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Then we in-vest gate Seiberg-Witten map, which defines the equivalence relation of the gauge symmetries in different spaces. We also analyse the U gauge electromagnetic field in noncommutative space. Finally we discuss the ambiguities of Seiberg-Witten map, and find although that some part of the ambiguities can be absorbed by gauge transformation and hence negligible, the existence of the matter field makes some other parts difficult to be absorbed by gauge transformation or field redefinition.

然后利用Seiberg-Witten映射，我们研究了不同空间的规范对称性的等价问题，并对非对易空间的电磁场进行了微扰分析，最后我们对Seiberg-Witten映射的松动进行了讨论，研究发现这些松动中的一部分可以由规范变换吸收因而是可以忽略的，由于物质场的存在，另外一部分很难用规范变换和场的重定义吸收。

In these programs, the author stores the non—zero elements of the global stiffness matrix in line, and solve the equations using Gauss—Seidel Method, which can not only save numerous internal spaces, so it improves the ability to deal with the giant engineering problem greatly, but also promote the efficiency to solve the equations.

本文作者独立编制了三维弹性损伤有限元程序，该程序采用一维索引压缩存贮整体刚度矩阵配合松驰法解方程，既大大节省了计算机内存从而为扩大解题规模提供了可能，同时又提高了解题效率。

But in this paper, it is proven that the upper bound has nothing to do with the viscosity of the fluid by using decomposing theorems of spaces, Gauss's formula and Sobolev Space methods.

文中利用空间分解定理、高斯公式及Sobolev空间方法证明了不可压缩粘性流体定常旋转流在Sobolev空间［H1］3中存在一个与流体粘度无关的上界。

He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues.

然后，他开始与线性方程组，矩阵和高斯消去的讨论，并着手讨论向量空间，线性映射，标产品，决定因素和特征。

The gear pump is the volume pump's one kind, by two gears, the pump body and the first top plate is composed two seal spaces, when gear-driven, the gear withdraws the side the space volume to increase since childhood, the vacuum production, inhales the liquid, the gear meshing side's space's volume is small from the big change, but pushes in the liquid in the pipeline.

齿轮泵是容积泵的一种，由两个齿轮、泵体与前后盖组成两个封闭空间，当齿轮转动时，齿轮脱开侧的空间的体积从小变大，形成真空，将液体吸入，齿轮啮合侧的空间的体积从大变小，而将液体挤入管路中去。

It is a main task of General Topology to compare different spaces.

一般拓扑学的一个主要任务是不同拓扑空间类的比较，映射直接建立了不同拓扑空间类的联系，是实现该任务的重要工具。

Since 1960s, the investigations of generalized metric spaces is an active direction of General Topology all the time.

从20世纪60年代起，广义度量空间理论一直是一般拓扑学中活跃的研究方向。

We first give the expression of the generalized inverse of the perturbed operatorin Hilbert spaces.

1.6中我们第一次给出Hilbert空间扰动算子广义逆的表达式，这个结果就是在矩阵问题中也未被得到过。

The result is generalized to a sequence of Banach spaces.

由此得出：Banach空间X上n次积分C-半群序列S强收敛于Banach空间X上n次积分C-半群S的充分条件是其生成元序列A强收敛于A，并将这一结论推广到一般的Banach空间序列上。

In chapter 2, we first give a generalized type of the Prigons fixed point theorem and a straightforward proof of it. At the same time, we give a positive answer to an open question about convergency of iterative sequence for generalized contraction which is bring forwarded by Frigon in [3]. Secondly, by using the introduced function φ, we establish some fixed point theorems for the set-valued φ-contraction mappings in gauge spaces. These results unify and generalize the corresponding results of Cain and Nashed [1], Figon [3], Espinola and Kirk[4], Tarafder [5] et al.

第二章，首先给出Frigon不动点定理[3]的推广及其简洁证明，同时对Frigon在文[3]中提出的关于广义压缩迭代序列收敛性的公开问题给出肯定的回答；然后，通过引入φ函数，在度规空间上建立集值φ-压缩映射的不动点定理，统一并推广了Cain和Nashed[1]，Figon[3]，Espinola和Kirk[4]，Tarafder[5]等的相应结果。

"Please accept this talisman as a token of our thanks."

请收下这个护身符以表达我们的感谢。

If I have that magic, I will be more acceptable in the society.

如果我拥有那种魔法，我将更加被这个社会所容纳。