931 research outputs found
Ball Packings with Periodic Constraints
We call a periodic ball packing in d-dimensional Euclidean space periodically (resp. strictly) jammed with respect to a period lattice Λ if there are no nontrivial motions of the balls that preserve Λ (resp. that maintain some period with smaller or equal volume). In particular, we call a packing consistently periodically jammed (resp. consistently strictly jammed) if it is periodically (resp. strictly) jammed on every one of its periods. After extending a well-known bar framework and stress condition to strict jamming, we prove that a packing with period Λ is consistently strictly jammed if and only if it is strictly jammed with respect to Λ and consistently periodically jammed. We next extend a result about rigid unit mode spectra in crystallography to characterize periodic jamming on sublattices. After that, we prove that there are finitely many strictly jammed packings of m unit balls and other similar results. An interesting example shows that the size of the first sublattice on which a packing is first periodically unjammed is not bounded. Finally, we find an example of a consistently periodically jammed packing of low density δ=4π/6√3+11+ε≈0.59, where ε is an arbitrarily small positive number. Throughout the paper, the statements for the closely related notions of periodic infinitesimal rigidity and affine infinitesimal rigidity for tensegrity frameworks are also given.National Science Foundation (U.S.) (Cornell University. Research Experience for Undergraduates. Grant DMS-1156350
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
Highly saturated packings and reduced coverings
We introduce and study certain notions which might serve as substitutes for
maximum density packings and minimum density coverings. A body is a compact
connected set which is the closure of its interior. A packing with
congruent replicas of a body is -saturated if no members of it can
be replaced with replicas of , and it is completely saturated if it is
-saturated for each . Similarly, a covering with congruent
replicas of a body is -reduced if no members of it can be replaced
by replicas of without uncovering a portion of the space, and it is
completely reduced if it is -reduced for each . We prove that every
body in -dimensional Euclidean or hyperbolic space admits both an
-saturated packing and an -reduced covering with replicas of . Under
some assumptions on (somewhat weaker than convexity),
we prove the existence of completely saturated packings and completely reduced
coverings, but in general, the problem of existence of completely saturated
packings and completely reduced coverings remains unsolved. Also, we
investigate some problems related to the the densities of -saturated
packings and -reduced coverings. Among other things, we prove that there
exists an upper bound for the density of a -reduced covering of
with congruent balls, and we produce some density bounds for the
-saturated packings and -reduced coverings of the plane with congruent
circles
New upper bounds on sphere packings I
We develop an analogue for sphere packing of the linear programming bounds
for error-correcting codes, and use it to prove upper bounds for the density of
sphere packings, which are the best bounds known at least for dimensions 4
through 36. We conjecture that our approach can be used to solve the sphere
packing problem in dimensions 8 and 24.Comment: 26 pages, 1 figur
Force balance in canonical ensembles of static granular packings
We investigate the role of local force balance in the transition from a
microcanonical ensemble of static granular packings, characterized by an
invariant stress, to a canonical ensemble. Packings in two dimensions admit a
reciprocal tiling, and a collective effect of force balance is that the area of
this tiling is also invariant in a microcanonical ensemble. We present
analytical relations between stress, tiling area and tiling area fluctuations,
and show that a canonical ensemble can be characterized by an intensive
thermodynamic parameter conjugate to one or the other. We test the equivalence
of different ensembles through the first canonical simulations of the force
network ensemble, a model system.Comment: 9 pages, 9 figures, submitted to JSTA
Ensemble Theory for Force Networks in Hyperstatic Granular Matter
An ensemble approach for force networks in static granular packings is
developed. The framework is based on the separation of packing and force
scales, together with an a-priori flat measure in the force phase space under
the constraints that the contact forces are repulsive and balance on every
particle. In this paper we will give a general formulation of this force
network ensemble, and derive the general expression for the force distribution
. For small regular packings these probability densities are obtained in
closed form, while for larger packings we present a systematic numerical
analysis. Since technically the problem can be written as a non-invertible
matrix problem (where the matrix is determined by the contact geometry), we
study what happens if we perturb the packing matrix or replace it by a random
matrix. The resulting 's differ significantly from those of normal
packings, which touches upon the deep question of how network statistics is
related to the underlying network structure. Overall, the ensemble formulation
opens up a new perspective on force networks that is analytically accessible,
and which may find applications beyond granular matter.Comment: 17 pages, 17 figure
Upper bounds for packings of spheres of several radii
We give theorems that can be used to upper bound the densities of packings of
different spherical caps in the unit sphere and of translates of different
convex bodies in Euclidean space. These theorems extend the linear programming
bounds for packings of spherical caps and of convex bodies through the use of
semidefinite programming. We perform explicit computations, obtaining new
bounds for packings of spherical caps of two different sizes and for binary
sphere packings. We also slightly improve bounds for the classical problem of
packing identical spheres.Comment: 31 page
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