1,614 research outputs found
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Mathematical optimization for packing problems
During the last few years several new results on packing problems were
obtained using a blend of tools from semidefinite optimization, polynomial
optimization, and harmonic analysis. We survey some of these results and the
techniques involved, concentrating on geometric packing problems such as the
sphere-packing problem or the problem of packing regular tetrahedra in R^3.Comment: 17 pages, written for the SIAG/OPT Views-and-News, (v2) some updates
and correction
Dense packing on uniform lattices
We study the Hard Core Model on the graphs
obtained from Archimedean tilings i.e. configurations in with the nearest neighbor 1's forbidden. Our
particular aim in choosing these graphs is to obtain insight to the geometry of
the densest packings in a uniform discrete set-up. We establish density bounds,
optimal configurations reaching them in all cases, and introduce a
probabilistic cellular automaton that generates the legal configurations. Its
rule involves a parameter which can be naturally characterized as packing
pressure. It can have a critical value but from packing point of view just as
interesting are the noncritical cases. These phenomena are related to the
exponential size of the set of densest packings and more specifically whether
these packings are maximally symmetric, simple laminated or essentially random
packings.Comment: 18 page
Pore-scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings
We propose a method for effectively upscaling incompressible viscous flow in
large random polydispersed sphere packings: the emphasis of this method is on
the determination of the forces applied on the solid particles by the fluid.
Pore bodies and their connections are defined locally through a regular
Delaunay triangulation of the packings. Viscous flow equations are upscaled at
the pore level, and approximated with a finite volume numerical scheme. We
compare numerical simulations of the proposed method to detailed finite element
(FEM) simulations of the Stokes equations for assemblies of 8 to 200 spheres. A
good agreement is found both in terms of forces exerted on the solid particles
and effective permeability coefficients
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
A Dense Packing of Regular Tetrahedra
We construct a dense packing of regular tetrahedra, with packing density .Comment: full color versio
On the hard sphere model and sphere packings in high dimensions
We prove a lower bound on the entropy of sphere packings of of
density . The entropy measures how plentiful such
packings are, and our result is significantly stronger than the trivial lower
bound that can be obtained from the mere existence of a dense packing. Our
method also provides a new, statistical-physics-based proof of the lower bound on the maximum sphere packing density by showing
that the expected packing density of a random configuration from the hard
sphere model is at least when the
ratio of the fugacity parameter to the volume covered by a single sphere is at
least . Such a bound on the sphere packing density was first achieved
by Rogers, with subsequent improvements to the leading constant by Davenport
and Rogers, Ball, Vance, and Venkatesh
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