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? ' M3 T% i" Z" A# C ' u/ B" Z. N4 Z9 d7 z

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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! i* a9 n' N% U, 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。

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。 0 }3 l* u3 z2 R0 T6 L7 j; T

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（a）How to define a mathematical term?

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is defined as & _: x6 G6 r7 O8 B is called ! L9 o k$ U+ Z' `. c ; }* K! B3 f8 \' e/ I# [

1. Something something " W& q; A5 t$ O

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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.

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4 C7 V+ p3 B" a is defined to be + d W( a/ w8 f3 l . X$ G. ]" t B is said to be 0 c* h' ]$ l: V1 }& v3 y

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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. 9 U9 l6 _4 ~" o$ `6 T+ J5 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. 6 ~# I( A, h+ k2 q; e1 ]

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Real numbers which are greater than zero are said to be positive. 2 Q) ]' y) T7 E( p# b

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define + S+ f) z, m/ k3 } call ! E& i$ g$ g, @# z

3. We something to be something. ! q4 i; x! ]1 b$ P$ P

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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： ! b' N. E1 f1 q

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is called ) L; ?; I, |0 X" M+ x5 I 8 \+ W: N1 ]' K5 \6 B3 K is said to be ( i& N" X) u; e 0 }; O: o$ u7 m0 w( v is defined as 8 M5 j8 _! A b) h) T is defined to be - {" I0 ]2 F# G1 J* U; R1 y

Let…, then… : f1 v) \0 Z) c, W

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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.

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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. 9 i, u" e5 F; S( m: E3 i

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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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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. # A1 W& _- a' z/ T1 ^0 l0 O

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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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& c$ X$ @8 K5 t% H t " K! j+ Q) Y6 p5 F5 G is called is said to be

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…. 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. : y0 p0 @! [1 f, m3 b. M

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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? . Q s3 Z5 } m+ m1 Z

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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。

1．某些定理可用简单句叙述。

The union of a finite number of closed sets is still a closed set. % P. T( k3 Q6 ?! Z0 ]0 T

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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：

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“Suppose…Then…”or“Let….Then…” ( P# }3 T q$ t* ?

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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]. ) B* S, [. g" L( F

Suppose that f(z) is analytic in a simply connected domain D, then for any closed simple curve C lying within D, we have ( i( H# l, S, o) b7 |2 H

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 6 U5 n. E5 I' f! W& V2 d4 i5 ]. ^

“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 8 K: d4 D ], B( G4 e- a

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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 . H4 m( ^0 n3 L

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“Let…. If…,then…”or . o8 I( M# ]5 u: S3 j

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“Suppose…. If…,then…” x; U7 i) L; T$ _. I( B

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. , T. X1 }. c0 k3 Z. z4 t" p& w+ `

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. 8 L o, Z7 f! U6 b V+ p

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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。

1．开门见山，说明文章内容，可用下面的句子起句：

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# X' M; b) f5 o 0 v/ w% @: D; l prove show present 1 a6 j" P$ Q% v+ _1 x7 t5 C develop / h3 Y+ U1 U K, @; g( m! x: Z generalize 2 k V+ @; e2 s5 [+ A( ?& u investigate 4 [1 Z0 w$ |: b) A' j: T V& i " H' e: R& }' f4 ~$ i

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5 q p% Y3 g& `1 Q2 Y2 A paper note

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3 L4 k. B" F; ~1 W1 d, D, W. J aim object & { r, {- A6 s t3 K+ u purpose

The of this is to … 4 G2 _- r4 Y+ Y3 ^6 N; @% v

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6 {5 O6 G' }& m2 D6 O; k! G prove show % f( N3 _; G4 y2 m0 F& p$ w7 f2 T b3 R present 4 e+ n: @9 O1 q& g" Z develop generalize 1 E& |0 B" {7 g0 ^7 u. ]. t investigate

It is the purpose of this paper to … 2 G/ o0 u& c* w; x

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; A1 w4 L4 g' Y is concerned ' X' k+ ^/ L0 A# \" E deals

This paper with…

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+ M) j" N% e8 {( t prove - T" G( C0 u/ @5 [$ g present propose to show

In this paper we … - {" U' f0 _! R1 t3 i

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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. ) Q& {/ I) n, ?+ M

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

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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： % j8 W0 X1 Z! u9 ^4 z& S5 s @

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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… ( O6 T8 e& z/ b6 w2 w" d

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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.

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. / K" Z( p5 J+ Q$ O7 E: ?

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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…. $ f5 }$ Z8 N# L% H% Y9 P

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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.

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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… - O6 Q8 E- n6 u8 N3 ^0 @

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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！