数学专业英语－(a) How to define a mathematical term?

hehe123

数学专业英语－(a) How to define a mathematical term?

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数学术语的定义和数学定理的叙述，其基本格式可归纳为似“if…then…”的格式，其他的格式一般地说可视为这一格式的延伸或变形。 ) a3 l. P) J+ G7 [0 h 2 I4 [5 j* I& c9 \: Y1 @ p. u

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如果一定语短语或定语从句，以界定被定义的词，所得定义表面上看虽不是“If……then……”的句型，而实际上是用“定语部分”代替了“If”句，因此我们可以把“定语部分”写成If句，从而又回到“If……then……”的句型。

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至于下面将要叙述的“Let…if…then”，“Let and assume…, If…then…”等句型，其实质也是基本句型“If……then……”的延伸。 * p# `8 p6 F. R/ \1 N9 U+ h0 O. U

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有时，在定义或定理中，需要附加说明某些成份，我们还可在“if…then…”句中插入如“where…”等的句子，加以延伸（见后面例子）。 ! Y9 p8 ]3 l7 C' |

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总之，绝大部分（如果不是全部的话）数学术语的定义和定理的叙述均可采用本附录中各种格式之。

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（a）How to define a mathematical term?

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is defined as ' k# ^7 q; ^, e$ K is called $ ~# e7 f i, e8 m7 m; B

1. Something something

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The union of A and B is defined as the set of those elements which are in A, in B or in both. ! w q* q" |* S7 F0 T% g

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The mapping , ad-bc 0, is called a Mobius transformation.

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is defined to be 5 a6 {3 Z! ^1 X ?+ ?4 ?/ P! z 7 j9 g! @, `3 U3 H is said to be 1 F: b% T: }5 S% o: w+ C) z

2. Something something(or adjective)

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The difference A-B is defined to be the set of all elements of A which are not in B.

A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. * J' ]3 ~4 U: C1 f% W* A

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Real numbers which are greater than zero are said to be positive.

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define 3 Y3 x0 h/ p2 b1 S call , {% a1 Y$ F0 |* i, J

3. We something to be something.

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We define the intersection of A and B to be the set of those elements common to both A and B. $ E, {# q7 T9 R* u5 n7 M+ z

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We call real numbers that are less than zero (to be) negative numbers. 4 ~! D# [# D; C/ G% Y, L- o# j

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4. 如果在定义某一术语之前，需要事先交代某些东西（前提），可用如下形式： : P" w8 I9 w* \4 a5 B

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& v, b+ [; F# C" J is called + f- t a, a" K4 v9 a- | is said to be 6 k |- a% D$ ~ is defined as : z; K) e' V, B 2 T% R9 F% q0 f; R) a/ E is defined to be + v( S) K4 G9 q6 a/ A

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Let…, then… / i. q! B- | C$ m! t

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Let x=( ) be an n-tuple of real numbers. Then the set of all such n-tuples is defined as the Euclidean n-space R.

Let d(x,y) denote the distance between two points x and y of a set A. Then the number ; I6 u9 e2 m+ N4 @! d

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D= 5 g: W; d, b8 k& O& Y2 p

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is called the diameter of A. 1 D$ S; a( Y6 |( u$ H

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5．如果被定义术语，需要满足某些条件，则可用如下形式： % W s9 T* `0 z+ h

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is called 8 r }- i8 g4 ^& H$ Z/ A5 } is said to be ) g" v1 u3 x: A5 f8 q: ~ : |1 C) d1 j( P$ q0 X- \2 t is defined as ; I, i1 A- i( r% ?( T; n is defined to be , ^- B% a- D) J F* h/ e+ q

If…, then… 3 y- p$ K/ v- }! g0 I; k- @

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If the number of rows of a matrix A equals the number of its columns, then A is called a square matrix. . [; ~: l+ G4 O: `1 u

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If a function f is differentiable at every point of a domain D, then it is said to be analytic in D.

6.如果需要说明被定义术语应在什么前提下，满足什么条件，则可用下面形式：

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is called is said to be

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Let Suppose

…. If…then… … " F9 m* x! A2 k

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Let f(z) be an analytic function defined on a domain D (前提条件). If for every pair or points , and in D with , we have f( ) f( ) (直接条件)，then f(z) is called a schlicht function or is said to be schlicht in D.

