数学专业英语[1]－The Real Number System

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Notes

1. If U = { u ₁ , u ₂ ,…}is a collection of points in a linear

space V , then the (linear) span of the set U is the set of all points of the form ∑ c _i u _{i , w}where c _i∈ K ,and all but a finite number of scalars c _I are 0.

意思是:如果 U = { u ₁ , u ₂ ,…}是线性空间 V 的点集,那么集 U 的(线性)生成是所有形如 ∑ c _i u _i 的点集,这里 c _i ∈ K ,且除了有限个 c_i 外均为0.

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2. A finite set { u ₁ , u ₂ ,…, u _k} is said to be linearly independent if the only way to write 0 = ∑ c _i u _I is by choosing all the c _i= 0.

这一句可以用更典型的句子表达如下: A finite set { u ₁, u ₂ ,…, u _k } is said to be linearly independent in V if ∑c _i u _i is by choosing all the c _i = 0.

这里independent 是形容词,故用linearly修饰它. 试比较F(x) is a continuous periodic function.这里periodic 是形容词但它前面的词却用continuous 而不用continuously,这是因为continuous 这个词不是修饰periodic而是修饰作为整体的名词periodic function.

3. Then these define isomorphisms of V onto Kⁿ and W onto K^M respectively, and these in turn induce a linear transformation A between these.

这里第一个these代表前句的两个基(basis);第二个these代表isomorphisms;第三个these代表什么留给读者自己分析.

4. The least satisfactory aspect of linear algebra is still the theory of determinants-

意思是:线性代数最令人不满意的方面仍是有关行列式的理论.least satisfactory 意思是:最令人不满意.

5. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.

意思是:如果方阵是三角形的,即所有在主对角线上方的元素均为零,那末det( A ) 刚好就是对角线元素的乘积.这里meaning that 可用that is to say 代替,turns out to be解为”结果是

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Exercise

I. Answer the following questions:

1. How can we define the linear independence of an infinite set?

2. Let T be a linear transformation (T: V → W ) whose associated matrix is A.Give a criterion for the non-singularity of the transformation T.

3. Where is the entry a₄₅ of a m -by- n matrix( m>4; n>5) located ?

4. Let A , B be two rectangular matrices.Under what condition is the product matrix well-defined ?

II.Translate the following two examples and their proofs into Chinese:

1.Example1. Let u_k= t^k ,k=0,1,2,... and t real. Show that the set ｛u ₀,u₁,u₂,…｝ is independent.

Proof: By the definition of independence of an infinite set, it suffices to show that for each n ,the n+1 polynomials u₀,u₁,...,u_n are independent.A relation of the form ∑ⁿ_k=0 c_ku_k=0 means ∑ⁿ_k=0 c_k t^k=0 for all t.When t=0,this gives c₀ =0.Differentiating both sides of ∑ⁿ_k=0 c_k t^k =0 and setting t=0,we find that c₁=0.Repeating the process,we find that each cocfficient is zero

2. Example 2. Let V be afinite dimensional linear space, Then every finite basis for V has the same number of elements.

Proof: Let S and T be two finite bases for V. Suppose S consists of k elemnts and T consists of m elements.Since S is independent and spans V ,every set of k+1 elements in V is dependent.Therefore every set of more than k elements in V is dependent. Since T is an independent set , we must have m<k. The same argument with S and T interchanged shows that k<m. Hence k=m.

III.Translate the following sentences into English:

1.设 A 是一矩阵。若其行数 n 等于其列数 m ，则称 A 是一方阵；若n ≠m，则称A 是一矩形阵。

2.设 T 是一线性变换，则 T 的特征值刚好是多项式 P（λ）=det（T- λE ）的根。

3.形如Σⁿ_k=1c_ik x_k =c_i i=1，2，...,m 的一组方程称为 m 个线性方程，n 个未知数的方程组。若所有 c_i=0，则称上述方程组为一齐次方程组。

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数学专业英语－Statistics $ d% J" o+ B' q7 { 7 W. @: z4 P R7 O1 o

The term statistics is used in either of two senses.In common parlance it is generally employed synonymously with the word data.Thus someone may say that he has seen”statistics of industrial accidents in the United States.” It would be conducive to greater precision of meaning if we were not to use statistics in this sense,but rather to say “data (or figures ) of industrial accidents in the United States.”

