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? 6 z) b* c. S& }+ n" Y6 c& e

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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。 ( \& E+ S/ ?* M4 ?9 a

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。 ) |* s9 B1 g) N0 x

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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）。

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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 ( J8 D% S0 Y3 a7 q& I: C, a0 U / Z7 V# H" W4 q 0 x4 \8 l# h/ e& t is called ) E. S* X, B6 e ( P$ Y: J3 {9 w

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. # n: g5 u9 h% K

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

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2 X. ?3 [# c5 r! [ is defined to be is said to be

2. Something something(or adjective) 1 W6 v; k2 z% S

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The difference A-B is defined to be the set of all elements of A which are not in B. ( m! h0 O& l' h7 C2 Z

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.

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define call & E6 Y" k7 [/ `' r, j R# `; f+ G

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3. We something to be something. W7 Q. \* F/ \ e& B) e& ?: P7 E# q

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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.

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： ) \5 b8 A; d: F" {" D" n( k

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is called . m$ q3 {) C! W % X3 C& J' M8 w7 |; J / V! e; H3 n9 Z* V/ w is said to be is defined as 8 a8 L7 f) n% K & c- T* A+ s0 ~4 }$ E x 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. : ^$ I! Q+ V, Y

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Let d(x,y) denote the distance between two points x and y of a set A. Then the number

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is called the diameter of A. 1 F- K) I0 X7 P( k I& n1 x

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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： 7 P8 [+ m+ a2 I1 a3 J8 V `

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is called m; y9 I1 e' H3 ]( R: |4 B is said to be $ z/ K2 H2 |) @6 U 2 N' N4 C3 @& Z" Z9 ? is defined as $ m, ?( ~1 C4 R. y8 k1 J6 C% {" O ; W, s2 ]6 X! [+ S is defined to be % F% P b( r: ^) x% x* e . K& E6 M9 r% e3 B6 }) G

If…, then… # n0 b! z/ {! {5 ~% {, J D4 {

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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. ' U* @' H, R4 n c" w3 j. m

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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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, B1 [- H X+ I is called is said to be

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- M7 ?7 P3 a) q Let ' V7 U/ g1 i1 r- O0 Z) V. k5 } Suppose

…. If…then… … $ P( ]1 w6 v, T" T

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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： 1 h5 Y0 l" S9 m0 [1 P; y

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? * b: ~/ A) X0 ^

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。 # W; T3 X! k' x# e

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1．某些定理可用简单句叙述。 e- \& A0 \4 F; p5 Q! Y$ `

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The union of a finite number of closed sets is still a closed set. ( X- Y5 P4 h$ d. ~* s; c% v8 k. ?' Y& \% X

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The space (E,f) is complete. $ ?0 ]1 G L9 _/ [6 U

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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： . G0 E2 d: s8 E6 w& j: B) l

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“Suppose…Then…”or“Let….Then…” ' a) _& p% E9 z; G3 a+ z+ j

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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 . W/ Y/ U$ C1 V6 w

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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 5 H9 k! q ^" j" Q) M

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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

“Let…. If…,then…”or

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“Suppose…. If…,then…” " ~9 h8 x1 }' @* y0 ^

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. " b |# I2 O2 M3 c% F0 Y2 M6 A6 x

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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：

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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.

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hehe123

数学专业英语－（c）How to write an abstract? 2 f7 f; |; ?: K4 {! l * R: P* c2 r/ ^

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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。 ! @' b" R$ }1 w. E- c$ x3 [5 V 5 _! P5 g8 D2 E! Z% ]

1．开门见山，说明文章内容，可用下面的句子起句： * k; u+ {4 F& _0 z( V, R6 R

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5 G) G+ m* s5 ]2 g2 | prove show , S4 b; @- A/ ], K) E present develop generalize 6 ?. F( ~4 w9 ?0 l investigate - v2 l. g+ S& `) |1 C 0 y3 W' p9 P% _' Y ; N) `0 _" I5 C9 B4 d% k" M2 ` a! H

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$ T* m/ D# o; W- N" \ paper note

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The of this is to … $ \* {0 @) m3 p8 H# O

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) j! s; `2 T& K4 y 3 D+ z* p$ a* m i' X$ F prove 1 k, F: G7 c6 t# z% M, w3 |+ `1 b show * L, k$ Q8 x# @1 Q9 C+ H9 o6 W present $ \' n" `& ~/ i2 M develop generalize investigate

It is the purpose of this paper to …

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is concerned $ v6 q6 M+ Q5 H7 v3 ?# b& S deals

This paper with…

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prove 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。 3 t" A( S+ p+ a( t% Y, C

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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. 1 i$ i R, M& m7 a

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

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： 8 n# {! S- h5 i7 ]4 N

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… $ n6 N1 ?: @9 ?' D

In this paper we shall prove several theorems which are generalizations to the results given by… % F7 X5 D. M! n$ N* e1 q

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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…

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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.

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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.

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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. 7 H1 P: @. M, d0 [& j4 f

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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…

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( )= . , f3 p2 v7 L+ \) w' \3 N

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！