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?

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。 2 [0 k/ ~' v1 B7 a , }1 Y) z( B0 m3 r2 E

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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。 8 Z R* s8 E! o, [# l

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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。 6 s A6 z f/ r" ^

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（a）How to define a mathematical term?

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- W- H2 p1 }/ v5 E1 |& o5 g' q is defined as ! C! u$ Z6 M2 z+ X. [) ^& @, y( w* s' A 0 ~9 c1 e! }, Y5 j7 P. {+ y9 z is called

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.

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

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7 P4 l7 ]% S/ C, T$ S& ?$ h8 | is defined to be d$ ^. }2 i" s6 W# P is said to be ! }0 Y3 |8 ?1 |( p8 U5 @9 X8 a

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. 8 U9 J6 G) P0 Z

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define t0 s; _9 m; e4 A7 v1 k# ]/ n & Y+ b, x1 j0 g x, R% W2 D; _ 2 H8 s1 U6 I1 w& x" ^ call & u1 P! o& J7 a + R( P: ~, f8 o- a! J, O

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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.

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：

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is called - x t$ J! Z' X" @) A6 ` , n- p+ J4 V1 Q is said to be : M K/ P: s/ Q7 [ & n, O, M! u( x' N ' }. p5 F( T$ P3 A8 q1 }5 p is defined as 6 t7 a2 W. t, u9 C" T2 V * K) S3 j; |+ ^8 L1 ? }3 i is defined to be 5 ]3 ]+ z3 [2 ]/ x: M; q 4 i6 ?7 \# `- |( z4 B* ]# k4 y

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. 8 \8 g' ?( j; W: n

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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： ( t* }3 [. X2 v' m

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$ E8 N) ^) i+ _, P( D) { is called ( K% [2 E* J, S8 A6 h. h - i3 t. A7 x- W is said to be ) n- w& t8 \; H' ` 7 I" q" Z: v% _/ s( @ - ?( ]7 f4 N% v7 X# A; C is defined as is defined to be ; b2 X& U7 r' p+ m3 m/ N" j / [* m0 _9 L. N

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.

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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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Let b+ F1 r' ^3 q6 M: A- j) ^1 n 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. * w+ c8 z! i# s# @

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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： $ X" k- }& E" \/ Z- U+ W

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。 - M2 w# m) [' Z; _

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. 3 q" R2 f3 T! w' g. L

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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：

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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].

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 0 a$ |4 _6 ~! X5 ^

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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

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 , e: v ]. O e, R& n( d

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“Suppose…. If…,then…”

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. . Z s, m: z6 ]! R

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： / q( o1 g+ k# i9 S, n- t

“Let…, and assume….If…then…” 8 G& F' M3 ]5 t5 M% O J) ~

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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. ! H* D- [- @% C; r& P

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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。 4 \5 P: S# g, d# J h * R2 F! R4 w. Q: p) c; M, n3 x$ H2 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：

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3 Z- }4 y/ j9 L5 I prove " u/ {) A8 b% T( e6 B# \7 K show . U' P+ Z) ]! O7 B' n, _$ w present develop generalize 9 x$ C) m9 e* N! E* [# D$ N investigate 7 `: m l& v, X# `

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aim object # c) \; ~( p: L) v purpose

The of this is to …

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1 T3 z& l4 y5 S5 p/ H, ]* p prove show % \$ @( E9 c7 d2 d3 Y8 @0 A! z, V2 V3 ~ present . V; d- {3 `% {' M7 B9 H0 Z develop generalize ~; m1 }: D. ^+ a! g investigate

It is the purpose of this paper to … ) V o! F: z/ A( P5 L0 z

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is concerned deals

This paper with… ) y V5 [* F5 ~

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In this paper we … 5 e( J8 u q) _( E. v$ C

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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。 % \: V' T4 {* q) c6 J* z) G1 G

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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. # f8 k% z) \/ h: T& E( E

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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： - i" I5 O, e- [" Q

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… 1 `& C( Y5 d, Z) t. ^/ v6 g$ `. z

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In this paper we shall prove several theorems which are generalizations to the results given by…

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…

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

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. . _2 I& I( }/ d" T- `0 T1 B

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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. ! C# R+ L$ v' c8 U

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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. " r3 M! G0 w5 T4 [) h6 b; V$ g5 h3 n2 q

This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form…. ; T# P, x% G; u

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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. N0 I$ a5 N# L6 q

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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… " ^ o" R* ]& N1 W

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The aim of this paper is to try to minimize the functional . y: p7 f5 @6 y. T* D4 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！