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According to classical interpolatory theory, two generalized sampling theorems are proposed and two interpolation formula which have stable and highly accurate recovery results are put forward.

根据经典的插值理论提出并证明了二个广义采样定理，在该定理中给出了具有稳定的、高精度恢复结果的采样和内插方式。

Compared with recovery algorithms which are based on Fourier transform, the interpolation formula of generalized sampling theorem one has much more highly accurate recovery result.

同其他基于Foureir变换理论的恢复算法进行了比较，计算结果表明：在相同条件下，定理一内插方式具有高得多的恢复精度。

The experimental results, show that the RBF neural network is suitable for solving the problem of scattered data interpolation on a surface, and is more efficient than traditional methods. Furthermore, it can be easily generalized to solve the scattered data interpolation problem in higher dimensions.

从应用结果来看，RBF网络适合于解决曲面离散点集的光滑插值问题，比传统的样条方法更有效、更方便，具有较好的使用价值，并且可以很容易地推广到求解高维散乱数据插值问题之中。

This thesis consists of five chapters. At first, We introduce the study history as well as present situation on KKM theory. Secondly, We discuss the invariability of transfer open valued multimap under the restriction of upper and lower semicontinuity on generalized convex spaces. Then, We obtain new KKM type theorems from the classical KKM theorem by applying upper and lower semicontinuity. We also obtain intersection theorems by applying the KKM type theorems for a multimap from a topological space to another topological space. Further, We obtain Ky-Fan type coincidence theorem by applying known variational results on the KKM type theorems. Finally, We give an application to new theorems.

本文共分五章，首先回顾了KKM理论的研究历史和现状，其次讨论了在一般化凸空间上转移开值映射在上、下半连续映射下的不变性，然后在上、下半连续映射下，以古典的KKM定理为基础得到新的KKM型定理，并利用从一个拓扑空间到另一个拓扑空间的集值映射及KKM型定理得到相交定理，进一步利用已知的KKM型定理的变形结论得到Ky-Fan型重叠定理，最后给出新定理的一个应用。

These methods are usually encountered when there is no information available concerning the network interzonal distances, travel times, or generalized costs.

如果不能得到小区之间的距离，出行时间或者运输成本，这种方法就有困难。

Finally, a basic lemma was given to reveal the relation between the solution set of a centain integral equations and the generalized cyclic invariant subspace. This lemma will play an important role in the determination of controllability.

然后以一个基本引理的形式揭式某一积分方程的解集与广义循环不变子空间之间的联系，这个引理将在能控性的判定中起关键作用。

Secondly, the concept of column space, cyclic invariant subspace and generalized cyclic invariant subspace were introduced.And some basic properties, such as separation lemma, were presented.

其次，引入列空间、循环不变子空间和广义循环不变子空间等基本几何概念，给出一些有关概念的基本性质，特别是分离引理。

In this thesis,on thebasis of this generalized invariant theory,the invariant-related unitary transformation methodis developed and used to investigate the problems as follows

本论文在推广了的不变量理论的基础上，进一步发展了与不变量有关的幺正变换方法，并应用这一方法对如下问题进行了深入的研究

The works completed in this thesis make it clear that the invariant-related unitarytransformation method,developed on the basis of the generalized invariant theory,is a usefultool for the treatment of the time-dependent systems.

这些工作表明，在推广了的不变量理论基础上建立起来的与不变量有关的幺正变换方法是研究与时间有关问题的有力工具。

In order to overcome the handicap to extract line-segments from natural hand-printed Chinese characters, the extension is made to other structural features, and the generalized connected background region is defined, and its extraction and recognition algorithm are developed. This recognition algorithm is based on matrix singular decomposition and invariant transformation theory.

针对从自然手写体汉字中难以提取到笔段特征的难题，对汉字的其它结构类特征进行了扩充，提出了汉字的广义背景连通域的概念，并给出了提取广义背景连通域特征的算法以及基于矩阵奇异值分解与不变性变换的广义背景连通域特征的匹配算法。

The network space is the typical form of abstract space when it comes to the digital stage.

网络空间是抽象空间数字化阶段的典型形式。

The Notarial System,as the system of judicial certification in our country,is both the constructor and the defender of social credit system,It has its special value and function in the social honesty.

公证制度作为我国司法证明制度，既是社会信用体系的建设者，同时也是社会信用的捍卫者，在诚信中有其特殊的价值和功能。