数学英语翻译.doc

数学英语（下）

3.1.1
1.调和函数，即拉普拉斯方程的调解，在数学、物理学、工程学的许多领域中扮演着一个至关重要的角色。
Harmonic functions—the solutions of Laplace’s equation—play a crucial role in many areas of mathematics, physics, and engineering.
2.本书的主要目的是让学习调和函数更容易些。
The main purpose of our text, then, is to make learning about harmonic functions easier.
3.唯一的准备条件是在实分析、复分析以及泛函分析的基本理论上有坚实的基础。
The only prerequisite for the book is a solid foundation in real and complex analysis, together with some basic results from functional analysis.
4.为完整起见，我们包含了一些低年级研究生课程中经常忽略的分析学的专题。
For completeness, we include some topics in analysis that frequently slip through the cracks in a beginning graduate student’s curriculum, such as real-analytic functions.
5.本书的作者除了写这本书，还开发了一个软件包去处理调和数论中出现的许多表达式。
In addition to writing the text, the authors have developed a software package to manipulate many of the expressions that arise in harmonic function theory.
6.附录B告诉读者如何免费得到我们的软件包。
Appendix B explains how readers can obtain our software package free of charge.

3.1.2
1.在过去三十年间拓扑向量空间领域取得了许多进展。
During the last three decades much progress has been made in the field of topological vector spaces.
2.在本书中作者打算写出这些出现在拓扑向量空间、序拓扑向量空间、拓扑基和拓扑代数中的反例。
In this book the author has attempted to give due recognition to the authorship of most of the counterexamples presented in this book.
3.对于本书中的大部分反例，作者不打算给出其原作者的规属认定。
He makes no attempt to give due recognition to the authorship of most of the counterexamples presented in this book.
4.本书假定读者已掌握一份拓扑学的知识，读者可参见B[18]了解一份拓扑学相关的知识。
It is assumed that the reader is familiar with general topology. The reader may refer to B[18] for information about general topology.
5.在本书末的参考文献中，参考书与论文分别开来，参考文献中的书籍记为B[]，论文记为P[]。
The books and papers are listed separately in the bibliography at the end of the book. Any reference to a book is indicated by writing B[] and to a paper by P[].

3.2.4
1.一个集合S上的关系指的是S到它自身的一种对应，例如“亲戚”就是所有的人组成的集合上的一种关系（如果能提供一份完整的家族谱的话）。
By a relation on a set S we mean a correspondence of S with itself. E.g., ‘being related’ is a relation on the set of all humans (provided that we are equipped with an exhaustive genealogy).
2.最后这一点说明，我们总必须确定我们正在讨论的集合。
This last point illustrates that we must always specify the set on which we are operating.
3.S上的一个满足反身性、对称性以及传递性的关系被称为等价关系。
A relation on S which is reflexive, symmetric and transitive is called an equivalence on S.

3.3.2
1.非零非单位元的整数P，如果它的所有因子是+P和单位元，那么就称它为素数。
An integer p that is not zero or a unit is said to be prime if its only divisors are + p and the units.
2.这个过程将结束当且仅当某个步骤中至少一个m是素数。
This process will terminate if and only if at some step at least one of the m’s is a prime number.
3.所以上述过程一定会在有限步后停止，而且m一定有一个素因子。
Thus the above process must terminate after a finite number of steps, and m must have a prime divisor.
4.我们下一个定理表明在求一个给定整数m的正因子时，哪些正整数需要思考。
Our next theorem indicates which positive integers need to be considered when one seeks the positive divisors of a given integer m.
5.全体正整数的几何是可数无限集。
The set of positive prime numbers is countably infinite.
6.在把正整数分解为素数幂的乘积时，可以找到与素数相关的这条性质的一个很重要的应用。
A very important application of this property of prime numbers is found in the factorization of all positive integers as products of powers of prime numbers.
7.在本教材的其余部分，我们将广泛使用素数的这些性质以及不可约的多项式的类似性质。
Throughout the remainder of this next we shall make extensive use of the properties of prime numbers and the analogous properties of irreducible polynomials.

3.3.3
1.在第1节中，整数已被用来定义有理数，代数数，超越数和实数。
The positive integers have been used in Chapter 1 to define rational, algebraic, transcendental, and real numbers.
2.我们的目的有3个方面：一是了解多项式的基本性质；二是得到多项式与其他常用函数的关系，三是引入一些在第四章方程理论讨论中需要的概念。
Our purpose is threefold: to understand the basic properties of polynomials, to see the relationships between polynomials and other common functions, and to introduce a few concepts that will be needed in our discussion of the theory of equations in Chapter 4.
3.在前面两章中，我们主要关注的是数：整数、有理数、实数及复数。
In the first two chapters, we have been primarily concerned with numbers; integers, rational numbers, real numbers, and complex numbers.
4.这个新的符号也可被看成变量，它从复数集的一个子集中取值。
The new symbols may be considered simply as symbols without any sets of values or assumed relations.
5.一些复数和这些新符号的乘积称作单项式。
The product of any set of complex numbers and the new symbols is called a monomial.

6.因为未定元不取值，所以关于未定元x的两个多项式，相等当且仅当对应的x的幂的系数相等。
Since indeterminates do not take on numbers as values, two polynomials in an indeterminate x are equal if and only if the coefficients of corresponding powers of x are equal.

3.36
1、n维向量的全体称为n维向量空间或者简称为n维向量，这个空间记为Vn。
The collection of all n-dimensional vectors is called the vector space of n-tuples, or simply n-space. We denote this space by Vn.
2、前面的三条性质可由定义直接推出，所以留作练习。
The first three properties are easy consequences of the definition and are left as exercises.
3、等号成立当且仅当其中一个向量是另一个向量的纯量倍数。
The equality sign holds if and only if one of the vectors is a scalar multiple of the other.
4、对于我们下面要讨论的一个概念即向量长度或者范数的性质，柯西—施瓦兹不等式有重要的应用。
The Cauchy-Schwarz inequality has important applications to the length or norm of a vector, a concept which we discuss next.
5、点乘积的基本性质可导出导数的相应性质。
The fundamental properties of the dot product lead to corresponding properties of norms.

3.4.3
1、这本书是关于被称为图论的一个离散分析。
This book is about a branch of discrete mathematics called graph theory.
2、所以图论被许多人看作是离散数学中最重要且充满活力的领域之一。
Therefore graph theory is considered by many to be one of the most important and vibrant fields within discrete mathematics.
3、这个问题极具历史意义，因为它是第一个用图论的知识表达且解决的问题。
This problem has historical significance, as it was the first problem to be stated and then solved using what is now known as graph theory.
4、这个问题是找到一条通过每座桥正好一次，环游哥尼斯堡的路径。
The problem was to make a round trip through downtown Konigsberg, traversing each bridge exactly once.
5、正如我们将在第五章看到的一样，欧拉用每个顶点有偶数条边通过的方法，对于这类问题什么时候有解给出了一个确切刻画，特别证明了哥尼斯堡问题无解。
As we shall see in Chapter 5, Euler gave an exact characterization of when such problem have solutions, in term of each dot having an even number of line-ends going into it, thereby showing in particular that the problem of the bridges of Konigsberd has no solution.
6、人们可以通过检验这种“网络图”发现有用的信息。
One can discover interesting information by examining this “Web graph”.
7、网络社区信息能被用在营销目的，或者用作检查某一产业公司间的关系。
Web community information can be used for marketing purposes or for examining the relationships among companies in a given industry.