On the densest packing of polycylinders in any dimension

Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $\pi/\sqrt{12}$ in any dimension.Comment: Edited to reflect acknowledgements in the published versio

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

Market partitioning and the geometry of the resource space

Operations such as integration or modularization of databases can be considered as operations on database universes. This paper describes some operations on database universes. Formally, a database universe is a special kind of table. It turns out that various operations on tables constitute interesting operations on database universes as well.

Bounds on packing density via slicing

This document is composed of a series of articles in discrete geometry, each solving a problem in packing density. • The first proves a local upper bound for the packing density of regular pentagons in R2. By reducing a nonlinear programming problem to a linear one, computational methods show that the conjectured global optimal solution is locally optimal. • The second proves an upper bound for the packing density of finite cylinders in R3. Using a measure theoretic approach to estimate boundary error, the first bound that is asymptotically sharp with respect to the length of the cylinder is found. This gives the first sharp upper bound for the packing density of half-infinite cylinders as a corollary. • The third proves an upper bound for the packing density of infinite polycylinders in Rn. Using transversality and a dimension reduction argument, an existing result for R3 is applied to Rn. This gives the first non-trivial sharp upper bound for the packing density of any object in dimensions four and greater

Quantification of marine sediment properties from planar and volumetric pore geometries

Pore geometry and topology are important determinants of sediment physical properties, such as porosity and permeability. They also influence processes that occur in the sediment, such as acoustic propagation, attenuation, and dispersion, single- and multi-phase fluid flow, and hydrodynamic dispersion. This study uses images to evaluate pore geometry and topology of ooid (subspherical particles) and siliclastic (angular quartz) sand that was collected from the marine environment south of Bimni Bahamas and Ft. Walton Beach, FL, respectively. Image analysis techniques and predictive tools enable insight into the relationships among sediment pore geometry, topology, and physical properties for these differently shaped sands. High frequency acoustics utilize short wavelength signals to evaluate sediments. Correspondingly short length scales are then needed for sedimentary property predictions, which is possible with planar and volumetric image analysis of sand. This data was compared to data obtained by direct large scale measurements (e.g., water weight loss, constant head permeability) were made. Mean porosity differed by as much as 0.04 and mean permeability showed good agreement and differed by a factor of 2. Given that the image analysis predictions were made from much smaller samples (~equivalent to the length scale of the high acoustic frequencies used) than the bulk samples, a sediment characterization at acoustically relevant length scales is possible. It was also demonstrated that for these homogeneous sands (i.e., ooids and quartz) two-dimensional pore geometry and topology are quite similar to three-dimensional pore geometry and topology (i.e., pore connectivity). Additionally it was determined that pore network models typically overestimate the topology and therefore, in order to match image and bulk predictions of sediment properties, these models must underestimate the conductance of individual pore throats (i.e., conductive element in sand). Typically pore throats are depicted as straight cylinders. Image data suggests that pore throats are better represented by biconical shapes where conductance is as much as 3 times higher than conductance within the straight cylinders. These findings indicate that increased realism in pore throat shape (higher conductivity) and in topology (fewer pore throats) may significantly influence network model evaluations of fluid flow or acoustic propagation in marine sand