141 research outputs found
New extremal singly even self-dual codes of lengths and
For lengths and , we construct extremal singly even self-dual codes
with weight enumerators for which no extremal singly even self-dual codes were
previously known to exist. We also construct new inequivalent extremal
doubly even self-dual codes with covering radius meeting the
Delsarte bound.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1706.0169
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
Self Assembly Problems of Anisotropic Particles in Soft Matter.
Anisotropic building blocks assembled from colloidal particles are attractive building blocks for self-assembled materials because their complex interactions can be exploited to drive self-assembly. In this dissertation we address the self-assembly of anisotropic particles from multiple novel computational and mathematical angles.
First, we accelerate algorithms for modeling systems of anisotropic particles via massively parallel GPUs. We provide a scheme for generating statistically robust pseudo-random numbers that enables GPU acceleration of Brownian and dissipative particle dynamics. We also show how rigid body integration can be accelerated on a GPU. Integrating these two algorithms into a GPU-accelerated molecular dynamics code (HOOMD-blue), make a single GPU the ideal computing environment for modeling the self-assembly of anisotropic nanoparticles.
Second, we introduce a new mathematical optimization problem, filling, a hybrid of the familiar shape packing and covering problem, which can be used to model
shaped particles. We study the rich mathematical structures of the solution space and provide computational methods for finding optimal solutions for polygons and convex polyhedra. We present a sequence of isosymmetric optimal filling solutions for the Platonic solids. We then consider the filling of a hyper-cone in dimensions two to eight and show the solution remains scale-invariant but dependent on dimension.
Third, we study the impact of size variation, polydispersity, on the self-assembly of an anisotropic particle, the polymer-tethered nanosphere, into ordered phases. We show that the local nanoparticle packing motif, icosahedral or crystalline, determines the impact of polydispersity on energy of the system and phase transitions. We show how extensions of the Voronoi tessellation can be calculated and applied to characterize such micro-segregated phases. By applying a Voronoi tessellation, we show that properties of the individual domains can be studied as a function of system properties such as temperature and concentration.
Last, we consider the thermodynamically driven self-assembly of terminal clusters of particles. We predict that clusters related to spherical codes, a mathematical sequence of points, can be synthesized via self-assembly. These anisotropic clusters can be tuned to different anisotropies via the ratio of sphere diameters and temperature. The method suggests a rich new way for assembling anisotropic building blocks.Ph.D.Applied Physics and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91576/1/phillicl_1.pd
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