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hehe123

7. 如果被定义术语需要满足几个条件（大前提，小前提，直接条件）则可用如下形式：

suppose assume

Let…and …. If…then…is called…

Let D be a domain and suppose that f(z) is analytic in D. If for every pair of points and in D with , we have f( ) f( ), then f(z) is called a schlicht function.

Notes:

(a) 一种形式往往可写成另一种形式。

Let{ }be a sequence of sets. If for all n, then{ }is called an ascending or a non-decreasing sequence.

我们可用一定语短语来代替“If”句，使其变为“Let……then”句

Let{ }be a sequence of sets with for all n, then{ }is called an ascending or a non-decreasing sequence.

(b) 注意“Let”，“suppose”（“assume”），“if”的使用次序，一般来说，前面的可用后面的替换，但后面的用前面的替换就不好了，如上面句子可改写为：

Suppose{ }is a sequence of sets. If , then{ }is called an ascending sequence.

Let{ }be a sequence of sets and suppose that then{ }is called an ascending sequence.

但下面的句子是错误的（至少是不好的句子）；

If{ }is a sequence of sets, and let , then{ }is called an ascending sequence.

(c) 在定义一些术语后，往往需要用符号来表达，或者需要对句中某些成份作附加说明，这时我们需要把定义句扩充，扩充的办法是在定义的原有结构中，插入一个由连接词引导的句子，这类连接词或短语经常是“and”，“where”，“in this (that) case”…请参看PARTIA第一课注1和第二课注4、5、6。

If every element of a set A also belongs to another set B, then A is said to be the subset of B, and we write

A real number is said to be a rational if it can be expressed as the ratio of two integers, where the denominator is not zero.

(d) 在定义中，“if”句是关键句，且往往比较复杂，要特别注意在一些定义中，“if”句又有它自己的表达格式，读者对这类句子的结构也要掌握，下面我们以函数极限定义中的“if”句的结构作为例子加以说明：

If for every ＞0, there is (there exists) a ＞0, such that ＜ whenever 0＜ ＜ , then we say f(x) has a limit A at the point a.

上面是函数极限的定义，其中的“if”句是它的典型结构，凡与极限相关的概念，如连续，收敛，一致连续，一致收敛等定义均有类似结构。例：

A sequence of functions { } is said to have the Cauchy property uniformly on a set E if for any ＞0, there is an N such that ＜ whenever n,m＞N.

当然，极限定义还有其他表达形式但基本结构是一样的，只不过对句中某些部分用等价的语法结构互作替换而已。

下面是函数极限定义中“if”句的另一些表达式，读者可把这些句子和原来的句子作比较。

If, given any ＞0, there exists a ＞0, such that ＜ whenever (if,for) 0＜ ＜ ,…

If, corresponding to any ＞0, a ＞0 can be found such that ＜ whenever 0＜ ＜ ,…

If, for every ＞0,there is a ＞0, such that 0＜ ＜ implies ＜ .

hehe123

数学专业英语－（b）How to state a theorem? ; q1 t8 ~) F7 q8 }8 u+ K# U

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定理叙述的格式，基本上与数学术语的定义一样，只不过在术语的定义中，“then”句有比较固定的格式，而定理的“then”句则随其结果而变吧了。 . E0 x% R* V2 n0 M 2 S, j: E; g- }0 N: P; i

1．某些定理可用简单句叙述。

The union of a finite number of closed sets is still a closed set. ( G, R. t+ g# W5 p( @' F4 `, c

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The space (E,f) is complete.

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2. 如果定理的结论是在一定前提下得到的，则可用下面形式：

“Suppose…Then…”or“Let….Then…” , [2 _1 e( A- S5 K, c

Let f(x) be a continuous function defined on[a,b]. Then f(x) attains its maximum and minimum on [a,b].

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Suppose that f(z) is analytic in a simply connected domain D, then for any closed simple curve C lying within D, we have

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3. 如果定理的结论在一定假设条件下成立，则可用下面的形式 ! F: [% T% ]8 u4 y/ J

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“If…, then…” 9 ~3 x. T; d( f% F* Z6 E* Z

If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 2 W! f9 X( e6 M' C4 W

4. 如果定理的结论除了在一定条件下，还需在一定前提下才成立，这时可用如下形式

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“Let…. If…,then…”or

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“Suppose…. If…,then…” # Q8 ~0 j6 B' u4 n( e2 E2 {. u4 _

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. & X- }# `% s; h% }$ P

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5. 如果定理的结论在不同层次的几种条件下面成立，可用如下形式： ' @0 ^' b. }% l' B% ^% n# D

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“Let…, and assume….If…then…” 1 v. ?* h- b6 g! r8 H& W' l) ^( d

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Let f(x) be defined on open interval I, and assume that f(x) has a relative maximum or a relative minimum at an interior point c of I. If the derivative f’(c) exists, then f’(c)=0.

