Andrey O. Matveev
A three-tope committee K* for a simple oriented matroid M is a 3-subset of its maximal covectors such that every positive halfspace of M contains at least two topes from K*. We consider three-tope committees as the vertex sets of triangles in graphs associated with the topes and enumerate them making use of the properties of the poset of convex subsets of the ground set of M.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The principle of inclusion-exclusion is applied to subsets of maximal covectors contained in halfspaces of a simple oriented matroid and to convex subsets of its ground set for enumerating tope committees.
Oriented matroids can serve as a tool of modeling of collective decision-making processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids. A tope committee for an oriented matroid is a subset of its maximal covectors such that every positive halfspace contains more than half of the maximal covectors from this subset. For a large subfamily of oriented matroids their committee structure is quite rich; for example, any maximal chains in their tope posets provide one with information sufficient to…
Let the sign components of the maximal covectors of a simple oriented matroid M be represented by the real numbers -1 and 1. Consider the vertex set V(R) of a symmetric cycle R of adjacent topes in the tope graph of M as a subposet of the tope poset of M. If B is the bottom element of the tope poset then B is equal to the unweighted sum of the members of the set min V(R) of minimal elements of the subposet V(R); if B is the positive tope then the set min V(R) is a critical tope committee for the acyclic oriented matroid M.
A tope committee K* for a simple oriented matroid M is a subset of its maximal covectors such that every positive halfspace of M contains more than half of the covectors from K*. The structures of the family of all committees for M, and of the family of its committees that contain no pairs of opposites, are described. A Farey subsequence associated with the elements of the m-th layer of the Boolean lattice of rank 2m is explored.
For a symmetric 2t-cycle in the tope graph of a simple oriented matroid M on the ground set {1,...,t}, where t is even, we describe decompositions of topes and subtopes of M with respect to the subtopes corresponding to the edges of the symmetric cycle.
If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T;R) of V(R) such that T is the sum of the topes of Q(T;R). If |Q(T;R)|>3, then the decomposition Q(T;R) of the tope T with respect to the symmetric cycle R satisfies certain Dehn-Sommerville type relations.
If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T,R) of V(R) such that T is the sum of the topes of Q(T,R). If for decompositions Q(T',R') and Q(T",R") with respect to symmetric cycles R' and R" in the tope graphs of two simple oriented matroids, whose ground sets have the cardinalities of opposite parity, we have |Q(T',R')|>3 and |Q(T",R")|>3, then these decompositions satisfy a certain orthogonality relation.
If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the positive tope, is a critical committee for M that votes for the base tope. We consider the sequence zR of poset ranks of the elements from the vertex sequence of R as a fragment of a signal with period 2t and relate the number of members of the committee min V(R) to the magnitudes of [t/2] components, with odd indices, of the discrete Fourier transform of the …
Colorful circuits and colorful topes in oriented matroids
本文给出有向拟阵中彩色Carathéodory定理圆锥版本的简短证明，并建立关于拓扑的彩色定理，其方法基于Sperner引理与Meshulam引理的共同推广。
We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Furthermore, we obtain polynomial time recognition a…
Unlabeled sample compression schemes for oriented matroids
本研究针对样本压缩猜想，利用有向拟阵的度量与组合结构，构建了以VC维为界的有向拟阵拓扑位类的无标记样本压缩方案，结果可推广至仿射及具角剥离性质的有向拟阵复形的拓扑位，主要工具为特定有向拟阵规划的解。
We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in $\mathbb{R}^{2^t}$, with respect to a distinguished symmetric $2\cdot 2^t$-cycle in its hypercube graph of topes $\boldsymbol{H}(2^t,2)$. We seek interpretations of such decompositions in the context of subset families on the ground set $E_t:=\{1,\ldots,t\}$ and of the families of their blocking sets, in the context of clutters on $E_t$ and of their blockers.
Filtrations of Tope Spaces of Oriented Matroids
本文比较了定向排列体拓扑空间的三种滤子，展示了它们在Z/2Z上的同构性，并探讨了双Varchenko-Gelfand度滤子在排列体扇上的Z-符号协层应用。
A Generalization of a Theorem of Mandel
本文将Mandel关于定向拟阵余向量集由顶集和复合条件确定的定理推广到条件定向拟阵，改用面对称条件，从而便于用计算机计算表示几何构型，并以超平面排列的公寓为例说明。
The components of K*-vectors associated to a simple oriented matroid M are the numbers of general or special tope committees for M. Using the principle of inclusion-exclusion, we determine how the reorientations of M on one-element subsets of its ground set affect K*-vectors.
Magnitude and motivic zeta functions of matroids
本文证明简单有向拟阵的顶部图幅值等于其底层拟阵动机zeta函数在$x=-1$的特殊化，细化Las Vergnas--Zaslavsky定理，构造反例反驳Koizumi--Liu猜想，并建立Varchenko--Gelfand滤过以恢复完整动机zeta函数。
Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of un…
We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for cones in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest…
On the geometry of flag Hilbert-Poincaré series for matroids
本研究将粗旗希尔伯特-庞加莱级数定义拓展至拟阵，结合有向拟阵顶的几何组合工具证明其分子系数下界为欧拉多项式及等式成立条件，并给出任意秩拟阵不可定向性的充分判据。
Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplan…
Vertex-shellings of Euclidean Oriented Matroids
本文证明欧几里得有向拟阵程序的余圈偏序的线性扩展可诱导拓扑扫描与壳化，并扩展至整体顶点壳化，联系可堆叠带状剖分及Mandel猜想反例。
Labeled sample compression schemes for complexes of oriented matroids
本文证明 VC 维为 d 的定向拟阵复形的 tope 集合存在大小为 d 的恰当有标号样本压缩方案，推广了多类已有结果，并借助组合结构与偏立方体理论推进样本压缩猜想。
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recove…
Canonical forms of oriented matroids
本文提出通过构造定向交错体 tope 的 canonical 形式，为 matroid 振幅理论提供了基础输入。
We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…
Finitary affine oriented matroids
本文开创仿射定向拟阵的公理化研究，建立子式、重定向等基本概念，证明有限情形下的可剥壳性，并利用群作用与多重Tutte多项式推广环面排列的计数性质。
Neither simpliciality nor mutation connectivity: conjectures of Las Vergnas and Cordovil-Las Vergnas fail
本文构造反例否证两个猜想：一是存在无单纯拓扑的秩7、24元素简单有向拟阵；二是均匀有向拟阵的突变图在无穷多参数下不连通。
Oriented Matroids and Combinatorial Neural Codes
本文将偏序体理论与组合神经码理论联系起来，证明了判断组合码是否凸是NP难问题。
Cell structure of mediangle graphs
本文证明 mediangle 图（中位数图和 Coxeter 图的共同推广）可赋予可缩胞腔复形结构，其胞腔是单形与单形有向拟阵的乘积，并将二部 mediangle 图识别为有限有向拟阵复形的 tope 图。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。