Related papers: Pattern Recognition on Oriented Matroids: Topes an…

Estimating the linear dimensionality of a data set in the presence of noise is a common problem. However, data may also be corrupted by monotone nonlinear distortion that preserves the ordering of matrix entries but causes linear methods…

We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical…

We show that for any regular matroid on $m$ elements and any $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a circuit in the matroid is bounded by…

We prove a general result concerning cyclic orderings of the elements of a matroid. For each matroid $M$, weight function $\omega:E(M)\rightarrow\mathbb{N}$, and positive integer $D$, the following are equivalent. (1) For all $A\subseteq…

A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies…

For a lattice polytope $P$, the rank of $P$ is defined by $F-(\dim P+1)$, where $F$ is the number of facets of $P$. In this paper, we study matroid polytopes with small rank. More precisely, we characterize matroid independence polytopes…

Positroids are matroids realizable by real matrices with all nonnegative maximal minors. They partition the ordered matroids into equivalence classes, called positroid envelope classes, by their Grassmann necklaces. We give an explicit…

Let $G=(V,A)$ be a digraph. For $X\subseteq V$, the subdigraph of $G$ induced by $X$ is denoted by $G[X]$. A subset $I$ of $V$ is an interval of $G$ if for every $a,b \in I$ and $x \in V \setminus I$, $(x,a) \in A$ if and only if $(x,b) \in…

This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear…

The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on…

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they…

Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented…

In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial…

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave: 1. The matroid complex of independent subsets of E. 2. The broken circuit complex of…

This paper investigates the principal minors of a parameterized Hermitian Laplacian matrix for oriented graphs. Particularly, we focus on the properties of the matrix for parameters chosen as Galois conjugates of a primitive $p$th root of…

In this note we characterize tropical bases as sets of circuits that by orthogonality determine the set of cocircuits of a simple matroid. Furthermore, we show that any circuit, which itself is closed, must be contained in any tropical…

Given a graph G with a distinguished vertex s, the critical group of (G,s) is the cokernel of their reduced Laplacian matrix L(G,s). In this article we generalize the concept of the critical group to the cokernel of any matrix with entries…

The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…

We introduce the poset of biflats of a matroid $M$, a Lagrangian analog of the lattice of flats of $M$, and study the topology of its order complex, which we call the biflats complex. This work continues the study of the Lagrangian…

Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$ and vertex conductance $\Psi^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap…