1、摘要论文摘要的写法不象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，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含一些 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开门见山，说明文章内容，可用下面的句子起句：The aim(or object, purpose) of this paper(or note) is to proveThe aim(or object,
2、purpose) of this paper(or note) is to showThe aim(or object, purpose) of this paper(or note) is to presentThe aim(or object, purpose) of this paper(or note) is to developThe aim(or object, purpose) of this paper(or note) is to generalizeThe aim(or object, purpose) of this paper(or note) is to invest
3、igateIt is the purpose of this paper to proveIt is the purpose of this paper to showIt is the purpose of this paper to presentIt is the purpose of this paper to developIt is the purpose of this paper to generalizeIt is the purpose of this paper to investigateThis paper is concerned withThis paper de
4、als withIn this paper we proveIn this paper we presentIn this paper we propose to show2如果需要简略回顾历史，然后再说明自己文章的内容，则可参考采用下面的句子：The problemwas first treated byand laterimproved by. The purpose of this paper is to prove that it holds in a more general case. first raised the problem which was later partly
5、solved by . We now solve this problem in the case of 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，则可参考采用下面的句子：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 This paper in
6、tends to remove some unnecessary assumptions(e. g. , regularity) from the paper onThis paper deals with generalizations of the following problemThis paper improves the result of onby weakening the conditions例：It is the purpose of the present paper to point out that certain basic aspects of informati
7、on-processing systems possess dynamical analogy, and to show that these analogies can be exploited to obtain deeper insights into the behaviour of complex systems.We present a general comparision principle for systems of boundary value problems and employ this result for proving existence and unique
8、ness of solutions, stability and existence of periodic solutions for non-linear boundary value problems.We proved a theorem for generalized non-expansive mapping in locally convex spaces and extend the results of Kirk and Kaun. We also obtain a theorem which generalizes the results of Brouder.This p
9、aper is concerned with the existence of multiple solutions of boundary problems for the non-linear differential equation of the formThis paper is concerned with the question of local uniqueness of solutions of Cauchy Problem for elliptic partial differential equations with characteristics of multipl
10、icity not greater than 2.The object of this paper is to investigate the behaviour at the boundary of solutions to the uniformly semi-linear equationThe aim of this paper is to try to minimize the functionalover the class F of all absolutely continuous functions f(x) which satisfy the boundary condit
11、ions .数学专业英语(a) How to define a mathematical term? _9 Q8 f 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“ifthen”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。 : l: l4 J: # f$ A+ E#6 + Y 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“Ifthen”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“Ifthen”的句型。 至于下面将要叙述的“Letifthen”，“Let and assume, Ifthen”
12、等句型，其实质也是基本句型“Ifthen”的延伸。 / h* y* A/ s/ w: i 有时，在定义或定理中，需要附加说明某些成份，我们还可在“ifthen”句中插入如“where”等的句子，加以延伸（见后面例子）。 : a0 i* |1 s; R/ z“ “ A3 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。 （a）How to define a mathematical term?1. Something something # L# % u- S/ u7 p / T) b4 |6 R: I+ J“ k9 s0 o9 3 z % N! |“ P
13、/ a* The union of A and B is defined as the set of those elements which are in A, in B or in both. ; u; The mapping , ad-bc 0, is called a Mobius transformation. L* T* b+ d o; f( m“ ( u, V“ t3 D8 M is defined to be is said to be # Q! 9 ?; 0 b! I 2. Something something(or adjective) ! G; The differen
14、ce A-B is defined to be the set of all elements of A which are not in B. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number. Real numbers which are greater than zero are said to be positive. 7 l8 U q+ B) S U2 , # P, D% ? call ) o7 p. c3 E3 C3 m* ?5
15、 j . O( N/ M1 : X“ S + _ - u8 j# O) M# e4 F is called 9 t2 w* O# v5 e m+ D Let, then 3 : t$ S9 Z2 b 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. , _; l7 f) g+ N“ / t Let d(x,y) denote the distance between two points x and y of a se
16、t A. Then the number D= is called the diameter of A. 5如果被定义术语，需要满足某些条件，则可用如下形式： is said to be 4 r3 L y; S* L# ? j 5 n9 z+ . j6 T L r9 K. j% c“ is defined as + j4 A5 r# i. J6 J$ a 0 u! d/ b. 2 V1 + n4 v is defined to be + n : R |0 b8 J 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： 4 e: p: D# U! is calledis sa
17、id to beLetSuppose. Ifthen + a4 C- G# p+ $ B0 K! k 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. 6 z! g3 n8 I5 x . 如果被定义术语需要满足几个条件（大前提，小前提，
18、直接条件）则可用如下形式： supposeassumeLetand . Ifthenis called is defined as $ * M is defined to be - R) 0 F, J8 m+ _8 K0 O- A* U“ l 7 Q; 1 s! t5 G0 q 0 C 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： . n% _% 9 x+ I( “SupposeThen”or“Let.Then” Let f(x) be a continuous function defined ona,b. Then f(x) attains its maximum and m
19、inimum on a,b. 9 p+ v, a0 B7 T- s# O/ e* A. S 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 5 o- C7 N. Q/ P3 D5 ?$ t3 N! ?$ Z “If, then” : ) o% O. X7 w, I9 5 x If P(z) is a non-constant polynomia
20、l 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 q- n6 “Let. If,then”or : T7 X% x/ J$ W, S “Suppose. If,then” Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real. ! 5 x9 # g: D, D“ y 5. 如果定理的结论在不同层次的
21、几种条件下面成立，可用如下形式： 6 D# M B7 “ S! Z8 u “Let, and assume.Ifthen” 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. 数学专业英语（c）How to write an abstract?* t U/ : Z0 s* t6 g1 c* s
22、4 f% n: d7 K F 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。 The problemwas first treated byand laterimproved byThe purpose of this paper is to prove that it holds in a more general case. first raised the problem which was later partly solved byWe now solve this problem in the case of 3.如果文章推广了别人的结果，或减弱了别人结
23、果中的条件，则可参考采用下面句子： 0 & The purpose of this paper is to generalize the results obtained byto a more general case,i.e., 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
24、 paper on This paper deals with generalizations of the following problem $ 6 # y$ C7 A+ x & s9 d! n h& c“ g This paper improves the result ofonby weakening the conditions 1 N% T D! j& H& % C, l0 B! h: g: h 例： * j& s8 ?# 2 s( i& It is the purpose of the present paper to point out that certain basic a
25、spects 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. ( V& # Q9 q1 V0 n6 b J& b) We present a general comparision principle for systems of boundary value problems and emplo
26、y this result for proving existence and uniqueness of solutions, stability and existence of periodic solutions for non-linear boundary value problems. P, _7 r% N“ Q/ P$ We proved a theorem for generalized non-expansive mappings in locally convex spaces and extend the results of Kirk and Kaun. We als
27、o 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. This paper is concerned with the question of local uniqueness of solutions of Cauchy Problem for el
28、liptic partial differential equations with characteristics of multiplicity not greater than 2. - m8 k u8 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 over the class of all absolutely continuous functions f(x) which satisfy the boundary conditions f( )= ,f( )= .
