有向图半群的超压缩性质及其在耦合振子中的应用
Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators
摘要：
我们知道，由无向图Laplacian算子给出的生成元为无穷小的半群的超压缩性质可以通过对无穷赋权图几何的理解得到.本文通过对有向图的几何结构进行类似的考察，将这些结果推广到由有向图拉普拉斯算子给出无穷小生成元的半群.特别地，我们引入了适当的命名法来讨论无限有向图的几何，并给出了将无向图Laplacian的超压缩性质推广到有向簇的超压缩性质的充分条件.这种有向图拉普拉斯算子经常可以在耦合振子的研究中观察到，最近的工作明确了同步模式与同耦合振子系统和无向图半群的超压缩性质之间的联系。因此，在这项工作中，我们证明了我们的结果有向图半群的适用性，通过扩展上述调查超出了理想情况下的同耦合振子。
英文摘要：
It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generator given by a directed graph Laplacian operator, through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficient conditions to extend ultracontractive properties of undirected graph Laplacians to those of the directed variety. Such directed graph Laplacians can often be observed in the study of coupled oscillators, where recent work made explicit the link between synchronous patterns to systems of identically coupled oscillators and ultracontractive properties of undirected graph semigroups. Therefore, in this work we demonstrate the applicability of our results on directed graph semigroups by extending the aforementioned investigation beyond the idealized case of identically coupled oscillators.
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