probability has value of 1, the answer is an unqualified “yes” or certainty.

这里unqualified解作：“绝对的”，“十足的”。如 an unqualified certainty (绝对的肯定)； An unqualified success (彻底胜利)。注意 qualified 常解作：“有资格的”，“合格的”。如 a qualified technician (合格的技术员)； qualified examination (资格考试，美国高等学校研究生院的一种考试)。

4． If the probability is a half, then the chances are even that the question has an affirmative answer.

意思是：如果概率是一半的话，那么问题有肯定答案的机会是对等的。注意这里even作“对等”解。

5． The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of probability rather than on intuition.

意思是：这对你越不可信，我们就要强调这种论点：宁可依靠概率论而不愚信直观，这里make the point that 意思是：“主张；强调；视~~为重要” .

hehe123

Exercise

1. Translate the following passage into Chinese

The origin of the theory of probability goes Bach to the mathematical problems connected with dice throwing that were discusses in letters exchanged by B.Ppascal and P.de Fermat in the 17^th century. These problems were principally concerned with concepts, such as permutations, combinations, and binomial coefficients, whose theory was established about the same time. This elementary theory of probability was later enriched by the work of scholars such as Jacob Bernoulli, A.de Moivre, T.Bayes, L, de Buffon, Danial Bcrnoulli, A, M, Legendre, and J.L. Lagrange. Finally, P.S. Laplace completed the classical theory of probability in his book “Throrie analytique des probabilities” (1812). In this work, Laplace not only systemized also greatly extended previous important results by introducing new methods such as the use of difference equations and generating functions. Since the 19^th century, the theory of probability has been extensively applied to the natural sciences and even to social sciences.

2. Translate the following sentences into Chinese:

1. The term random process is use to describe process that gives rise to one of a number of admitted possible outcomes but which outcome cannot be predicted with any certainty in advance.

2. Tow events A and B in a probability model with sample space and probability function P are said to be independent if

P (A B) =P(A) ·P(B)

3. Describe briefly the kind of logic developed in the theory of Probability.

4. Translate the following sentences into English (make use of the phrase or the phrases in the bracket):

设X=[a b], A X (A X) 是一开集， 又设a A 令r=sup{ : [a+ ] A}, 求证a+r …A. (这一部分不用翻译， 仅需翻译下下面证明部分)

证明：（1）若论不成六，即是说a+r A，则由于A是开集，存在 >0使得[a+r, a+r+ ] A, 从而[(a,a+r+ ) A, 这与r 的定义矛盾。（~~~would not hold, or~~~were false, or were not true; contrary to）

(2)若a+r A,则由于A是开集，存在 >0使得[a+r,a+r+ ] A，由这推出[a,a+r+ ] A,这是不可能的。故a+r A. (this implies)

(3)若论断是错的，则由于A是开集，存在 >0使得[(a+r,a+r+ ) A,从而[a,a+r+ ] A,这就导至与r是 的上确界这一事实相矛盾结论。（leads to contradiction to the that）