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? ! z" @9 w `( Q$ C( }2 _ 6 q P$ _& T6 a& T

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。 F8 v5 T$ F8 {& a, b7 C8 e* o5 v 9 T- h- w1 S: U4 t1 P1 c ^! z+ Q* } ) e; ^! J7 G% Q, i) I% }

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。 8 l {* H6 I" N) x0 L1 I8 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。 + g- V; {7 }% c' ]3 G' l6 t

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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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is defined as / m3 ?- u) n7 z5 O9 S: r. H 8 L7 C5 V3 c9 e is called

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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. $ I$ z$ J Z( F& i

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

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is defined to be * |! S0 V2 S5 A % o' z4 |) I3 ]2 ?8 A z is said to be y; z7 T+ j* `: l ! ] p/ f+ I! G* K

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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. ! P) I: a: f- d2 ?0 s

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Real numbers which are greater than zero are said to be positive. . T' c. Z8 f3 y9 d) y: J

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, h( P: x x' p! i' V, M+ F define ' j6 a+ v$ q& J9 M1 v4 R; h call . H- M3 C2 N8 ?& }, N

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.

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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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0 y9 `5 ~& n& s: Q, h8 \' G is called / `! K9 V" Y R ( D7 J; N+ }/ U6 ~ is said to be + b$ ^9 t% ^$ t/ A; m* x . q8 p& p7 i0 n* m& V" p0 X is defined as + u( R+ d% o. {; A: O8 W6 q- S( f( F is defined to be . s" m5 T3 K) `, }

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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 / Y2 m: u! P. i2 X( S

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D= * t! y! I) x5 Y+ v8 y- F" D" m

is called the diameter of A. + w/ s3 z( t0 K; J2 W" y& K

5．如果被定义术语，需要满足某些条件，则可用如下形式： , u. \+ e" d D g3 D

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is called # F( ^9 u$ v& k3 \/ O 4 L0 L1 F3 z6 M/ j is said to be is defined as 1 K0 i4 `4 k. f D" }2 `# C/ H8 h" [$ f& T is defined to be + [/ S6 O, V+ {& V7 D& J. k

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If…, then… 9 \$ r c0 i/ ?5 `

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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： ' H) y" H6 z0 f3 g

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$ f' V- P( a. ^. h* U" U7 Y # V4 @" Q) }, |* }5 r& v+ R9 A: q is called is said to be

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# ~! I% J+ k5 Z$ f- y1 w! @. |5 W" k* `0 | 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.

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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： : h$ c" D7 w& K7 [" D

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? , M, ~( x& B: ~8 P3 O( |4 _

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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。 - l! z" K H# I/ S* G% A& \ - {4 K- P; \& n 8 ]' X% [) E4 n

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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. 4 H# f8 v3 O3 z2 }5 j, N

The space (E,f) is complete. " e/ [) z: w4 ?2 z2 I+ M5 n6 y/ f4 W

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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： 9 s* F2 J0 O, r4 |' _5 k- n

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“Suppose…Then…”or“Let….Then…”

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 + m5 B1 Y f6 F/ h& p4 V

“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 4 A: M+ h3 L3 r3 r6 p: y" k9 ^8 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

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“Let…. If…,then…”or

“Suppose…. If…,then…”

Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. 7 U% d1 |( Z7 Q/ T& ^

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： # H0 e- }2 C8 v; g, x" X

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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. ! S+ h) A9 n! l6 ^6 \

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hehe123

数学专业英语－（c）How to write an abstract? % g; G9 ^) C v% x3 Z0 J

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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。 % I( `, @" b3 j: Y 2 d3 M. A8 A- F" h7 u0 u) o+ X5 G

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1．开门见山，说明文章内容，可用下面的句子起句： " i% c' K- a5 ^8 ^- U

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prove show 8 T9 E- O* N+ ^# B* G8 I: a present develop generalize investigate

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f' T [$ Y% s" q4 \7 i" } paper note

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aim object " C: H8 L- x0 X S" ~ purpose

The of this is to … * ^8 u8 X; R7 _( G" |5 D

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1 Y0 G$ j. z. z* ?1 X prove a! G% a d [0 d show present develop " b4 f% Z# o: G0 R% v( {0 V7 d generalize 6 b6 U9 Y! X' N) v/ N* ~7 p investigate

It is the purpose of this paper to …

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This paper with… ' T U0 y2 c; t% @

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) f' x Z) p# D" T1 O5 b* x, u+ F prove present propose to show

In this paper we … 0 H8 U- B$ c$ D+ N/ O1 T. A& C8 q! I

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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。 - s [8 g) b. [1 K

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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 … - u* e6 [, K" h _! C4 O

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：

The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,…

In this paper we shall prove several theorems which are generalizations to the results given by… + `5 }0 \; G0 ~9 \; }

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… & G3 l0 ~* K& U4 u; o* L0 M

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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… # t- k2 ]# R( {2 F) |

例：

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. : n& r0 E+ ?( b9 c2 a4 z2 m6 @

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. ! `8 ^4 ?, S1 B% o$ X" n

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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…. 7 ~1 a$ H6 a, j% ?

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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.

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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…

The aim of this paper is to try to minimize the functional " m+ I7 y; a( B$ A0 }, z

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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！