Related papers: Orientable arithmetic matroids
Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given…
A unified approach to parametrization of the mixing matrix for $N$ generations is developed. This approach not only has a clear geometrical underpinning but also has the advantage of being economical and recursive and leads in a natural way…
A fundamental theorem of matroid theory establishes that a transversal matroid is representable over fields of any characteristic. It was proved in 1970 by Piff and Welsh: their proof is elegant and concise and, moveover, constructive.…
We show that qubit and chirotope concepts are closely related. In fact, we prove that the qubit concept leads to a generalization of the chirotope concept, which we call qubitope. Moreover, we argue that a possible qubitope theory may…
We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of…
We present a comprehensive overview of Ardila and Develin's (generic) tropical oriented matroids, as well as many related objects and their axiomatics. Moreover, we use a unifying framework that elucidates the connections between these…
There are many similarities between the theories of matroids and $q$-matroids. However, when dealing with the direct sum of $q$-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable…
We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindstr\"om valuations and Frobenius flocks of such matroids. Building on work by Evans…
In a previous work, we gave a construction of (not necessarily realizable) oriented matroids from a triangulation of a product of two simplices. In this follow-up paper, we use a variant of Viro's patchworking to derive a topological…
Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been…
An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.
Using Tutte's combinatorial definition of a map we define a $\Delta$-matroid purely combinatorially and show that it is identical to Bouchet's topological definition.
Is it possible to define cryptomorphic axiom systems for infinite oriented matroids by lifting some of the axiom systems for finite oriented matroids to the infinite setting while not losing duality in the process? We show that the answer…
Efficient deterministic algorithms to construct representations of lattice path matroids over finite fields are presented. They are built on known constructions of hierarchical secret sharing schemes, a recent characterization of…
At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid,…
We consider the problem of determining whether the union of two infinite matroids is a matroid. We introduce a superclass of the finitary matroids, the nearly finitary matroids, and prove that the union of two nearly finitary matroids is a…
We show that, given a rank 3 affine root system $\Phi$ with Weyl group $W$, there is a unique oriented matroid structure on $\Phi$ which is $W$-equivariant and restricts to the usual matroid structure on rank 2 subsystems. Such oriented…
We give a characterization of a matroid to be paving, through its set of hyperplanes and give an algorithm to construct all of them.
We introduce the notion of irregular vertex (operator) algebras. The irregular versions of fundamental properties, such as Goddard uniqueness theorem, associativity and operator product expansions are formulated and proved. We also give…