Related papers: Perpendicular dissections of space
We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection…
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs…
A simple geometric mechanism: "the locus of intersections of perpendicular bisectors and normal lines", often arises in many guises in Nonlinear Sciences. In this paper, a new application of this simple geometric mechanism is given. Namely,…
Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line…
In this paper, we present a combinatorial characterization of the hyperplanes associated with non-singular hermitian varieties ${H}\left(s, q^2\right)$ in the projective space $\mathrm{PG}\left(s,q^2\right)$ where $s\geq3$ and $q>2$. By…
Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$, $\partial D_1 \subset \Pi _1$, $\partial D_2 \subset \Pi _2$, $\Pi_1$ and $\Pi _2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$).…
A discriminantal hyperplane arrangement B(n,k,A) is constructed from a given (generic) hyperplane arrangement A, which is classified as either very generic or non-very generic depending on the combinatorial structure of B(n,k,A). In…
The "Perpendicular Bisectors Construction" is a natural way to seek a replacement for the circumcenter of a noncyclic quadrilateral in the plane. In this paper, we generalize this iterative construction to a construction on polytopes with…
In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…
In this paper, we count the number of independent sets of a type of graph $G(\mathcal{A},q)$ associated to some hyperplane arrangement $\mathcal{A}$, which is a generalization of the construction of graphical arrangements. We show that when…
Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result…
We study imbedded hypersurfaces in spacetime whose causal character is allowed to change from point to point. Inherited geometrical structures on these hypersurfaces are defined by two methods: first, the standard rigged connection induced…
Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and…
Let $\text{PG}(n,q)$ be the Desarguesian projective space of dimension $n$ over the finite field of order $q$. The \emph{linear representation} of a point set $\mathcal{K}$ in a hyperplane at infinity of $\text{PG}(n,q)$ is the point-line…
Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to…
Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…
We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [Wa96],…