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- h# P) e" B1 ^

0 _5 H7 l+ @8 [9 [

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。

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。 8 B9 L% w% P. a7 ]( ^6 l% S

: ^1 s. B; w$ w* a, y

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。

( w8 Z) `- m% R) J4 V0 J

( {$ E* i# X& v( S% C _* l

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）。

6 E& N; G+ u. F4 r

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 a, b. v R: r1 J

+ M. s( W; K4 o6 {# M

1 |% F$ v" z R: V+ r

& u- g$ \+ R \: E. b

（a）How to define a mathematical term?

) M2 a9 L- V( x* G, k

: y0 {$ l U4 d* r

is defined as ! I# Q( R' Z v0 V% n3 A/ R is called 9 a/ c" a; \7 _6 E& f% B* ^

1. Something something ) F% J1 J& k2 k. F

- j( ^: @5 K: j7 |6 _( ^

& o. B# M1 s: Q

The union of A and B is defined as the set of those elements which are in A, in B or in both. : q T2 V" ?0 n ]9 c2 N$ [

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

" z9 z0 d+ D* i/ ~( b' i2 X

" d3 g k5 I8 y+ ^' Y

7 p! G: \, m( A9 B

is defined to be & y* A$ F( O6 S8 x \6 a is said to be ]; E5 k# ?; H& R Z8 A/ ~1 d

2. Something something(or adjective)

- M2 T! d8 [( \4 t6 C- w' q

7 a% g; _9 {2 U6 o

9 ?' a9 y7 j. m" p* ~( F

6 M1 }1 R' c0 C& w. m

The difference A-B is defined to be the set of all elements of A which are not in B. . V, N: u4 X4 Y' ]' k6 B

! N2 q/ a$ E1 W* t x

A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. * V! M2 [6 v) C5 @

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

define : e0 C! z0 b, ] . f4 d8 t1 H5 E6 R call ( S4 P* G7 {4 c- r2 j

# J \" |3 G8 c+ G. m

3. We something to be something. 7 F% k/ W3 L3 n: Z5 [- n+ S

0 J& k# f# ]8 f& D+ M

1 ^- _3 T) S9 E; b* G( x& T) g

1 z2 H3 s5 T$ { H

; O" ]5 p+ m/ Q/ a3 M1 I

We define the intersection of A and B to be the set of those elements common to both A and B.

5 z1 V5 x2 A2 Z/ [# r8 ?4 j

We call real numbers that are less than zero (to be) negative numbers.

! k/ ~( q7 u# l2 s

! w, U% \4 P7 x1 R5 Y- V

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：

, l+ h$ g- a( y+ ^; H) y0 ]

7 O2 K! u* @; {7 l6 z$ w

: S8 t9 `& Z o is called . O* ]6 k, m: K( q( b is said to be ! _3 N6 n4 s$ x: j % A2 ]- g/ B8 r* c! W8 E 1 w ^3 f3 w+ ?5 B3 c. d! y0 [ is defined as is defined to be 1 F& M3 A5 D0 ~7 M: Z 1 T( {5 K3 T5 H' V+ K; I' \

Let…, then…

" k8 f9 M" O. K' o7 R+ p1 r

) K6 G: y9 M% `9 T

3 L8 O# E- ^, j# u7 P. z

4 P& E$ g: @6 Z! O2 \

4 ?! S7 I+ \6 c+ L

, S, y$ U5 f5 b

+ \: V, X2 ?! D, K' a

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. 1 Y& X, W4 V1 I! ]* a$ }" B

0 c- ~8 \9 c- e4 b8 Q# L+ F

/ w2 S+ G6 O$ F( | k3 ?

Let d(x,y) denote the distance between two points x and y of a set A. Then the number 9 Z( [7 i& ^* o' s

( }* z" @6 r5 D: i( F! y3 h! m

D= 5 H# x% J7 z' V1 V

) i# l$ A8 _' _$ K L" R

; s, N) o) `5 Z5 Q0 c

is called the diameter of A. 1 l& z+ w1 g2 p. M

# L/ |/ _, l8 }+ f4 o" x+ _

5．如果被定义术语，需要满足某些条件，则可用如下形式：

2 c3 A, t# x3 ?4 g6 _0 Z

2 q8 v" R3 L5 W' h$ T6 ?) d0 T

8 ~- H9 |* G, q! x( t2 k is called 8 U) z W' x( p4 v' h m8 _9 p 6 k1 H. L y$ ~; m( Z! s 7 S) E* i* B8 L- M: n is said to be ) Y# u, e0 B3 a' R: z is defined as is defined to be - B/ J8 Z; i) _

3 [- O* n; v7 G) p) }, r

If…, then… ' B; G) a2 Y4 E; @

: q- a" U9 y6 i: X4 I" _

. i( |8 m6 N+ }2 w' j" }7 d

: ?/ l h) u. H5 H& o

; D: s: t: b0 e$ ?1 O% p

' e1 K3 S8 x- K' h6 q. w6 I( h

/ F% g% |6 h* a' X! R

4 F H: b+ S. q8 l8 D

* A. l, B) r. D2 s6 O: I" \

1 U( k/ B7 _" Q$ P) \ x

3 Z; a9 p/ Q- E4 z3 y

# S: } S4 B: T

If the number of rows of a matrix A equals the number of its columns, then A is called a square matrix. y* q, n! b4 u e$ R, }

If a function f is differentiable at every point of a domain D, then it is said to be analytic in D.

