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? / p" D( [, I$ }9 Q0 t5 Z8 S

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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。 5 k6 ^% i; l3 |- y

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）。 7 s: t, g, r) b- {4 H: f; e3 x6 Z

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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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/ i9 `/ t, _% ?3 Z is defined as ; x+ D4 _; u" V. Q& x3 z' ?3 C1 ] , u" D7 A5 e) M( d# g1 \8 a$ }! b 8 u3 I2 G9 {8 X) q _ is called 2 \9 u4 V" U1 l2 Y7 l: l' F . d8 e( k" B5 S8 r& L% v3 g1 z

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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. / G! ]$ G2 z! b X- O& S! z! V

The mapping , ad-bc 0, is called a Mobius transformation. & r+ x) p) k# y$ H5 E" q) o. P

0 R% \3 a8 S" ?+ O% |% @* W( E( \( \9 A0 Z- N

3 ?+ W. P$ h. }% O+ x is defined to be # U1 L* l5 r5 V7 l% a 2 `$ {: U4 x+ ~8 \ is said to be

2. Something something(or adjective) $ ]4 Z3 n+ g% y) Z2 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.

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. 7 P" d3 }/ n" L' J: }

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6 r' ]0 t" M2 ^' E; g! a define 7 e1 D+ ^: L Y9 U9 M+ w6 w! k0 A 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. . ]8 J+ \, G" S' ~9 }8 L1 E

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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： + r! b+ C; W" u8 Y7 ~: |' \

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. [* `- b' f, ~9 w) F- X3 {! N0 A" k is called / d8 i* y7 N/ G0 ~# _( J is said to be * {1 ?8 }! i( C" I is defined as - q( x: y& Z1 g$ K is defined to be

Let…, then…

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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

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D=

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is called the diameter of A. ; ?6 E5 |% D: N0 X( c7 }

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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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/ r6 B- h8 j; n! k( s is called + I6 S) t# e2 a, e is said to be 8 {8 G2 C" {. I% F! e6 V: _ 4 D' y8 k/ v- I# _5 ~ 6 d0 \" C3 W0 O @" } \ is defined as / H) Y4 u1 e4 r* j 9 T5 X3 o* |& J1 L- Y1 V0 J & U2 i$ a& |0 W1 |$ d5 {2 S, n" z is defined to be : t/ P$ o" [4 y

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If…, then… ' B7 ]6 z% U, A: \7 @5 L0 G; N/ x" Z

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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.

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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： - X( {1 A% q1 L9 x) C

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1 V! L) ?' N. q is called 1 s) Q W1 e7 J4 ?1 `$ [& H% @ is said to be

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. k+ z3 W5 J) f& ?1 Y ! v! m; T& w# k; E7 N0 u5 s 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. h+ L x) H$ G$ s: ~

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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： ! c2 S% _8 c) G* t

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? ; h2 R4 |% n! m) o/ j, a8 Y b

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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。

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.

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The space (E,f) is complete.

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：

“Suppose…Then…”or“Let….Then…” % p( w- [7 \8 U+ Q* M c

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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 ' E! J( m) Z8 }6 M) s$ A% G7 C

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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 , O4 o: w2 C% R, M% m# c. ?. i5 g+ T

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“If…, then…” ' A9 x, Z6 p, I; h

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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

“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. 4 B9 q& H& J2 x* ]: x: P

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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：

“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. 5 a+ |2 ~5 A& ^# u% H, }

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hehe123

数学专业英语－（c）How to write an abstract? ' K. k6 C* ~! I" s " ?3 ^6 L7 m* d |7 t7 K2 a/ 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。 ; P% A7 s: s0 O7 ~; I7 x4 N

1．开门见山，说明文章内容，可用下面的句子起句： 8 w6 i' @) I5 z- Q$ {& b

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, q' O" ?% o$ _6 C3 F2 v" C 0 B1 K6 a0 Y1 L prove @: O3 `! R& C; x show . C, h. O" G% S6 S7 V present develop & G! v5 l k. o2 U generalize investigate 3 N8 P5 k# o6 q! W 6 W2 g- A m+ P; r5 t$ H& l

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paper 1 S0 R' C8 b! E note

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The of this is to …

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prove 7 J& d! d& ~7 v: i6 `6 [# l show present develop generalize ( O7 b6 @ b* ~. ]" f. T" [ investigate

It is the purpose of this paper to …

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& \9 V5 b8 Q* C" o3 n8 D! ] # n4 e7 m+ X$ b$ |: e) r, M! U is concerned deals

This paper with… 8 D& b! J# k' n8 @

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In this paper we …

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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。 0 y, W" ~- B8 d' v, @, R

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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. ; S+ `8 U6 m/ e

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…first raised the problem which was later partly solved by…We now solve this problem in the case of …

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.,… 3 c, n- N7 x, h0 I. D

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In this paper we shall prove several theorems which are generalizations to the results given by… ( ?0 w* ^/ ~; k* k

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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… + F: d6 D* ^, p, _5 S$ d

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This paper improves the result of…on…by weakening the conditions… - _2 Z3 P! R8 }& q* {

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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. 1 j# p7 ]8 Q) u5 a9 R7 j2 s

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. 1 J- {( `) ?. C; L& S! u* j. u

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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. & {# A- p6 z/ g7 f

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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…. , V! ]! }5 Z) n/ k' q5 \2 P1 A

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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. 9 r M' q; ?5 ]' n. K0 o) ]

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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… 5 y7 ?) l7 P) Z5 |2 ?$ O1 N2 E

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！