Largest Empty Circle Centered on a Query Line

The Largest Empty Circle problem seeks the largest circle centered within the convex hull of a set $P$ of $n$ points in $\mathbb{R}^2$ and devoid of points from $P$. In this paper, we introduce a query version of this well-studied problem. In our query version, we are required to preprocess $P$ so that when given a query line $Q$, we can quickly compute the largest empty circle centered at some point on $Q$ and within the convex hull of $P$. We present solutions for two special cases and the general case; all our queries run in $O(\log n)$ time. We restrict the query line to be horizontal in the first special case, which we preprocess in $O(n \alpha(n) \log n)$ time and space, where $\alpha(n)$ is the slow growing inverse of the Ackermann's function. When the query line is restricted to pass through a fixed point, the second special case, our preprocessing takes $O(n \alpha(n)^{O(\alpha(n))} \log n)$ time and space. We use insights from the two special cases to solve the general version of the problem with preprocessing time and space in $O(n^3 \log n)$ and $O(n^3)$ respectively.Comment: 18 pages, 13 figure

Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=\Omega(\sqrt n)$, and at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr)$ edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Cellular Topological Packings in Early Embryos

At very early embryonic stages, when embryos are composed of just a few cells, establishing the correct packing arrangements (contacts) between cells is essential for proper development of the organism. As early as the 4-cell stage, the observed blastomere packings in different species are different and, in many cases, differ from the equilibrium packings expected for simple adherent and deformable particles.We use a novel 3D Voronoi-augnemted Langevin simulator to systematically study how the forces between blastomeres, their division rates, orientation of cell division, and embryonic confinement influence the final packing configurations. In the absence of physical confinement of the embryo, we find that blastomere packings are not robust, with multiple packing configurations simultaneously possible (degeneracy) and are very sensitive to parameter changes. Our results indicate that the geometry of the embryonic confining shell determines the packing configurations at the 4-cell stage, removing degeneracy in the possible packing configurations and overriding division rules in most cases.Furthermore, we use our simulator to study the robustness of the C. elegans early embryo to noise in division timing and angle. We find that there exists a range of timing and angular noise that the embryo is fully robust to and categorize the errors outside this regime as coming from mistimed divisions or misplaced cells. We also study how robust the embryo is to overall shifts in the timing offset between the AB and P1 divisions and find that even large changes can be non-lethal. Finally, we systematically investigate how robust the embryo is to deterministic shifts in division directions from the wildtype rules and find that the major source of lethal error is from offsets of more than 90 degrees to the P2-EMS division axis. Overall, our results demonstrate how confinement, division timing and division rules all contribute to ensuring robust development with confinement setting the overall packing topology and division timings and rules specifying where individual cells will go within that shape

Role of Shape in the Self-Assembly of Anisotropic Colloids.

Self-assembly is the process of spontaneous organization of a set of interacting components. We examine how particle shape drives the self-assembly of colloids in three different systems. When particles interact only via their shape, entropic crystallization can occur; we discuss a design strategy using the Voronoi tesslelation to create “Voronoi particles,” (VP) which are hard particles in the shape of Voronoi cells of their target structure. Although VP stabilize their target structure in the limit of infinite pressure, the self-assembly of the same structure at moderate pressure is not guaranteed. We find that more symmetric crystals are often preferred due to entropic contributions of several kBT from configurational degeneracies. We characterize the assembly of VP in terms of their symmetries and the complexities of the target structure and demonstrate how controlling the degeneracies through modifying shape and field-directed assembly can improve the assembly propensity. With the addition of non-adsorbing, polymers, hard colloids experience an attraction dependent on polymer concentration, the form of which is dictated by the colloid shape; we study a system of oblate, spheroidal colloids that self-assemble thread-like clusters. In both simulation and experiment the colloids condense into disordered droplets at low polymer concentrations; at higher concentrations we observe kinetic arrest into primarily linear clusters of aligned colloids. We show that the mechanical stabilty of these low-valence structures results from the anisotropic particle shape. Particle surfaces can be patterned with metal coatings, introducing enthalpic attraction between particles; we study a system of prolate spheroidal colloids, half-coated in gold. We show with experiments and computer simulations that Janus ellipsoids can self-assemble into self-limiting one-dimensional fibers with shape-memory properties, and that the fibrillar assemblies can be actuated on application of an external alternating-current electric field. Actuation of the fibers occurs through a sliding mechanism (allowed by the curved ellipsoidal surface) that permits the reversible elongation of the Janus-ellipsoid chains by ~36%. In each case, we find shape plays a critical role. By understanding and isolating its impact, we enhance shape's utility as a parameter for the design of self-assembling colloids.PhDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111630/1/baschult_1.pd

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

Double barred galaxies at intermediate redshifts: A feasibility study

Despite the increasing number of studies of barred galaxies at intermediate and high redshifts, double-barred (S2B) systems have only been identified in the nearby (z<0.04) universe thus far. In this feasibility study we demonstrate that the detection and analysis of S2Bs is possible at intermediate redshifts (0.1 < z < 0.5) with the exquisite resolution of the Hubble Space Telescope Advanced Camera for Surveys (HST/ACS). We identify barred galaxies in the HST/ACS data of the Great Observatories Origins Deep Survey (GOODS) using a novel method. The radial profile of the Gini coefficient -- a model-independent structure parameter -- is able to detect bars in early-type galaxies that are large enough that they might host an inner bar of sufficient angular size. Using this method and subsequent examination with unsharp masks and ellipse fits we identified the two most distant S2Bs currently known (at redshifts z=0.103 and z=0.148). We investigate the underlying stellar populations of these two galaxies through a detailed colour analysis, in order to demonstrate the analysis that could be performed on a future sample of intermediate-redshift S2Bs. We also identify two S2Bs and five S2B candidates in the HST/ACS data of the Cosmic Evolution Survey (COSMOS). Our detections of distant S2Bs show that deep surveys like GOODS and COSMOS have the potential to push the limit for S2B detection and analysis out by a factor of ten in redshift and lookback time (z=0.5, t=5Gyr) compared to the previously known S2Bs. This in turn would provide new insight into the formation of these objects.Comment: 9 pages + 10 figures. Accepted for publication in MNRAS. Main change from version 1 is an extension of the introduction/motivation and discussion section. A full resolution version including colour figures is available at http://www.astro.unibas.ch/~tlisker/papers/lisker2006_s2b.pd

On Degeneracy of Lower Envelopes of Algebraic Surfaces

Abstract We analyze degeneracy of lower envelopes of algebraic surfaces. We focus on the cases omitted in the existing complexity analysis of lower envelope