Continuous-time Dynamic Shortest Paths with Negative Transit Times

We consider the dynamic shortest path problem in the continuous-time model because of its importance. This problem has been extensively studied in the literature. But so far, all contributions to this problem are based on the assumption that all transit times are strictly positive. However, in order to study dynamic network flows it is essential to support negative transit times since they occur quite naturally in residual networks. In this paper we extend the work of Philpott [SIAM Control Opt., 1994, pp. 538-552] to the case of arbitrary (also negative) transit times. In particular, we study a corresponding linear program in space of measures and characterize its extreme points. We show a one-to-one correspondence between extreme points and dynamic paths. Further, under certain assumptions, we prove the existence of an optimal extreme point to the linear program and establish a strong duality result. We also present counterexamples to show that strong duality only holds under these assumptions

Nash Equilibria and the Price of Anarchy for Flows Over Time

We study Nash equilibria in the context of flows over time. Many results on static routing games have been obtained over the last ten years. In flows over time (also called dynamic flows), flow travels through a network over time and, as a consequence, flow values on edges change over time. This more realistic setting has not been tackled from the viewpoint of algorithmic game theory yet; but there is a rich literature on game theoretic aspects of flows over time in the traffic community. We present a novel characterization of Nash equilibria for flows over time. It turns out that Nash flows over time can be seen as a concatenation of special static flows. The underlying flow over time model is a variant of the so-called deterministic queuing model that is very popular in road traffic simulation and related fields. Based upon this, we prove the first known results on the price of anarchy for flows over time

The Kepler Pixel Response Function

Kepler seeks to detect sequences of transits of Earth-size exoplanets orbiting Solar-like stars. Such transit signals are on the order of 100 ppm. The high photometric precision demanded by Kepler requires detailed knowledge of how the Kepler pixels respond to starlight during a nominal observation. This information is provided by the Kepler pixel response function (PRF), defined as the composite of Kepler's optical point spread function, integrated spacecraft pointing jitter during a nominal cadence and other systematic effects. To provide sub-pixel resolution, the PRF is represented as a piecewise-continuous polynomial on a sub-pixel mesh. This continuous representation allows the prediction of a star's flux value on any pixel given the star's pixel position. The advantages and difficulties of this polynomial representation are discussed, including characterization of spatial variation in the PRF and the smoothing of discontinuities between sub-pixel polynomial patches. On-orbit super-resolution measurements of the PRF across the Kepler field of view are described. Two uses of the PRF are presented: the selection of pixels for each star that maximizes the photometric signal to noise ratio for that star, and PRF-fitted centroids which provide robust and accurate stellar positions on the CCD, primarily used for attitude and plate scale tracking. Good knowledge of the PRF has been a critical component for the successful collection of high-precision photometry by Kepler.Comment: 10 pages, 5 figures, accepted by ApJ Letters. Version accepted for publication

Photometric Variability in Kepler Target Stars: The Sun Among Stars -- A First Look

The Kepler mission provides an exciting opportunity to study the lightcurves of stars with unprecedented precision and continuity of coverage. This is the first look at a large sample of stars with photometric data of a quality that has heretofore been only available for our Sun. It provides the first opportunity to compare the irradiance variations of our Sun to a large cohort of stars ranging from vary similar to rather different stellar properties, at a wide variety of ages. Although Kepler data is in an early phase of maturity, and we only analyze the first month of coverage, it is sufficient to garner the first meaningful measurements of our Sun's variability in the context of a large cohort of main sequence stars in the solar neighborhood. We find that nearly half of the full sample is more active than the active Sun, although most of them are not more than twice as active. The active fraction is closer to a third for the stars most similar to the Sun, and rises to well more than half for stars cooler than mid K spectral types.Comment: 13 pages, 4 figures, accepted to ApJ Letter

Continuous and Discrete Flows Over Time: A General Model Based on Measure Theory

Network flows over time form a fascinating area of research. They model the temporal dynamics of network flow problems occurring in a wide variety of applications. Research in this area has been pursued in two different and mainly independent directions with respect to time modeling: discrete and continuous time models. In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each suitable subset a real value, interpreted as the amount of flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem