数学专业英语[1]－The Real Number System

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数学专业英语－Groups and Rings . p+ h* W$ k# s) @6 i" D1 Z ( `1 L! y$ V5 z/ h+ S0 t 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. ; R0 j2 U% K) T9 b5 Z4 a, @ # e: Z8 _- _1 q ~9 |' s, \: _ * Q: e, s2 F0 H* {! `6 \- ~% W0 b ; p0 y/ w+ }% R* ~ Groups " c$ J% ^6 Z- ^- z5 c' ~ ! C# w" _+ r( \ 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 5 J' {0 R d: o9 M/ `$ T' u aо(bоc)=(aоb)оc - M" n* L [/ }6 q' u# K) l for every a, b, c of G s; r/ _2 J9 b! ~" _% j( B4 v3 }6 W 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 8 X# U9 |& q/ { For every a and b of G. ' J. X F4 A, F9 o; L0 ] 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. % X3 H! _; ?, Y( f 8 V, C0 ^0 a. n+ V EQUIVALENCE $ A' T3 j/ c1 q+ p 7 |3 E$ N) t* X6 r+ S: `7 M4 n/ _ 6 \ p- M4 O1 w: c5 I p& y , {( f6 k2 H- L8 W8 n* w% N+ ~* S 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 9 B( w5 C* S% ]! w S : a a´ (a in G and a´ in G´) " @5 E8 q% }6 C; Q+ t7 H( s between G and G´ such that (aοb)´=a´ο b´ 2 Z2 V: a( K3 u$ P 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. , M2 D; b, @) h Z/ u 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 a 7 @; ~; E [% N9 E Of G, but other automorphisms may also exist. ; A" B5 n7 y3 [; c, z6 h4 D- F0 k+ p Rings + N. p7 A# ~4 a( E8 j ) x; ^; S5 E. k1 ^4 L5 W, q% A 9 Q' c* O8 V# x _& T4 I( E A ring is an additive abelian group B such that 2 ]3 v3 J0 o: v7 m0 } 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 ) Y3 p& l2 o, R1 j for every a, b, c of B. Ⅲ. The distributive laws a (b+c) = ab +ac (b+c) a=ba +ca 9 c& L5 j+ {2 y+ K hold for every a, b, c of B. The concept of equivalence again arises. We shall write ( z( c b- `! Q B ≌ B′ 6 D' O7 ]& H+ c5 ?! g ( a+ y# z# d$ [: F2 r) c, ] to mean that B and B′are equivalent. ; Y5 R+ b8 {% A! `2 G: B( }* w L6 Y( T 1 X- X, Q# |! B) l( p1 |/ E

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Vocabulary ; d' B4 u* W5 |8 V v

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

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The Two Basic Concepts of Calculus $ O2 c5 c! i- k9 O# F5 A2 c 0 \5 A, J# V5 x+ A

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.

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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?

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These examples,chosen from various fields,illustrate some of the technical questions that can be answered by more or less routine applications of calculus.

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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.

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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.

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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.

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Historical Background " S% S# g: C9 ~. f/ g% I

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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.

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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.

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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.

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Fig.2. The method of exhaustion applied to a semicircular region.

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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.

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Vocabulary h3 T1 V. c. M$ p* W

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

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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.

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Table 1.Distribution of stature in 117 males

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Absolute ( `! ^, w. l/ G+ x0 p9 x Height (m)	Working " g% Q% E6 ]2 ~" y' `# O, }9 m measurements ) X- y$ I! P: r. W4 {0 T* S with origin at 1.70(x)	# |: [- i- m! h9 U7 w Number of men 9 f T; O } q. `# \. y1 B observed(f)	Contributions 2 ^* d6 {* B: b to the sum (f x )	: q# i0 k0 @; ]* \ Contributions 6 [/ x4 b3 y; B! P to the sum of squares (f )
1.58 1.60 , C. X. r8 v& Z/ M 1.62 1.64 1.66 1.68 1.70 / V0 X4 O7 o: ]2 X; I: H0 Y 1.72 : |3 @9 Z+ |. i) t& Y. B& o# \1 G 1.74 1.76 ! H" S! s% S: \+ B p% H+ r 1.78 1.80 5 _/ n# j6 v+ Y3 ?" V, c 1.82 1.84	& E- E- x/ _/ e; \! y! m y －6 ) M8 h! R5 s8 N( c, Z+ j －5 －4 －3 6 p; c8 A: f( M% \ －2 －1 0 C5 N- q: Y- f 0 1 ) a. s5 A& t3 e' H* G- \ 2 3 ! K& a* X3 G* `* X y" M0 t 4 5 6 7	1 3 6 8 13 18 19 $ f0 I, y1 D! _8 \+ N! T 14 2 n5 b+ {' n6 i 14 / B8 y) S, N s q 9 , @) ^8 [ ~# C. P& s 5 4 # U# J' S6 ?5 f9 `4 o 2 1	－6 & M; M; Z9 _& E. V) G) i% `# z/ P9 p$ V －15 ( X/ R+ ?$ v- E2 b －24 －24 ; M# c- Z0 I! g －26 , X, \9 ~$ \7 M O+ o －18 8 `6 w' f0 H: x# p 0 ' I; z" Z0 M8 {% a1 C# R( ^- w 14 28 27 , @* N/ l, Y4 e2 a6 R 20 0 k" s( u' f2 x5 x" X% E% Z( p 20 12 ) N, J) |2 b8 ~( j, q* J: J 7	36 ' Y- l; |# V$ V7 s9 S/ s5 L 75 96 72 52 , P2 L: i! h! Q9 u 18 0 14 ( ~+ r1 r# H* S' d5 y K 56 6 ~! J- ? ~3 J- G 81 80 100 72 49
: B) z5 G- {% ^4 M Ｔotals		117	8 o; Y9 f, i( ^ a; [ ＋15	801

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.

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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

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Vocabulary

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

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数学专业英语－Operations Research 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. . N" E+ r2 E! N* T0 ?" |# P 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. 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. # H ?: @! K4 d3 t, V! O! @# R These four points are essential to any operations research, and are implicit in each of the three aforementioned definitions. 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 x& f% t0 `$ S9 w 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. 8 Z& g: n* P) j! F) z+ e% X8 S 0 b# X3 G8 `7 j: n; w: Y t' A; E2 A* Y! v- D8 {4 T. E + X c1 x; l2 b ——From Encyclopedic Dictionary + c8 w2 ?# ]/ Q6 c3 A. f3 z5 m* O of Mathematics 4 ^9 t- @! e o; A( }# a1 x; { 8 ]! v' s. Y9 { 3 f4 m5 v: ]' \3 p# k9 S9 g$ H8 w 8 t; Z" O1 }7 `5 K

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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

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Notes

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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.