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 [6 |+ l% q; m* W

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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。 , l" m- t) T: V8 L& d4 d' p" c

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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。 - x" S, T ]) Z& y

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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）。 * H! p1 |$ F8 Y9 `4 P G* ~

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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。 / R1 x1 t! j, n- G

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（a）How to define a mathematical term?

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is defined as 6 {6 n, z& O! ~9 u8 S is called : H8 n% A6 D1 z7 y

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1. Something something 9 k; [# ]& L# d; i; ~( f

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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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4 x' j: u* ?0 L7 t is defined to be $ _- i! y0 `2 |! q6 F , q* c* i3 R- H. Y is said to be * ? g' \2 e# z2 L5 T( f

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2. Something something(or adjective) 8 D5 c8 R' n1 E) O5 ? g) 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. + X/ J3 o3 b, W6 p

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* {5 |+ J3 `7 U( M) J define * _, e: G2 a/ T& i call # _; x8 V5 m: U* W: R

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3. We something to be something. ; m# Q s; w6 o. R

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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. : U9 Q3 W+ S7 |* r' z# p

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We call real numbers that are less than zero (to be) negative numbers. " c: \5 x6 J+ t1 B) |% _

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 3 D0 N. o6 m+ M& y is said to be 3 w; T- E3 Z% @. ?4 i: d0 \ ) x$ Y6 u+ R, T$ j* k is defined as & s% b& f5 I6 ], ]+ T3 h . B1 R v# x) T is defined to be

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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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is called the diameter of A.

5．如果被定义术语，需要满足某些条件，则可用如下形式： 2 i' N Z9 a$ @# c8 g; ~3 z! I- J

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4 }' _1 z( o% P0 j% k% ]3 i is called ' z( U3 I, V, ~- ?/ Z ) }2 {$ ?3 y6 u) O* {$ y ) _8 a9 R9 Z( F9 y, {) ? is said to be & s- i3 l d- U0 H1 D is defined as is defined to be ; W, n7 k5 s$ Q, v5 m+ b $ ?( O% c# M& U& M; i

If…, then… G: I! q4 r- |: ~

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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. - N5 [2 w, P7 O4 c: O

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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.

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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is called # K. ]8 R0 x) h! ~& r9 V is said to be

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# E' L. L M6 X- m9 ~ Let 7 }$ m& q! t ~; ?2 @+ x2 P Suppose

…. If…then… … 8 m9 s! F6 x6 R& D

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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. + j) P0 j" L& h; U. c- V

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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： & B* l9 Y. Z9 {: S

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? / o" X a) U" y% F

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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。 . r5 n8 c) \7 @7 H! o " A( a* e5 V/ V7 k/ C, t4 B

1．某些定理可用简单句叙述。 E5 @; j- ]: \1 B

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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： . N: g' @) x1 f' B/ O) R! Y) x

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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 4 S+ T5 e- w4 A* F& |9 R

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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…” ) H/ B- ^' o% P' P

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 ; U2 y- B9 a% D- B+ v

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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 7 h" l- X9 k% [6 ^

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“Let…. If…,then…”or 8 c8 d# U9 U/ e1 R) Q8 x

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“Suppose…. If…,then…” # W$ @( K# J3 C L) R6 Q- v

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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…”

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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. . Z: A1 Z# }6 j7 W

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hehe123

数学专业英语－（c）How to write an abstract?

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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。 $ j2 d- `* d5 Q# \) x* r$ I* i 5 c0 E' ^: G; W3 I

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1．开门见山，说明文章内容，可用下面的句子起句：

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/ H; _# Q2 U) i; @& u prove & ?, }, {" k: j4 Z show * R; W) B" X5 M& M8 B6 v! X present , ?' I1 r# Q* h, t# P8 X develop 8 }0 v' O. p3 M6 d9 x: } generalize : y* k6 n% G. D investigate

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" l8 F, Q5 \! r" V0 x8 u$ _ paper 4 h6 R' ^' M; X1 E- h+ C/ S note

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, D; N2 }$ w+ A$ t k; d- K! \ aim object purpose

The of this is to …

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a! v" b$ P6 P prove show present 2 b* s8 v H& n# I- k* g5 W! b develop ( T; v; l( w7 Y: d! | f generalize & K# W1 h) j, @$ y investigate

It is the purpose of this paper to …

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is concerned deals

This paper with… - t4 m' v& N( G, t F7 S0 I

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4 d+ K4 m$ _: T3 Z1 |+ m' U6 w prove 7 G3 |- R1 U+ [ present & q' v* v( b- x7 L t% t propose to show

In this paper we … ) ]. n3 `' N0 t6 [0 Q7 M* z

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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。 9 N! [ {1 u4 f" O9 h, z7 I

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 … 7 ], v1 x( G0 w

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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： 4 r7 \. g% p% Y- b9 _/ h+ g

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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… ! g0 e. @/ I# Q( S! `& F2 M- U

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… 1 e& y# G( x5 B9 o5 g; b5 g

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This paper deals with generalizations of the following problem…

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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…. L8 F( y2 t F' k2 k

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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. / Z5 i) J2 `: g' |5 q6 F7 s7 U- F

The object of this paper is to investigate the behavior at the boundary of solutions to the uniformly semi-linear equation… ; \5 E H: I/ x" Z

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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！