1,051 research outputs found
Measurement of Indeterminacy in Packings of Perfectly Rigid Disks
Static packings of perfectly rigid particles are investigated theoretically
and numerically. The problem of finding the contact forces in such packings is
formulated mathematically. Letting the values of the contact forces define a
vector in a high-dimensional space enable us to show that the set of all
possible contact forces is convex, facilitating its numerical exploration. It
is also found that the boundary of the set is connected with the presence of
sliding contacts, suggesting that a stable packing should not have more than
2M-3N sliding contacts in two dimensions, where M is the number of contacts and
N is the number of particles.
These results were used to analyze packings generated in different ways by
either molecular dynamics or contact dynamics simulations. The dimension of the
set of possible forces and the number of sliding contacts agrees with the
theoretical expectations. The indeterminacy of each component of the contact
forces are found, as well as the an estimate for the diameter of the set of
possible contact forces. We also show that contacts with high indeterminacy are
located on force chains. The question of whether the simulation methods can
represent a packing's memory of its formation is addressed.Comment: 12 pages, 13 figures, submitted to Phys Rev
Geometrical relations between space time block code designs and complexity reduction
In this work, the geometric relation between space time block code design for
the coherent channel and its non-coherent counterpart is exploited to get an
analogue of the information theoretic inequality in
terms of diversity. It provides a lower bound on the performance of
non-coherent codes when used in coherent scenarios. This leads in turn to a
code design decomposition result splitting coherent code design into two
complexity reduced sub tasks. Moreover a geometrical criterion for high
performance space time code design is derived.Comment: final version, 11 pages, two-colum
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
The isodiametric problem with lattice-point constraints
In this paper, the isodiametric problem for centrally symmetric convex bodies
in the Euclidean d-space R^d containing no interior non-zero point of a lattice
L is studied. It is shown that the intersection of a suitable ball with the
Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among
all bodies with the same volume. It is conjectured that these sets are the only
extremal bodies, which is proved for all three dimensional and several
prominent lattices.Comment: 12 pages, 4 figures, (v2) referee comments and suggestions
incorporated, accepted in Monatshefte fuer Mathemati
Mesh ratios for best-packing and limits of minimal energy configurations
For -point best-packing configurations on a compact metric
space , we obtain estimates for the mesh-separation ratio
, which is the quotient of the covering radius of
relative to and the minimum pairwise distance between points in
. For best-packing configurations that arise as limits of
minimal Riesz -energy configurations as , we prove that
and this bound can be attained even for the sphere.
In the particular case when N=5 on with the Euclidean metric, we
prove our main result that among the infinitely many 5-point best-packing
configurations there is a unique configuration, namely a square-base pyramid
, that is the limit (as ) of 5-point -energy
minimizing configurations. Moreover,
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