“Statistics” also refers to the statistical principles and methods which have been developed for handling numerical data and which form the subject matter of this text.Statistical methods,or statistics, range form the most elementary descriptive devices, which may be understood by anyone , to those extremely complicated mathematical procedures which are comprehended by only the most expert theoreticians.It is the purpose of this volume not to enter into the highly mathematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.

Statistics may be defined as the collection, presentation, analysis, and interpretation of numerical data.The facts which are dealt with must be capable of numerical expression.We can make little use statistically of the information that dwellings are built of brick, stone, wood, and other materials; however, if we are able to determine how many or what proportion of,dwellings are constructed of each type of material, we have numerical data suitable for statistical analysis.

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Statistics should not be thought of as a subject correlative with physics, chemistry, economics, and sociology. Statistics is not a science; it is a scientific method. The methods and procedures which we are about to examine constitute a useful and often indispensable tool for the research worker. Without an adequate understanding of statistics, the investigator in the social sciences may frequently be like the blind man groping in a dark closet for a black cat that isn’t there. The methods of statistics are useful in an ever---widening range of human activities, in any field of thought in which numerical data may be had.

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In defining statistics it was pointed out that the numerical data are collected, presented, analyzed, and interpreted. Let us briefly examine each of these four procedures.

COLLECTION Statistical data may be obtained from existing published or unpublished sources, such as government agencies, trade associations, research bureaus, magazines, newspapers, individual research workers, and elsewhere. On the other hand, the investigator may collect his own information, going perhaps from house to house or from firm to firm to obtain his data. The first-hand collection of statistical data is one of the most difficult and important tasks which a statistician must face. The soundness of his procedure determines in an overwhelming degree the usefulness of the data which he obtains.

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It should be emphasized, however, that the investigator who has experience and good common sense is at a distinct advantage if original data must be collected. There is much which may be taught about this phase of statistics, but there is much more which can be learned only through experience. Although a person may never collect statistical data for his own use and may always use published sources, it is essential that he have a working knowledge of the processes of collection and that he be able to evaluate the reliability of the data he proposes to use. Untrustworthy data do not constitute a satisfactory base upon which to rest a conclusion.

It is to be regretted that many people have a tendency to accept statistical data without question. To them, any statement which is presented in numerical terms is correct and its authenticity is automatically established.

PRESENTATION Either for one’s own use or for the use of others, the data must be presented in some suitable form. Usually the figures are arranged in tables or presented by graphic devices.

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ANALYSIS In the process of analysis, data must be classified into useful and logical categories. The possible categories must be considered when plans are made for collecting the data, and the data must be classified as they are tabulated and before they can be shown graphically. Thus the process of analysis is partially concurrent with collection and presentation.

There are four important bases of classification of statistical data: (1) qualitative, (2) quantitative, (3) chronological, and (4) geographical, each of which will be examined in turn.

Qualitative When, for example, employees are classified as union or non—union, we have a qualitative differentiation. The distinction is one of kind rather than of amount. Individuals may be classified concerning marital status, as single, married, widowed, divorced, and separated. Farm operators may be classified as full owners, part owners, managers, and tenants. Natural rubber may be designated as plantation or wild according to its source.

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Quantitative When items vary in respect to some measurable characteristics, a quantitative classification is appropriate. Families may be classified according to the number of children. Manufacturing concerns may be classified according to the number of workers employed, and also according to the values of goods produced. Individuals may be classified according to the amount of income tax paid.

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Chronological Chronological data or time series show figures concerning a particular phenomenon at various specified times. For example, the closing price of a certain stock may be shown for each day over a period of months of years; the birth rate in the United States may be listed for each of a number of years; production of coal may be shown monthly for a span of years. The analysis of time series, involving a consideration of trend, cyclical period (seasonal ), and irregular movements, will be discussed.

In a certain sense, time series are somewhat akin to quantitative distributions in that each succeeding year or month of a series is one year or one month further removed from some earlier point of reference. However, periods of time—or, rather, the events occurring within these periods—differ qualitatively from each other also. The essential arrangement of the figures in a time sequence is inherent in the nature of the data under consideration.

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Geographical The geographical distribution is essentially a type of qualitative distribution, but is generally considered as a distinct classification. When the population is shown for each of the states in the United States, we have data which are classified geographically. Although there is a qualitative difference between any two states, the distinction that is being made is not so much of kind as of location.

