scholarly journals Covering random points in a unit disk

2008 ◽  
Vol 40 (01) ◽  
pp. 22-30
Author(s):  
Jennie C. Hansen ◽  
Eric Schmutz ◽  
Li Sheng
Keyword(s):  
Uniform Distribution ◽  
Unit Disk ◽  
Asymptotic Probability ◽  
Unit Disks

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let V n = {X 1, X 2, …, X n }, where X 1, X 2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in V n that cover all of V n .

scholarly journals Covering random points in a unit disk

2008 ◽  
Vol 40 (1) ◽  
pp. 22-30 ◽  
Author(s):  
Jennie C. Hansen ◽  
Eric Schmutz ◽  
Li Sheng
Keyword(s):  
Uniform Distribution ◽  
Unit Disk ◽  
Asymptotic Probability ◽  
Unit Disks

Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let Vn = {X1, X2, …, Xn}, where X1, X2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in Vn that cover all of Vn.

ON THE DISCRETE UNIT DISK COVER PROBLEM

2012 ◽  
Vol 22 (05) ◽  
pp. 407-419 ◽  
Author(s):  
GAUTAM K. DAS ◽  
ROBERT FRASER ◽  
ALEJANDRO LÓOPEZ-ORTIZ ◽  
BRADFORD G. NICKERSON
Keyword(s):  
Unit Disk ◽  
Constant Factor ◽  
Set Cover ◽  
Minimum Cardinality ◽  
Running Time ◽  
Approximation Factor ◽  
Set Cover Problem ◽  
Cover Problem ◽  
Unit Disks ◽  
Disk Cover

Given a set [Formula: see text] of n points and a set [Formula: see text] of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in [Formula: see text] is covered by at least one disk in [Formula: see text] or not and (ii) if so, then find a minimum cardinality subset [Formula: see text] such that the unit disks in [Formula: see text] cover all the points in [Formula: see text]. The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard. The general set cover problem is not approximable within [Formula: see text], for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is [Formula: see text]. The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time [Formula: see text].

scholarly journals The probability of intransitivity in dice and close elections

2020 ◽  
Vol 178 (3-4) ◽  
pp. 951-1009
Author(s):  
Jan Hązła ◽  
Elchanan Mossel ◽  
Nathan Ross ◽  
Guangqu Zheng
Keyword(s):  
Recent Result ◽  
Uniform Distribution ◽  
High Probability ◽  
Majority Vote ◽  
The Other ◽  
Simple Majority ◽  
Hurst Index ◽  
Analogous Model ◽  
Asymptotic Probability ◽  
Two Alternatives

Abstract We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for n-sided dice with pairwise ordering induced by the probability, relative to 1/2, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $$\{1,\ldots ,n\}$$ { 1 , … , n } and conditioned on the average of faces equal to $$(n+1)/2$$ ( n + 1 ) / 2 are intransitive with asymptotic probability 1/4. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to 0, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $$H\in (0,1)$$ H ∈ ( 0 , 1 ) . Second, we pose an analogous model in the context of Condorcet voting. We consider n voters who rank k alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the k alternatives is equal to $$2^{-k(k-1)/2}$$ 2 - k ( k - 1 ) / 2 , which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of “close to tied” for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.

scholarly journals Conflict-Free Coloring of Intersection Graphs

2018 ◽  
Vol 28 (03) ◽  
pp. 289-307 ◽  
Author(s):  
Sándor P. Fekete ◽  
Phillip Keldenich
Keyword(s):  
Graph Theory ◽  
Unit Disk ◽  
Wireless Networking ◽  
Worst Case ◽  
Intersection Graphs ◽  
Geometric Objects ◽  
Np Complete ◽  
Geometric Intersection Graphs ◽  
Unit Disks ◽  
General Graphs

A conflict-free[Formula: see text]-coloring of a graph [Formula: see text] assigns one of [Formula: see text] different colors to some of the vertices such that, for every vertex [Formula: see text], there is a color that is assigned to exactly one vertex among [Formula: see text] and [Formula: see text]’s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of [Formula: see text] geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in [Formula: see text] and in [Formula: see text] for disks or squares of different sizes; it is known for general graphs that the worst case is in [Formula: see text]. For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.

