Classifying Cantor Sets by their Fractal Dimensions

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.Comment: 10 pages, revised version. To appear in Proceedings of the AMS

Firefighting as a game

The Firefighter Problem was proposed in 1995 [16] as a deterministic discrete-time model for the spread (and containment) of a fire. Its applications reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. In this work, we study the problem from a game-theoretical perspective. Such a context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination. We model the Firefighter Problem as a strategic game where there is one player for each time step who decides where to place the firefighters. We show that the Price of Anarchy is linear in the general case, but at most 2 for trees. We prove that the quality of the equilibria improves when allowing coalitional cooperation among players. In general, we have that the Price of Anarchy is in T(n/k) where k is the coalition size. Furthermore, we show that there are topologies which have a constant Price of Anarchy even when constant sized coalitions are considered.Peer ReviewedPostprint (author’s final draft

Abundance Measurements of Titan's Stratospheric HCN, HC3_3N, C3_3H4_4, and CH3_3CN from ALMA Observations

Previous investigations have employed more than 100 close observations of Titan by the Cassini orbiter to elucidate connections between the production and distribution of Titan's vast, organic-rich chemical inventory and its atmospheric dynamics. However, as Titan transitions into northern summer, the lack of incoming data from the Cassini orbiter presents a potential barrier to the continued study of seasonal changes in Titan's atmosphere. In our previous work (Thelen et al., 2018), we demonstrated that the Atacama Large Millimeter/submillimeter Array (ALMA) is well suited for measurements of Titan's atmosphere in the stratosphere and lower mesosphere (~100-500 km) through the use of spatially resolved (beam sizes <1'') flux calibration observations of Titan. Here, we derive vertical abundance profiles of four of Titan's trace atmospheric species from the same 3 independent spatial regions across Titan's disk during the same epoch (2012 to 2015): HCN, HC$_3$N, C$_3$H$_4$, and CH$_3$CN. We find that Titan's minor constituents exhibit large latitudinal variations, with enhanced abundances at high latitudes compared to equatorial measurements; this includes CH$_3$CN, which eluded previous detection by Cassini in the stratosphere, and thus spatially resolved abundance measurements were unattainable. Even over the short 3-year period, vertical profiles and integrated emission maps of these molecules allow us to observe temporal changes in Titan's atmospheric circulation during northern spring. Our derived abundance profiles are comparable to contemporary measurements from Cassini infrared observations, and we find additional evidence for subsidence of enriched air onto Titan's south pole during this time period. Continued observations of Titan with ALMA beyond the summer solstice will enable further study of how Titan's atmospheric composition and dynamics respond to seasonal changes.Comment: 15 pages, 16 figures, 2 tables. Accepted for publication in Icarus, September 201

Finding secluded places of special interest in graphs.

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets

Finding secluded places of special interest in graphs

Finding a vertex subset in a graph that satisfies a certain property is one of the most-studied topics in algorithmic graph theory. The focus herein is often on minimizing or maximizing the size of the solution, that is, the size of the desired vertex set. In several applications, however, we also want to limit the “exposure” of the solution to the rest of the graph. This is the case, for example, when the solution represents persons that ought to deal with sensitive information or a segregated community. In this work, we thus explore the (parameterized) complexity of finding such secluded vertex subsets for a wide variety of properties that they shall fulfill. More precisely, we study the constraint that the (open or closed) neighborhood of the solution shall be bounded by a parameter and the influence of this constraint on the complexity of minimizing separators, feedback vertex sets, F-free vertex deletion sets, dominating sets, and the maximization of independent sets