26,473 research outputs found
On the density of sets of the Euclidean plane avoiding distance 1
A subset is said to avoid distance if: In this paper we study the number
which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, represents the highest proportion of the plane that can be filled by a
set avoiding distance 1. This parameter is related to the fractional chromatic
number of the plane.
We establish that and .Comment: 11 pages, 5 figure
On the density of sets avoiding parallelohedron distance 1
The maximal density of a measurable subset of R^n avoiding Euclidean
distance1 is unknown except in the trivial case of dimension 1. In this paper,
we consider thecase of a distance associated to a polytope that tiles space,
where it is likely that the setsavoiding distance 1 are of maximal density
2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n =
2, and for the Vorono\"i regions of the lattices An, n >= 2
Projecting Ising Model Parameters for Fast Mixing
Inference in general Ising models is difficult, due to high treewidth making
tree-based algorithms intractable. Moreover, when interactions are strong,
Gibbs sampling may take exponential time to converge to the stationary
distribution. We present an algorithm to project Ising model parameters onto a
parameter set that is guaranteed to be fast mixing, under several divergences.
We find that Gibbs sampling using the projected parameters is more accurate
than with the original parameters when interaction strengths are strong and
when limited time is available for sampling.Comment: Advances in Neural Information Processing Systems 201
Origin of Scaling Behavior of Protein Packing Density: A Sequential Monte Carlo Study of Compact Long Chain Polymers
Single domain proteins are thought to be tightly packed. The introduction of
voids by mutations is often regarded as destabilizing. In this study we show
that packing density for single domain proteins decreases with chain length. We
find that the radius of gyration provides poor description of protein packing
but the alpha contact number we introduce here characterize proteins well. We
further demonstrate that protein-like scaling relationship between packing
density and chain length is observed in off-lattice self-avoiding walks. A key
problem in studying compact chain polymer is the attrition problem: It is
difficult to generate independent samples of compact long self-avoiding walks.
We develop an algorithm based on the framework of sequential Monte Carlo and
succeed in generating populations of compact long chain off-lattice polymers up
to length . Results based on analysis of these chain polymers suggest
that maintaining high packing density is only characteristic of short chain
proteins. We found that the scaling behavior of packing density with chain
length of proteins is a generic feature of random polymers satisfying loose
constraint in compactness. We conclude that proteins are not optimized by
evolution to eliminate packing voids.Comment: 9 pages, 10 figures. Accepted by J. Chem. Phy
From isomorphism to polymorphism: connecting interzeolite transformations to structural and graph similarity
Zeolites are nanoporous crystalline materials with abundant industrial
applications. Despite sustained research, only 235 different zeolite frameworks
have been realized out of millions of hypothetical ones predicted by
computational enumeration. Structure-property relationships in zeolite
synthesis are very complex and only marginally understood. Here, we apply
structure and graph-based unsupervised machine learning to gain insight on
zeolite frameworks and how they relate to experimentally observed polymorphism
and phase transformations. We begin by describing zeolite structures using the
Smooth Overlap of Atomic Positions method, which clusters crystals with similar
cages and density in a way consistent with traditional hand-selected composite
building units. To also account for topological differences, zeolite crystals
are represented as multigraphs and compared by isomorphism tests. We find that
fourteen different pairs and one trio of known frameworks are graph isomorphic.
Based on experimental interzeolite conversions and occurrence of competing
phases, we propose that the availability of kinetic-controlled transformations
between metastable zeolite frameworks is related to their similarity in the
graph space. When this description is applied to enumerated structures, over
3,400 hypothetical structures are found to be isomorphic to known frameworks,
and thus might be realized from their experimental counterparts. Using a
continuous similarity metric, the space of known zeolites shows additional
overlaps with experimentally observed phase transformations. Hence, graph-based
similarity approaches suggest a venue for realizing novel zeolites from
existing ones by providing a relationship between pairwise structure similarity
and experimental transformations.Comment: 11 pages, 6 figure
Spherical sets avoiding a prescribed set of angles
Let be any subset of the interval . A subset of the unit
sphere in will be called \emph{-avoiding} if for any
. The problem of determining the maximum surface measure of a -avoiding set was first stated in a 1974 note by Witsenhausen; there the
upper bound of times the surface measure of the sphere is derived from a
simple averaging argument. A consequence of the Frankl-Wilson theorem is that
this fraction decreases exponentially, but until now the upper bound for
the case has not moved. We improve this bound to using an
approach inspired by Delsarte's linear programming bounds for codes, combined
with some combinatorial reasoning. In the second part of the paper, we use
harmonic analysis to show that for there always exists an
-avoiding set of maximum measure. We also show with an example that a
maximiser need not exist when .Comment: 21 pages, 3 figure
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