2,605 research outputs found
Connectivity in Sub-Poisson Networks
We consider a class of point processes (pp), which we call {\em sub-Poisson};
these are pp that can be directionally-convexly () dominated by some
Poisson pp. The order has already been shown useful in comparing various
point process characteristics, including Ripley's and correlation functions as
well as shot-noise fields generated by pp, indicating in particular that
smaller in the order processes exhibit more regularity (less clustering,
less voids) in the repartition of their points. Using these results, in this
paper we study the impact of the ordering of pp on the properties of two
continuum percolation models, which have been proposed in the literature to
address macroscopic connectivity properties of large wireless networks. As the
first main result of this paper, we extend the classical result on the
existence of phase transition in the percolation of the Gilbert's graph (called
also the Boolean model), generated by a homogeneous Poisson pp, to the class of
homogeneous sub-Poisson pp. We also extend a recent result of the same nature
for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the
so-called perturbed lattices are sub-Poisson. More generally, perturbed
lattices provide some spectrum of models that ranges from periodic grids,
usually considered in cellular network context, to Poisson ad-hoc networks, and
to various more clustered pp including some doubly stochastic Poisson ones.Comment: 8 pages, 10 figures, to appear in Proc. of Allerton 2010. For an
extended version see http://hal.inria.fr/inria-00497707 version
A random version of Sperner's theorem
Let denote the power set of , ordered by inclusion, and
let be obtained from by selecting elements
from independently at random with probability . A classical
result of Sperner asserts that every antichain in has size at
most that of the middle layer, . In this note
we prove an analogous result for : If then, with high probability, the size of the largest antichain in
is at most . This
solves a conjecture of Osthus who proved the result in the case when . Our condition on is best-possible. In fact, we prove a
more general result giving an upper bound on the size of the largest antichain
for a wider range of values of .Comment: 7 pages. Updated to include minor revisions and publication dat
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Counting inequivalent monotone Boolean functions
Monotone Boolean functions (MBFs) are Boolean functions satisfying the monotonicity condition for any . The number of MBFs in n variables is
known as the th Dedekind number. It is a longstanding computational
challenge to determine these numbers exactly - these values are only known for
at most 8. Two monotone Boolean functions are inequivalent if one can be
obtained from the other by renaming the variables. The number of inequivalent
MBFs in variables was known only for up to . In this paper we
propose a strategy to count inequivalent MBF's by breaking the calculation into
parts based on the profiles of these functions. As a result we are able to
compute the number of inequivalent MBFs in 7 variables. The number obtained is
490013148
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