Minimal Translation Covers For Sets of Diameter 1

In this note we prove that the smallest perimeter convex domain that is a translation cover for the collection of all planar sets of diameter 1 is the circle of diameter .     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

Rigidity of packings

In Ludwig Danzer’s Habilitatiionsschrift [L. Danzer, Endliche Punktmengen auf der 2-Sphäre mit möglichst grossem minimalabstand, Habilitationsschrift, Göttingen, 1963] he initiated a study of the local nature of the packings from the point of view of whether their density can be increased by a small perturbation of the packing configuration. This is an abbreviated biased introduction to the theory of such locally maximally dense packings of disks in various spaces from the point of view of the theory of tensegrity structures. This has applications to jamming of granular materials as well as leading to a better understanding of jammed packings.     

Relative Quantum Yield Measurements of Coumarin Encapsulated in Core-Shell Silica Nanoparticles

Fluorescent silica nanoparticles encapsulating organic fluorophores provide an attractive materials platform for a wide array of applications where high fluorescent brightness is required. We describe a class of fluorescent silica nanoparticles with a core-shell architecture and narrow particle size distribution, having a diameter of less than 20 nm and covalently incorporating a blue-emitting coumarin dye. A quantitative comparison of the scattering-corrected relative quantum yield of the particles to free dye in water yields an enhancement of approximately an order of magnitude. This enhancement of quantum efficiency is consistent with previous work on rhodamine dye-based particles. It provides support for the argument that improved brightness over free dye in aqueous solution is a more general effect of covalent incorporation of fluorescent organic dyes within rigid silica nanoparticle matrices. These results indicate a synthetic route towards highly fluorescent silica nanoparticles that produces excellent probes for imaging, security, and sensing applications.     

Modeling of semiconductor optical amplifier RIN and phase noise for optical PSK systems

Phase modulation schemes are attracting much interest for use in ultra-fast optical communication systems because they are much less sensitive to fibre nonlinearities compared to conventional intensity modulation formats. Semiconductor optical amplifiers (SOAs) can be used to amplify and process phase modulated signals, but with a consequent addition of nonlinear phase noise (NLPN). Existing SOA NLPN models are simplistic. In this paper we show that a more accurate model can be used, which results in simple expressions for SOA nonlinear noise, in particular when used to amplify differential phase shift keyed modulated data. The model is used to calculate the optical signal to noise ratio introduced by a power booster SOA and the first inline amplifier of a 40 Gb/s NRZ-DQPSK single channel link.

     

Uptake of small oxygenated organic molecules onto ammonium nitrate under upper tropospheric conditions

The uptake of formic (C1), propanoic (C3), butanoic (C4), and pentanoic (C5) acids onto ammonium nitrate (AN) has been investigated as a function of temperature and relative humidity using a Knudsen cell flow reactor coupled with FTIR-reflection absorption spectroscopy (FTIR-RAS). The uptake of acetone and methanol onto AN was also briefly studied. Initial uptake coefficients (gamma) were determined over the temperature range 200-240 K. Formic, propanoic, and butanoic acids exhibited efficient but temperature-dependent uptake on AN, with larger uptake coefficients observed at lower temperatures. Pentanoic acid was not taken up by AN under any of the conditions studied. Uptake of acetone and methanol onto AN was observed, but in insignificant amounts under atmospherically relevant conditions. Infrared spectra revealed that propanoic and butanoic acids ionized on the surface, despite the fact that the AN films were effloresced. Formic acid reacted with the AN film to produce ammonium formate and ionized nitric acid. Adding small amounts of water vapor (4% RH) to the chamber resulted in dramatically increased gamma values for all of the acids. Furthermore, the IR spectra showed the formation of a liquid layer when propanoic and butanoic acids adsorbed on the surface at RH = 20% and greater. Liquid water features were not observed at a similar relative humidity in the absence of the acids. These results show that small organic acids can be efficiently scavenged by AN and lead to enhanced water uptake under upper tropospheric conditions.