Related papers: Hyperplanes in Configurations, decompositions, and…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
It is shown that the structure of the generalized quadrangle of order two is fully encoded in the properties of the Desargues configuration. A point of the quadrangle is represented by a geometric hyperplane of the Desargues configuration…
This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe…
We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…
In [7, Papadima and Suciu, When does the associated graded Lie algebra of an arrangement group decompose? Comment. Math. Helv. {\bf 81:4} (2006), 859--875] it is proved that the holonomy Lie algebra of an arrangement of hyperplanes through…
We first prove a new separating hyperplane theorem characterizing when a pair of compact convex subsets $K, K'$ of the Euclidean space intersect, and when they are disjoint. The theorem is distinct from classical separation theorems. It…
Hyperplane arrangements form the latest addition to the zoo of combinatorial objects dealt with by polymake. We report on their implementation and on a algorithm to compute the associated cell decomposition. The implemented algorithm…
We establish a connection between linear codes and hyperplane arrangements using the Thomas decomposition of polynomial systems and the resulting counting polynomial. This yields both a generalization and a refinement of the weight…
Let $ V $ a vector space of dimension $n$. A $V$ family $ \{H_1, \ldots, H_p \} $ of vectorial hyperplanes being distinct two by two defines an arrangement $ {\cal A}_p = {\cal A} ( H_1, \ldots ,H_p ) $ of $ V $. For $ i \in \{ 1, \ldots, p…
Modeling potential alloys requires the exploration of all possible configurations of atoms. Additionally, modeling the thermal properties of materials requires knowledge of the possible ways of displacing the atoms. One solution to finding…
The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…
The aim of this paper is to investigate an attempt to build a binary classification algorithm using principles of geometry such as vectors, planes, and vector algebra. The basic idea behind the proposed algorithm is that a hyperplane can be…
This paper is a study of the so-called `ricochet configuration' (or $R$-configuration) which arises in the context of Pascal's theorem. We give a geometric proof of the fact that a specific pair of Pascal lines is coincident for a sextuple…
A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine hyperplanes. The regions are the connected components of the complement of these hyperplanes. By a theorem of Zaslavsky, the number of regions of a hyperplane…
In this paper, firstly, by a determinant of deformed Pascal's triangle, namely the normalized Hessenberg matrix determinant, to count Dyck paths, we give another combinatorial proof of the theorems which are of Catalan numbers determinant…
In this paper we develop a new geometric method to answer the log-concavity questions related to a nice class of combinatorial sequences arising from the Pascal triangle.
In previous work on Clebsch-Gordan coefficients, certain remarkable hexagonal arrays of integers are constructed that display behaviors found in Pascal's Triangle. We explain these behaviors further using the binomial transform and discrete…
We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which…
Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the…