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? ' L6 X7 d! s. g% Y- `: y. T 7 Z! @9 `+ ]( V4 e( ?

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。 : z5 B% ~+ P( E) g! P$ T

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。 % X5 x8 G Z/ V+ Z& }. D

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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。

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）。 / b# G) R" D, P# i4 u

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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。 1 \# Z+ Z# t, K

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（a）How to define a mathematical term?

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! {$ e' m5 |: Y8 q$ N' r( c1 T( @8 ]2 ^( q$ Z

9 c* S+ I/ ~9 [( Q4 Q2 ^ is defined as - x: Y; A2 o; d N3 J" b0 {, f n* W( L% C$ h% P& @. n% m$ [" F& J) } is called 3 T1 X( L' e) P6 Y( i" g8 |

1. Something something 8 t) |5 g6 ^9 L3 `

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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.

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

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: n* I- L% q8 H% u is defined to be # z5 O& m. P: ]: X! ~% u 9 w! b! K5 B4 f8 T * a; u! w+ E! I1 J0 H8 |: K- E6 v is said to be / Y4 [4 Y! x c, @. n( W% b7 j 4 s! w9 N: E) F5 g2 Z: a2 ?4 b; H

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2. Something something(or adjective) $ V- y# p! N6 ], U [+ @

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

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A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.

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Real numbers which are greater than zero are said to be positive. " p" Z, V: a8 s( h

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# g2 X: E2 \- q! _ define * g9 g0 N/ a3 {$ e7 b+ o . V( w' N' d, n9 z- ]3 V/ ? call & B2 a* D1 Z1 a3 `" _5 L8 h / L* f+ J1 l+ ?2 m

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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. 5 }3 U2 {, e- I a! `0 {2 ~

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We call real numbers that are less than zero (to be) negative numbers.

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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：

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% S. A- u5 Y: F! X: |* `- v/ ? s, L* c is called ( C* a& X% ?9 P& B+ o- K+ v% h 7 Z* n8 {0 [' h& T8 v is said to be # Q8 y/ d, q( |% s9 B3 |# F ) w) m( a) [5 A: ^ is defined as is defined to be

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Let…, then…

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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. " p6 S9 j) p; d: J: f0 v( [- K8 z

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Let d(x,y) denote the distance between two points x and y of a set A. Then the number % F% K2 X0 L' S) j6 e

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D=

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is called the diameter of A. 8 O' {5 Y5 ~4 J# {1 ]' O6 u3 s

5．如果被定义术语，需要满足某些条件，则可用如下形式：

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is called ; y: G( W0 C5 u0 F' O $ @9 K; I+ p4 Y6 K% _% w! `0 g 5 A! M7 V" b% ~! @ is said to be is defined as 3 [; l! D e9 A" t2 | is defined to be 5 x, b0 X U1 T" \9 t. L

If…, then… 2 U. B H5 T7 \7 i8 C" |' J$ d) u

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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. ) u8 z) B6 h* b# } W

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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. ! p/ E4 b, i; a* b2 O, z

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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! y( _% S X' H 4 t6 ?! s" Y9 z) s( v is called $ @: s1 A* g" f: T5 X! Z e S is said to be

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3 h0 Q6 X( E$ u0 Q: ]! A5 k. U Let Suppose

…. If…then… … # Z. x/ Q1 n2 S) p" D( M

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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. 9 o7 m+ x$ W+ [; z8 V. R

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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?

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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。 " X$ s7 W5 x" z, L: t( ] 5 W6 m# M% k% W8 s7 @

1．某些定理可用简单句叙述。

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The union of a finite number of closed sets is still a closed set. ' Y# [4 x, T1 R3 m4 g! G

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The space (E,f) is complete. % g3 V& L" m- C& m4 a

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： ! p# F* ^) @$ P1 Q

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“Suppose…Then…”or“Let….Then…”

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Let f(x) be a continuous function defined on[a,b]. Then f(x) attains its maximum and minimum on [a,b]. R% u' ?: \8 Q& f9 C- B8 ^: C" G, d

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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 2 s( D. a# E% X% L

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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

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“If…, then…” 0 N# w. B6 F) y# U; y

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 . \7 ?) l' k. b* p9 A5 a# a

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 $ F3 N$ k' ^! p6 [7 z( \8 Y

“Let…. If…,then…”or

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“Suppose…. If…,then…”

Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. 1 i- ~9 K* D+ m; `; {; Q

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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： 9 Q5 C5 Q5 c- D7 c

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“Let…, and assume….If…then…” & u! @8 E; d# [5 |" c0 `

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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. ( D, X! Z* U3 d3 f0 s

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hehe123

数学专业英语－（c）How to write an abstract? 4 d8 ]) n2 u, _' W

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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。

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1．开门见山，说明文章内容，可用下面的句子起句： + Q0 f* P' J* h' v5 Q3 D7 u

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9 d, ]/ d9 j+ \5 N/ Z" p" _ ! W9 _3 |( p: P( a) l prove , v4 C( D+ A0 n show : N. ?" P7 {7 x0 B7 Y$ U! ~ present 3 w$ A. m1 @' o* a develop generalize ! I4 K. m0 W4 ? | investigate - s0 Q6 D( e+ E& N6 b0 M , ^7 e# \; G1 a% [ R) ?& R" p' J @

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% Y3 q# j* N/ h& [; z! Y Q8 X+ M paper 7 Y* Q& d; r Z0 |( ? note

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aim 1 K F- a- I: b, t% t& K object 0 m! J; {; `& E, z# w6 r% b- P" w purpose

The of this is to …

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$ y- y3 }+ H3 m prove : |( h: k5 i# h, N& c9 R/ i1 f2 G show present 6 c! a, T" `) A1 g" @* ] develop generalize 5 s# w$ V+ E; |1 v Q( Y investigate

It is the purpose of this paper to …

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( z' P7 H7 }4 L* D is concerned # G; e" q o0 j3 M deals

This paper with…

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5 z5 f; ]3 K+ f$ D- g prove , y; G1 M) o" U% v5 f- ? present propose to show

In this paper we …

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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. ' ^( @ T1 b3 N: S) l) W

…first raised the problem which was later partly solved by…We now solve this problem in the case of …

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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： 9 ]' ^) {7 L `7 W W% O

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

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In this paper we shall prove several theorems which are generalizations to the results given by…

This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on…

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This paper deals with generalizations of the following problem… + u" U) Z r% F& T

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

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例： $ z4 H; ]+ u% b& o

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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. ! ~9 q' @" Y+ G" 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. : w* @+ X& `) ]) V/ |) N

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.

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This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form….

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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.

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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 3 F' {6 N/ I- ?9 ^+ L

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over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= . ! E, u6 l3 r( N$ M# L( I

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！