欧氏空间商空间上的速度对齐模型
A velocity alignment model on quotient spaces of the Euclidean space
摘要：
Cucker-Smale（CS）模型是一种速度对齐模型，该模型也被推广到一般流形上。我们对流形上的CS模型进行了修改，去掉了关于粒子位置的先验条件和关于通信函数的条件。由于最短测地线用于定义两个粒子之间的相互作用，如果存在两个或两个以上的最短测地线，则系统不是良好定义的。在本文中，而不是使用最短的测地线来定义两个粒子之间的相互作用，我们使用所有的测地线来定义一个相互作用。从这个假设出发，我们可以放宽先验条件和通信函数的条件。我们还解释了建议的模型和以前的模型之间的关系。最后，我们给出了一些特定流形（如平坦环面、平坦莫比乌斯带、平坦克莱因瓶）上的涌现行为。从这些结果中，我们可以讨论域的拓扑结构的影响。
英文摘要：
The Cucker-Smale(CS) model is a velocity alignment model, and this model also has been generalized on general manifolds. We modify the CS model on manifolds to get rid of a-priori condition on particles' positions and conditions on communication functions. Since the shortest geodesic is used to define an interaction between two particles, if there exist two or more than two shortest geodesics, then the system is not well-defined. In this paper, instead of using the shortest geodesic to define an interaction between two particles, we use all geodesics to define an interaction. From this assumption, we can relax the a-priori condition and conditions on communication functions. We also explain the relationship between the suggested model and previous models. Finally, we provide some emergent behaviors on some specific manifolds(e.g. flat torus, flat Möbius strip, and flat Klein bottle). From these results, we can discuss the effect of the topology of the domain.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
