Convex Rank Tests and Semigraphoids

Three Counterexamples on Semigraphoids

Semigraphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group. We resolve two problems on semigraphoids posed in Studeny's book, and we answer a related question by Postnikov, Reiner, and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semigraphoids.

Several Convex-Ear Decompositions

In this paper we give convex-ear decompositions for the order complexes of several classes of posets, namely supersolvable lattices with non-zero Mobius functions and rank-selected subposets of such lattices, rank-selected geometric lattices, and rank-selected face posets of shellable complexes which do not include the top rank. These decompositions give us many new inequalities for the h-vectors of these complexes. In addition, our decomposition of rank-selected face posets of shellable complexes allows us to prove inequalities for the flag h-vector of face posets of Cohen-Macaulay complexes.

Three notions of tropical rank for symmetric matrices

We introduce and study three different notions of tropical rank for symmetric and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull.

关于通过一点将某种类型秩3的均匀定向矩阵体约化的研究

On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point

本文研究了凸n边形内部区域与对称群S_n中特定类型的n循环的关系，发现并非所有区域标签的循环都出现在每个凸n边形中，进而定义了确定性和不定性循环，并探讨了其在均匀秩3凸定向矩阵体的一点扩展中的可约性。

On the dimension of max-min convex sets

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvability of linear systems and trapezoidal property.

倾斜理论中的扇与多面体III：秩3凸g-扇的分类

Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3

本文针对有限维代数的g-扇分类问题，利用将扇分解为卦限并分析迭代突变g-向量序列的方法，证明秩3的凸g-扇在同构意义下恰好有61个。