Related papers: Fermat-type arrangements
An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the…
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering…
We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis…
The leading terms in the long-range interaction potential between an arbitrary pair of matrix theory objects are calculated at one-loop order. This result generalizes previous calculations by including arbitrary fermionic background field…
We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any…
By generalizing the notion of linearization, a concept originally arising from microlocal analysis and symbolic calculus, to diffeological spaces, we make a first proposal setting for optimization problems in this category. We show how…
Like simpler graphs, nested (hypernodal) graphs consist of two components: a set of nodes and a set of edges, where each edge connects a pair of nodes. In the hypernodal graph model, however, a node may contain other graphs, so that a node…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…
Within the applications of spatial point processes, it is increasingly becoming common that events are labeled by marks, prompting an exploration beyond the spatial distribution of events by incorporating the marks in the undertaken…
Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is…
This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids,…
The operator $\Lambda_{\{2\},\{3\}}$ acting on line arrangements is defined by associating to a line arrangement \mathcal{A}, the line arrangement which is the union of the lines containing exactly three points among the double points of…
The conventional topological description given by the fundamental group of nematic order parameter does not adequately explain the entangled defect line structures that have been observed in nematic colloids. We introduce a new topological…
A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the…
Given a collection $L$ of line segments, we consider its arrangement and study the problem of covering all cells with line segments of $L$. That is, we want to find a minimum-size set $L'$ of line segments such that every cell in the…
In this paper we demonstrate that two common problems in Machine Learning---imbalanced and overlapping data distributions---do not have independent effects on the performance of SVM classifiers. This result is notable since it shows that a…
We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of…
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane…