数学专业英语[1]－The Real Number System

hehe123

Vocabulary

geology 地质学 decimal 小数，十进小数

biology 生物学 discipline 学科

social sciences 社会科学 contemporary 现代的

disk (disc) 圆盘 bacteria 细菌

isosceles triangle 等腰三角形 elastic 弹性的

perimeter 周长 impetus 动力

volume 体积 proportional to 与…成比例

center 中心 inscribe 内接

steepness 斜度 solid sphere 实心球

method of exhaustion 穷举法 refinement 精炼，提炼

polygon 多边形，多角形 cumbersome 笨重的，麻烦的

polygonal 多角形 fragmentary 碎片的，不完全的

approximation 近似，逼近 background 背景

hehe123

We shall begin by considering some simple continuously variable quantity like stature.We know this varies greatly from one individual to another ,and may also expect to find certain average differences between people drawn from different social classes or living in different geographical areas,etc.Let us suppose that a socio-medical survey of a particular community has provided us with a representative sample of 117 males whose heights are distributed as shown in the first and third columns of Table 1.

9 l$ U! a8 r! |' H( ]6 C1 j* N# ?6 Q. w

+ j" {. i* e( I$ L: \0 U. Z 6 |" g9 t4 i! w4 N: r% t

7 X6 k( d' P/ q6 _/ Q6 D/ c5 I E2 r1 x

Table 1.Distribution of stature in 117 males

% p# w4 _; |( [$ W' G, v$ v5 O% ?2 `7 q0 b' [& x3 j, O2 m3 A1 j, }; e! r y1 O, e6 J( L) n7 T3 i# ^2 m- o3 P- f/ L W- \9 C4 N6 O( Q" e( }! a2 q$ G1 y; A1 A- H: A; }2 Q

Absolute Height (m)	8 h5 U" e0 m" B9 n K: I2 k) p Working measurements with origin at 1.70(x)	7 y7 k/ Z; c7 p3 z! f Number of men 6 y, c/ C8 P# d' D5 W observed(f)	Contributions to the sum ' F$ K( |0 L- _8 q* r. R$ X! w (f x )	Contributions + ?$ I- E: N! n _' L! p to the sum of 0 K" R6 {+ A; b2 y; t: h1 D1 r squares (f )
1.58 " |! p9 O) p0 N1 _- h5 w+ E 1.60 ; S0 `# e- e: A3 k8 o' X1 \ 1.62 8 e# z M @5 j1 M 1.64 1.66 1.68 1.70 8 p% q4 z/ ~; a( \ 1.72 , X" C% E* C2 ^( G+ L# N 1.74 / J4 V6 D+ N7 o Y4 J 1.76 1.78 1.80 1.82 $ X7 U- E( S0 s; k F 1.84	－6 －5 －4 . e4 I( v) q! w' W. j3 B8 E －3 0 Y D7 }; ~, y. P% [. R －2 －1 0 $ i- H1 v& b, F0 w 1 % V2 J9 ^* y+ s 2 3 9 p% {6 t( \/ C7 G 4 8 ]) c% j' D1 D4 S8 T 5 6 7	0 q$ N0 k) o" \5 A2 W- l8 ]; H# k 1 3 6 8 8 @* `* n0 b& A. y9 s: s 13 0 D9 ]4 y" {, i% A2 u 18 19 $ y( J2 x2 \# i) @( W 14 4 V L5 y. R2 `: B 14 9 5 4 " G4 T1 c8 g4 Z( q1 ]) _, s 2 ( ?% @. Y4 c: y1 T3 H) L' S 1	－6 －15 －24 2 ]$ r1 {" ~) r$ q" W! J7 p; A0 | －24 －26 －18 0 - T( ~1 X: a% C/ c6 R6 r. c 14 28 7 J4 s# h6 [* H 27 8 c4 C3 m8 N: N. O! B5 E 20 4 V. k3 p/ T" c 20 12 7	4 ~/ @5 N* ?; B, E 36 75 96 72 2 Y8 N: P. i, M0 M* z+ h 52 - }$ c' F2 q2 o7 y. p6 Y 18 0 14 4 S* e4 t& j; W& x5 E3 U 56 81 80 , I; f3 n& }- s( g4 a3 {& P/ [# Y 100 + Q8 v" F% ^' t! z) Y3 i. s" i 72 # \+ @: a+ L+ d, E& X" G 49
Ｔotals	- N1 `: x) f. s( b: ]- n' } 8 Y2 o1 x9 R: |- W	& g* B3 x# y4 ~$ U( g 117	: f4 l* c0 T7 X; z6 i4 q) ?3 R ＋15	801

