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数学专业英语-(a) How to define a mathematical term?

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发表于 2004-11-27 13:39 |只看该作者 |倒序浏览
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数学专业英语-(a) How to define a mathematical term? ( Q+ k$ j% j" I, v# ^( I# n0 f# J

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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。 5 ^- v2 T* I% c% m9 K: G& { 1 j5 |( {" g) K( u( S& u+ l9 \; q( |4 i" O8 F n

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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

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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。 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)。 , d0 i @7 ^$ A% j

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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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aHow to define a mathematical term?

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is defined as . N; m4 Z5 i: B* ?

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is called 0 x+ @" J/ h0 O+ z; |& C

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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. 8 V, S0 ?0 t& s% z' ~: N

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The mapping , ad-bc 0, is called a Mobius transformation. ) E7 V9 l$ b4 L- c& ?

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is defined to be s# G, ^9 b5 ?, q9 U0 P

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is said to be . |" k, G: J6 t; r

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The difference A-B is defined to be the set of all elements of A which are not in B. 7 L5 H! H/ }/ p6 {0 X

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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

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Real numbers which are greater than zero are said to be positive. 6 E: o; _* @/ _5 g; q

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define / o6 V1 V6 I; L5 w7 s

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call 7 [. S) x% ^/ I' n& ]

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3. We something to be something. ; |! ^0 H2 i3 d/ W+ ~) o5 b& `5 i

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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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is called 9 |* G$ ~7 l0 s" C' {+ j

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is defined as : o7 i% G) U4 S( e( K; i

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is defined to be 4 B2 P' x/ _8 [

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Let…, then… 0 X8 e1 A" h2 K( a

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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- G: I) O% _0 }8 D

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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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D= * U& [. M5 m: ^9 i c5 S

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is called the diameter of A. 5 j6 A+ q7 W$ W7 I8 P9 X

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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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is called . i* G8 e$ x/ b" u1 d3 X# X

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is said to be [4 |" [, s0 Z

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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. " {# Z7 X" @ U& C

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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. c% A0 {6 n" |+ N

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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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is called

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is said to be

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Let

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Suppose

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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. 3 Y4 U2 x( ?7 @4 l A; I

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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: - q6 Q1 d( w) w4 u

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456

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 .

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数学专业英语-(b)How to state a theorem?

数学专业英语-(b)How to state a theorem?! D# K0 \ J3 k# @& @ 4 u% l9 _' j' C2 f: Y" R

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

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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: 3 |- _7 l: f8 ~2 ]; q2 q7 J9 c

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

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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 2 H% w4 ]! v; a3 ?) q

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“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

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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 * ]9 Y: i3 S- z2 ]3 S1 m+ ~( ~9 W

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“Let…. If…,then…”or 3 _/ `; E" `& [9 m& @

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“Suppose…. If…,then…” ' w/ X' H# C6 d" O# Z( e9 C6 P

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. , T& n# I8 p7 P$ f9 u9 z

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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( \

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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. . L: h5 e, E- s& b" [7 I+ \

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数学专业英语-(c)How to write an abstract?

数学专业英语-(c)How to write an abstract?! M$ b' n$ u0 F " }* B# r" s1 G' f3 W* @) \* }

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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 }' M" O5 [( E& \: Q

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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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prove

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show

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present

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develop

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generalize

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investigate

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paper

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note

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aim

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object

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purpose

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

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prove

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show

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present

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develop

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generalize

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investigate

It is the purpose of this paper to 5 M( p* ]5 _; f2 @* ~( O2 V3 G

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is concerned

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deals

This paper with… 4 \4 v) G9 a# T" Q1 }

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prove

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present

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propose to show

In this paper we … " \ g; l( {$ A0 | N' b4 _

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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。 4 z$ Q5 n4 j2 z5 \, 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. H" Z* `$ r: Z

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…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

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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: , L5 s* ^" z% m. G& m

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… ; I) W# S) P- L/ _! c* R h

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In this paper we shall prove several theorems which are generalizations to the results given by… 0 O" ?7 \" b# j( P; f

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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… ' ^! O# j! z* X9 y1 H' R& @ i2 S

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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…. - `( ^& t% C s

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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 * z% I' J- s3 |/ {( t- ?& [. ]2 q

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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+ @

点评

kittygoodice  很棒的东东  发表于 2016-1-20 20:08
天光li  ding~~~~~~  详情 回复 发表于 2014-2-6 20:32
mongo1992  顶一下  详情 回复 发表于 2013-1-19 09:59
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