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? 3 N( B. x2 W: K

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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。 7 m2 q0 f. N- S1 L% Z V" x6 m$ c- @! S2 G% G 1 x. U/ T# X9 I- n, o" J: G

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。 " H4 v& c: X( H% g1 P" e# G

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。 , f& T. f1 e# a5 H) b8 M, h- f

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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）。 ! q% M! I# M: C/ U9 {/ ]9 a9 ]. n

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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。 - X3 L; d' G0 Q* Z4 v, G! U* T" w) U

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（a）How to define a mathematical term?

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is defined as ; b$ q$ K5 u E1 M: q$ t ?' v8 _3 _1 J( o& u is called

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1. Something something p' q0 K: l" z1 A. 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. 9 X y# g: h) T" m1 Y. J

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

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is defined to be 6 G- V. s& B' A5 k# Z" k is said to be

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. 6 D4 O* T1 c' d7 F: w7 H

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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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& ]9 t- h# i: z6 C define call 4 Y7 H9 }$ E1 g/ b1 C : [1 q3 W) _7 |

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.

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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： 0 r, f/ z$ ?+ c7 A( ]. s

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is called + o# x- a9 k! `; u ) Y8 h4 o) Q% O' ~ is said to be 0 D8 F7 D: z3 S# u# P# I, s. K , A! Y& e% o' k. F2 \5 e H0 k : Z3 v( l5 j& i0 v is defined as ' F- M6 O5 X3 _- ~ 0 R; _" q* Z8 s is defined to be 9 V) j( B4 a% }* h4 d; C$ B8 m c3 x1 h 6 n- Y1 l |# {. l2 Z& H

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Let…, then… 4 h: w2 `/ E" a+ h$ d

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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. $ Q- U9 G& G, W w; @2 v* y' `

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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. ( [( T0 X+ i8 N. L3 O

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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： - I9 [7 i% t: M3 z+ i1 |# M |

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is called 3 Q3 T; ]+ A1 Q* X' H8 | ( Y7 d+ O, l! p7 c. v is said to be 0 Q- a0 p/ o" ~" P5 A/ O is defined as - C# ]9 @& Z/ g- ~ % R. A# s" v5 g8 F& X% ` is defined to be

If…, then… . K; R, ^ P# W, s4 Y8 \6 D+ i! H! A

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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. * X) N3 z* D7 j# y2 o W% a; }/ s$ U

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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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1 p" I0 Q7 U- f' L8 ^, x/ y is called + G3 c/ Q2 m" G is said to be

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Let * F9 p! ]) F6 h2 s: f( y+ x 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. , S/ [6 Z. w N I: r; g

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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? 1 c4 a* p) Y2 @, e3 q: z- h {, f4 w0 Q8 w% q0 J+ p9 E9 |

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。 / \* ^5 [* a; ^: k! m

1．某些定理可用简单句叙述。 $ L: a9 Z) ^" n" n) Y

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The union of a finite number of closed sets is still a closed set.

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The space (E,f) is complete. 9 D* {4 [# k! @ l& q) A" ?" |

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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：

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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]. 6 A- |! m( o4 B1 V

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+ p; O. R1 F

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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 ( v4 ^9 b: w, x9 t2 l& e" u

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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 ( ?7 I8 X7 T0 \6 f7 W

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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

“Let…. If…,then…”or

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“Suppose…. If…,then…” , @# o3 R9 H- V4 t

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…” 2 ~# t& s d$ V% x& R5 V8 c6 O

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. 8 G% s( r* ~7 J L. a3 l- u

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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。 1 B+ P2 f2 S$ } + A. X8 H$ z& X7 }, ^+ b6 m 0 }; `$ s& L) X

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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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+ `( @, D' }: F7 H3 N7 K8 | prove show present develop generalize investigate * z* W. N& W8 G ; i S& S3 f3 J$ B. F ) S, d! K; S6 m8 `+ \) i/ P. }4 b

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aim object purpose

The of this is to … : k% L! J4 N7 h2 L# A

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8 n1 s g" m5 F2 A. Y prove & c. u- C! [6 W' O show present develop generalize investigate

It is the purpose of this paper to … ; E/ f) o# n u1 Z' C

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" D$ F7 p! M) l" w5 D/ x is concerned deals

This paper with… . A- T6 p! S/ k/ ?0 f

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prove present ; v/ B$ K( c! q4 Z6 y" @( l propose to show

In this paper we … 8 |, v& _( n3 |# ~+ K' [: E

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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. 4 z" @- p: m, B$ `. U& J/ [' q

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…first raised the problem which was later partly solved by…We now solve this problem in the case of … 4 R; b; `9 z( \9 K

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： 2 k, S; H6 M ?' q/ p

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…

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This paper deals with generalizations of the following problem…

This paper improves the result of…on…by weakening the conditions… ( U# l! }# J# ~1 i

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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.

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.

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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.

This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form…. 3 @+ @1 n" w1 h

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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…

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The aim of this paper is to try to minimize the functional ; Q" l, H4 A3 ~3 j! O

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over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= . ! o4 s! R% \+ X/ M

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！