数学专业英语[1]－The Real Number System

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Vocabulary 4 k$ X" s% O8 w# r! j0 i

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

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Notes y1 {' p+ J' H. h9 y& {" M( U' p- I

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。

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Exercise

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)

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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.

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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.”

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.”

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.

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----Weaver, W.

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Vocabulary ( B* M V6 k; c, \

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

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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” .

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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）

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数学专业英语－The Role of Mathematics in Economics % z! ?8 [ o- m* r

Economics is a mathematical discipline. This assertion may seem strange to the traditional political economist, but mathematical methods were introduced at an early stage (Cournot,1838) in the two-hundred-year history of our subject and have been steadily growing in significance .At the present time and, essentially, since the end of World WarⅡ,mathematical methods have become predominant in American economics. The mathematical approach was originally inspired in Europe and England but it has flowered in America, with no little stimulus from European immigrants. The mathematical approach is steadily gaining favor throughout the world, especially because the younger generation in developing economics is embracing the new methods and because the socialist countries have shed a previous bias against the use of mathematical methods in economics. It is clear that the future development of economics will see continued and increasing use of mathematics, although it would be rash to assume that the future course of economic analysis will be predominantly mathematical as it has been in the last twenty years.

The Economic Problem " L" R$ F' d1 o/ P; b) L 2 `, w4 K7 ?: K6 ^

A favored definition of economics (Lionel Robbins, 1932) is “…the science which studies human behavior as a relationship between ends and scarce means which have alternative uses.” Whether or not we accept this definition as bracketing all of economics, it is a good starting point for our discussion of the role of mathematics .I might want to sharpen this definition by noting that economists try to select among alternative uses of scarce resources in such a way as to make the most efficient (or lease wasteful) employment of resources to achieve stated ends.

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Stated in this way, we see clearly that economics involves optimization, and this is the engine that produces principles of economic analysis. We have either a maximum problem or a minimum problem, which is a compelling reason for the use of mathematics. An abstract economy is viewed as consisting of numerous consuming and producing units, who make optimal decisions about their own economic behaviour, given market prices, and then interact with one another to clear supply and demand in markets to determine prices.

Economic theory usually begins with an analysis of the individual consumer who attempts to maximize his satisfaction, subject to a budget constraint (or to minimize budget outlays for the attainment of any given level of satisfaction).The theory then takes up the analysis of producers who strive to maximize profits, Subject to a technological constraint (or minimize cost for reaching a given output level, subject to a technological constraint). These are the typical optimization problems of economics.

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The standard mathematical formulations of these problems are as follows. The consumer problem is to maximize a utility function

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of quantities of goods and services consumed, subject to the requirement of living within a fixed income

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where are the respective prices of the goods and services consumed. The producer problem is to maximize income minus production costs

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where are inputs of factors of production and are their costs, subject to the restraint imposed by a technical production function

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of the quantities of goods and services produced and of the production factor inputs .

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These two formulations pose the economic problem as the maximization of utility (satisfaction), subject to a budget constraint, and the maximization of profit, subject to a technologicsi constraint.We could also formulate minimum problems that seek minimum production costs for producing a given combination of outputs and the least-cost budget to achieve a given level of utility.

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Treatment of optimization problems

The consequences of these maximization or minimization problems have been enormous for economics in building a set of rules of behavior. Nearly all economic truths have some root in these or closely related propositions. The original mathematical attack was quite straightforward. Assume that and are smooth continuous functions (with first and second deriva tives), and optimize according to the rules of the differential calculus, given market prices. The necessary and sufficient conditions for optimization define the well –known demand and supply functions of economics of economics and establish many properties of these functions.

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For the problem as I have stated it, these solutions are well established and have been in the literature of economics for more than fifty years. Refined points are made from time to time but the ramifications of this theory were made clear in mathematical treatments by Pareto (1896), Slutsky (1915), Fisher (1892), Hotelling (1932), Frisch (1932), Hicks and Allen (1943), Samuelson (1947).

In the 1930's, and again after World War Ⅱ, these problems received extended mathematical treatment ,The extensions were to optimize over time either continuously or in finite incremental periods and to enlarge the number of side conditions. In stochastic models (i.e., those that incorporate chance), uncertainty about future conditions such as price can be introduced. Also, we can allow for the accumulation of tiny neglected factors that always influence human decisions.

The subjective nature of the utility function, led to analysis of conditions in which the results of optimization would be invariant under transformations of the function and to study of the possibility of deriving a utility function, starting from objective demand functions. The latter problem became known as the integrability problem.

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It may be remarked that the early development of mathematical economics followed the steps of physics and engineering. There are many analogies between the classical methods of mathematical economics and the laws of mechanics, thermodynamics, and similar branches of science. In some cases, there was a tendency to draw strict analogies that could hardly be rationalized in terms of economic behavior.

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An idea that received much encouragement from J. Von Neumann was that mathematical economics should draw upon different branches of mathematics that were more suited to the peculiar nature of the economic problem and economic variables. It was even suggested that new mathematical methods might be developed that would be tailored to economics .In the sense that mathematicians of the eighteenth and nineteenth centuries developed methods that were suited to the problems of physics, we might hope that modern mathematicians would receive inspiration from problems of economics, and social sciences generally. To some extent, this development has occurred in linear programming and optimization theory for situations in which the ordinary methods of differential calculus do not apply. It is up to the mathematicians themselves, however, to decide the significance of this line of development in modern mathematics.

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----------Lawrence R.Klein

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Vocabulary

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predominant 主导的

economic analysis 经济分析

scarce resource 不充足资源

outlay 开支、费用

income 收入

proposition 命题

smooth 光滑

increment 增量

incremental 增量的

stochastic 随机的

derive 推出

thermodynamics 热动力学

rationalize 合理化

market price 市场价格

supply 供应，供给

demand 需求

budget 预算

budget outlay 预算开支

profit 利益，利润

cost 成本

goods 货物

services 服务

ramification (of the theory ) 理论的）细节

side condition 附属条件

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Notes

1. 象two-hundred-year ( 两百年)这样的复合词，year 不用复数。例如：Five-year-plan (五年计划)

2. The mathematical approach was originally inspired in Europe and England but it has flowered in America with no little stimulus from European immigrants.

意思是：这种数学方法创于欧洲大陆和英国，但是已经在美洲（美国）开花，当然少不了欧洲移民的激励。这里flower作为动词用；而且 with no little stimulus 是一种肯定语气。

3. Whether or not we accept this definition as bracketing all of economics, it is a good starting point for our discussion of the role of mathematics.

意思是：不管我们是否接受这个定义作为概括所有经济学，它都是我们用讨论数学（用于经济学）作用的一个良好起点，这里bracketing 作为“概括”解.

4. An abstract economy is viewed as … who make optimal decisions about their own economic behaviour, given market prices, and then interact with one another to clear supply and demand in markets to determine prices.

意思是：抽象经济可以看成由许多个消费和生产单位所组成，这些单位（的决策者）就他们自己的经济行为——给出市场价格——作出最优决策，然后相互去特约市场的供需交换，以便确定价格。这里

1） who 可理解为units 的决策人的关系代词

2） given market prices 是economic behaviour的同位语。

3） one another 是指units 之间，而不是指market prices 之间

4） clear 这一词用于商业上其意思是：“卖光，买光，交换，清理”等。

5. New mathematical methods might be developed that would be tailored to economics.

tailor 是“裁缝”的意思，这里作动词用，意思是：“使其适用于经济学”

6. Up to 作“取决于”解。