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 prove…
The aim(or object, purpose) of this paper(or note) is to show…
The aim(or object, purpose) of this paper(or note) is to present…
The aim(or object, purpose) of this paper(or note) is to develop…
The a im(or object, purpose) of this paper(or note) is to generalize…
The aim(or object, purpose) of this paper(or note) is to investigate…
It is the purpose of this paper to prove…
It is the purpose of this paper to show…
It is the purpose of this paper to pres ent…
It is the purpose of this paper to develop…
It is the purpose of this paper to generalize…
It is the purpose of this paper to investigate…
This paper is concerned with…
This paper deals with…
In this paper we prove…
In this paper we present…
In this p aper we propose to show…
2．如果需要简略回顾历史，然后再说明自己文章的内容，则可参考采用下面的句子：
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.
…first raised the problem which was later partly 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 intends to remove some unnecessary assumptions(e. g. , regularity) from the paper on…
This paper deals with generalizations of the following problem…
This paper improves the result of …on…by weakening the conditions…
例：
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 behaviour of complex systems.
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 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 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 elliptic partial differential equations with characteristics of multiplicity not greater than 2.
The object of this paper is to investigate the behaviour 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 F of all absolutely continuous functions f(x) which satisfy the boundary conditions .
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。
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。
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。
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）。
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?
The union of A and B is defined as the set of those elements which are in A, in B or in both.
The mapping , ad-bc 0, is called a Mobius transformation.
2. Something something(or adjective)
The difference 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.
3. We something to be something.
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.
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：
Let…, then…
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.
Let d(x,y) denote the distance between two points x and y of a set 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：
If…, then…
If the number of rows of a matrix A equals the number of its columns, then A is called a square matrix.
If a function f is differentiable at every point of a domain D, then it is said to be analytic in D. 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：
is called
is said to be
Let
Suppose
…. If…then… …
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 beschlicht in D.
. 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：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 ＜ .
数学专业英语－（b）How to state a theorem?
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。
1．某些定理可用简单句叙述。
The union of a finite number of closed sets is still a closed set.
The space (E,f) is complete.
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：
“Suppose…Then…”or“Let….Then…”
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…”
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
“Let…. If…,then…”or
“Suppose…. If…,then…”
Let , , , be four distinct points. If all these four points lie on a circle, then the cross-ratio( , , , ) is real.
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：
“Let…, and assume….If…then…”
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. I f the derivative f’(c) exists, then f’(c)=0.
数学专业英语－（c）How to write an abstract?
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。
1．开门见山，说明文章内容，可用下面的句子起句：
paper
note
aim
object
purpose
The of this is to …
It is the purpose of this paper to …
is concerned
deals
This paper with…
In this paper we …
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 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.
…first raised the problem which was later partly 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 intends to remove some unnecessary assumptions (e.g., regularity) from the paper on…
This paper deals with generalizations of the following problem…
This paper improves the result of…on…by weakening the conditions…
例：
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.
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.
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 elliptic partial differential equations with characteristics of multiplicity not greater than 2.
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( )= .