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? $ h% b( I% q. x3 ^ Q

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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。 3 e1 h$ w. ]7 D& L 5 a; R' Q7 D, a# L) Q8 f6 L # y$ C& y5 z/ R4 o$ D6 f' Y5 @

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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。 1 V- ?: o: x0 g! j: v) o4 ~

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）。

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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。 ]2 n9 a0 f% ?! V) C; E5 N

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（a）How to define a mathematical term?

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* h) ?8 L5 n, S' P4 Z, ~ I& J is defined as is called

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1. Something something 7 b& T3 Y; y0 _

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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. 6 L) |( x9 I! ~2 Y0 O' d

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is defined to be " I; X) }( h2 F& Y9 C- L, a + ]) _. d, P# ?. r) z" h is said to be 2 w/ J7 }7 U6 T, o5 f. j( ^5 Q

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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. 1 ]! B/ `! r$ |: |3 I

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A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. - v' ~1 I4 H. z1 A' b

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

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define 6 B) M9 p% Q* e% O7 V8 n1 E ( H" ?6 N6 K7 Z call $ B. G; G" o* p) J& @9 P

3. We something to be something. 9 u0 A0 N: y2 C+ S) W( L

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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.

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

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 . Y/ E! `5 X: X 6 E. H9 l4 ?- v1 R is said to be + X% F' w) k1 j1 Y & {3 y( U X, E6 @ {2 E is defined as 5 q& G) P2 N" V* a9 ~( z8 H% X , g6 ]" N# g4 O6 ~ is defined to be

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Let…, then… * L2 y0 E7 J0 n- I/ s; A4 d

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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

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D=

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is called the diameter of A.

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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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is called 5 Z+ w& v6 y6 X. |( { is said to be ' | }9 C( t5 c* w5 `3 u5 h4 C& r $ p7 F7 |) ?3 C8 | is defined as ' l! F( Y7 W) {; j! Q5 ? is defined to be % E$ E4 ?* N/ e3 J) y: M# }

If…, then… 4 v2 o" l) Z% U+ O

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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- _9 c1 T2 U- q) `

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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.

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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： 2 x: x7 u2 ]7 O1 v, e

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: c0 G+ y' O8 W+ b! x8 h8 d- p is called is said to be

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Let Suppose

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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. ! w3 [. h( k5 F4 H J- |2 e! k. T

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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? 4 k/ h7 k8 v+ q. s; e" M) t

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。 6 ~* x0 p5 J; p3 w' A) c

1．某些定理可用简单句叙述。 : B: I' j" p# Q6 ^$ d' a

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The union of a finite number of closed sets is still a closed set.

The space (E,f) is complete. . J* M" N* {2 k' `5 S$ e

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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： 3 x, g3 f& @) _9 u& r- M7 y

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“Suppose…Then…”or“Let….Then…” ! Z( C1 `/ ^; H9 j) Q9 W

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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 / f- y% f# u: I

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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 8 M5 f' `9 p3 R9 z6 M

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“If…, then…”

If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 * ~! o; Z- \+ P0 p

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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 : P) p9 h4 z# m; D0 f3 z

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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. ) r b- V! a3 M

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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： ) F0 d( _. @' m( [2 r+ `" u# T

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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. # h8 ]5 P# t9 p# v) g( P5 `# q2 H

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hehe123

数学专业英语－（c）How to write an abstract? ; m/ |6 t/ t5 ?% q5 ?* U( }0 }6 z( z

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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。 7 ]. g! p# W% m; \& P

1．开门见山，说明文章内容，可用下面的句子起句： 9 z# f! l! Q% R

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8 Z2 L- s& s) x5 L prove x* n3 ^( r5 D: L* ^ show present develop 2 K$ U/ [8 g6 \) w" S6 [' ~- X generalize % P0 F1 E! ?+ g investigate 6 X9 \# e, J& |+ ~. l " }) h- L8 L9 }! l( z4 ~! O! ~9 a " k+ ^' D: ? B1 y- M6 x6 {

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. e3 g$ d$ t* a. W C9 ^$ l , J7 g8 m* g- e' M: l& t paper . w5 @4 f8 Z2 z note

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The of this is to …

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prove show present develop generalize investigate

It is the purpose of this paper to … 5 c# ]) o* u9 R. ~' f& C% V

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This paper with… * |0 y% c ]; W+ E1 l

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8 I! U% W8 P9 a' G prove present propose to show

In this paper we … " g' V% a" H2 i& C {, {

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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. 6 I* c f* q% ?5 m2 ?) X: e( V

…first raised the problem which was later partly solved by…We now solve this problem in the case of … , r5 i, q: T, G6 R( G3 T7 A9 |9 F i

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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： ' A8 {: D- [8 d% e6 I, 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… ; ~( H, ?" @( I/ [( Y

This paper improves the result of…on…by weakening the conditions… ) W6 C/ [! V) S" F6 r. G; T" a

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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.

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. " L2 P# B p# j$ X

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. 9 W' b! {7 ]7 i8 s3 |) x9 o% N0 \/ U

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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…. 7 _6 M! P* P& O2 @- f2 P8 @) C! q

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. " q' a6 u# J7 c1 N" i

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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

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！