A statistical approach to persistent homology

Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying distribution.Comment: 30 pages, 2 figures, minor changes, to appear in Homology, Homotopy and Application

Eccentricity Evolution for Planets in Gaseous Disks

We investigate the hypothesis that interactions between a giant planet and the disk from which it forms promote eccentricity growth. These interactions are concentrated at discrete Lindblad and corotation resonances. Interactions at principal Lindblad resonances cause the planet's orbit to migrate and open a gap in the disk if the planet is sufficiently massive. Those at first order Lindblad and corotation resonances change the planet's orbital eccentricity. Eccentricity is excited by interactions at external Lindblad resonances which are located on the opposite side of corotation from the planet, and damped by co-orbital Lindblad resonances which overlap the planet's orbit. If the planet clears a gap in the disk, the rate of eccentricity damping by co-orbital Lindblad resonances is reduced. Density gradients associated with the gap activate eccentricity damping by corotation resonances at a rate which initially marginally exceeds that of eccentricity excitation by external Lindblad resonances. But the corotation torque drives a mass flux which reduces the density gradient near the resonance. Sufficient partial saturation of corotation resonances can tip the balance in favor of eccentricity excitation. A minimal initial eccentricity of a few percent is required to overcome viscous diffusion which acts to unsaturate corotation resonances by reestablishing the large scale density gradient. Thus eccentricity growth is a finite amplitude instability. Formally, interactions at the apsidal resonance, which is a special kind of co-orbital Lindblad resonance, appears to damp eccentricity faster than external Lindblad resonances can excite it. However, apsidal waves have such long wavelengths that they do not propagate in protoplanetary disks. This reduces eccentricity damping by the apsidal resonance to a modest level.Comment: Submitted to Ap

THE EFFECTS OF HOUSING PRICES, WAGES, AND COMMUTING TIME ON JOINT RESIDENTIAL AND JOB LOCATION CHOICES

A utility maximizing framework is used to model how wages, housing prices, and commuting time affect joint decisions of where to live and where to work. The implied multinomial logit model yields plausible estimates of the role of economic variables on joint residence/job location choices.Labor and Human Capital,

Label-free optical detection of single enzyme-reactant reactions and associated conformational changes

Monitoring the kinetics and conformational dynamics of single enzymes is crucial in order to better understand their biological functions as these motions and structural dynamics are usually unsynchronized among the molecules. Detecting the enzyme-reactant interactions and associated conformational changes of the enzyme on a single molecule basis, however, remain as a challenge with established optical techniques due to the commonly required labeling of the reactants or the enzyme itself. The labeling process is usually non-trivial and the labels themselves might skew the physical properties of the enzyme. Here we demonstrate an optical, label-free method capable of observing enzymatic interactions and the associated conformational changes on the single molecule level. We monitor polymerase/DNA interactions via the strong near-field enhancement provided by plasmonic nanorods resonantly coupled to whispering gallery modes in microcavities. Specifically, we employ two different recognition schemes: one in which the kinetics of polymerase/DNA interactions are probed in the vicinity of DNA-functionalized nanorods, and the other in which these interactions are probed via the magnitude of conformational changes in the polymerase molecules immobilized on nanorods. In both approaches we find that low and high polymerase activities can be clearly discerned via their characteristic signal amplitude and signal length distributions. Furthermore, the thermodynamic study of the monitored interactions suggests the occurrence of DNA polymerization. This work constitutes a proof-of-concept study of enzymatic activities via plasmonically enhanced microcavities and establishes an alternative and label-free method capable of investigating structural changes in single molecules

Multi-scale Optics for Enhanced Light Collection from a Point Source

High efficiency collection of photons emitted by a point source over a wide field-of-view (FoV) is crucial for many applications. Multi-scale optics over improved light collection by utilizing small optical components placed close to the optical source, while maintaining a wide FoV provided by conventional imaging optics. In this work, we demonstrate collection efficiency of 26% of photons emitted by a point-like source using a micromirror fabricated in silicon with no significant decrease in collection efficiency over a 10 mm object space.Comment: 4 pages, 4 figure