数学专业英语[1]－The Real Number System

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Exercise

Ⅰ .Give an example of a typical optimation problem of Economics so as to show that Economics needs mathematics.

Ⅱ. Translate the following passage into Chinese:

Economic analysis has, in the last twenty years, become predominantly mathematical. This is particularly true in the United States, where doctoral candidates now substitute various courses in mathematics for at least some of the traditional foreign language requirement. Economic problems involving optimal decisions by government and business or stable growth of an economy have analogies in problems of physics and engineering that have long been successfully treated mathematically, But economics has outgrown the days when it merely aped the physical sciences in applying mathematics. The author suggests that in the coming era economics may call forth its own branch of mathematics or provide inspiration for great new mathematical discoveries.

Ⅲ. Translate the following sentences into English (make use of the phrase in bracket and see whether one can be replaced by the other or not):

1. 求在下列限制条件下，函数F(x, y) 的最大值。（Subject to ）

2. 设 其中 是一集合， 是实数集，若 满足如下条件：

（Ⅰ） ;(Ⅱ)当且仅当x=y时， ;(Ⅲ)对称性： (Ⅳ)三角不等式： 其中 .则称 是一距离函数.（Satisfy the following condition(s)）.

3.设 是定义在区间Ⅰ的一个连续函数，则在区间Ⅰ是有界闭的假设下，我们可以断言， 在Ⅰ上一致连续（Under the assumption (hypothesis); claim）

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数学专业英语－Polya’s Craft of Discovery . L! e7 p* f. E- L& t" O3 M* X

George Polya has a scientific career extending more than seven decades. Abrilliant mathematician who has made fundamental contributions in many fields. Polya has also been a brilliant teacher, a teacher’s teacher and an expositor. Polya believes that there is a craft of discovery. He believes that the ability to discover and the ability to invent can be enchanced by skillful teaching which alerts the student to the principles of discovery and which gives him an opportunity to practise these principles.

In a series of remarkable books of great richness, the first of which was published in 1945. Polya has crystallized these principles of discovery and invention out of his vast experience, and has shared them with us both in precept and in example.These books are a treasure-trove of strategy, know-how, rules of thumb, good advice, anecdote, mathematical history, together with problem after problem at all levels and all of unusual mathematical interest. Polya places a global plan for “How to Solve It” in the endpapers of his book of that name:

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HOW TO SOLVE IT

First: You have to understand the problem.

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Second: Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

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Third: Carry out your plan.

Fourth: Examine the solution obtained.

These precepts are then broken down to “molecular” level on the opposite endpaper. There, individual strategies are suggested which might be called into play at appropriate momentsm, such as:

If you cannot solve the proposed problem, look around for an appropriate related problem.

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Work backwards

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Work forwards

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Narrow the condition

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Widen the condition

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Seek a counter example

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Guess and test

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Divide and conquer

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Change the conceptual mode

Each of these heuristic principles is amplified by numerous appropriate examples.

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Subsequent investigators have carried Polya’s ideas forward in a number of ways. A.H.Schoenfeld has made an interesting tabulation of the most frequently used heuristic principles in college-level mathematics. We have appended it here.

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Frequently Used Heuristics

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Analysis

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1) Draw a diagram if at all possible

2) Examine special cases:

a) Choose special values to exemplify the problem and get a “feel” for it.

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b) Examine limiting cases to explore the range of possibilities

c) Set any integer parameters equal to 1,2,3,…,in sequence, and look for an inductive pattern.

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3) Try to simplify the problem by

a) exploiting symmetry, or

b) “Without Loss of Generality” arguments (including scaling)

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Exploration

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1) Consider essentially equivalent problems:

a) Replacing conditions by equivalent ones.

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b) Re-combining the elements of the problem in different ways.

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c) Introduce auxiliary elements.

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d) Re-formulate the problem by

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I) change of perspective or notation

II) considering argument by contradiction or contrapositive

III) assuming you have a solution , and determining its properties

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2) Consider slightly modified problems:

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a) Choose subgoals (obtain partial fulfillment of the conditions)

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b) Relax a condition and then try to re-impose it .

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c) Decompose the domain of the problem and work on it case by case .

