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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? % J$ g* P/ T5 [) A6 c4 l# z6 b- }) R' i$ j: m

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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。 4 e7 f6 d( u% {5 v 7 R9 n( `& G* ]0 Y; ]8 }; M 5 i. z- P# J: V5 R! I, b- z

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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。 1 m: O- B4 t2 G+ \

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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。 1 D) a7 j, Y3 n% J/ a) a

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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)。 5 T5 A, l, g; i1 u& d) [' d

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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。 % L z% y i6 O5 ^4 P

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aHow to define a mathematical term?

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is defined as % O0 X* }( f: l5 A

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1. Something something 4 K* ?, ~( }" N$ n( ]. ?

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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 @! [ Q* d+ j: L& e2 S

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The mapping , ad-bc 0, is called a Mobius transformation. 9 W/ ], O% f6 c, s9 ^6 ^

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is defined to be - S9 M8 ?2 j/ c6 x% I, ~

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is said to be 9 D( V: @" M F4 m

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2. Something something(or adjective) & h6 Y- J4 s: H2 y9 Q

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The difference A-B is defined to be the set of all elements of A which are not in B. 3 H& M! |# m5 s0 _

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A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. 5 _1 \! Y2 Q S: g

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Real numbers which are greater than zero are said to be positive. * A- C' O% |, H/ ?2 P0 K

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define 5 ^9 [. `9 j0 K8 K! h: H

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call 0 y, [7 k( s' `

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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 s* g: ]! v) K7 j [* a

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We call real numbers that are less than zero (to be) negative numbers. % y% L4 i( y; s: }4 s0 P

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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: + d) L4 U' S, c

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is said to be 2 d9 Z: P6 s# L$ V. W8 x* f% g

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is defined as # e O; I* x8 g7 }8 W! q0 `. p

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is defined to be ( `5 h( T# F' T

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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. 9 R4 q- F; D0 w

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Let d(x,y) denote the distance between two points x and y of a set A. Then the number % g- n4 o7 E/ y3 {7 p

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is called the diameter of A. ' i7 t, n2 K/ l% j v+ d$ [7 y5 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: " z8 }* y- V- V

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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. , z0 |+ j a; d* I( m8 ?8 G4 y( t

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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: $ ?/ U# W5 m7 e0 L# m

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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 ^' z6 n. W' D- N- o' E) U

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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: 3 s5 h( G* |3 \

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?; o2 O: n' v. V4 K# Z. S$ ` 9 F3 V4 ?% ?9 Q" m: g

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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。 % n5 B; c8 c. E, J M! x 9 i$ S7 ]( P& r2 C* O1 Y. I5 m3 a

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The union of a finite number of closed sets is still a closed set. ! N d& E) P9 |' K* D

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The space (E,f) is complete. - _6 w* q; W" w6 _2 S

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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: : P+ y* w5 V' w/ I& l+ j

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“Suppose…Then…”or“Let….Then…” 4 d7 l' I5 j8 Y- i5 J$ C' O8 c5 Z" g6 z

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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]. $ O# |" ^( [; l0 V1 ~

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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 * ~' |0 D5 b5 S \. k9 l2 q

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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 ! Y0 Z) V% H, e/ Y

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“If…, then…” ( b% |( V$ @ H+ D Z; n

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If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 + P: b8 D" m( v& y1 c9 U# K

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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 % x& J0 x/ _% z% m

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“Let…. If…,then…”or ' Q: L T+ @4 c' z( Q5 K

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“Suppose…. If…,then…” 5 F, f! w! M+ k, B* h% P3 [. G

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Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. , G( l5 h* U: y4 S

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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: 6 N S& G' D4 J! p' U. ^9 B+ `: T

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“Let…, and assume….If…then…” 5 T# ?8 I. a- L$ e5 Y+ Q

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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. 7 s: ] \- K2 n8 Y5 P% j

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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?, H" o+ {! E& Y3 K , T' e$ T1 O/ y5 l7 `; i- L& `

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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。 $ t# p- J$ H0 l' c, c$ S8 g( [: H. R& T) ~. M * _! }0 p; F3 ~' {2 D7 }; W

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1.开门见山,说明文章内容,可用下面的句子起句: & m7 Q. y. E# [

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prove

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show

( Y% n( a, L. _. m0 a0 }! B$ \* S/ j# b

present

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develop

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generalize

/ f: i5 h& l |& P1 K) g

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 … % h+ i# G S3 W- p2 H

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# T& C" |' k0 S8 |

prove

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show

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present

" A6 j5 Z5 J. S

develop

* p2 c W: y# A' X1 p( o

generalize

6 ~ b- D! A9 V n7 J% s0 H

investigate

It is the purpose of this paper to ; E( y9 w* I; H' g

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is concerned

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deals

This paper with… * q- F# V; B0 o5 `6 C' }& w

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prove

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present

% b) C& u4 p! b+ W2 y, Z

propose to show

In this paper we … # z$ w8 @3 B* e0 L+ J. @

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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。 1 t. ]& L3 D; }" O

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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. ) P' P+ T9 c$ d9 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 … % y* z* } G7 U5 J

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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: l9 h/ [3 z" a, d" C: u- n

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The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… $ S; l% c$ m' S0 U

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In this paper we shall prove several theorems which are generalizations to the results given by… 5 n2 [4 f% {5 E& F

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This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… . }! t; d+ C' G

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This paper deals with generalizations of the following problem… 5 o1 ~& N) I/ p3 F2 O4 ?

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This paper improves the result of…on…by weakening the conditions… % N, \3 _* z* f* J+ z

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例: $ H X L* X# L/ f

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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. # \" ^ D$ z1 D1 v# ?

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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. * }/ j" s: g; R1 F2 q C0 }& m, ?

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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. / B7 F: Z% n$ u

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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 [6 k Y% h1 H( k7 d4 v9 I' O

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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. / _7 m Q) d; P; c; u

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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… 6 c6 m; r( ?+ F: t. p; S5 t

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The aim of this paper is to try to minimize the functional - v+ v- [$ T: r, q

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over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= . " U: c4 U) v$ t4 r! H. g

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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