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? 4 s$ t; Z% J3 t& ]$ T

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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 X/ S* N8 \9 r0 |; c# O$ s * U2 E- `) ~3 T4 B # N& |! W' w- k( z, O% v

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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。

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。 6 }7 L o, V/ d

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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）。

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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。 7 m1 K7 a9 @; ^7 b' s% f( y) ^

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（a）How to define a mathematical term?

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is defined as 0 O5 Q' H" P! g* ? is called ) D6 m8 u4 E: I7 x5 g

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. , T. E& U, z) b' j' M3 }

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

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is defined to be & L4 u# N& ^7 _7 ]" u* P is said to be 1 w% T( p' D$ ~! V, ^

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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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: ^; n1 }* O0 e; D/ D define $ Z: R+ T3 b$ d- a, ~6 }) F- b / j- `. i8 b, m/ m* r6 j call + D! N- x7 |0 y- V! n; s4 P4 z( l

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3. We something to be something. 8 h$ }- w$ _/ b4 m) }/ L7 G8 ~$ g

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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. & D4 y9 v- G+ e: a* y3 b

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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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; f/ K' [4 X, P# T is called is said to be M# ^$ V8 U% F! ]& G$ x, Q5 r' N # }4 [% e" T' F+ b5 B# Z is defined as is defined to be 1 {) o4 C/ E/ K4 L1 `2 A3 E

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Let…, then… $ d+ q3 ^: @3 a0 I; @" o- x

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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. 1 u2 v) [( Q' m" B; w- |: z

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Let d(x,y) denote the distance between two points x and y of a set A. Then the number 3 C. c) {% ]. k L- ~ v' u

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D=

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is called the diameter of A. 0 {9 ~7 F2 i0 V& F& S% m* `

5．如果被定义术语，需要满足某些条件，则可用如下形式： 6 p2 l% g% \+ V+ G$ ~3 z1 X

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. N: F2 Y' D/ M0 F is called ( Y3 F$ Z; l8 J8 N5 O: } is said to be is defined as 1 _( L- Z( y+ k % F4 `" @0 Y4 M; B7 q 5 M7 y& x: v5 c; _6 R is defined to be - i2 Y( I) P0 H$ m6 k

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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. ; g5 N. S+ H2 ~- n/ O6 f5 ]. A6 c3 h

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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. # X8 [: R& f& T' f$ d

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： 1 Y6 h+ U6 \1 Y2 J' N( P$ h$ v

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is called is said to be

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…. If…then… … ; L) r( ^9 z% @' p7 R4 N! 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.

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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? K( `! n; j' F6 x# k! b4 ~ , W) |1 P1 Y9 g* s- k+ b

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。

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1．某些定理可用简单句叙述。

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The union of a finite number of closed sets is still a closed set. , \6 w7 G& J2 p2 W' ~7 ]8 e" x

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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： * A" G: H0 T2 h& a4 N* `) ~3 ~

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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…” 4 R: O% g' E3 {( T; `! Q2 _6 O b

If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 , m7 u* n% l& W

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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 & {: k4 U) Y( T! I. q3 ~0 b) Q" |# o

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“Let…. If…,then…”or ! P4 f8 M) Q; M; k

“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. 6 M* V/ q( d0 t2 U2 A

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： ' Q/ }5 x9 r0 [5 w

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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. ; F" W# t8 q" V/ a$ ?; p

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hehe123

数学专业英语－（c）How to write an abstract? - J$ b# A4 y' U* ?% x" L5 V3 }* ?# o - K% k6 ?: r3 U v6 t

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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 b1 A4 C$ X0 Z0 ^7 I; {& o% D # b! s' X; z7 I' A2 X: N8 ^8 ]

1．开门见山，说明文章内容，可用下面的句子起句：

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4 B( h! ^8 Z; s$ N4 h7 B7 E ; }7 g* @4 u+ F' M# Z2 A prove show / ~8 q( p' u1 r) d* i6 u present 2 e& \* `3 n: g" t( l% P develop 6 U0 M: w) h' f T% \6 ^ generalize ) x: }9 J P. u7 s investigate " O' E; S! o! i- G9 Z: K

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paper note

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aim object # J% _7 ` M- N/ m+ s' v purpose

The of this is to …

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7 {" Y7 c% J8 k2 ?: ] prove show present ( y% ^0 P4 U2 E y0 J develop # m3 w, x0 I" b7 e' C7 N generalize investigate

It is the purpose of this paper to …

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, P" f5 I7 c$ d. J, H( c N 9 K9 ?, n- I0 L( S' X' z, F is concerned deals

This paper with…

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, y7 n! K/ U# O1 l9 o ' i3 s2 ^8 V) G% h9 v: }" _ prove present ; ~1 M1 B. O3 a7 p propose to show

In this paper we … % k# x7 U2 ^# [$ t5 a

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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.

…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： ; D0 N) X7 ^% B- S' k4 N+ \ \

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… 7 J% y+ T# d& E7 q2 a

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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… U% R0 s, V3 a3 b3 ~& z3 ~

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This paper deals with generalizations of the following problem… 7 e$ \% e9 D! Z: _+ `

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

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例：

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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.

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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…. - i: X0 H! M4 V$ b1 v9 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. 2 V& \& ^ G. g7 D* M& t) E6 G8 a

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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！