! {. a( |2 Q" {* f0 F8 }/ t/ @, F

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：

/ g3 Z! o. f4 Y& {

, K; A% ^! q8 [% c; _

" V# c" H/ D; ^# h) M

! W6 }7 s# ~6 D& i9 U w4 H" L0 B1 E9 r& S' C9 w

8 g/ {: o. A! C9 Y! @3 d9 F) H) a0 k is called is said to be

- |" [1 H' f# x0 }+ X- z) B2 F! s/ h8 ]% N9 Q+ v9 e+ m: I# F/ `

Let Suppose

…. If…then… …

3 a+ i3 t$ Z; X' t

% O# C: F4 S: k: v+ m5 X4 {

( U5 W1 F7 D9 S8 a

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. 9 ]# T `' C. j2 _* R- a5 \5 l

$ `3 k; @8 `8 p. |$ Y. e P

2 O0 o- M, B* L) I

, H q+ e/ @3 w. `3 _4 A

5 e( M6 I# l) U A7 j, T% x9 g

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： * B! | k( Y9 Y' |0 |( O

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? ' _: Q% k8 N6 h8 U2 z0 \1 ` 9 e1 d7 P, g3 G! p

' E9 `6 z' l5 L4 r0 H

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。 ) I8 H8 b( s# \( @ ! R/ a1 j& M0 J4 x' F5 p$ R3 n

1．某些定理可用简单句叙述。 9 h7 i0 P* i1 l0 V

$ Y6 X: l8 Z P' M

The union of a finite number of closed sets is still a closed set. : E4 ?; q$ p. Y& T

/ l# n8 \, E7 b6 h0 Q

The space (E,f) is complete. 8 z' D- _% f( v' r- j x! L6 n' z

- W3 q& j3 @. L* S

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： + f S: W2 {4 d; M

0 X. `9 @, @5 s# G6 Z6 w

“Suppose…Then…”or“Let….Then…”

. g; O) |+ t5 U9 \) B( B3 s

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

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…” - z- @! C' D" U3 }

4 _/ I: I9 W; `4 d

If P(z) is a non-constant polynomial then there is a complex number c with P(c)=0 / i9 H+ w1 p! w* p# R+ I4 o8 j6 c

^# Z/ m8 K& O4 Y2 f

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

/ H5 y1 W- ^: m5 e# N$ R& U

# _ b3 k; c6 ?3 q) _4 w1 g! ]

“Let…. If…,then…”or

“Suppose…. If…,then…”

8 V9 {3 P* ]% ]* M( g" x* l

6 S1 U7 T; b# J% ?( h; g

Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real.

- j& f3 Z& G0 o1 C

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： b' j4 R. C! y% G9 r

]% t U! v+ c0 ^+ `2 G& E1 a% h

“Let…, and assume….If…then…”

* t6 R2 R6 v$ v/ `

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. 4 \: ]" U2 V) U" m# z) @; V

! {% [4 e1 I' r, e

/ v1 H9 d) F. T* j0 I% ^

# s& c* T/ o0 m- `$ W* k

4 [/ r9 M" g+ d' H

hehe123

数学专业英语－（c）How to write an abstract? 5 f [% r) D. u4 W: d $ S" }; Y" u* x

3 A/ G- B9 K9 I% K$ I

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。 $ b$ M) m- f1 @2 S 4 e/ U5 O# W9 } X" _; G

1．开门见山，说明文章内容，可用下面的句子起句： - `. ~- Z/ [1 H% d) z+ j8 A

9 |7 Z$ V2 Z0 E5 ?" P

~/ c" R+ s8 w8 T5 I) @

9 v6 t+ i% s/ ?9 k* N1 A/ N' Y0 {. S0 }: U+ g4 J( N2 [, ]. I

- S7 X: U L+ U0 v) U; l8 J 4 J/ ~4 t2 l. }( C3 Y+ s( I" k5 W prove & N. D. c) [0 a. S' C; \ n8 c- n$ q show present develop $ ?/ ?1 v! D* y( V" U, u+ g6 b generalize investigate

