数学专业英语[1]－The Real Number System

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数学专业英语－The Role of Mathematics in Economics 1 x( Z) X4 C) m, l7 f6 P

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.

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The Economic Problem 6 F) }( k' j/ y- [' Z0 j) u7 Y5 } Q

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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.

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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.

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

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

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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.

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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.

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

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。

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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 + o+ T8 F6 E* d L; l; o1 ~ 2 G* D4 j' n8 P$ \ e& F$ b

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.

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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:

HOW TO SOLVE IT

First: You have to understand the problem.

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.

Third: Carry out your plan.

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Fourth: Examine the solution obtained.

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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:

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

Work backwards

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Work forwards

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Narrow the condition

Widen the condition

Seek a counter example

Guess and test

Divide and conquer

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 7 B* n% u/ E+ P, t2 x

Analysis

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1) Draw a diagram if at all possible

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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

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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

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a) exploiting symmetry, or

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

Exploration - g4 F; k' U! H

1) Consider essentially equivalent problems:

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a) Replacing conditions by equivalent ones.

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

c) Introduce auxiliary elements.

d) Re-formulate the problem by

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I) change of perspective or notation

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II) considering argument by contradiction or contrapositive

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

2) Consider slightly modified problems:

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

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

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

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3) Consider broadly modified problems:

a) Construct an analogous problem with fewer variables .

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

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c) Try to exploit any related problems which have similar

I) form

II) “givens”

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III) conclusions

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

Verifying your solution , e+ w5 V8 D( }

1) Does your solution pass these specific tests:

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a) Does it use all the pertinent data?

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b) Does it conform to reasonable estimates or predictions?

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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?

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

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Vocabulary - h1 }& p6 A2 L$ W' g. e6 s" ~5 L

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”。

lipu_2003

好的，谢谢了

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lipu_2003

好的，谢谢了，你真伟大，有用

lipu_2003

好的，谢谢了，你真伟大，有用

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你要是发成文档就更好了

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