系统发育系统的紧化与电网络物种
Compactifications of phylogenetic systems and species of electrical networks
AI总结：
提出圆形电网络与圆形平面分裂系统间的图映射，将其扩展至空间紧化以保留仙人掌结构，并通过组合物种实现对诱导系统发育分裂系统的优雅枚举。
摘要：
我们描述了新的空间与映射。我们的图映射是圆形电网络空间与圆形平面分裂系统之间的视觉与数值对应关系。当限制在平面圆形电网络情形时，该图映射能找到与对偶网络的Kalmanson电阻距离唯一关联的分裂系统，这与系统发育学中常见的诱导分裂系统相匹配。这种对应关系被扩展到各自空间的紧化上，将仙人掌网络映射到本文定义的仙人掌分裂系统。该图映射同时保留了网络组件和仙人掌结构，从而允许通过组合物种对诱导的系统发育分裂系统进行优雅的枚举。我们引入了圆形平面电网络和圆形分裂系统的全局空间。这些新空间同样是CW复形，但每个空间的0-胞腔由Bell数计数，而非Catalan数。作为物种，这两类全局仙人掌被视为以互补的方式进行组合。
英文摘要：
We describe new spaces and maps. Our graphical map is a visual and numerical correspondence between spaces of circular electrical networks and circular planar split systems. When restricted to the planar circular electrical case, this graphical map finds the split system uniquely associated with the Kalmanson resistance distance of the dual network, matching the induced split system familiar from phylogenetics. This correspondence is extended to compactifications of the respective spaces, taking cactus networks to the cactus split systems defined herein. The graphical map preserves both network components and cactus structure, allowing an elegant enumeration of induced phylogenetic split systems via combinatorial species. We introduce the global spaces of circular planar electrical networks and circular split systems. These new spaces are also CW complexes, but the 0-cells of each are counted by the Bell numbers as opposed to the Catalan numbers. As species, the two sorts of global cacti are seen to be compositions in complementary ways.
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