3) Consider broadly modified problems:

a) Construct an analogous problem with fewer variables .

b) Hold all but one variable fixed to determine that variable’s impact .

c) Try to exploit any related problems which have similar

I) form

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II) “givens”

III) conclusions

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Remember: when dealing with easier related problems , you should try to exploit both the RESULT and the METHOD OF SOLUTION on the given problem .

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Verifying your solution

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1) Does your solution pass these specific tests:

a) Does it use all the pertinent data?

b) Does it conform to reasonable estimates or predictions?

c) Does it withstand tests of symmetry, dimension analysis , or scaling?

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2) Does it pass these general tests?

a) Can it be obtained differently?

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b) Can it be sudstantiated by special cases?

c) Can it be reduced to known results?

d) Can it be used to generate something you know?

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Vocabulary

craft 技巧

enchance 增强

alert 警觉，机警

precept 箴言，格言

treasure trove 宝藏

anecdote 轶事，趣闻

auxiliary 辅助的

appropriate 适当的

heuristic 启发式的

amplified 扩大，详述

append 附加，追加

exploration 探查，细查

perspective 透视

contrapositive 对换的

relax 放松

decompose 分解

pertinent 适当的

substantiate 证实，证明

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Notes

1．A brilliant mathematician who has made fundamentral contributions in many fields,Polya has also been a brilliant teacher, a teacher’s teacher, and an expositor.

意思是：Polya，一个在许多领域中都作出重要贡献的数学家，也是一位出色的教师，教师的教师和评注家。这里Polya是a brilliant mathematician 的同位语

2．…which alerts the student to the principles of discoveries…

这里alert的意思是：“使机警，使注意”。因此，本句意思是：这种熟练（有技巧的）的教学可使学生机敏地注意到这些发现原则……

3．Polya has crystallized these principles of discoveries out of his vast experience,…

意思是：Polya从他的浩瀚的经验中，把这些发现原则提炼得更加具体而明朗。

4．Rules of thumb以经验为基础的规则，方法。

5．There,individual strategies are suggested, which might be called into play at appropriate moments,such as…

意思是：在那里，提供了许多个别的策略，它们在适当的时刻就会发挥作用，例如……这里call into play意思是：“发挥作用”。

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Exercise 0 C6 [2 t: |4 B4 p& y

I．Translate the following sentences into Chinese ( pay attention to the phrases underlined:

1. Note that a+ib=c+id means a=c and b=d

2. We recall that log z: C－{0} C is an inverse for when is restricted to a strip

3. Notice that if ,angles need not be preserved.

4. To show that the test fails when ,observe that, by elementary analysis, and but diverges while converges.

5. To prove the results of this section, we shall use the techniques developed in the last section.

6. We can deduce, in a way similar to the way we deduced theorem A, the following theorem.

7. We are now in a position to draw important consequences from Cauchy’s theorem.

8. We are now in a position to prove easily an otherwise difficult theorem stating that any polynomial of degree n has a root.

9. Unless otherwise specified (stated), curves will always be assumed to be continuous and piecewise differentiable.

10. We shall prove a theorem that appears to be elementary and that the student has, in the past, taken for granted.

11. The solution to this differential equation is unique up to the addition of a constant.

12. The function that maps the simply connected domain onto the unit disc is unique up to a Mobius transformation.

II．Translate the following passages into Chinese:

1． If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standing point, not only is this problem frequently more accessible to our investigation ,but at the same time we come into possession of a method which is applicable also to related problems.

2． In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends then, on finding out these easier problems, and on solving them by means of methods as perfect as possible.

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数学专业英语－How to Write Mathematics?

How to Write Mathematics?

------ Honesty is the Best Policy

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The purpose of using good mathematical language is, of course, to make the understanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the sense of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss.

The emphasis in the preceding paragraph, while perhaps necessary, might seem to point in an undesirable direction, and I hasten to correct a possible misinterpretation. While avoiding pedantry and fuss, I do not want to avoid rigor and precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way to get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his sympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation.”

Here is the sort of the thing I mean by less than complete honesty. At a certain point, having proudly proved a proposition P, you feel moved to say: “Note, however, that p does not imply q”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be perfectly pure, but the reader may feel cheated just the same. If he knew all about the subject, he wouldn’t be reading you; for him the nonimplication is, quite likely, unsupported. Is it obvious? (Say so.) Will a counterexample be supplied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean that you have tried to derive q from p, you failed, and you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into your confidence.

