Enumerating projections of integer points in unbounded polyhedra

An algebraic approach to enumerating non-equivalent double traces in graphs

Recently designed biomolecular approaches to build single chain polypeptide polyhedra as molecular origami nanostructures have risen high interest in various double traces of the underlying graphs of these polyhedra. Double traces are walks that traverse every edge of the graph twice, usually with some additional conditions on traversal direction and vertex neighborhood coverage. Given that double trace properties are intimately related to theefficiency of polypeptide polyhedron construction, enumerating all different possible double traces and analyzing their properties is an important step i…

积木建造：在铺砌中枚举多形

Building with Blocks: Enumerating Polyforms on Tilings

本文提出了一种算法，用于计算在周期性二维和三维铺砌中，n个单元格结构的数量，并探讨了其在艺术和谜题设计中的应用。

无界多面体的剥壳

Shellings of Unbounded Polyhedra

研究无界多面体边界复形的剥壳性，先研究紧化，利用多面体结果得热带超曲面剥壳性，探讨多面体复形对偶性与剥壳性相互作用，转化到离散莫尔斯理论，揭示正则细分紧支撑特性。

Vertex-Facet Incidences of Unbounded Polyhedra

How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circul…

Computing the bounded subcomplex of an unbounded polyhedron

We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results.

超平面避难题与多面体中整数点计数

Hyperplanes Avoiding Problem and Integer Points Counting in Polyhedra

本文研究了在给定超平面集合中寻找最小范数向量的问题，提出了一种确定性多项式时间算法，改进了整数点计数算法的复杂度。