52,263 research outputs found
On interference among moving sensors and related problems
We show that for any set of points moving along "simple" trajectories
(i.e., each coordinate is described with a polynomial of bounded degree) in
and any parameter , one can select a fixed non-empty
subset of the points of size , such that the Voronoi diagram of
this subset is "balanced" at any given time (i.e., it contains points
per cell). We also show that the bound is near optimal even for
the one dimensional case in which points move linearly in time. As
applications, we show that one can assign communication radii to the sensors of
a network of moving sensors so that at any given time their interference is
. We also show some results in kinetic approximate range
counting and kinetic discrepancy. In order to obtain these results, we extend
well-known results from -net theory to kinetic environments
Threshold functions and Poisson convergence for systems of equations in random sets
We present a unified framework to study threshold functions for the existence
of solutions to linear systems of equations in random sets which includes
arithmetic progressions, sum-free sets, -sets and Hilbert cubes. In
particular, we show that there exists a threshold function for the property
" contains a non-trivial solution of
", where is a random set and each of
its elements is chosen independently with the same probability from the
interval of integers . Our study contains a formal definition of
trivial solutions for any combinatorial structure, extending a previous
definition by Ruzsa when dealing with a single equation.
Furthermore, we study the behaviour of the distribution of the number of
non-trivial solutions at the threshold scale. We show that it converges to a
Poisson distribution whose parameter depends on the volumes of certain convex
polytopes arising from the linear system under study as well as the symmetry
inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
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