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There is nothing wrong with often derided “obvious” and “easy to see”, but there are certain minimal rules to their use. Surely when you wrote that something was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still think that something was obvious? (A few months’ ripening always improves manuscripts.) When you explained it to a friend, or to a seminar, was the something at issue accepted as obvious? (Or did someone question it and subside, muttering, when you reassured him? Did your assurance demonstration or intimidation?) the obvious answers to these rhetorical questions are among the rules that should control the use of “ obvious”. There is the most frequent source of mathematical error: make that the “ obvious” is true.

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It should go without saying that you are not setting out to hide facts from the reader: you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them either. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the important and minimize the trivial. The reason saying that they are obvious is to put them in proper perspecti e for the uninitiated. Even if your saying so makes an occasional reader angry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must obey the rules. Don’t let the reader down; he wants to believe in you. Pretentiousness, bluff, and concealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor themselves, but quite properly, the author. Complete honesty makes for greatest clarity.

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---------Paul R.Haqlmos

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vocabulary

flashy 一闪的 counter-example 反例

unobtrusiveness 谦虚 dictum 断言；格言

forestall 阻止，先下手 deride嘲弄

anticipate 预见 subside沉静

pedantry 迂腐；卖弄学问 mutter出怨言，喃喃自语

fuss 小题大做 intimidation威下

reconcilable 使一致的 rhetorical合符修辞学的

gloss 掩饰 pretentiousness自命不凡

alleviate 减轻，缓和 bluff 欺骗

implication 包含，含意 concealment隐匿

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notes

1. 本课文选自美国数学学会出版的小册子How to write mathematics 中Paul R.Halmos. 的文章第9节

2. The purpose is smooth the reader’ way, to anticipates his difficulties and to forestall them. Clarity is what’s wanted, not pedantry; understanding, not fuss.

意思是：目的是为读者扫清阅读上的障碍，即预先设想读者会遇到什么困难，并力求避免出现这类困难。我们需要的是清晰明了，而不是故弄玄虚。这里fuss的意思是 “小题大做”。Understanding后面省去is what’s wanted以避免重复。

3. I do not mean to advise a young author to be ever so slightly but very very cleverly dishonest and to gloss over difficulties.

意思是：我的意思是青年的作者绝不可有哪怕只是些少，当却是掩饰得非常巧妙得虚伪，我也劝告他们不要去掩饰困难。

4．Here is the sort of thing I mean by than complete honesty.

意思是：这就是我所认为的不够完全诚实的那类事情（东西）。注意：Here is 的意思是：“这里就是---”，然后把要说的事情在随后给出，若用This is the sort of thing---一般是当你把要说的事情已经说了然后用指示代词This来概括所说的事，注意这一区分。

5．In any event: take the reader into your confidence.

意思是：在任何情况，要敢于对读者讲出真相。这里take---into one’s confidence意思是：“对---吐露秘密；把---当成心腹朋友”。

6．Don’t let the reader down..

意思是：不要使读者丧气。这里down是形容词。

7．Complete honesty makes for greatest clarity.

意思是：彻底的诚实就是最大的明嘹。 Make for 是“有助于”的意思。这样简洁而又充满哲理的句子还有 Emphasize the important minimize the trivial.

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Exercise + [1 l; J; @1 T7 w

(Miscellaneous Exercises)

Ⅰ.Fill in each blank with a suitable word.

1. is called the ________ ________of .

2.p (x)= is a_________of_________n.

3. called a __________ _________.

4. is called the________of .

5.The graph of (p>0) is a___________.

6. then is called a________ _________sequence.

7. is a_________of two equations with three_______.

8.Numbers such as and πare called________numbers.

9.The relation between the celements of a set of real numbers denoted

by<(or<;>;>) is called an_________relation.

10.The relation between sets, denoted by is called an_________relation.

Ⅱ.Each ofthe following sentences is grammatically wrong. Correct these sentences.

