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

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数学专业英语－(a) How to define a mathematical term? ' s2 k4 W" {/ z8 B6 X9 _

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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。 # T/ ]+ W! a- F- e; {* d& y( s

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）。 ! H f; {; S! m6 l9 K

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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。 - P% m8 f1 g1 q F7 S

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（a）How to define a mathematical term?

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is defined as * w, u$ T/ F1 O' f" x* M1 U ; |/ U& v6 h2 X3 ? is called 0 j* U3 ]+ e0 @, o2 R& ~% m

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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. 2 N' ]1 n; v/ f: z: i/ m+ l

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

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is defined to be + E* \/ E7 B- L, V# f is said to be 4 c& @3 d% S5 r; E/ `4 }8 U

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.

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Real numbers which are greater than zero are said to be positive. ! P* `, O1 ^/ v2 b

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) W; q1 I& {' y; W define 2 p6 Z0 O/ v/ B2 Y8 K1 I# @, W B& H H* u7 h$ G/ `8 a; } 9 w6 s1 Z3 T% l6 I call D# s ] f; g9 p5 ? 7 t, }. V" z& S8 d

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. 7 i: T1 x1 m9 Z* L, ^, q. _5 T

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We call real numbers that are less than zero (to be) negative numbers. 3 ~0 ]* E1 a$ a N- o' H# R$ p! g

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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： % l9 w% [; B* q# Z* i4 l3 U

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is called is said to be 5 j7 P. ?3 N% K% M3 Q' ]) B* h* [ u 9 e {' U) ~, I+ [. W is defined as 9 d. g' h, u | 4 Q# c. _$ g- a7 a4 ?* Y* b is defined to be / P6 X- K5 ?0 g- O7 o& V 1 [8 B' Z4 h. ]& o! N' ^

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. 8 Y) v0 X3 |( u8 P6 m

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Let d(x,y) denote the distance between two points x and y of a set A. Then the number $ u% C8 p- y+ v) g; K

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D=

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is called the diameter of A. & m+ L8 H5 O$ ^( V* k& z$ m

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 : Y9 E6 N* N' O k7 f is said to be 4 v6 [; o: P* j is defined as 2 e4 R" i( P3 |+ e is defined to be

If…, then… N8 d0 S* X! r1 V+ u- J7 n# t W

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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.

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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. $ d6 _% `' x8 K( |1 t4 {6 m; W

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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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Let Suppose

…. If…then… … 9 \% M6 ?; Y/ t5 P

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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? , U6 W: \/ k2 l8 x Z" f$ `( z9 H

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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。 z+ A7 A9 a5 A, V/ d% R & X# f0 i" Y0 a$ w- i5 m . z! {+ T) o- @; I/ F$ }5 a+ K

1．某些定理可用简单句叙述。 1 V) B7 f( g, t, k9 h( L# Z3 Y. b

The union of a finite number of closed sets is still a closed set. 3 J. f. r+ B7 `+ c6 R

The space (E,f) is complete. & x4 d# Z* Y0 P# ?

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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： ( x+ h+ d" ` g! Y4 M

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“Suppose…Then…”or“Let….Then…” 6 w3 F7 r5 v( E: l

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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].

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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 3 x' E" `* N2 c. [% P# `3 q

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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 3 @0 x, t/ y' O* `& \9 T

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“If…, then…” , [* z* l3 Y" F4 G8 }/ }3 Q, s& O6 _2 ?

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 * ^8 B+ M/ E5 q% @6 [8 i" Q

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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. + |4 m9 Q" O, ^7 K* j f- {

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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： : b2 c( o" Y2 H5 r4 @$ @1 j5 H4 s

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“Let…, and assume….If…then…”

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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. n+ v. l0 J% S U Y

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hehe123

数学专业英语－（c）How to write an abstract? " Y- `6 n8 l9 s, F % V1 X2 _1 Z, m) {) ~0 e/ N$ Q

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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。 % {: y' O/ c& y4 ~' ?; R

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1．开门见山，说明文章内容，可用下面的句子起句：

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; R/ p; m( M; v4 f prove , W6 g- X5 P) Y/ g6 O) r show 0 V2 V; J6 u1 w$ g$ O present 3 @4 b2 O* Q6 T. w develop 7 O4 _1 \% _1 i7 x4 Z generalize ) _) I" T/ ]& ~+ P' C investigate / F% ]1 a' J" x: ]) D. m* G: ]- R

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paper ( M4 M1 I9 J8 q0 ]' k: ]; b7 D; O note

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aim ! I7 N& c5 ?8 ~9 Y) T. @/ ] object / I+ X0 [. y/ {9 V; @ purpose

The of this is to …

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% @/ p- @+ f" |( e prove : {* P: M3 c$ ^1 y7 C show 8 y8 L) i) ?/ t1 G present develop generalize : Z9 h2 f. g( y+ n( P0 i3 {3 G investigate

It is the purpose of this paper to … 9 S) {9 Q) a3 u z4 H- r2 C4 M

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+ L6 c- E( z( I8 \8 S6 N ' @9 C9 c/ |" H2 ~ is concerned deals

This paper with… $ J* H8 k* O7 D3 A }0 l

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: U7 G* M/ a2 L5 R5 Y$ v prove 6 O# y- W! z% F# B 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.

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … % B7 x% e6 M# B2 u4 {

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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： y& \8 ]5 u; Z }2 Y& W

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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… . R5 \$ X) |% @6 x* a) r- F6 n

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… 8 w$ d3 E# E+ W" W0 L

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This paper deals with generalizations of the following problem… `4 \+ Z% Q- P) O) A3 \+ R7 i

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This paper improves the result of…on…by weakening the conditions… 7 O$ g; f0 o& G. l0 _

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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. D3 S6 F7 M# [! E

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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. 4 J) z* r: A5 O b# k0 Q8 L

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. * {7 o+ e; G! l8 q( f9 w

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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…. ( P. d6 G& {; T

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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. 2 @7 {! C1 m/ I8 n( w: q! V

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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

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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！