The presentation of classified data in tabular and graphic form is but one elementary step in the analysis of statistical data. Many other processes are described in the following passages of this book. Statistical investigation frequently endeavors to ascertain what is typical in a given situation. Hence all type of occurrences must be considered, both the usual and the unusual.

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In forming an opinion, most individuals are apt to be unduly influenced by unusual occurrences and to disregard the ordinary happenings. In any sort or investigation, statistical or otherwise, the unusual cases must not exert undue influence. Many people are of the opinion that to break a mirror brings bad luck. Having broken a mirror, a person is apt to be on the lookout for the unexpected”bad luck “ and to attribute any untoward event to the breaking of the mirror. If nothing happens after the mirror has been broken, there is nothing to remember and this result (perhaps the usual result )is disregarded. If bad luck occurs, it is so unusual that it is remembered, and consequently the belief is reinforced. The scienticfic procedure would include all happenings following the breaking of the mirror, and would compare the “resulting” bad luck to the amount of bad luck occurring when a mirror has not been broken.

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Statistics, then, must include in its analysis all sorts of happenings. If we are studying the duration of cases of pncumonia, we may study what is typical by determining the average length and possibly also the divergence below and above the average. When considering a time series showing steel—mill activity, we may give attention to the typical seasonal pattern of the series, to the growth factor( trend) present, and to the cyclical behaviour. Sometimes it is found that two sets of statistical data tend to be associated.

Occasionlly a statistical investigation may be exhaustive and include all possible occurrences. More frequently, however, it is necessary to study a small group or sample. If we desire to study the expenditures of lawyers for life insurance, it would hardly be possible to include all lawyers in the United States. Resort must be had to a sample;and it is essential that the sample be as nearly representative as possible of the entire group, so that we may be able to make a reasonable inference as to the results to be expected for an entire population. The problem of selecting a sample is discussed in the following chapter.

Sometimes the statistician is faced with the task of forecasting. He may be required to prognosticate the sales of automobile tires a year hence, or to forecast the population some years in advance. Several years ago a student appeared in summer session class of one of the writers. In a private talk he announced that he had come to the course for a single purpose: to get a formula which would enable him to forecast the price of cotton. It was important to him and his employers to have some advance information on cotton prices, since the concern purchased enormous quantities of cotton. Regrettably, the young man had to be disillusioned. To our knowledge, there are no magic formulae for forecasting. This does not mean that forecasting is impossible; rather it means that forecasting is a complicated process of which a formula is but a small part. And forecasting is uncertain and dangerous. To attempt to say what will happen in the future requires a thorough grasp of the subject to be forecast, up-to-the-minute knowledge of developments in allied fields, and recognition of the limitations of any mechanica forecasting device.

INTERPRETATION The final step in an investigation consists of interpreting the data which have been obtained. What are the conclusions growing out of the analysis? What do the figures tell us that is new or that reinforces or casts doubt upon previous hypotheses? The results must be interpreted in the light of the limitations of the original material. Too exact conclusions must not be drawn from data which themselves are but approximations. It is essential, however, that the investigator discover and clarify all the useful and applicable meaning which is present in his data.

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Vocabulary

Statistics统计学 in tables 列成表

Statistical 统计的 tabular列成表的

Statistical data统计数据 sample样本

Statistical method统计方法 inference推理，推断

Original data原始数据 reliance信赖

Qualitative定性的 forecasting预测

Quantitative定量的 in common parlance按一般说法

Chronological年代学的 conducive有帮助的

Time series时间序列 grope摸索

Cyclical循环的 akin to类似

Period周期 apt to易于

Periodic周期的 undue不适当的

Prognosticate预测 sociology社会学

Authenticity可信性，真实性 phase相位；方面

Synonymously同义的 categories范畴，类型

Correlative相互关系的，相依的 concurrent会合的，一致的，同时发生的

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Notes ; ?3 c% _ l) J1 K/ c

1. It is the purpose of this volume not to enter into the highly mathematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.

意思是：本书的目的并不是要深入到这个论题的有关高深的数学与理论的方面，而是要讨论它的更为初等和更为常用的方面，not…but rather 意思是“不是…而是”，而rather than意思是“宁愿…而不”，两者意思相近但有差别（主要表现为强调哪方面的差别）。

2. Without an adequate understanding of statistics, the investigator in the social sciences may frequently be like the blind man groping in a dark closet for a black cat that isn’t there.