1. Let is a continuous function defined on[a,b].

2. Differentiating both sides of with respect to x, the equation becomes y’=—x/y

3. Take the derivatives of both sides of the equation ,we get x+yy’=0.

4. The primtive of here C is a constant.

5. We say that has a limit A at if approaches to A when X tends to .

Ⅲ.Translate the following sentences into Chinese (pay attention to the phrases underlined):

1. We are now in a position to prove the main theorem.

2. An analogous argument gives a proof of the corresponding theorem for decreasing functions.

3. An immediate consequence of Bolzano’s theorem is the intermediate-value theorem for continuous functions.

4. We claim that has no real solution, In fact if is a real solution, then we have which is impossible.

5. It is clear that the method described above also applies to the general case.

6. It is easy to show that has derivatives up to order n at the point x=0,where n>1.

Ⅳ.Translate the following passage into Chinese:

1. It is helpful to introduce the words”local”and“global”to contrast two types of situations that frequently arise. If we are considering a given set D, then we say that any specific property holds“locally”at of D if it is tre at if and at all points near ；thus there will be an open ball B apout and the property will hold for all .On the other hand, a property that holds at all points in D is said to hold “globally”in D.

2. The study of sequences is concerned primarily with the following type of question:if each term of a sequence has a certain property, such as continuity, differentiability or integrability, to what extend is this property transferred to the limit function?

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数学专业英语－How to Organize a paper (For Beginers)? + _8 f% L' H) _! g

The usual journal article is aimed at experts and near-experts, who are the people most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There are many good ways of organizing a paper which can be learned by studying papers of the better expositors. The following suggestions describe a standard acceptable style.

Choose a title which helps the reader place in the body of mathematics. A useless title: Concerning some applications of a theorem of J. Doe. A. good title contains several well-known key words, e. g. Algebraic solutions of linear partial differential equations. Make the title as informative as possible; but avoid redundancy, and eschew the medieval practice of letting the title serve as an inflated advertisement. A title of more than ten or twelve words is likely to be miscopied, misquoted, distorted, and cursed.

The first paragraph of the introduction should be comprehensible to any mathematician, and it should pinpoint the location of the subject matter. The main purpose of the introduction is to present a rough statement of the principal results; include this statement as soon as it is feasible to do so, although it is sometimes well to set the stage with a preliminary paragraph. The remainder of the introduction can discuss the connections with other results.

It is sometimes useful to follow the introduction with a brief section that establishes notation and refers to standard sources for basic concepts and results. Normally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.

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The section following the introduction should contain the statement of one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a major theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts.

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Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by J.E.Littlewood, in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed. To achieve conceptual proofs, it is often helpful for an author to adopt an initial attitude such as one would take when communicating mathematics orally (as when walking with a friend). Decide how to state results with a minimum of symbols and how to express the ideas of the proof without computations. Then add to this framework the details needed to clinch the results.

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Omit any computation which is routine (i.e. does not depend on unexpected tricks). Merely indicate the starting point, describe the procedure, and state the outcome.

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It is good research practice to analyze an argument by breaking it into a succession of lemmas, each stated with maximum generality. It is usually bad practice to try to publish such an analysis, since it is likely to be long and uninteresting. The reader wants to see the path-not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice later on.

The rudiments of grammar are important. The few lines written on the blackboard during an hour’s lecture are augmented by spoken commentary, and aat the end of the day they are washed away by a merciful janitor. Since the published paper will forever speak for its author without benefit of the cleansing sponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless appropriate.

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The barbarism called the dangling participle has recently become more prevalent, but not less loathsome. “Differentiating both sides with respect to x, the equation becomes---”is wrong, because “the equation” cannot be the subject that does the differentiation. Write instead “differentiating both sides with respect to x, we get the equation---,” or “Differentiation of both sides with respect to x leads to the equation---”

Although the notion has gained some currency, it is absurd to claim that informal “we” has no proper place in mathematical exposition. Strict formality is appropriate in the statement of a theorem, and casual chatting should indeed be banished from those parts of a paper which will be printed in italics. But fifteen consecutive pages of formality are altogether foreign to the spirit of the twentieth century, and nearly all authors who try to sustain an impersonal dignified text of such length succeed merely in erecting elaborate monuments to slumsiness.

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A sentence of the form “if P,Q” can be understood. However “if P,Q,R,S,T” is not so good, even if it can be deduced from the context that the third comma is the one that serves the role of “then.” The reader is looking at the paper to learn something, not with a desire for mental calisthenics.