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? ( Q+ k$ j% j" I, v

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。 1 j5 |( {" g) K( u( S& u+ l

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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。 & d. I5 x% `, M8 ^( G' L0 b4 d

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。 5 r% z7 d1 ]/ Y4 H) U

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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。 * ^2 r2 g$ o! }

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（a）How to define a mathematical term?

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; Z8 J- u0 d; Z6 u3 i1 o2 \1 @% ~7 _ is defined as * G: T' I$ n( ]2 H- X9 U 5 T8 ~2 K2 v: V% M& T is called 0 x+ @" J/ h0 O+ z; |& C ) ]- O$ ~4 m4 z( h4 U! X1 V

1. Something something 3 U1 f: T: s" T

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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.

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

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is defined to be 5 H6 q9 i) o$ x! ~1 s) X is said to be . |" k, G: J6 t; r

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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.

A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. 8 C# e6 E! X1 a4 H

Real numbers which are greater than zero are said to be positive.

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4 P, I9 Z% D+ V7 r- X) ~1 y define call 7 [. S) x% ^/ I' 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. 8 U1 t$ Q: t- M* L3 {% J

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We call real numbers that are less than zero (to be) negative numbers. ! x: ?5 h2 J6 R% T7 Q6 K

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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： ' }; j5 Q; g- m! u. a; H' k( j2 s5 F

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5 |* g2 J6 p. Z; ~ is called is said to be + ?" c& O& O @! x4 O/ o4 M ; ?8 F+ |, g, `( G7 y0 r is defined as : o7 i% G) U4 S( e( K; i 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 5 y7 U5 X' s; I2 k

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is called the diameter of A.

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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： ' j/ R& v* V7 g

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# M8 l8 [( i- v6 E8 e is called 6 P- j" m0 a( a) u is said to be 4 f; Q2 k4 q' y3 S ! T q, R' R; q, h/ b is defined as 8 Y& D" V% V# r/ w0 F T+ |0 @5 X/ Z# D is defined to be - \+ u* ^7 l; R# ~ % B2 O3 U" z7 Q3 g. m: P2 C

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If…, then… " w& s: a6 v2 T1 o4 l- g1 o

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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： # q9 q6 _- L1 U1 `: f

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8 ~7 B A' V' U : }. \! A4 n- j1 } is called 1 |6 _$ X4 z4 ~- i) O is said to be

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0 w) W& y0 I i2 x/ t3 G2 T; q % n! k( b& }) J% W4 F5 ~ 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：

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。 " X* a1 H0 J' h7 _" K& ^ ( g% t4 U' o0 f6 F 8 C0 }) A3 Y3 e& V) M

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1．某些定理可用简单句叙述。 . P: u1 U7 o+ l1 r: e" c% O

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The union of a finite number of closed sets is still a closed set. % u0 [5 |# y6 d F

The space (E,f) is complete. " l/ @: g% }: \# J1 y, I M

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： 3 |- _7 l: f8 ~2 ]; q2 q7 J9 c

“Suppose…Then…”or“Let….Then…” ; T7 a& B/ l" Z. r& G z2 P1 l- {

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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

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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

“If…, then…” , M0 p- W9 w) A) ~9 S

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 / l2 R* z; V5 B+ w' c+ h A' M- O" i4 i

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.

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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： " y. q$ c0 Y7 C5 J& \& x5 u7 \6 [9 j

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“Let…, and assume….If…then…” 9 _7 a8 C4 q7 |1 n( \

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?

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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。 " z# c3 p+ G' o/ k6 ] 9 v9 R4 B' d0 X- y3 }

1．开门见山，说明文章内容，可用下面的句子起句： 2 O* X. ]: S- H# T( k

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$ R8 B7 B: @* f" X2 Y) ~$ ~ prove show present develop generalize investigate / \( y) U1 P8 v2 k/ r

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paper note

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& J4 L& u C: l% U7 N7 z aim object purpose

The of this is to … 5 b/ T3 a$ y6 e J) S1 r

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prove show & ]6 D" d4 z0 T present 6 @. R4 R# D; {3 I, G4 h. c8 Z) t' J. I develop generalize # c: o R3 R* Z investigate

It is the purpose of this paper to …

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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。

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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 … 1 Z3 c, i0 s5 }) g [6 A

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…

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… ( H B. ?* y: d$ {+ K1 M7 Q& Y

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This paper deals with generalizations of the following problem… ' `5 Q5 ? R, m/ ~3 M

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

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例： : E0 f( ~; l- b6 C/ h3 p& t) X: X% L

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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. 2 a8 c9 h3 G% s' S

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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. + o2 j& L" b. b* w) q$ S

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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. 8 J! O7 y: |# P& R- I# {5 Z

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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….

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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. . P ?; S- a3 C5 i2 h9 I8 i

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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… 4 `, s( Q' a0 ^

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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( )= . 3 v& F" W+ p+ @

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！