Related papers: Construction of the Voronoi Diagram and Secondary …
This paper presents how the space of spheres and shelling may be used to delete a point from a $d$-dimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we…
We study error-correcting codes in the space $\mathcal{S}_{n,q}$ of length-$n$ multisets over a $q$-ary alphabet under the deletion metric, motivated by permutation channels in which ordering is completely lost and errors act only on symbol…
We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $\Psi^j$, $j\in…
Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in…
The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very…
Let $S$ be a set of $n$ points in $\mathbb{R}^2$. Our goal is to preprocess $S$ to efficiently compute the smallest enclosing disk of the points in $S$ that lie inside an axis-aligned query rectangle. Previous data structures for this…
$ \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}}…
Two-dimensional mixtures of dipolar colloidal particles with different dipole moments exhibit extremely rich self-assembly behaviour and are relevant to a wide range of experimental systems, including charged and super-paramagnetic colloids…
Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods. While abstract models lack predictive capabilities at the cellular level, explicit network representations…
This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A…
We consider special geometry of the vector multiplet moduli space in compactifications of the heterotic string on $K3 \times T^2$ or the type IIA string on $K3$-fibered Calabi-Yau threefolds. In particular, we construct a modified dilaton…
This article represents the computational model for spacial addresation of the sensors in the dynamically changing real-time internet of things system. The model bases on the Voronoi diagrams as a basic data structure. Problem - the correct…
Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains…
We define a map S: D^2 x D^2 --> D^2 x D^2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We…
The restricted Delaunay triangulation of a closed surface $\Sigma$ and a finite point set $V \subset \Sigma$ is a subcomplex of the Delaunay tetrahedralization of $V$ whose triangles approximate $\Sigma$. It is well known that if $V$ is a…
We present a new open source code for massive parallel computation of Voronoi tessellations(VT hereafter) in large data sets. The code is focused for astrophysical purposes where VT densities and neighbors are widely used. There are several…
We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the…
In this paper, we study the shape reconstruction problem, when the shape we wish to reconstruct is an orientable smooth d-dimensional submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the…
Given a lattice $L$, a full dimensional polytope $P$ is called a {\em Delaunay polytope} if the set of its vertices is $S\cap L$ with $S$ being an {\em empty sphere} of the lattice. Extending our previous work \cite{DD-hyp} on the {\em…
The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms, ranging over all surfaces. More precisely, we consider pseudo-Anosovs F:S to S with |chi(S)| log(lambda(F))…