Subsets Of Spheres Research Articles

Optimized packing unequal spheres into a multiconnected domain: mixed-integer non-linear programming approach

The problem of packing unequal spheres into a multiconnected domain (container) is considered. Given a set of spheres, the objective is to maximize the packing factor. The problem is considered as a knapsack problem and modelled as a mixed-integer non-linear programming. Characteristics of the model are indicated. We propose a new solution method based on a combination of a branch-and-bound approach and the known local optimization method. The search procedure is represented by a tree which allows handling all possible subsets of spheres. We develop a set of truncation rules to reduce the number of variants under test. The local optimization algorithm proceeds from the assumption of spheres radii being variable. A number of numerical examples are given.

Disordered strictly jammed binary sphere packings attain an anomalously large range of densities

Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. Here we report that disordered strictly jammed binary packings (packings that remain mechanically stable under general shear deformations and compressions) can be produced with an anomalously large range of average packing fractions 0.634≤φ≤0.829 for small to large sphere radius ratios α restricted to α≥0.100. Surprisingly, this range of average packing fractions is obtained for packings containing a subset of spheres (called the backbone) that are exactly strictly jammed, exactly isostatic, and also generated from random initial conditions. Additionally, the average packing fractions of these packings at certain α and small sphere relative number concentrations x approach those of the corresponding densest known ordered packings. These findings suggest for entropic reasons that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. We also identify an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded). The backbone packing fraction is about 0.624 over the majority of the α-x plane, even when large numbers of small spheres are present in the backbone. Over the (relatively small) area of the α-x plane where the backbone is not roughly constant, we find that backbone packing fractions range from about 0.606 to 0.829, with the volume of rattler spheres comprising between 1.6% and 26.9% of total sphere volume. To generate isostatic strictly jammed packings, we use an implementation of the Torquato-Jiao sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)], which is an efficient producer of inherent structures (mechanically stable configurations at the local maxima in the density landscape). The identification and explicit construction of binary packings with such high packing fractions could have important practical implications for granular composites where density is critical both to material properties and fabrication cost, including for solid propellants, concrete, and ceramics. The densities and structures of jammed binary packings at various α and x are also relevant to the formation of a glass phase in multicomponent metallic systems.

A nonsmooth program for jamming hard spheres

We study packings of $$n$$ hard spheres of equal radius in the $$d$$ -dimensional unit cube. We present a nonsmooth function whose local extrema are the radii of jammed packings (where no subset of spheres can be moved keeping all others fixed) and show that for a fixed number of spheres there are only finitely many radii of such jammed configurations. We propose an algorithm for the maximization of this maximal radius function and present examples for five to eight disks in the unit square and four to six spheres in the unit cube. The method allows straightforward generalization to packings of spheres in other compact containers.

Extension theorems for spheres in the finite field setting

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields. In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, previous results by the authors in [Iosevich and Koh, Illinois J. of Mathematics: 2008]. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.

Actin- and protein-4.1-containing filaments link nuclear pore complexes to subnuclear organelles in Xenopus oocyte nuclei.

We imaged the interiors of relatively intact Xenopus oocyte nuclei by field emission scanning electron microscopy (feSEM) and visualized a network of filaments that attach to nuclear pore complexes and extend throughout the nucleus. Within the nucleus, these 'pore-linked filaments' (PLFs) were embedded into spherical structures 100 nm to approximately 5 microm in diameter. A subset of spheres was identified as Cajal bodies by immuno-gold labeling; the rest were inferred to be nucleoli and snurposomes both of which are abundant in Xenopus oocyte nuclei. Most PLFs were independent of chromatin. The thickness of a typical PLF was 40 nm (range, approximately 12-100 nm), including the 4 nm chromium coat. PLFs located inside the nucleus merged, bundled and forked, suggesting architectural adaptability. The PLF network collapsed upon treatment with latrunculin A, which depolymerizes actin filaments. Jasplakinolide, which stabilizes actin filaments, produced PLFs with more open substructure including individual filaments with evenly-spaced rows of radially projecting short filaments. Immuno-gold labeling of untreated oocyte nuclei showed that actin and protein 4.1 each localized on PLFs. Protein 4.1-gold epitopes were spaced at approximately 120 nm intervals along filaments, and were often paired ( approximately 70 nm apart) at filament junctions. We suggest that protein 4.1 and actin contribute to the structure of a network of heterogeneous filaments that link nuclear pore complexes to subnuclear organelles, and discuss possible functions for PLFs in nuclear assembly and intranuclear traffic.

Tame Subsets of Spheres in E 3

Tame Subsets of Spheres in E 3

Tame subsets of spheres in 𝐸³

Tame subsets of spheres in 𝐸³

Tame Surfaces and Tame Subsets of Spheres in E 3

Tame Surfaces and Tame Subsets of Spheres in E 3

Tame surfaces and tame subsets of spheres in 𝐸³

Tame surfaces and tame subsets of spheres in 𝐸³

On extending a homeomorphism between two subsets of spheres

On extending a homeomorphism between two subsets of spheres