意思是：没有对统计学的适当的理解，社会科学的研究人员就会常常像一个盲人在一个暗室里摸索一只并不存在的黑猫一样，这里investigator 指一般研究人员，请比较与researcher，scientist等词的异同。

3. It is essential that he have a working knowledge of the processes of collection and that he be able…

这里essential是虚拟形容词，所以后面用he have；he be able…have和be前面省去了should。

4. The distinction is one of kind rather than of amount.

意思是：这种区别是类型的而不是数量的。

5. …the distinction that is being made is not so much of kind as of location.

参看第二课注2关于not so much…as的解释，并比较注4和注5两句。

6. （To attempt to say what will happen in the future） requires…of any mechanical forecasting device.

括号里的部分为此句的主语，谓语是requires， 后面是三个并列宾语短语。

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Exercise

I. Translate the following passages into Chinese:

1. Statistical methods, or statistics, range from the most elementary descriptive devices which may be understood by anyone, to those extremely complicated mathematical procedures which are comprehended by only the most expert theoreticians.

2. Although a person may never collect statistical data for his own use and may always use published sources, it is essential that he have a working knowledge of the processes of collection and that he be able to evaluate the reliability of the data he proposes to use.

II Answer the following questions according to the content of this lesson.

1. In defining statistics, what are the four procedures about the numerical data?

2. How are the numerical data presented usually?

3. Give examples for qualitative differentiation and quantitative classification of statistical data .

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Linear Programming is a relatively new branch of mathematics.The cornerstone of this exciting field was laid independently bu Leonid V. Kantorovich,a Russian mathematician,and by Tjalling C,Koopmans, a Yale economist,and George D. Dantzig,a Stanford mathematician. Kantorovich’s pioneering work was motivated by a production-scheduling problem suggested by the Central Laboratory of the Leningrad Plywood Trust in the late 1930’s. The development in the United States was influenced by the scientific need in World War II to solve logistic military problems, such as deploying aircraft and submarines at strategic positions and airlifting supplies and personnel.

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The following is a typical linear programming problem:

A manufacturing company makes two types of television sets: one is black and white and the other is color. The company has resources to make at most 300 sets a week. It takes $180 to make a black and white set and $270 to make a color set. The company does not want to spend more than $64,800 a week to make television sets. If they make a profit of $170 per black and white set and $225 per color set, how many sets of each type should the company make to have a maximum profit?

This problem is discussed in detail in Supplementary Reading Material Lesson 14.

Since mathematical models in linear programming problems consist of linear inequalities, the next section is devoted to such inequalities.

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Recall that the linear equation lx+my+n=0 represents a straight line in a plane. Every solution (x,y) of the equation lx+my+n=0 is a point on this line, and vice versa.

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An inequality that is obtained from the linear equation lx+my+n=0 by replacing the equality sign “=” by an inequality sign < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to) is called a linear inequality in two variables x and y. Thus lx+my+n≤0, lx+my+n≥0 are all linear liequalities. A solution of a linear inequality is an ordered pair (x,y) of numbers x and y for which the inequality is true.

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EXAMPLE 1 Graph the solution set of the pair of inequalities

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SOLUTION Let A be the solution set of the inequality x+y-7≤0 and B be that of the inequality x-3y +6 ≥0 .Then A∩B is the solution set of the given pair of inequalities. Set A is represented by the region shaded with horizontal lines and set B by the region shaded with vertical lines in Fig.1. Therefore the crossed-hatched region represents the solution set of the given pair of inequalities. Observe that the point of intersection (3.4) of the two lines is in the solution set.

Generally speaking, linear programming problems consist of finding the maximum value or minimum value of a linear function, called the objective function, subject to some linear conditions, called constraints. For example, we may want to maximize the production or profit of a company or to maximize the number of airplanes that can land at or take off from an airport during peak hours; or we may want to minimize the cost of production or of transportation or to minimize grocery expenses while still meeting the recommended nutritional requirements, all subject to certain restrictions. Linear programming is a very useful tool that can effectively be applied to solve problems of this kind, as illustrated by the following example.