2 W: f8 v8 l) H: g7 g1 w; C

! P: x D$ ^1 W0 u8 y

7 Z: n! A$ P. ~0 a

We shall assume that the original measurements were made as accurately as possible,but that they are given here only to the mearest 0.02 m (i.e.2 cm).Thus the group labeled “1.66” contains all those men whose true measurements were between 1065 and 1067 m.One si biable to run into trouble if the exact methods of recording the measurements and grouping them are not specified exactly.In the example just given the mid-point of the interval labeled”1.66” m.But suppose that the original readings were made only to the nearest 0.01 m (i.e. 1 cm )and then “rounded up “to the nearest multiple of 0.02 m.We should then have “1.65”, which covers the range 1.645 to 1.655,included with “1.66”.The interval “1.66”would then contain all measurements lying between 1.645 m and 1.665 m .for which the mid-point is 1.655 m. The difference of 5 mm from the supposed value of 1.66 m could lead to serious inaccuracy in certain types of investigation.

A convenient visual way of presenting such data is shown in fig. 1, in which the area of each rectangle is ,on the scale used, equal to the observed proportion or percentage of individuals whose height falls in the corresponding group.The total area covered by all the rectangles therefore adds up to unity or 100per cent .This diagram is called a histogram.It is easily constructed when ,sa here ,all the groups are of the same width.It is also easily adapted to the case when the intervals are uneqal, provided we remember that the areas of the rectangles must be proportional to the numbers of units concerned.If, for example, we wished to group togcther the entries for the three groups 1.80,1.82 and 1.84 m,totaling 7 individuals or 6 per cent of the total,then we should need a rectangle whose base covered 3working groups on the horizontal scale but whose height was only 2 units on the vertical scale shown in the diagram.In this way we can make allowance for unequal grouping intervals ,but it is usually less troublesome if we can manageto keep them all the same width.In some books histograms are drawn so that the area of each rectangle is equal to the actual number (instend of the proportion) of individuals in the corresponding group.It is better, however, to use proportions, sa different histograms can then be compared directly.

hehe123

The general appearance of the rectangles in Fig.1 is quite striking ,especially the tall hump in the centre and the rapidly falling tails on each side.There are certain minor irregularities in the pattern, and these would, in general ,be more ronounced if the size of the sample were smaller. Conversely, weth larger samples we usually find that the set of rectangles presents a more regular appearance. This suggests that if we had a very large number of measurements ,the ultimate shape of the picture for a suitably small width of rectangle would be something very like a smooth curve,Such a curve could be regarded as representing the true ,theoretical or ideal distribution of heights in a very (or ,better,infinitely)large population of individuals.

What sort of ideal curve can we expect ? There are seveala theoretical reasons for expecting the so-called Gaussiao or “normal “curve to turn up in practice;and it is an empirical fact that such a curve lften describes with sufficient accuracy the shape of histograms based on large numbers of obscrvations. Moreover,the normal curve is one of the easiest to handle theoretically,and it leads to types of statistical analysis that can be carried out with a minimum amount of computation. Hence the central importance of this distribution in statistical work .

The actual mathematical equation of the normal curve is where u is the mean or average value and is the standard deviation, which is a measure of the concentration of frequency about the mean. More will be said about and later .The ideal variable x may take any value from to .However ,some real measurements,like stature, may be essentially positive. But if small values are very rare ,the ideal normal curve may be a sufficiently close approximation. Those readers who are anxious to avoid as much algebraic manipulation as possible can be reassured by the promise that no derect use will be made in this book of the equation shown. Most of the practical numerical calculations to which it leads are fairly simple.

Fig. 1 shows a normal curve, with its typocal symmetrical bell shape , fitted by suitable methods to the data embodied in the rectangles. This is not to say that the fitted curve is actually the true, ideal one to which the histogram approxime.tes; it is merely the best approximation we can find.

