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? 0 X, P. b4 v; _4 g) G. l/ Y

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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。 / ^, X Q4 y; I% D5 B5 W- f

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）。 - q/ q1 z- G. l U

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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 ; n9 C! d3 V& u2 R' A/ m# p% _ , h6 }$ r" ~" O' i' o, J6 } ! ~ i; z4 S& w! _0 o7 G is called

1. Something something 1 N. W$ R4 u8 e8 g0 W

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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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6 w- g# A1 H# d) @% i7 h# p, U3 t. Y8 Q6 c0 T: Z( X" j' P

& L9 b: S5 Z: @! w3 W3 F is defined to be 5 e; x0 a* v5 d4 m4 H. t 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. 5 I" z' r& i) V3 b/ z+ _ i

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define ! B Y* c+ h5 c7 Q$ T , H6 b9 l; d4 a/ ?# `0 X7 s ) P; `+ f# Q* y7 T0 i% L F6 Y; Z call I+ g* e& Q; i* k3 E9 Y2 k

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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. 8 v- c+ k; e; ?, P6 {3 v8 O) w2 H

We call real numbers that are less than zero (to be) negative numbers. ) A1 Y8 |, C8 M; X) t8 C

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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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is called is said to be H& o3 V- j4 H& B is defined as & k# ^: V R, h2 [ is defined to be 6 R( C, b, E/ K7 w7 F

Let…, then… : B9 E& {' ]) L+ _

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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. ) L7 N- F5 q: S0 E

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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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7 z* g, Z8 \% H1 H is called is said to be 2 m' d u: r3 ^1 s( b1 t # \1 T( R6 f. h4 ?3 `$ | is defined as 8 h5 F8 J/ {3 a9 `) X5 g 1 j, o3 [, P2 n is defined to be % ]0 Y% u9 Q# |1 W . k7 A/ S$ @5 Z

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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. 0 R- {; ~/ p; C& [, r1 {0 ~" ?

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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+ }0 {: i/ s; R

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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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2 C3 A; T- s: q$ @# |( D( t is called is said to be

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…. If…then… … - `. x( q& O0 ~) o$ u( [

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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. + a% K4 o8 M3 n/ y

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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： : l9 A ?' V! K- D% R, 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? 6 x0 Y, [2 ]: P$ |

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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。 ( o$ X1 i, `, f, X6 U+ n _" _6 E

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1．某些定理可用简单句叙述。 3 Q: X4 X1 q- s. i* a4 ~& _% A

The union of a finite number of closed sets is still a closed set. , M% s1 o) x# i9 C4 I! Y- l2 i& e* }7 l- o

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The space (E,f) is complete. 7 Q+ f- w, v" A# n* L

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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： ) L% o! C; }) k& E, L9 m

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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…” 6 `& B! s, ^& s% O6 x& ?5 _

If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 * X& ~( t% v2 A8 y7 p

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 ! P, s* F9 p, t+ ?3 P

“Let…. If…,then…”or

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“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. + H4 ~1 f3 c( M; p1 E

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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： ( S& v4 }+ C. l2 t7 S O) |/ A

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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? / {. A {/ q1 q B/ p- J r ) d- v3 Q. Q/ N! b5 z. 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。 $ T& Q# a: V" D$ H; t; Z& _

1．开门见山，说明文章内容，可用下面的句子起句：

( P c( R. `% \; [' ]3 h2 ]' W5 y prove show 9 [5 f( `0 ~. K, @ present $ e* R( g6 j" P develop generalize investigate 4 ~' x. H+ M d / |% L! r! F% e

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, B4 f6 a* W2 ?- |/ {+ U paper note

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The of this is to … ; o \" `- i4 L* G

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9 F8 M$ c/ M' q prove show + T2 c$ d; M, c/ g3 m" A& F present , e7 v! y& M7 F7 ]: n develop + A+ k! e# t5 `# X( J" w2 g generalize investigate

It is the purpose of this paper to … 6 w0 R2 K. G4 K9 U* h9 Z. Q4 D

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# l$ ~* ^/ i/ _( s! o3 \+ K: d is concerned deals

This paper with…

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prove present propose to show

In this paper we … y$ b0 O7 ?% |, u7 Z8 y

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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. 4 s2 Y' z+ s* t. z9 b; g/ W

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … 5 x9 l0 p+ e% g* j4 ?

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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： $ I( G3 j) K/ j1 e8 S+ `- K9 h

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… * b. `; w5 d) V6 h6 O! v; A, y

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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… 8 z A1 ^1 I, ?1 E- c

This paper improves the result of…on…by weakening the conditions… ( Y3 N4 N+ v1 R6 D. f

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例： : @0 y/ C& S% e! v, \

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.

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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…. + U; J f' L% y: T+ `, y8 s

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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… ; I" O4 T) [7 E! @8 i9 [5 N% z

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！