相关论文: Matroids in OSCAR
Geometry is a fundamental part of robotics and there have been various frameworks of representation over the years. Recently, geometric algebra has gained attention for its property of unifying many of those previous ideas into one algebra.…
We discuss what is special about the reproducibility of workflows in computer algebra. It is emphasized how the programming language Julia and the new computer algebra system OSCAR support such a reproducibility, and how users can benefit…
The theory of rough sets is concerned with the lower and upper approximations of objects through a binary relation on a universe. It has been applied to machine learning, knowledge discovery and data mining. The theory of matroids is a…
A combinatorial neural code $\mathscr C\subseteq 2^{[n]}$ is convex if it arises as the intersection pattern of convex open subsets of $\mathbb R^d$. We relate the emerging theory of convex neural codes to the established theory of oriented…
Systems of cooperation and interaction are usually studied in the context of real or complex vector spaces. Additional insight, however, is gained when such systems are represented in vector spaces with multiplicative structures, i.e., in…
The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the $n^{th}$ object from…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We…
The intersection data of a hyperplane arrangement is described by a geometric lattice, or equivalently a simple matroid. There is a rich interplay between this combinatorial structure and the topology of the arrangement complement. In this…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which…
In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be…
The main goal of the paper is the discussion of a deeper interaction between matrix theory over polynomial rings over a field and typical methods of commutative algebra and related algebraic geometry. This is intended in the sense of…
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…
From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a…
In this paper, we provide a unified definition of mediated graph, a combinatorial structure with multiple applications in mathematical optimization. We study some geometric and algebraic properties of this family of graphs and analyze…
In various subjects including mathematics, one can hope to use mathematical thinking well when the right kinds of algebraic structure to consider can be discovered or spotted. Therefore, it would help to understand kinds of algebraic…
A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We…
Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point…