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 h6 k9 S; Q9 e

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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。 ; h8 b: S6 z6 K) p+ A6 t7 B # |, k1 H& J8 W: T8 k* H

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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。

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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。

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（a）How to define a mathematical term?

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, p+ @. v: Z+ v7 A* l) ]; C( N2 l, h is defined as 4 `* m) x8 L; d" [$ L8 z! U: u/ t is called ' H6 {( d# C9 x' f8 \) R

1. Something something 0 T# C! `' j: |( `. L3 {

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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 Y: L) D' ]% J

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The mapping , ad-bc 0, is called a Mobius transformation. . l# C( A- S* c0 R+ ~9 f

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; j7 |( v3 w; Y" j2 \ is defined to be 4 N8 O8 {& E6 N0 S; Q% I9 _$ A $ {3 M* ]. D/ z3 F : X5 i3 I5 G' }& z is said to be & _. d8 \* I6 U% R3 N' ?

2. Something something(or adjective) 8 @9 N8 V/ c; d7 m6 c! x

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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.

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; h, _* x! I% N4 ^8 | define 5 R+ D8 X( p) g x $ M3 f, ?0 b1 m9 K- M6 T - X/ g+ |; z, o' q1 s3 Y+ R call 8 C( L1 U, O. t4 w

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3. We something to be something. & y. b: e* M0 w. G; `* X# I) G8 D

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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. ) O7 n% n6 \8 M9 a/ Y

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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 l9 u: M# p4 j% G6 g is said to be + ]0 ?! J1 W/ ? is defined as is defined to be ) J% o% E7 i, w4 o% t g' j

Let…, then… B% O% C* [; ^9 b

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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.

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. + w1 ~8 f% R. w, I/ A3 h9 I8 t

5．如果被定义术语，需要满足某些条件，则可用如下形式：

is called is said to be 1 r9 o- @5 O* C. t) r3 m : ^9 V7 I( f8 J+ i+ J + w6 T! w$ Q# ~ is defined as P7 n: B+ D6 f4 K+ u. m, b k is defined to be D1 t6 C5 e6 }& |

If…, then… & [9 }# k9 [7 q* S: x1 G3 q

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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：

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4 n7 {' o8 \. ~' x6 s+ v6 b8 \ 9 o+ |5 S" A! y is called 8 [6 |, d2 d# J2 ? is said to be

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8 R- {+ ~5 r; a. [: B 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. 1 S K& ? n% ~3 r: h, N! C

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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： # R/ u5 r, c3 q* 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? * n2 U, Z, f- \( ~5 t5 c7 ^" w

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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。

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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.

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： ; w2 ]$ u/ g0 v0 ^* a4 ^0 ?% q

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“Suppose…Then…”or“Let….Then…” 3 o) X! [ D0 Z8 }0 ^8 R

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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 ; t% d+ G; V. G9 \# E* B, t

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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 + S# K3 B% I6 H7 V5 l

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“If…, then…” - Y; W5 A( {; e; [& X

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 q% U# d% ?# |, [7 |/ `1 E( {

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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 / \) Q1 q$ I5 n- @, [

“Let…. If…,then…”or 7 _5 }: {. _0 F2 z

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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. I, t, j! o; }: u7 m2 T0 @$ B. o

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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： $ {$ @7 E6 G M& F! ^

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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? 3 d9 ?; e3 f, U * Q8 p& E$ x. L' |' b8 u( W5 S

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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。 , E/ m7 l$ |- ]2 s9 ~% |

1．开门见山，说明文章内容，可用下面的句子起句： # b T) E; i6 Y" _9 U/ z7 g

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* c& r. S0 F1 F r 3 C2 [9 X) C6 @' s prove 4 E3 f8 ?9 P* K% \ show present . v+ b7 S6 {" X4 U4 L4 e: ~ develop 0 F. Q e# G5 O3 v generalize investigate

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paper ! @4 |& Y& B5 F note

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The of this is to … 0 p8 j/ {2 @+ m& P7 `; n

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: w5 ]) V! X5 x: {2 Y+ }* t; ^ prove show present $ M( q" Z- b! P( w8 b% i develop 6 Z4 N5 B: S) N generalize investigate

It is the purpose of this paper to …

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) T5 A* a8 h9 u1 d is concerned 0 ^0 {- g8 m0 T deals

This paper with… ! [( B Y& F: H" L

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( M N# }$ a0 ]6 r3 [6 I % k+ s) X9 Z: i4 M7 S prove present ; F8 m" t& c( Q% Y$ s9 ` propose to show

In this paper we … 4 i# _. l7 w0 U7 L5 V% 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。 4 c+ p; {3 ~) f$ t/ i! c

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. % f' O: T4 _0 d" x% H b) D, a

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … 7 A6 Q: T: U! M) `0 [. D

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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：

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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… # d. b! A/ v/ V0 G4 p3 \

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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… & G& m; K; x1 k2 W

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

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例： * q0 Z. p" U+ a- M2 t9 U9 }& R; r/ n

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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. + C5 G- @4 N& ~) f% n

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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. 0 E/ y. X/ l: p& F& k3 N1 R5 d

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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. / ^9 j, W5 C) M0 I: k

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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…. + x$ Y6 u: l" V8 l8 b" V

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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. 4 C/ C/ L6 b5 S% P+ o6 t) T7 t+ b! r

The object of this paper is to investigate the behavior at the boundary of solutions to the uniformly semi-linear equation…

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！