AN IMPROVED LINE-SEPARABLE ALGORITHM FOR DISCRETE UNIT DISK COVER

2010 ◽  
Vol 02 (01) ◽  
pp. 77-87 ◽  
Author(s):  
FRANCISCO CLAUDE ◽  
GAUTAM K. DAS ◽  
REZA DORRIGIV ◽  
STEPHANE DUROCHER ◽  
ROBERT FRASER ◽  
...  
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Unit Disk ◽  
Time Algorithm ◽  
Minimum Cardinality ◽  
Practical Solution ◽  
Cover Problem ◽  
Unit Disks ◽  
Set Of Points ◽  
Disk Cover

Given a set [Formula: see text] of m unit disks and a set [Formula: see text] of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset [Formula: see text] to cover [Formula: see text]. This problem is NP-hard [14] and the best previous practical solution is a 38-approximation algorithm by Carmi et al. [5]. We first consider the line-separable discrete unit disk cover problem (the set of disk centers can be separated from the set of points by a line) for which we present an O(n( log n + m))-time algorithm that finds an exact solution. Combining our line-separable algorithm with techniques from the algorithm of Carmi et al. [5] results in an O(m2n4) time 22-approximate solution to the discrete unit disk cover problem.

Levels taxation countries the EU and Ukraine’s

2019 ◽  
pp. 131-160
Author(s):  
Helena Borzenko ◽  
Tamara Panfilova ◽  
Mikhail Litvin
Keyword(s):  
European Union ◽  
Uniform Distribution ◽  
Tax Burden ◽  
Tax System ◽  
Tax Legislation ◽  
Improvement Process ◽  
Tax Systems ◽  
Historical Comparative ◽  
Different Levels ◽  
The Eu

Purpose articles rassm and experience and benefits systems taxation countries European Union, manifestation iti the main limitations domestic taxlegislation and wired STI their comparisons. In general iti ways the provisiontax reporting countries Eurozone in the appropriate organs, dove STI need theintroduction Ukraine electronic methods receiving and processing such reports.define iti key directions reforming domestic tax legislation. Methodology research is to use aggregate methods: dialectical, statistical, historical, comparative. Scientific novelty is to are provided recommendations for improvement ofefficiency systems taxation of our states in international ratings characterizingtax institutions country. Therefore, despite some problems in legislation heldcomparative study systems taxation EU and Ukraine. Conclucions Coming fromof this, the main directions reforming tax systems Ukraine, in our opinion,today should become: improvement process administration, reduce scales evasiontaxes, provision more uniform distribution tax burden between taxpayers, themaximum cooperation tax bodies different levels as well adjustment systemselectronic interactions tax authorities and payers, tax system must contain ascan less unfounded benefits, consistent with the general by politics pricing.

Statistical Justification of Pearson's Criterion for Testing a Complex Hypothesis on the Uniform Distribution

2018 ◽  
pp. 45-53
Author(s):  
T. V. Oblakova
Keyword(s):  
Uniform Distribution ◽  
Degrees Of Freedom ◽  
Likelihood Function ◽  
Random Number Generator ◽  
Maximum Likelihood Estimates ◽  
Chi Square ◽  
Main Hypothesis ◽  
Pearson Criterion ◽  
Complex Hypothesis ◽  
Uniform Law

The paper is studying the justification of the Pearson criterion for checking the hypothesis on the uniform distribution of the general totality. If the distribution parameters are unknown, then estimates of the theoretical frequencies are used [1, 2, 3]. In this case the quantile of the chi-square distribution with the number of degrees of freedom, reduced by the number of parameters evaluated, is used to determine the upper threshold of the main hypothesis acceptance [7]. However, in the case of a uniform law, the application of Pearson's criterion does not extend to complex hypotheses, since the likelihood function does not allow differentiation with respect to parameters, which is used in the proof of the theorem mentioned [7, 10, 11].A statistical experiment is proposed in order to study the distribution of Pearson statistics for samples from a uniform law. The essence of the experiment is that at first a statistically significant number of one-type samples from a given uniform distribution is modeled, then for each sample Pearson statistics are calculated, and then the law of distribution of the totality of these statistics is studied. Modeling and processing of samples were performed in the Mathcad 15 package using the built-in random number generator and array processing facilities.In all the experiments carried out, the hypothesis that the Pearson statistics conform to the chi-square law was unambiguously accepted (confidence level 0.95). It is also statistically proved that the number of degrees of freedom in the case of a complex hypothesis need not be corrected. That is, the maximum likelihood estimates of the uniform law parameters implicitly used in calculating Pearson statistics do not affect the number of degrees of freedom, which is thus determined by the number of grouping intervals only.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

A Time Petri Net Model Based on Uniform Distribution

2010 ◽  
Vol 33 (5) ◽  
pp. 900-907 ◽  
Author(s):  
Li PAN ◽  
Zhi-Jun DING ◽  
Gang CHEN
Keyword(s):  
Uniform Distribution ◽  
Petri Net ◽  
Model Based ◽  
Time Petri Net
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