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? o" D8 a$ [3 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。 . r1 i n2 u& }6 r u! n; Y

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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。 8 \1 u2 R4 Z% r/ Q( Q

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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）。 % I5 {; {2 t0 d# r. v7 V \

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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 + [4 f6 l S5 e6 m 7 X/ K2 Y1 J- J; z9 V# [/ \! t is called 6 A7 E5 ^/ M0 x" e8 T" _' r$ f # w! J# v" O; b% f

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1. Something something " N" ?: D, ?( N9 ^+ ~0 c: U

The union of A and B is defined as the set of those elements which are in A, in B or in both. " X6 i/ F- U& K1 g

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The mapping , ad-bc 0, is called a Mobius transformation. 0 ~- U6 `: \! D5 q

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is defined to be ! a) j, H. j6 V$ z9 x4 ]9 t ; p( C( J+ M3 ^3 `7 j9 n, w+ _ is said to be 7 S; G+ a! b D9 D- H

2. Something something(or adjective) ' z4 i- n$ H' q4 [, D

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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. ( }9 T9 t& W: Y5 e

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define 7 V/ h G+ p, E$ A- J. w* R 9 g" d: e# P* i5 X& F$ W+ p call 3 F; H5 ~ w( h# q

3. We something to be something. $ _2 _$ V7 y4 Z/ B5 H

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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. # c8 [2 h& h K$ _% x3 a* ~9 @

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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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6 r/ {# O/ }- j* S3 R0 u7 h' [ is called 2 Z- @) l$ J/ M* E ; n. E( s% c i# A4 z is said to be " l7 i8 T; S4 s0 s is defined as 4 {4 V! o, ?# Z k' m% W7 ]7 f 6 c/ R: u, |* o * u* p. ^9 ~* e/ [, R is defined to be + Z+ U# a4 C2 ]( \& [ p$ h! y

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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. 1 P& T7 M8 q( Q; |* n; \/ 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 . L4 N) x/ E* ]- J+ I8 g

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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： : m5 ~- H {: y* m& \2 S+ W

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is called . n( ]7 y' E2 v {: a+ S; y1 e ' v: ?6 `$ c1 O7 F) K' O % i! u8 ?$ [2 `8 [4 ?5 l$ f/ M1 O is said to be " X0 z# t7 d( }' q: T& { is defined as $ ], n6 ^1 I+ `0 G2 @4 { is defined to be 7 k) k9 b! T6 c; a 3 b: q6 r" w5 j9 n9 s! y( F

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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.

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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is called ; a8 ]4 p) O t8 j. }5 \ is said to be

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1 J, i }- J; x/ o Let Suppose

…. If…then… … # Q. R/ M* l6 w

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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. 7 ]3 F5 M) s3 u5 x2 l3 f

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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? ; e4 k) D: I. [$ N/ ] + X2 e% W4 C; X1 Q0 E1 [

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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。 2 k+ `2 k8 k* @: k" q2 a$ K* H

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.

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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： ' Y5 _& h8 i. m( X3 F' k7 L/ X) g

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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]. + @) D/ b# H+ J' E* J( F- ~ q

Suppose that f(z) is analytic in a simply connected domain D, then for any closed simple curve C lying within D, we have / _+ Q7 b2 H% \: z( p7 R

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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 ) T, }. p0 ?& B. `

“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 N8 X' [& F/ [4 }# D

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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 9 Z5 F: U; r0 x& _

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“Let…. If…,then…”or

“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.

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：

“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? 9 k8 |, k3 e9 u7 y, ^0 @9 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。 . S6 T) `8 G9 H4 p$ s. y% Z! k 8 q% x% d0 y6 U n& I

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1．开门见山，说明文章内容，可用下面的句子起句： 5 |) i4 v8 M) p4 E; H3 p

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\3 @% u( a3 J6 J4 w 9 E M+ I* l5 d2 a8 j s0 R prove show present & |$ X5 R) {* ]9 D' D develop 2 Y. k' J3 i. _# A6 L8 y generalize investigate 5 }! J6 Z# G1 s

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. h! k2 `5 S8 q: G' r8 }7 _8 g paper * w$ O- a5 N4 K- }( q0 c9 W note

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aim object % I$ o% _2 [4 d! ^3 m# W7 N purpose

The of this is to … 2 L; L0 R* ?$ |) u% ]

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6 t2 N' L1 h* ^3 U% ? prove 9 }% |1 {! ?; g% }* `$ b show % o9 i, @2 z, I6 }4 ] present develop generalize ' }6 I& J( G' E: N investigate

It is the purpose of this paper to …

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; a* r* H; h) `. j, A $ s$ U0 V( K+ L, h is concerned 8 `+ M3 w0 p* I6 k. ?" C; e' j deals

This paper with…

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# \8 ?2 \3 @8 n3 U, ^ prove ' ?: k0 D% y% L( D present 2 Q7 {! ^+ [9 _) O' G/ U5 m propose to show

In this paper we … 1 i0 R; {9 \" P0 B7 m

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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。 + [- E/ a5 w1 V+ ]. {

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 … 2 z! x5 y9 Y* H3 e7 c% E

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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：

The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… : s- n! X! z/ N4 Q6 J0 L' b# b u

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

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… ' p3 K7 \7 M- C: u W( M7 p

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This paper deals with generalizations of the following problem… . Y* g& `" i# z) s7 _+ ^

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

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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.

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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. " ]' k4 I7 Y$ y# ^; Y) _8 V. ]# m

This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form…. 1 w9 t/ M- L; [# U/ Z

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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. ) z2 Q& a0 S* g" G# z

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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 ! K9 t, Q* ^) E0 d

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over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= . ; R) x7 E( A/ g: S. S5 _- o# _5 s

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！