931 research outputs found

    Ball Packings with Periodic Constraints

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    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

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    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

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    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 P\cal P with congruent replicas of a body KK is nn-saturated if no n−1n-1 members of it can be replaced with nn replicas of KK, and it is completely saturated if it is nn-saturated for each n≥1n\ge 1. Similarly, a covering C\cal C with congruent replicas of a body KK is nn-reduced if no nn members of it can be replaced by n−1n-1 replicas of KK without uncovering a portion of the space, and it is completely reduced if it is nn-reduced for each n≥1n\ge 1. We prove that every body KK in dd-dimensional Euclidean or hyperbolic space admits both an nn-saturated packing and an nn-reduced covering with replicas of KK. Under some assumptions on K⊂EdK\subset \mathbb{E}^d (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 nn-saturated packings and nn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a d+2d+2-reduced covering of Ed\mathbb{E}^d with congruent balls, and we produce some density bounds for the nn-saturated packings and nn-reduced coverings of the plane with congruent circles

    New upper bounds on sphere packings I

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    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

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    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

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    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 P(f)P(f). 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 P(f)P(f)'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

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    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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