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? % J$ g* P/ T5 [) A

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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。

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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。

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）。 5 T5 A, l, g; i1 u& d) [' d

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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。 % L z% y i6 O5 ^4 P

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（a）How to define a mathematical term?

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is defined as ! z$ U7 n( Y' v% `+ ^1 M; e is called ; r/ s) r4 t) f! a

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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. 9 @! [ Q* d+ j: L& e2 S

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

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/ y6 E% o0 B- b5 B8 r& | is defined to be - S9 M8 ?2 j/ c6 x% I, ~ is said to be 9 D( V: @" M F4 m ; N. S7 K) U8 v) T

2. Something something(or adjective) & h6 Y- J4 s: H2 y9 Q

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The difference A-B is defined to be the set of all elements of A which are not in B. 3 H& M! |# m5 s0 _

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. * A- C' O% |, H/ ?2 P0 K

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2 u4 ^6 _" d2 U/ E0 e ]5 @ define 5 ^9 [. `9 j0 K8 K! h: H : p; R2 T, h4 z( x 4 O& X i4 F: q5 H3 m9 h call

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. % y% L4 i( y; s: }4 s0 P

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 0 c) x" W9 j/ L4 O! \8 B6 S 1 w" H ?; F9 p7 W; @ is said to be 2 d9 Z: P6 s# L$ V. W8 x* f% g 1 d( V1 E- F% r3 V: D is defined as # e O; I* x8 g7 }8 W! q0 `. p : _; z7 v' a. n + J. G: U+ X6 ^$ l( s4 o/ q$ U is defined to be

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Let…, then… 1 E+ F/ [" ~- L6 N9 j

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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 % g- n4 o7 E/ y3 {7 p

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D= & T) c0 B$ K0 [; s) b

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is called the diameter of A. ' i7 t, n2 K/ l% j v+ d$ [7 y5 n

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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： " z8 }* y- V- V

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is called is said to be 7 A: e7 h2 ]8 H is defined as - E6 d: r: O( z 8 n9 S$ |6 b3 X' X is defined to be 0 k* n% Q& w+ ~9 U) ^2 s

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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. $ z6 c0 ~1 H2 ^, s4 u

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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.

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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： $ ?/ U# W5 m7 e0 L# m

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x, @* i- ~0 X6 W L2 ? is called , Z; |4 G- D; O9 w- w# W9 V( | is said to be

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…. If…then… … ( A& ~/ u0 g o. h

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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： 3 s5 h( G* |3 \

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? 9 F3 V4 ?% ?9 Q" m: g

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。 9 i$ S7 ]( P& r2 C

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. ! N d& E) P9 |' K* D

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： : P+ y* w5 V' w/ I& l+ j

“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].

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…”

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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 ' Q: L T+ @4 c' z( Q5 K

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“Suppose…. If…,then…” 5 F, f! w! M+ k, B* h% P3 [. G

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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：

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“Let…, and assume….If…then…” 5 T# ?8 I. a- L$ e5 Y+ Q

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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. 7 s: ] \- K2 n8 Y5 P% j

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hehe123

数学专业英语－（c）How to write an abstract? , T' e$ T1 O/ y5 l7 `; i- L& `

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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。 $ t# p- J$ H0 l' c, c

1．开门见山，说明文章内容，可用下面的句子起句：

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8 k1 G1 x% ?+ }: S prove 8 G/ a1 N2 Y: o2 g' o4 M show ( Y% n( a, L. _. m0 a0 }! B$ \* S/ j# b present develop 4 _! ^8 R" [0 U2 L generalize investigate $ X4 S( ?) U1 v . q( s7 I- s% U; a# b

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2 y1 u) n# e0 J, d2 K 4 o8 T; f9 f2 q: k paper note

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aim object 8 u) K0 }; ?) x, Z: y purpose

The of this is to … % h+ i# G S3 W- p2 H

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4 x" Z9 F& K9 q, n% F& Q # T& C" |' k0 S8 | prove ) H9 J+ z7 q0 b7 c5 e3 |6 c% J show present develop * p2 c W: y# A' X1 p( o generalize 6 ~ b- D! A9 V n7 J% s0 H investigate

It is the purpose of this paper to … ; E( y9 w* I; H' g

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& T" ?3 M- @+ h M6 C$ k is concerned # r% s$ e/ K D ?9 H1 F deals

This paper with… * q- F# V; B0 o5 `6 C' }& w

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prove present propose to show

In this paper we … # z$ w8 @3 B* e0 L+ J. @

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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.

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … % y* z* } G7 U5 J

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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： l9 h/ [3 z" a, d" C: u- n

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,…

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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…

This paper deals with generalizations of the following problem…

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This paper improves the result of…on…by weakening the conditions… % N, \3 _* z* f* J+ z

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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. * }/ j" s: g; R1 F2 q C0 }& m, ?

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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…. 7 [6 k Y% h1 H( k7 d4 v9 I' O

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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 m Q) d; P; c; u

The object of this paper is to investigate the behavior at the boundary of solutions to the uniformly semi-linear equation… 6 c6 m; r( ?+ F: t. p; S5 t

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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！