Kinetic Euclidean 2-centers in the black-box model

We study the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_max on the maximum displacement of any point within one time step. We show how to maintain a (1 + e)-approximation of the Euclidean 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. In many cases --namely when the distance between the centers of the disks is relatively large or relatively small-- the solution we maintain is actually optimal

Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$

Dilute Wet Granulates: Nonequilibrium Dynamics and Structure Formation

We investigate a gas of wet granular particles, covered by a thin liquid film. The dynamic evolution is governed by two-particle interactions, which are mainly due to interfacial forces in contrast to dry granular gases. When two wet grains collide, a capillary bridge is formed and stays intact up to a certain distance of withdrawal when the bridge ruptures, dissipating a fixed amount of energy. A freely cooling system is shown to undergo a nonequillibrium dynamic phase transition from a state with mainly single particles and fast cooling to a state with growing aggregates, such that bridge rupture becomes a rare event and cooling is slow. In the early stage of cluster growth, aggregation is a self-similar process with a fractal dimension of the aggregates approximately equal to D_f ~ 2. At later times, a percolating cluster is observed which ultimately absorbs all the particles. The final cluster is compact on large length scales, but fractal with D_f ~ 2 on small length scales.Comment: 14 pages, 20 figure

Feedback and the Structure of Simulated Galaxies at redshift z=2

We study the properties of simulated high-redshift galaxies using cosmological N-body/gasdynamical runs from the OverWhelmingly Large Simulations (OWLS) project. The runs contrast several feedback implementations of varying effectiveness: from no-feedback, to supernova-driven winds to powerful AGN-driven outflows. These different feedback models result in large variations in the abundance and structural properties of bright galaxies at z=2. We find that feedback affects the baryonic mass of a galaxy much more severely than its spin, which is on average roughly half that of its surrounding dark matter halo in our runs. Feedback induces strong correlations between angular momentum content and galaxy mass that leave their imprint on galaxy scaling relations and morphologies. Encouragingly, we find that galaxy disks are common in moderate-feedback runs, making up typically ~50% of all galaxies at the centers of haloes with virial mass exceeding 1e11 M_sun. The size, stellar masses, and circular speeds of simulated galaxies formed in such runs have properties that straddle those of large star-forming disks and of compact early-type galaxies at z=2. Once the detailed abundance and structural properties of these rare objects are well established it may be possible to use them to gauge the overall efficacy of feedback in the formation of high redshift galaxies.Comment: 16 pages, 12 figures. Accepted for publication in MNRAS. Minor changes to match published versio