54,715 research outputs found
Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types
The goal of this paper to determine the optimal horoball packing arrangements
and their densities for all four fully asymptotic Coxeter tilings (Coxeter
honeycombs) in hyperbolic 3-space . Centers of horoballs are
required to lie at vertices of the regular polyhedral cells constituting the
tiling. We allow horoballs of different types at the various vertices. Our
results are derived through a generalization of the projective methodology for
hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian
density upper bound for "congruent horoball" packings of remains
valid for the class of fully asymptotic Coxeter tilings, even if packing
conditions are relaxed by allowing for horoballs of different types under
prescribed symmetry groups. The consequences of this remarkable result are
discussed for various Coxeter tilings.Comment: 26 pages, 10 figure
The Complete Jamming Landscape of Confined Hard Discs
An exact description of the complete jamming landscape is developed for a
system of hard discs of diameter , confined between two lines separated
by a distance . By considering all possible local
packing arrangements, the generalized ensemble partition function of jammed
states is obtained using the transfer matrix method, which allows us to
calculate the configurational entropy and the equation of state for the
packings. Exploring the relationship between structural order and packing
density, we find that the geometric frustration between local packing
environments plays an important role in determining the density distribution of
jammed states and that structural "randomness" is a non-monotonic function of
packing density. Molecular dynamics simulations show that the properties of the
equilibrium liquid are closely related to those of the landscape.Comment: 5 Pages, 4 figure
Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra
The determination of the densest packings of regular tetrahedra (one of the
five Platonic solids) is attracting great attention as evidenced by the rapid
pace at which packing records are being broken and the fascinating packing
structures that have emerged. Here we provide the most general analytical
formulation to date to construct dense periodic packings of tetrahedra with
four particles per fundamental cell. This analysis results in six-parameter
family of dense tetrahedron packings that includes as special cases recently
discovered "dimer" packings of tetrahedra, including the densest known packings
with density . This study strongly suggests that
the latter set of packings are the densest among all packings with a
four-particle basis. Whether they are the densest packings of tetrahedra among
all packings is an open question, but we offer remarks about this issue.
Moreover, we describe a procedure that provides estimates of upper bounds on
the maximal density of tetrahedron packings, which could aid in assessing the
packing efficiency of candidate dense packings.Comment: It contains 25 pages, 5 figures
Dense periodic packings of tori
Dense packings of nonoverlapping bodies in three-dimensional Euclidean space
are useful models of the structure of a variety of many-particle systems that
arise in the physical and biological sciences. Here we investigate the packing
behavior of congruent ring tori, which are multiply connected nonconvex bodies
of genus 1, as well as horn and spindle tori. We analytically construct a
family of dense periodic packings of unlinked tori guided by the organizing
principles originally devised for simply connected solid bodies [Torquato and
Jiao, PRE 86, 011102 (2012)]. We find that the horn tori as well as certain
spindle and ring tori can achieve a packing density higher than the densest
known packing of both sphere and ellipsoids. In addition, we study dense
packings of cluster of pair-linked ring tori (i.e., Hopf links).Comment: 15 pages, 7 figure
Densest local packing diversity. II. Application to three dimensions
The densest local packings of N three-dimensional identical nonoverlapping
spheres within a radius Rmin(N) of a fixed central sphere of the same size are
obtained for selected values of N up to N = 1054. In the predecessor to this
paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305
(2010)], we described our method for finding the putative densest packings of N
spheres in d-dimensional Euclidean space Rd and presented those packings in R2
for values of N up to N = 348. We analyze the properties and characteristics of
the densest local packings in R3 and employ knowledge of the Rmin(N), using
methods applicable in any d, to construct both a realizability condition for
pair correlation functions of sphere packings and an upper bound on the maximal
density of infinite sphere packings. In R3, we find wide variability in the
densest local packings, including a multitude of packing symmetries such as
perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest
local packings of N spheres near a central sphere to minimal-energy
configurations of N+1 points interacting with short-range repulsive and
long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that
they are in general completely different, a result that has possible
implications for nucleation theory. We also compare the densest local packings
to finite subsets of stacking variants of the densest infinite packings in R3
(the Barlow packings) and find that the densest local packings are almost
always most similar, as measured by a similarity metric, to the subsets of
Barlow packings with the smallest number of coordination shells measured about
a single central sphere, e.g., a subset of the FCC Barlow packing. We
additionally observe that the densest local packings are dominated by the
spheres arranged with centers at precisely distance Rmin(N) from the fixed
sphere's center.Comment: 45 pages, 18 figures, 2 table
Dense semantic labeling of sub-decimeter resolution images with convolutional neural networks
Semantic labeling (or pixel-level land-cover classification) in ultra-high
resolution imagery (< 10cm) requires statistical models able to learn high
level concepts from spatial data, with large appearance variations.
Convolutional Neural Networks (CNNs) achieve this goal by learning
discriminatively a hierarchy of representations of increasing abstraction.
In this paper we present a CNN-based system relying on an
downsample-then-upsample architecture. Specifically, it first learns a rough
spatial map of high-level representations by means of convolutions and then
learns to upsample them back to the original resolution by deconvolutions. By
doing so, the CNN learns to densely label every pixel at the original
resolution of the image. This results in many advantages, including i)
state-of-the-art numerical accuracy, ii) improved geometric accuracy of
predictions and iii) high efficiency at inference time.
We test the proposed system on the Vaihingen and Potsdam sub-decimeter
resolution datasets, involving semantic labeling of aerial images of 9cm and
5cm resolution, respectively. These datasets are composed by many large and
fully annotated tiles allowing an unbiased evaluation of models making use of
spatial information. We do so by comparing two standard CNN architectures to
the proposed one: standard patch classification, prediction of local label
patches by employing only convolutions and full patch labeling by employing
deconvolutions. All the systems compare favorably or outperform a
state-of-the-art baseline relying on superpixels and powerful appearance
descriptors. The proposed full patch labeling CNN outperforms these models by a
large margin, also showing a very appealing inference time.Comment: Accepted in IEEE Transactions on Geoscience and Remote Sensing, 201
On the local zeta functions and the b-functions of certain hyperplane arrangements
Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef and
F. Loeser for topological local zeta functions assert that (the real part of)
the poles of these local zeta functions are roots of the Bernstein-Sato
polynomials (i.e. the b-functions). We prove these conjectures for certain
hyperplane arrangements, including the case of reduced hyperplane arrangements
in three-dimensional affine space.Comment: 21 pages, minor changes, to appear in J. London Math. So
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