51,383 research outputs found
Simple structures axiomatized by almost sure theories
In this article we give a classification of the binary, simple,
-categorical structures with SU-rank 1 and trivial pregeometry. This is
done both by showing that they satisfy certain extension properties, but also
by noting that they may be approximated by the almost sure theory of some sets
of finite structures equipped with a probability measure. This study give
results about general almost sure theories, but also considers certain
attributes which, if they are almost surely true, generate almost sure theories
with very specific properties such as -stability or strong minimality.Comment: 27 page
Optimal subgraph structures in scale-free configuration models
Subgraphs reveal information about the geometry and functionalities of
complex networks. For scale-free networks with unbounded degree fluctuations,
we obtain the asymptotics of the number of times a small connected graph
occurs as a subgraph or as an induced subgraph. We obtain these results by
analyzing the configuration model with degree exponent and
introducing a novel class of optimization problems. For any given subgraph, the
unique optimizer describes the degrees of the vertices that together span the
subgraph.
We find that subgraphs typically occur between vertices with specific degree
ranges. In this way, we can count and characterize {\it all} subgraphs. We
refrain from double counting in the case of multi-edges, essentially counting
the subgraphs in the {\it erased} configuration model.Comment: 50 pages, 2 figure
Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed or localized vectors
The celebrated Marchenko-Pastur theorem gives the asymptotic spectral
distribution of sums of random, independent, rank-one projections. Its main
hypothesis is that these projections are more or less uniformly distributed on
the first grassmannian, which implies for example that the corresponding
vectors are delocalized, i.e. are essentially supported by the whole canonical
basis. In this paper, we propose a way to drop this delocalization assumption
and we generalize this theorem to a quite general framework, including random
projections whose corresponding vectors are localized, i.e. with some
components much larger than the other ones. The first of our two main examples
is given by heavy tailed random vectors (as in a model introduced by Ben Arous
and Guionnet or as in a model introduced by Zakharevich where the moments grow
very fast as the dimension grows). Our second main example is given by vectors
which are distributed as the Brownian motion on the unit sphere, with localized
initial law. Our framework is in fact general enough to get new correspondences
between classical infinitely divisible laws and some limit spectral
distributions of random matrices, generalizing the so-called Bercovici-Pata
bijection.Comment: 40 pages, 10 figures, some minor mistakes correcte
Approximation Algorithms for the Joint Replenishment Problem with Deadlines
The Joint Replenishment Problem (JRP) is a fundamental optimization problem
in supply-chain management, concerned with optimizing the flow of goods from a
supplier to retailers. Over time, in response to demands at the retailers, the
supplier ships orders, via a warehouse, to the retailers. The objective is to
schedule these orders to minimize the sum of ordering costs and retailers'
waiting costs.
We study the approximability of JRP-D, the version of JRP with deadlines,
where instead of waiting costs the retailers impose strict deadlines. We study
the integrality gap of the standard linear-program (LP) relaxation, giving a
lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as
well as an upper bound and approximation ratio of 1.574. The best previous
upper bound and approximation ratio was 1.667; no lower bound was previously
published. For the special case when all demand periods are of equal length we
give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness
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