The mormal curve used above is the curve we have chosen to represent the frequency distribution of stature for thr ideal or infinitely large population. This ideal poplation should be contrasted with the limited sample of obsrever. Values that turns up on any occasion when we make actual measurements in the real world. In the survey mentioned above we had a sample of 117 men .If the community were sufficiently large for us to collect several samples of this size, we should find that few if any of the corresponding histograms were exactly the same ,although they might all be taken as illustrating the underlying frequency distribution. The differences between such histograms constitute what we call sampling variation, and this becomes more prominent at the size of sample decreases

hehe123

Vocabulary $ a0 o4 l4 }0 ?

Socio-medical survey 社会医疗调查表 visual 可见的。

distribute 分布（动词） percentage 百分比

distribution 分布（名词） individual 个人，个别

histogram 直方图，矩形图 mean 平均值，中数

hump 驼背，使隆起 standard deviation 标准差

normal distribution 正态分布 sample varianice 样本方差

hehe123

f' N3 O! c* w5 {* S& G& n+ r. L9 ^ ^$ ?

数学专业英语－Operations Research $ A( g* Z( q& K" y2 ?5 p The start of operations research took place in a military context in the United Kingdom during World War Ⅱ, and it was quickly taken up under the name operations research (OR) in the United States. After the war it evolved in connection with industrial organization, and its many techniques allowed for expanding areas of application in the United States, the United Kingdom, and in other industrial countries. It is, however, not easy to give a precise definition of operations research, There are three different representative definitions. According to the classical definition, due to P. M. Morse and G. E. Kimball, operations research is a scientific method of providing executives with a quantitative basis for decisions regarding operations under their control. The second definition, due to C. W. Churchman, R. L. Ackoff, and E. L. Arnoff, is as follows: operations research in the most general sense can be characterized as the application of scientific methods, techniques, and tools to the operations of systems so as to provide those in control with optimum solutions to problems. $ J* R6 z. F! A; R1 q: m8 @5 g# R As the third definition we mention the suggestion due to S. Beer: operations research is the attack of modern science on problems of likelihood (accepting mischance) that arise in the management and control of men, machines, materials, and money in their natural environments. Its special technique is to invent a strategy of control by measuring, comparing, and predicting probable behaviour through a scientific model of a situation. These three definitions have several common features. In the first place, operations research serves executives by providing partial observations and advice which they can use in judging a situation. Second, the applicability of operations research is limited to areas where scientific methods can be successfully applied. This is the reason why operations research would not be considered to extend beyond only partial observation and advice. A fundamental requirement for a scientific approach is that it must have a mathematical model whose validity can be tested by actual data, Third, any operation should satisfy three necessary conditions in order that it may be an object of scientific approach: (1) the operation should be defined objectively; (2) the results, consequences, and effects of its application should be objectively measurable; (3) the operation should be capable of repetition. Fourth , operations research should aim at finding a practical strategy. Although operations research is based on scientific methodology, it does not aim at establishing general scientific assertions that are valid for all situstions. 8 N. \# O, I" Q3 T3 N! X2 k3 b These four points are essential to any operations research, and are implicit in each of the three aforementioned definitions. ! B2 n) N% R' x) i. d On the other hand these three definitions emphasize differently some specific features of operations research, according to their historical positions. In comparion with the first definition, the second makes clearer the place where operations research is applied by pointing out that it is concerned with the operations of systems, and, instead of the vague mention of quantitative basis for decisions in the first definition, it states that operations research seeks optimum solutions, reflecting a stage where optimum solutions were sought by applications of mathematical programming techniques. In the third definition of operations research the notion of system is defined explicitly, the notion of operation is defined to be its special technique, and the objectives of operations research are given. It is clearly asserted that operations research belongs to the methodology of applied sciences. In operations research, operations and systems are dealt with in their intimate interconnection. The methodology of operations research therefore relies on an overall approach for which interdisciplinary cooperation is indispensable and in which the operations research team plays an important role. 2 d% W q- Q5 ~ In applying the operations research approach to the circumstances with which we are concerned, we concentrate our interest on mutual relationships among input and output characteristics. A black-box method by which the interrelation between input and output can be clarified without entering the actual mechanism of the transformation yielded by the system or by its subsystems plays a fundamental role in operations research. The following are major phases of an operations research project: (1) formulating the problem; (2) constructing a mathematical model to represent the system under study and deriving a solution from the model; (3) testing the model and the solution derived form it; (4) the implementation stage of establishing controls over the solution and putting it to work. It is important to construct a model of information communication in connection. With a mathematical model of any problem in operations research. Process of aliocation, competition, queuing, inventory, and production appear frequently in the mathematical models of operations research. - [9 x& H8 w. N6 f" C8 }6 w& W : F/ N- ]. v% }- [3 {% x" J* q ——From Encyclopedic Dictionary of Mathematics : g! F- E9 q2 C. w' H _' ]6 d2 K/ V% ?* L

