ACM sets of points in multiprojective space

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.

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

Computing the shape of a planar points set

In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and Roberts relating the Hilbert function of $${\mathbb{X}}$$ and $${\mathbb{X}} \backslash \{P\}$$ via the degree of a separator, and Abrescia, Bazzotti, and Marino relating the degree of a separator to shifts in the minimal multigraded free resolution of the ideal of 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.

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

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.

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results.

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.

Fix an integer s. Let \(f:[0,1)^s \rightarrow {\mathbb {R}}\) be an integrable function. Let \({\mathscr {P}}\subset [0, 1]^s\) be a finite point set. Quasi-Monte Carlo integration of f by \({\mathscr {P}}\) is the average value of f over \({\mathscr {P}}\) that approximates the integration of f over the s-dimensional cube. Koksma–Hlawka inequality tells that, by a smart choice of \({\mathscr {P}}\), one may expect that the error decreases roughly \(O(N^{-1}(\log N)^s)\). For any \(\alpha \ge 1\), J. Dick gave a construction of point sets such that for \(\alpha \)-smooth f, convergence rate \(O(N^{-\alpha }(\log N)^{s\alpha })\) is assured. As a coarse version of his theory, M-Saito-Matoba introduced Walsh figure of Merit (WAFOM), which gives the convergence rate \(O(N^{-C\log N/s})\). WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of order \(N^{-\alpha }\) with \(\alpha >1\), for several test integrands for \(s=4\) and 8.

The Quintile Share Ratio in location analysis

Discoveries of gaps in data have been important in astrophysics. For example, there are kinematic gaps opened by resonances in dynamical systems, or exoplanets of a certain radius that are empirically rare. A gap in a data set is a kind of anomaly, but in an unusual sense: Instead of being a single outlier data point, situated far from other data points, it is a region of the space, or a set of points, that is anomalous compared to its surroundings. Gaps are both interesting and hard to find and characterize, especially when they have non-trivial shapes. We present in this paper a statistic that can be used to estimate the (local) "gappiness" of a point in the data space. It uses the gradient and Hessian of the density estimate (and thus requires a twice-differentiable density estimator). This statistic can be computed at (almost) any point in the space and does not rely on optimization; it allows to highlight under-dense regions of any dimensionality and shape in a general and efficient way. We illustrate our method on the velocity distribution of nearby stars in the Milky Way disk plane, which exhibits gaps that could originate from different processes. Identifying and characterizing those gaps could help determine their origins. We provide in an Appendix implementation notes and additional considerations for finding under-densities in data, using critical points and the properties of the Hessian of the density.