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?

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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。 9 ~/ ?1 y! e" C3 \ Q8 M4 V$ T5 o0 r, [

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。 % ?' ^& z: w5 j

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）。 ! c1 g6 `) `( U5 E" K, ?: R8 r0 }" l! ~

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。 # K) |; y! J9 y& U# d

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（a）How to define a mathematical term?

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is defined as * |4 W# N" [" T* R ' P, [* L, x+ @+ K+ M is called

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1. Something something 2 A0 t4 j) [5 s2 O' S

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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.

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

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* X% v) t% ~4 [1 i is defined to be ' ~1 ?5 G E# N 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. $ ~$ P4 ?/ S7 k2 U2 e, H

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

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* x: B# X& I$ q define ; P2 l* d+ v4 v$ O8 X% ]! P c6 B$ d ! m2 X; g, | x7 Z call , p+ e4 l _4 Y' j

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3. We something to be something.

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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 7 ]$ g8 K; J4 v4 V4 A3 o4 p 4 a1 C# Z% ~2 q; Z" T is said to be is defined as " U# v% Z$ N) X5 Q( s$ ?* Z 1 n3 ~8 `# m& c" X( o2 J5 L0 T 3 n9 ^0 H4 {% b' V4 D is defined to be / r. c5 \) V% O' ^6 b 2 |3 z/ C9 u& g5 V: f

Let…, then… $ u4 Y D/ 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. / I J0 y7 `- u) J3 w

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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.

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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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+ C d% H+ z; [+ i" k# s is called 4 `# ?, w$ X0 a is said to be 6 _1 T* f8 Y" e! m, j! m' n ) u$ r5 ~6 s" I7 ~7 P; N is defined as ; U% c- e+ u1 w3 \& Y: I+ x " x6 k/ T2 M* g) u3 G- Q8 U is defined to be

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. * f* R# V- y8 P) ?* 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： & S4 [; P8 G e' @

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" N# d' T5 B% Q Let Suppose

…. 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. ( a2 s- z6 t0 `, h7 i

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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/ Q. N$ Q+ _2 W! V* g0 r9 M 5 ~' i; _2 w2 d4 W% C2 [: o, a- v

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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。 # Q/ l4 `8 u' j( {7 c N $ Y$ }6 R* |* r4 v h " T" r9 C. {% V+ }

1．某些定理可用简单句叙述。 3 i ?7 M Z! E% R4 O

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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…”

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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

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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…” " A7 n8 G9 {, `4 U+ k1 }! `2 p

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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 * M' O, p x5 y" H% f% e

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“Let…. If…,then…”or * R0 ], r0 @; h. A. _: g4 C8 k) S

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“Suppose…. If…,then…” 9 W% Z* R# Y4 u; T. n: A

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real.

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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： 9 H' a) X7 m$ o" o1 N7 E

“Let…, and assume….If…then…”

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. * F% |6 W1 Z9 x, p

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hehe123

数学专业英语－（c）How to write an abstract? 0 \& v) m0 Q! I7 ^, H. Y7 l# Y - r! ]9 D6 o' C5 ?% Y) }6 j, w: F' T1 n

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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 }, B1 `0 x( k

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1．开门见山，说明文章内容，可用下面的句子起句： 7 k2 @5 F1 |! v" w& o! D! E

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0 t8 H- C$ x% P prove + g p/ c1 B$ } show present 1 O/ S$ S" b c5 r7 A9 u develop 6 v% L' C. v( }1 W generalize investigate ' A' p, u8 D! e

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paper 2 t, X2 ]/ |# R! N8 }4 S9 v note

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The of this is to …

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prove / Y" d5 }5 ~: j: a: S7 o* h& O show 9 q& F! W& |) E, b present develop generalize investigate

It is the purpose of this paper to … . [3 z/ A. j; M' w' ~8 D5 S

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This paper with… % t" a6 v# B+ _

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prove present propose to show

In this paper we … 6 |! s8 O, K' S. q

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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。

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. 7 [$ w9 ^' v8 S$ ` V2 n- V

…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： ! Z6 Q4 `" u* V( @3 @! C

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… $ z6 h0 M2 B9 Q! Q; e8 [' L

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In this paper we shall prove several theorems which are generalizations to the results given by… # |5 s# c) D, |6 t' f

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… # _6 r4 Y% `: D8 u D. V

This paper deals with generalizations of the following problem… , I- `3 N3 ?& V N

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This paper improves the result of…on…by weakening the conditions… " o7 t2 ~) }: E5 C$ J

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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. 0 Z' x( K( x0 T' y; k8 w- a

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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. 6 B, ^7 m: }' [" p5 I/ h

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. / h# [; n! t1 g8 x Z0 D. b

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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

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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！