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hehe123

数学专业英语－（c）How to write an abstract? 8 h9 t$ g" J% Y- p& c 7 L- K7 ^+ [3 |* V {: b, L* I

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论文摘要的写法不像数学术语的定义和数学定理的叙述那样。有一定的格式可循，但对于初学者来说仍有一些常见的句子可加以摹仿。现略举一些这样的句子，并附上一些论文摘要作为例子，供读者参考。需要指出的是，我们这里所举的例句对普遍的文章均适合，比较抽象，具体的论文摘要除了可用上下面某些句子外，必须有具体内容，更确切地说摘要中要包括一些 key words 以说明该文涉及的内容，但一般不要在摘要中引用文献。 : {- k3 d; u w$ g5 o

1．开门见山，说明文章内容，可用下面的句子起句： 2 F, u( G& \* U- l' O7 X

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1 O* Y' Q0 t4 ]; o prove 1 d$ Y% Y) j- k, L- P3 S: A4 } show present 8 `5 ?5 e. D- x, D) q4 n develop generalize investigate ) ^$ S! |: G: V5 S

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- n$ R$ m# s/ I- Z paper & V4 C5 S/ L+ T b0 z7 ~: `8 j note

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The of this is to … 8 r/ y0 l- V9 T# ]& Y

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It is the purpose of this paper to … # @8 {2 c# a6 @+ t* m! F

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: A) G3 ?: n I; Y is concerned deals

This paper with… 3 q( L; d% [% D! ]* Q3 R7 |0 t

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In this paper we … " n' s* ~. z! c, F3 x( `

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2.如果需要简略回顾历史，然后再说明自己文章的内容，则可参考采用下面句子。

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The problem…was first treated by…and later…improved by…The purpose of this paper is to prove that it holds in a more general case. * t/ c* F4 s7 g# D! D. |

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … 2 C! W. ]7 C$ g# H* z* K

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3.如果文章推广了别人的结果，或减弱了别人结果中的条件，则可参考采用下面句子： ( ~$ }5 ?9 k( d( V2 O/ g

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,…

In this paper we shall prove several theorems which are generalizations to the results given by… % w0 C1 M; U; \ n) v: a( O/ u* C

This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… ; T1 N( `9 s0 i X, ~0 u

This paper deals with generalizations of the following problem… 3 [+ F! @* |' p, x# N1 O

This paper improves the result of…on…by weakening the conditions…

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例：

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It is the purpose of the present paper to point out that certain basic aspects of information-processing systems possess dynamical analogy, and to show that these analogies can be exploited to obtain deeper insights into the behavior of complex systems. }+ c9 I/ J d: [! w

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We present a general comparision principle for systems of boundary value problems and employ this result for proving existence and uniqueness of solutions, stability and existence of periodic solutions for non-linear boundary value problems.

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We proved a theorem for generalized non-expansive mappings in locally convex spaces and extend the results of Kirk and Kaun. We also obtain a theorem which generalizes the results of Brouder. 8 k; L) e: g# }; u7 Q, q. W

This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form…. 5 L5 K$ [7 V' T6 r

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This paper is concerned with the question of local uniqueness of solutions of Cauchy Problem for elliptic partial differential equations with characteristics of multiplicity not greater than 2. # c. G% I F* a

The object of this paper is to investigate the behavior at the boundary of solutions to the uniformly semi-linear equation…

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The aim of this paper is to try to minimize the functional " l* V2 L" q* h' E. h* O

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over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= . 9 i* Y0 q7 l- p

风月晴

顶一下

布赖

尽管我还没写过英语论文，但还是十分感谢你

satre

参加MCM比赛就可以有锻炼的机会了!

rankvic

我不懂英语

summit001

对学习数学专业英语还是有一定的用处的.

谢了~!!

[em01][em01]

hxo1202

顶！