( I9 v/ w4 j( {6 k( H

paper note

# D2 t, ^- u7 q |& Q0 H# c% D5 W" [$ E( R0 h2 q) b) @: I% a! k' T; F

aim ' f E1 b" o. R6 ?$ m V object + g1 S0 \4 K% ~5 }+ i' w3 d" b purpose

The of this is to … J2 W0 ]% t- ~3 o

4 z2 h$ m e3 X# p

, ]3 `* f* Y+ {, f

, j R; v% g0 j. f) G

. J4 ~6 W. c( J; L0 t: V

3 A% }+ p) x7 [0 ~+ \3 p% `4 @

7 k( w' L; l. A* K

' f& ^& }& q& x A2 k9 }

" p' W* N5 e, o+ W5 v7 c. u) D5 F9 t

" o1 r+ F1 V6 ^8 I

$ K: R7 x7 C6 p- P+ O

5 v+ a- j4 R2 p, i) |5 k

8 _7 c. ?% ~ R- x* N. [% ]

: Y. x: h. t; t. J) c

prove show present + q8 q1 G. \7 \. `0 k2 | develop / |6 g" H ~+ X$ x9 S generalize ( w% r! z# w% k9 o5 x$ o5 C investigate

It is the purpose of this paper to … : a9 }+ V. H1 U# L$ T$ u

, d" F1 }# ^8 `3 A' Y+ X6 O# |1 @

+ r, S5 z& _3 p+ ]0 d; i0 g0 @

, X t2 [6 Z+ P* U

% m2 t# o4 r2 d

5 j( [8 o/ ~9 I/ J

0 Z9 B b0 J9 [' H) e

1 Q+ j+ s& y4 r' N

, e( A9 x- u6 t- f3 y1 q

4 x$ b2 Z+ ], V& N. a

2 ]! i. }5 ^9 V3 C8 F5 c; L

, |4 S. _' B9 O

' f, W( g1 \4 u7 i9 H4 _

" i3 @; Z' o9 E) a1 h6 g" E

$ ^/ k' H4 o* j8 x

0 \3 K4 B# ]7 [2 F8 g! B; n$ K6 M u9 Y9 ?0 B

is concerned deals

This paper with… 9 u N- ^% f* T

4 D) r5 m4 S: m8 t: s$ q5 f* I( \$ w! |

5 r" ~+ ~+ b, v' b! T5 { D

1 R& M7 m6 b+ g8 v

6 V& A% F" e0 J

5 Y! `$ c. T T7 c: [1 S; E1 t

0 L" I- c& c b4 j1 }. K* y& ?

prove & Z" ?0 Y, F8 [( A. q( @( ~# b present 0 U/ @, J% L: j" Y1 g# r propose to show

In this paper we …

2 O) ~, y/ s5 v

0 z5 i- S! j, E% ? b3 H

* ] |% ~( X% t

" }1 [6 ]: z' O' k

" Y( b6 i$ [: Y( A; D$ v* p e% H6 E

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。 # @9 h& b- j) P# J& a

A7 g: Z+ L3 ]: O% j7 ^( t

+ {4 O+ G/ Z* N! E% M

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.

2 G8 n3 l# J/ R3 k, w0 O1 a

$ y$ `8 b1 r7 b- }

…first raised the problem which was later partly solved by…We now solve this problem in the case of … * B( V. k4 }) K" ]. M+ |/ p# I

v. |4 c% R1 P2 a, ~+ x! G

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：

3 j0 w# W% O }% D

# \0 Y4 ~* ~3 K( A8 |4 G9 k

The purpose of this paper is to generalize the results obtained by…to a more general case,i.e.,… 9 X. L( [3 W! v) _$ S

$ X4 a% ~( u$ ~3 ^# ]) Q+ D

In this paper we shall prove several theorems which are generalizations to the results given by… 8 S G& i' S% E5 ?9 Q. p

5 Y4 t/ q/ I' @3 R4 ]. d" b' N

) d- O6 R1 X8 |0 n" L( u1 ~9 n0 a

This paper intends to remove some unnecessary assumptions (e.g., regularity) from the paper on… 0 w0 J& Y1 m1 B) t+ m

% k# z, D4 |- x6 H4 l% A! K

This paper deals with generalizations of the following problem…

5 {* ~/ C: B% X

5 P" F2 a, O* ?4 q! s

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

J/ u& W9 x( [1 m

例：

" a8 s8 a8 K I4 w. P @

" M5 H; g. z. q: T! V

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.

8 u8 O, h( Z% ?( Q2 G

C2 D2 K) A+ `5 p# X7 |- E

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.

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.

( O' g: Z K% c6 Q* \& Z

This paper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the form…. 1 M7 ^, |- }" V. D1 l% r; i# b+ S, ^

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.

, A% g6 x4 }2 F+ k( {8 m

The object of this paper is to investigate the behavior at the boundary of solutions to the uniformly semi-linear equation…

The aim of this paper is to try to minimize the functional " a. V) {! g( j8 E1 q/ e

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！