EXAMPLE 2 Maximize the function f(x,y)=5x+7y subject to the constraints

x≥0 y≥0

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x+y-7≤0

2x-3y+6≥0

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SOLUTION First we find the set of all possible pairs(x,y) of numbers that satisfy all four inequalities. Such a solution is called a feasible sulution of the problem. For example, (0,0) is a feasible solution since (0,0) satisfies the given conditions; so are (1,2) and (4,3). / ^1 R1 O3 Z6 J/ H " G0 c. t, y, @

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Secondly, we want to pick the feasible solution for which the given function f (x,y) is a maximum or minimum (maximum in this case). Such a feasible solution is called an optimal solution. ; P* @7 ^2 { y- b) n1 A3 ~& S

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Since the constraints x ≥0 and y ≥0 restrict us to the first quadrant, it follows from example 1 that the given constraints define the polygonal region bounded by the lines x=0, y=0,x+y-7=0, and 2x-3y+6=0, as shown in Fig.2.

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Fig.2.

Observe that if there are no conditions on the values of x and y, then the function f can take on any desired value. But recall that our goal is to determine the largest value of f (x,y)=5x+7y where the values of x and y are restricted by the given constraints: that is, we must locate that point (x,y) in the polygonal region OABC at which the expression 5x+7y has the maximum possible value.

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With this in mind, let us consider the equation 5x+7y=C, where C is any number. This equation represents a family of parallel lines. Several members of this family, corresponding to different values of C, are exhibited in Fig.3. Notice that as the line 5x+7y=C moves up through the polygonal region OABC, the value of C increases steadily. It follows from the figure that the line 5x+7y=43 has a singular position in the family of lines 5x+7y=C. It is the line farthest from the origin that still passes through the set of feasible solutions. It yields the largest value of C: 43.(Remember, we are not interested in what happens outside the region OABC) Thus the largest value of the function f(x,y)=5x+7y subject to the condition that the point (x,y) must belong to the region OABC is 43; clearly this maximum value occurs at the point B(3,4).

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Fig.3.

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Consider the polygonal region OABC in Fig.3. This shaded region has the property that the line segment PQ joining any two points P and Q in the region lies entirely within the region. Such a set of points in a plane is called a convex set. An interesting observation about example 2 is that the maximum value of the objective function f occurs at a corner point of the polygonal convex set OABC, the point B(3,4).

The following celebrated theorem indicates that it was not accidental.

THEOREM (Fundamental theorem of linear programming) A linear objective function f defined over a polygonal convex set attains a maximum (or minimum) value at a corner point of the set.

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We now summarize the procedure for solving a linear programming problem:

1. Graph the polygonal region determined by the constraints.

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2. Find the coordinates of the corner points of the polygon.

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3. Evaluate the objective function at the corner points.

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4. Identify the corner point at which the function has an optimal value.

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Vocabulary

linear programming 线形规划 quadrant 象限

objective function 目标函数 convex 凸的

constraints 限制条件，约束条件 convex set 凸集

feaseble solution 容许解，可行解 corner point 偶角点

optimal solution 最优解 simplex method 单纯形法

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Notes

1. A Yale economist, a Stanford mathematician 这里Yale Stanford 是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)

2. subject to some lincar conditions 解作“在某些线形条件的限制下”。试比较下面各词组在用法上的异同： subject to; under the condition(s) of; satisfying the condition(s).

3. celebrated theorem 意思为“著名定理”。

4. Since…it follows…与Notice…it follow…都是数学中常用的句型，以表达“根据什么，可得什么”这一意思，请参看附录III。

5. 本课多次用到recall, observe, notice, remember等词，用以提醒读者一些已知的事实或定理，读者可从这些例句中体会这些词的用法。请参看附录III。

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Exercise

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I.Translate the following passage into Chinesre:

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Although satisfactory for two-variable linear programs, the corner-point method is not computationally effective when extended beyond three-variable situations. Its usefulness is restricted to introducing the general idea of linear programming, and the need for more poweful solution techniques is obvious. The simplex method, devised by George Dantzig in the late 1940’s, was, central to the explosion in linear programming applications that occurred in the 1950’s and 1960’s. Although more powerful techniques exist for solving special problems, the simplex method is still the most computationally effective general method available for solving the widest variety of linear programming problems.

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II.Read the content of this lesson carefully and complete the following sentences:

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1. Linear programming problems consist of

2. The set of feasible solutions of a linear programming problem is

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3. A polygonal region is a region bounded by

4. A convex set in a plane is . \# }" J1 }- u E# C

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III.Translate the following sentences into English:

1. 点(3,4)是两条直线x+y-7=0和2x-3y+6=0的交点。

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2. (3,4)是例2的最优解。

3. 计算目标函数在偶角点处的值，然后进行比较，求出目标函数的最大值或最小值。

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