155 research outputs found
On contact numbers of totally separable unit sphere packings
Contact numbers are natural extensions of kissing numbers. In this paper we
give estimates for the number of contacts in a totally separable packing of n
unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page
Density bounds for outer parallel domains of unit ball packings
We give upper bounds for the density of unit ball packings relative to their
outer parallel domains and discuss their connection to contact numbers. Also,
packings of soft balls are introduced and upper bounds are given for the
fraction of space covered by them.Comment: 22 pages, 1 figur
Contact numbers for congruent sphere packings in Euclidean 3-space
Continuing the investigations of Harborth (1974) and the author (2002) we
study the following two rather basic problems on sphere packings. Recall that
the contact graph of an arbitrary finite packing of unit balls (i.e., of an
arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is
the (simple) graph whose vertices correspond to the packing elements and whose
two vertices are connected by an edge if the corresponding two packing elements
touch each other. One of the most basic questions on contact graphs is to find
the maximum number of edges that a contact graph of a packing of n unit balls
can have in Euclidean 3-space. Our method for finding lower and upper estimates
for the largest contact numbers is a combination of analytic and combinatorial
ideas and it is also based on some recent results on sphere packings. Finally,
we are interested also in the following more special version of the above
problem. Namely, let us imagine that we are given a lattice unit sphere packing
with the center points forming the lattice L in Euclidean 3-space (and with
certain pairs of unit balls touching each other) and then let us generate
packings of n unit balls such that each and every center of the n unit balls is
chosen from L. Just as in the general case we are interested in finding good
estimates for the largest contact number of the packings of n unit balls
obtained in this way.Comment: 18 page
Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps
We apply the semidefinite programming approach developed in
arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps.
We compute new upper bounds for the one-sided kissing number in several
dimensions where we in particular get a new tight bound in dimension 8.
Furthermore we show how to use the SDP framework to get analytic bounds.Comment: 15 pages, (v2) referee comments and suggestions incorporate
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
IST Austria Thesis
We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications
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