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? . G; Y' ?# D& O/ a! d

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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。 . A8 p& q F1 Z( P

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。

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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）。 7 g( a* _1 E# M2 B# q

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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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: B; e& F0 b4 N- a" x7 a is defined as 9 |; G; d2 P3 e7 ^; t5 F b is called

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1. Something something ( [% w# w! f3 R/ }$ q) ~$ h

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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.

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The mapping , ad-bc 0, is called a Mobius transformation. 8 e" L' L& w' i6 L+ x5 N

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: r6 y; E( M: ^% e8 K! a. \ is defined to be & R7 u" f5 u k6 I! p" k- B is said to be & k1 Z9 D1 |9 _2 i' p

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.

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A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. ' N: D( F) Q: h% e- ~0 h8 _0 @

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

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* Y1 \! n" A7 R2 ~" ]5 v, r define m+ ^ E; e4 G3 y' F( l8 V: Y call 3 q Y; E* Q+ ~3 V5 Q, Z' P 9 K2 C& o9 o- n) E+ g9 v

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. 6 K+ g6 c% x) l. M6 \

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We call real numbers that are less than zero (to be) negative numbers. 2 X Y5 U+ X5 M1 h2 S+ s! S+ p6 H

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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is called 1 J# l- |7 \) u is said to be # ]+ T$ J# K0 V) t2 ]8 V% b# j is defined as + f/ S# P- s- c4 p1 [ s, @: m' a9 s is defined to be 2 b$ A/ Q0 D7 p8 Y

Let…, then… # w) }) o3 b9 `4 z2 b0 ^; L

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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. 9 D. @+ s6 {3 O- F- u

Let d(x,y) denote the distance between two points x and y of a set A. Then the number G% T/ a V+ [! y9 @$ g. s" p

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D=

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is called the diameter of A. ' V' J0 ~7 h& H0 Y% {6 t

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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：

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8 h) ~* h. H' R( ~# F# J7 |4 t is called 8 `. W7 e4 M% K* y* V 7 V" ~. \3 ]" @( ]/ J: k is said to be . J" D$ r& x* d is defined as is defined to be 1 J2 x" z- e5 t3 ~6 A1 P! Y

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If…, then…

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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. / e+ V4 W+ p( x4 A* _5 y) ?

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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： ' h" U. M- U* |+ R) f9 ^

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4 x1 Z1 p. B- D% V5 R! ^ u Let ( v& F/ a, W7 x& G0 S5 k Suppose

…. If…then… …

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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. ; N$ X: \0 x7 ~

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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? 1 C# K3 m& \) M" p3 G; V

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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。 * ^0 D0 S: q" ?) \ , p' O7 r" J5 q. Y

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.

The space (E,f) is complete. % y% q/ S. ]- N5 k' }

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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： * ~9 n; l+ r, D5 ^ e; i

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“Suppose…Then…”or“Let….Then…” ; i+ o: j" i5 s3 i8 `9 J

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

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“If…, then…”

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 ! U K X- o3 K% ]3 \* [* {

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…”

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real.

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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： # o# B: ?5 V* u

“Let…, and assume….If…then…” % @6 x2 D M( x% F

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?

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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。 5 L" w/ K3 D2 n3 G3 J7 i : K7 K8 M& N, \8 h

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1．开门见山，说明文章内容，可用下面的句子起句： 0 b0 [) M1 J- I; p/ A2 u6 C

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prove 7 \; }3 T; {6 J5 ]+ ? show present develop generalize & U& M; O, {4 ^8 P& F& T4 J' z investigate " Y! t% U1 x% N

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paper note

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The of this is to …

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1 v5 q; N: d3 D! L! I prove show present develop generalize investigate

It is the purpose of this paper to … 5 {3 @1 S, l5 o" O

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$ B7 P: _# f) Z% A/ z! } is concerned deals

This paper with… / Y4 C+ l: J, F" ] r$ t" f# Q

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prove 5 r; R$ t% p' N+ K$ J2 ] present 7 c6 H8 b0 f3 t v7 ?0 R o8 A" | 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。

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.

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … 0 \ B$ e& e6 |1 y, s2 C( c

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： 3 A- V( Y) O! H" z- Y6 B9 |/ K' W

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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… $ u1 P9 C, x7 R$ i/ u# l+ W

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… 3 G7 v, B9 M; s! T

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

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例：

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. 1 v' W$ h% X$ z' A

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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. 6 a* q2 q: N! F- C

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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. : X* h2 V5 u& A

This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form…. 3 P$ Z. y, {# E

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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. 1 z; A1 T& u" e/ \; c8 ]

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 ; h/ _& \/ s7 N; o

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over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= .

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！