The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3). We also show that this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.Comment: 12 pages, 5 figure

From the Big Bang to the Multiverse: Translations in Space and Time

Since 2004, I have been collaborating with artist Josiah McElheny on the design of cosmological sculptures, inspired originally by the chandeliers of the Metropolitan Opera House in New York. This article describes the science behind the four works that have emerged from this collaboration to date: An End to Modernity (2005), The Last Scattering Surface (2006), The End of the Dark Ages (2008), and Island Universe (2008). These works incorporate idealized representations of many fundamental aspects of contemporary cosmology, including expansion of the universe, the last scattering surface, cosmic microwave background anisotropies, the growth and morphological transformation of galaxies, the rise and fall of the quasar population, the development of large scale structure, and the possibility that our universe is one of many cosmic islands in an eternally inflating multiverse. A companion article describes the history of the collaboration.Comment: From "Josiah McElheny: A Prism," edited by L. Neri and J. McElheny, published by Skira/Rizzoli, New York, 2010. More information and higher resolution images available at http://www.astronomy.ohio-state.edu/~dhw/McElhen

Behavior of the lean methane-air flame at zero-gravity

A special rig was designed and constructed to be compatible with the NASA Lewis Research Center Airborne Research Laboratory to allow the study of the effect of gravity on the behavior of lean limit in a standard 50.4 mm (2 in.) internal diameter tube when the mixtures are ignited at the open end and propagate towards the closed end of the tube. The lean limit at zero gravity was found to be 5.10% methane and the flame was found to extenguish in a manner previously observed for downward propagating flames at one g. It was observed that g-jitter could be maintained at less than + or 0.04 g on most zero g trajectories. All of propagating lean limit flames were found to be sporadically cellularly unstable at zero g. There was no observable correlation between the occurrence of g-jitter and the lean limit, average propagation speed of the flame through the tube or the occurrence of cellular instability

Fragility and hysteretic creep in frictional granular jamming

The granular jamming transition is experimentally investigated in a two-dimensional system of frictional, bi-dispersed disks subject to quasi-static, uniaxial compression at zero granular temperature. Currently accepted results show the jamming transition occurs at a critical packing fraction $\phi_c$. In contrast, we observe the first compression cycle exhibits {\it fragility} - metastable configuration with simultaneous jammed and un-jammed clusters - over a small interval in packing fraction ($\phi_1 < \phi < \phi_2$). The fragile state separates the two conditions that define $\phi_c$ with an exponential rise in pressure starting at $\phi_1$ and an exponential fall in disk displacements ending at $\phi_2$. The results are explained through a percolation mechanism of stressed contacts where cluster growth exhibits strong spatial correlation with disk displacements. Measurements with several disk materials of varying elastic moduli $E$ and friction coefficients $\mu$, show friction directly controls the start of the fragile state, but indirectly controls the exponential slope. Additionally, we experimentally confirm recent predictions relating the dependence of $\phi_c$ on $\mu$. Under repetitive loading (compression), the system exhibits hysteresis in pressure, and the onset $\phi_c$ increases slowly with repetition number. This friction induced hysteretic creep is interpreted as the granular pack's evolution from a metastable to an eventual structurally stable configuration. It is shown to depend upon the quasi-static step size $\Delta \phi$ which provides the only perturbative mechanism in the experimental protocol, and the friction coefficient $\mu$ which acts to stabilize the pack.Comment: 12 pages, 10 figure

Surface guidance of stem cell behavior: Chemically tailored co-presentation of integrin-binding peptides stimulates osteogenic differentiation in vitro and bone formation in vivo

Surface modification stands out as a versatile technique to create instructive biomaterials that are able to actively direct stem cell fate. Chemical functionalization of titanium has been used in this work to stimulate the differentiation of human mesenchymal stem cells (hMSCs) into the osteoblastic lineage, by covalently anchoring a synthetic double-branched molecule (PTF) to the metal that allows a finely controlled presentation of peptidic motifs. In detail, the effect of the RGD adhesive peptide and its synergy motif PHSRN is studied, comparing a random distribution of the two peptides with the chemically-tailored disposition within the custom made synthetic platform, which mimics the interspacing between the motifs observed in fibronectin. Contact angle measurement and XPS analysis are used to prove the efficiency of functionalization. We demonstrate that, by rationally designing ligands, stem cell response can be efficiently guided towards the osteogenic phenotype: In vitro, PTF-functionalized surfaces support hMSCs adhesion, with higher cell area and formation of focal contacts, expression of the integrin receptor a5ß1 and the osteogenic marker Runx2, and deposition a highly mineralized matrix, reaching values of mineralization comparable to fibronectin. Our strategy is also demonstrated to be efficient in promoting new bone growth in vivo in a rat calvarial defect. These results highlight the efficacy of chemical control over the presentation of bioactive peptides; such systems may be used to engineer bioactive surfaces with improved osseointegrative properties, or can be easily tuned to generate multi-functional coatings requiring a tailored disposition of the peptidic motifs.Peer ReviewedPostprint (author's final draft