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? 2 w1 M a4 C3 R9 X ) ~) s5 p& B' V! F, G% p; Y2 ^

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 Q# F7 I5 m& Q5 W$ ] ( L0 w+ m) }1 g; e' `* n# l; P: _

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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。 : Z/ Z' K: O6 y

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）。 & k. c; ~, \2 [+ A! @* F# U; K

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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。 . K) I" {/ ]* f8 L7 ~# e

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（a）How to define a mathematical term?

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( r5 ]% M8 ^9 ~5 S. } is defined as 5 f7 n5 B3 C1 y$ ~ , i# A1 ?$ Y- h' ^8 z* S is called

1. Something something ' ^# O4 T: M4 \3 v4 }

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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. % h. O; @1 z7 v! C0 z- E3 b2 _9 R2 p

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The mapping , ad-bc 0, is called a Mobius transformation. 7 C& D7 q Z$ n

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is defined to be 7 I8 w& A( L# a. D d! Z3 N # s- _' p0 A. J) c. `$ H4 _2 N/ S is said to be ' k6 I4 |6 s( E2 e* i

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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) `& U) G. g+ `. H+ W7 [% H define - e+ y j/ ?; R7 [0 X$ N6 E ~# h call & H1 U7 m! S9 i* z0 j4 N

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3. We something to be something. 2 L: [! |7 n8 U, P# q4 U$ J

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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.

We call real numbers that are less than zero (to be) negative numbers. 2 Q9 Y( [: p7 W! U4 A/ N9 W- O

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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： : n; o1 n/ t" x

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F2 W ]" n1 M( u; F is called 2 X$ v! B# d/ @ is said to be & ^, ]4 P7 ~- J2 [+ c6 e0 w - W( H6 e- P; z/ Q/ ? is defined as $ y9 U z. z( ]8 c ; j' v; S1 }9 }1 H1 t9 V 5 w& L* M4 G- Q, m# G! [ is defined to be % m& m8 M" v* K5 y0 f, n1 Y; R% B/ W5 Q B7 g' n: n6 u( J4 g, R

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Let…, then… + W( L: A7 `* u

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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 # H# Y/ }2 W1 u) c7 z$ W

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is called the diameter of A. , {9 D7 s7 v" H

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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6 t+ [* T0 p% z* B/ q( s is called + B# R8 Y/ S5 s6 r; [& f. \ 5 H9 r S1 {( \( L is said to be / F: P1 J+ Z& C2 k& z9 B ) d! F9 w2 L- A0 n3 g is defined as 6 i) q" ~- ?3 l' \4 b9 e4 s is defined to be [9 U- x5 g) @/ m# t

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.

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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. 5 `* H) ^4 r- Z) |% l* I8 s

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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： ' D8 n, o! p8 }7 k& ?9 }6 t( V

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is called is said to be

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Let Suppose

…. If…then… … 7 f- W4 N" H/ i Y

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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. 2 V8 `5 K7 [8 q/ C* ~: 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?

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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。 3 c0 C/ Q5 ?6 Z) ]% x$ g8 ^2 V " R3 R8 N b% u6 Q$ {; u

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.

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：

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“Suppose…Then…”or“Let….Then…” 6 w/ J8 [* V; D. _6 W

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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]. % w/ v$ Z, C$ i! p' k4 \9 n2 ]1 K

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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…”

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 8 ? H s4 R" A7 w) R4 p

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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 " a1 M i. a6 g, Y5 K

“Let…. If…,then…”or ; K1 e$ o. X1 |1 s# W- [4 b

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“Suppose…. If…,then…” " s# c- {- _$ H2 {

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： % n2 K, b; |# a2 _1 J! c2 n% C

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“Let…, and assume….If…then…” ; M( {* j( j/ }. X; ~6 \% R" ?/ ?

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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? 7 x! \+ y8 y7 t7 }3 L8 k+ c' D

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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。 # ^4 B! M6 g# F& i k. i & Z6 t. I& H2 G6 N

1．开门见山，说明文章内容，可用下面的句子起句： - u& t e, ?7 L; Y2 C

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% `. ^( M4 u$ A- x paper / ]" N l9 C3 e) N' R3 n note

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- s6 n6 P/ o, E8 D aim object purpose

The of this is to … 9 _3 O& @1 L+ [: J- H

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4 y. O+ T1 S. j( `0 I prove show * H2 C/ @ B, _ present + U# R6 O+ O& a) J1 _- y develop 5 @0 N& l7 b7 n' H generalize * Q+ Q T7 M0 a7 B investigate

It is the purpose of this paper to … ) a5 P- G) g6 G* ?" Q

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This paper with…

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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。

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 …

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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.,… - t7 w; u5 A, H

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In this paper we shall prove several theorems which are generalizations to the results given by… + c7 [) j1 p) j4 x7 \

This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… ! |! F: w% Z* e- x, x

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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… ! j6 ~2 X9 d( }

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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. 8 W3 d! ?: u# w2 Q- v% W6 v& G

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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 r/ I# w' T5 c4 E3 q- L/ 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…. 2 U+ l7 p' Z0 R9 W4 a

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. ' @/ B1 V. N4 Q' U9 H

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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 * d* U' G% N+ Z1 H

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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！