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? 2 j' ^: g( D5 D5 N1 U- q ( v$ \7 v, V6 P. _

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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。 ( U" t/ M! `: J# n

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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。

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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）。 ' C. `" r b4 I) M

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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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" R5 \& z0 q3 O/ f is defined as ) n7 I8 k; L/ J1 z0 p d is called # x8 P( r) y7 C# c/ p

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1. Something something ! L8 t3 L/ J8 j0 {1 k

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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. : v7 n$ j4 S3 l5 @& T1 c3 o

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$ @% |8 f9 n- T% D& W& | d is defined to be s% b/ `( V9 I2 D* f is said to be

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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.

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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.

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/ G* L* Q& x. `* t. G define call % w! F, O- T. H( [$ y8 h# Y

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3. We something to be something. - X- J1 C" e. v8 x

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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. ( a) ]! |$ j! I. Q- F

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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： $ ?8 l6 Q* ?" Q3 S

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: X/ v; g5 }7 R9 Z is called 6 @$ v9 z9 n* [6 {" l& ? # ?' g$ a/ h1 O8 S/ C 9 a! I; C) m, c2 Q6 J9 w" Y$ f is said to be 3 u& j: _' \7 \! ?0 d" p2 A 2 J {7 ^8 M# D) I& r0 z7 P is defined as $ P0 H0 Q! X# |2 j 0 o% t& b! w$ r" i2 F! c is defined to be

Let…, then… % o2 W6 i: b; H2 Y9 z

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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. 3 U" c% K; V! \$ D4 \- X5 M+ \

Let d(x,y) denote the distance between two points x and y of a set A. Then the number . G7 h0 @7 I2 F& Q/ {2 i7 H

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is called the diameter of A. + R3 P- n9 D8 G l

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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 |4 C3 d" b, }0 X; |3 { M 9 O. O, s' h" q7 M# r is said to be 1 U$ P+ \+ }. ~# d( N3 r; a/ ~: Z - I7 P+ ^3 _+ U0 D) w: g+ L8 ~ is defined as % o. {8 f& W" q d * m9 h$ Z* D# }3 w" U2 C5 r ; a M4 S' s7 ]& ^; T is defined to be ! z0 M' D2 |. d6 S# W" N! Z' @

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.

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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. 7 Y: N# P1 K- ^! e

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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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…. 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.

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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： . \. U( y; |. j9 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? 5 L, F: f8 p, R/ O: w

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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。 L5 {5 z! e# ~, C2 _

1．某些定理可用简单句叙述。 ) {1 G E/ p7 A& Y W

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

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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…” 7 [; h" Z% p( v

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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 6 ~* G: q) n Y. o5 D ?5 y0 J, D) e$ O8 B

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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

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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…”

Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. " p5 j/ V- n2 u' k' R, G

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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： & O7 v3 \+ R) H3 L7 g$ i) O5 F7 Y

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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? E$ \3 o8 Y& U; s' Z3 |: ~

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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。 + l+ }; V+ l/ D6 z4 o

1．开门见山，说明文章内容，可用下面的句子起句： 3 }+ E5 T% s6 p$ U) W& Q. ^

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& E9 f1 } B, V prove show present 9 r4 h' Y5 q$ y* J! Q' k$ D develop ! q6 O8 ~. M6 H9 r! ~8 x generalize investigate " |* U; M. k# b( Q. a 9 U( S$ N! U; a( [+ Z& c

paper note

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aim object 3 O* b5 h+ Y0 v% a' V8 c purpose

The of this is to …

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0 `, p7 T. Z5 K+ B5 o prove 7 V" a0 Z- o* ?3 T- y9 H" i& \* D show present develop generalize , l+ {" j+ D5 b. C6 f investigate

It is the purpose of this paper to …

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! i- c/ Z4 b0 [6 a! N% a " N' t& J$ \4 s! }+ [3 s$ `9 `( Y/ G is concerned ' w, W q* P0 w- s' [ deals

This paper with… ) g! K! i# ^) L* Y3 ~/ ^

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% P& `! w0 I( D* `* a5 ` prove 5 H+ J- \, _ f1 J# f% u present propose to show

In this paper we … / F5 E6 b5 g8 v& M+ E

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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。 5 V) u6 I/ X6 u+ v7 v' X

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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. ( p3 x/ D2 D+ D6 y

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … 2 ?, y- L* w+ S8 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：

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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…

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… ' V! j# R* m4 q1 L2 S, b

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This paper deals with generalizations of the following problem…

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

例： : \; a7 Y6 s$ `: h% }) O4 n/ {

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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.

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. 5 _: W2 _( v) }

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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.

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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 5 z8 M$ {8 y# |7 W

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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！