论文标题
非多功能矢量场的庞加莱紧凑型
Poincaré compactification for non-polynomial vector fields
论文作者
论文摘要
在这项工作中，开发了一个理论上的框架，将庞加莱压实技术应用于本地Lipschitz连续矢量场。事实证明，这些向量场在n维球体中是可压缩的，尽管压实的向量场可以在赤道中相同null。此外，对于对半球的固定投影，向量场的所有压缩在赤道上并非相同的无效。同样，还获得了确定压缩矢量场赤道不变性的条件。根据作者的知识，这是首次研究本地Lipschitz连续矢量场的庞加莱紧凑型。这些结果被说明了这些结果，将它们应用于矢量场的某些家族，例如多项式矢量场，定义为均匀函数和由分段线性函数定义的矢量字段定义的矢量字段。
In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.