hehe123

Vocabulary

Operations research (OR) 运筹 interdiscipline 交叉学科

Executive 行政人员 interdisciplinary cooperation 交叉学科的

likelihood 似然 合作

scientific approach 科学方法 black-box method 黑箱方法

methodology 方法论 implementation stage 实现阶段

aforementioned 前述的 queue 排队

mathematical programming 数学规划

hehe123

Notes

6 t% @& u- \$ l* F' S3 X6 P. _

1. Operations Research运筹学. 运筹学是第二次世界大战期间,为解决后勤供应问题而发展起来的一门学科,它运用最优化技术去解决管理和决策问题.

2. According to the classical definition, due to P. M. Morse and G. E. Kimball, operations research is a scientific method of providing executives with a quantitive basis for decisions regarding operations under their control.

意思是:根据P. M. Morse和G. E. Kimball提出的古典定义,运筹学是一种科学方法,它提供行政人员一种定量基础,以便他们对所控的操作进行决策,这里due to是”归功于” “由…提出”之意,providing…with…for…是”提供…给…用于…”之意.

3. …instead of the vague mention of quantitative basis for decisions in the first definition, it states that …by applications of mathematical programming techniques.

意思是:代替第意个定义中对于决策的定量基础那种模糊的提法,它(第二个定义)阐明了运筹学用于寻求最优解,反映了运用数学规划方法求最优解的阶段.这里reflecting至句子结束一段,属独立分词结构,用以补充说明it states that…的句子.

4. The methodology of operations research therefore relies on…the operations research team plays an important role.

意思是:因而运筹学的方法论依赖于…一种全面的研究,对这种研究来说,各交叉学科的合作是不可避免的,而且,在这种研究中,运筹学小组起了重要的作用.注意:前后两个which都是approach的关系代词,很容易误认为第二个which 是cooperation的关系代词,虽然这在意思上说得过去,但从语法结构上却不然.

5. A black-box method by which the interrelation between input and output …plays a fundamental role in operations research.

意思是:黑箱方法不需引进由系统或它的子系统所产生的变换的确切机制而能阐明输入和输出的相互关系,这种方法在运筹学中起了重要的作用,注意这一句中的主语A black-box and method和谓语plays相隔甚远.

hehe123

Exercise ; R: v/ F+ O8 c/ A

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 ) a* _$ t/ G& L; a' m 5 ^) g* _3 h X4 A) M6 l& N1 K

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

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.

# K4 a/ F3 [* P3 @) R

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.

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.

1 L% Y1 I7 }3 [. h

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

, `! B5 R9 f# B+ R) c

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

2. The Konigsberg Bridge Problem

1 \& ~& m$ ~, v& U9 s! y g

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.

) f+ Q7 @2 V3 g* H. s6 ]. D

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.

1 e6 o7 ?: Y4 c3 i- |' ^) w* h

; ~1 x' f1 A; } 5 f+ @$ }( t w3 O$ d4 {0 P. m" d% L

3 `# v2 E* ], ~ G4 v

) m1 d' j' c" g

- o5 ~4 Q' I& Z

. u4 N/ }0 }4 E4 T

Fig .1 The city of Konigsberg

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.

m1 W9 L' F3 l7 X( x* G

/ ~5 f- U6 C9 F! L5 s* v. t

# q9 l, j, S6 L# b6 y5 e

2 y" p$ \2 U2 s4 ]

4 e! z; |4 h9 A ?0 h' h7 R

Fig .2 A mathematical model for the Konigsberg bridge problem

) U. z' S$ L3 z5 l3 i! B( I% G

' v: M: E: r2 E* H( ^# |

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.

: F! j$ I" I6 |, A2 @: z

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.