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 $k$

A method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure

Environmental Influences on the Morphology and Dynamics of Group Size Haloes

We use group size haloes identified with a ``friends of friends'' (FOF) algorithm in a concordance $\Lambda \rm{CDM}$ GADGET2 (dark matter only) simulation to investigate the dependence of halo properties on the environment at $z=0$. The study is carried out using samples of haloes at different distances from their nearest massive {\em cluster} halo. We find that the fraction of haloes with substructure typically increases in high density regions. The halo mean axial ratio $$ also increases in overdense regions, a fact which is true for the whole range of halo mass studied. This can be explained as a reflection of an earlier halo formation time in high-density regions, which gives haloes more time to evolve and become more spherical. Moreover, this interpretation is supported by the fact that, at a given halo-cluster distance, haloes with substructure are more elongated than their equal mass counterparts with no substructure, reflecting that the virialization (and thus sphericalization) process is interrupted by merger events. The velocity dispersion of low mass haloes with strong substructure shows a significant increase near massive clusters with respect to equal mass haloes with low-levels of substructure or with haloes found in low-density environments. The alignment signal between the shape and the velocity ellipsoid principal axes decreases going from lower to higher density regions, while such an alignment is stronger for haloes without substructure. We also find, in agreement with other studies, a tendency of halo major axes to be aligned and of minor axes to lie roughly perpendicular with the orientation of the filament within which the halo is embedded, an effect which is stronger in the proximity of the massive clusters.Comment: 11 pages, 12 figures, accepted for publication in MNRA