相关论文: Matroids in OSCAR

Apolarity is an important tool in commutative algebra and algebraic geometry which studies a form, $f$, by the action of polynomial differential operators on $f$. The quotient of all polynomial differential operators by those which…

We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples…

The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in…

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…

In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of…

We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory…

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…

Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of…

We describe the C program mori.x. It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties. The ambient toric…

Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and…

It is possible to write the indicator function of any matroid polytope as an integer combination of indicator functions of Schubert matroid polytopes. In this way, every matroid on $n$ elements of rank $r$ can be thought of as a lattice…

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…

We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified" algorithm whose performance matches previous results developed…

A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of…

This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher…

We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…

If $G$ is a looped graph, then its adjacency matrix represents a binary matroid $M_{A}(G)$ on $V(G)$. $M_{A}(G)$ may be obtained from the delta-matroid represented by the adjacency matrix of $G$, but $M_{A}(G)$ is less sensitive to the…

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time…

In exploratory factor analysis, rotation techniques are employed to derive interpretable factor loading matrices. Factor rotations deal with equality-constrained optimization problems aimed at determining a loading matrix based on measure…

We introduce the notions of a commutative square ring $R$ and of a quadratic map between modules over $R$, called $R$-quadratic map. This notion generalizes various notions of quadratic maps between algebraic objects in the literature. We…