The Quintile Share Ratio in location analysis

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalisation of the concept of the median of a (finite) set of points on the real line. In this paper, we present an algorithm for computing a centerpoint of a set of n points in the plane. The algorithm has complexity O(n) which significantly improves the O(n log3 n) complexity of the previously best known algorithm. We use suitable modifications of the ham-sandwich-cut algorithm and the prune-and-search technique to achieve this improvement.

On the boundary of a finite set of points in the plane

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

If $$\mathbb{X}$$ is a finite set of points in a multiprojective space $$\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } $$ withr ≥ 2, then $$\mathbb{X}$$ may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM

Computing the shape of a planar points set

Structuring finite sets of points is at the heart of computational geometry. Such point sets arise naturally in many applications. Examples in R3 are point sets sampled from the surface of a solid or the locations of atoms in a molecule. A first step in processing these point sets is to organize them in some data structure. Structuring a point set into a simplicial complex like the Delaunay triangulation has turned out to be appropriate for many modeling tasks. Here we introduce the flow complex which is another simplicial complex that can be computed efficiently from a finite set of points. The flow complex turned out to be well suited for surface reconstruction from a finite sample and for some tasks in structural biology. Here we study mathematical and algorithmic properties of the flow complex and show how to exploit it in applications.

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a continuous region (\({Imprecise}\) model) or a finite set of points (\({Indecisive}\) model). Given a set of inexact points in one of \({Imprecise}\) or \({Indecisive}\) models, we wish to provide a lower-bound on the diameter of the real points.

The discrete multi-agent rendezvous problem we consider in this paper is concerned with a specified set of points in the plane, called dwell-points, and a set of mobile autonomous agents with limited sensing range. Each agent is initially positioned at some dwell-point, and is able to determine its distances to dwell-points within sensing range, and also the bearings of those dwell-points with respect to a local reference of the agent. An agent can also determine if each dwell-point within sensing range is occupied by at least one agent. We do not, however, assume that an agent can count the number of agents occupying a dwell-point. We say the agents have rendezvoused if all the agents have moved to a set of dwell-points all within sensing radius of each other. The goal is to devise distributed motion strategies for each agent which will cause all the agents to rendezvous in the sense just defined, without any active communication among the agents. We propose a distributed motion rule and use graphs to characterize a class of dwell-points in the plane for which rendezvous is guaranteed under the rule.

Let X be a curve of genus g>1 over Q whose Jacobian J has Mordell–Weil rank r and Néron–Severi rank ρ. When r<g+ρ−1, the geometric quadratic Chabauty method determines a finite set of p-adic points containing the rational points of X. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of p-adic heights and p-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of p-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.

Multiple voting location problems

Construction of a minimum-volume ellipsoid enclosing a given finite set of points and construction of a maximum-volume simplex inscribed in a convex hull of a fine set of points are considered. Properties of these objects are presented as well as a relation between them that provides the possibility to improve geometrical algorithms for finding the objects.

Given a finite set of points in the projective plane, we use the module of Kähler differentials to investigate the configurations of these points. More precisely, depending on the values of the Hilbert function of the module of Kähler differential 3-forms, we determine whether the set of points lies on a nonsingular conic, on two different lines, or on a single line.

A procedure is given for translating a rectangular grid in such a way that a finite set of points in ℝN is approximated as good as possible by points of the translated grid.