论文标题
切断的角落
Cutting Corners
论文作者
论文摘要
我们定义并研究了一类有限类型（SFT）的子缩短，这些子类型由相同形状的允许模式的家族定义，在这些家族中，对于形状的任何内容，都会减去角落，填充角度的方法是相同的。主要结果是，对于这样的SFT，凸形形状的本地合法模式在全球合法上是合法的，并且有一项措施在所有凸组集合中均匀地进行样品。在适当的可计算性假设下，可以采样此措施，并有效，有效地计算和枚举法律配置。我们通过示例表明，这些子迁移不必通过转移持续操作来允许一组（更一般的Unital Magma或Quasigroup）结构。我们的凸度方法是公理的，只需要“相对于形状中的中值”的抽象凸几何形状。我们在几个组上构建了这种凸几何形状，尤其是所有强烈的多环类和自由组。我们还展示了一些用于采样有限模式的方法，一种方法基于订购，一种基于旧的新“独立集”的基础。我们还展示了与戈特萨克和卡普兰斯基的猜想的联系。
We define and study a class of subshifts of finite type (SFTs) defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. The main results are that for such an SFT, a locally legal pattern of convex shape is globally legal, and there is a measure that samples uniformly on all convex sets. Under suitable computability assumptions, this measure can be sampled, and legal configurations counted and enumerated, effectively and efficiently. We show by example that these subshifts need not admit a group (more generally unital magma or quasigroup) structure by shift-commuting continuous operations. Our approach to convexity is axiomatic, and only requires an abstract convex geometry that is "midpointed with respect to the shape". We construct such convex geometries on several groups, in particular all strongly polycyclic groups and free groups. We also show some other methods for sampling finite patterns, one based on orderings and one based on contructing new "independent sets" from old. We also show a link to conjectures of Gottshalk and Kaplansky.
