An Optimal Algorithm for the Separating Common Tangents of two Polygons

We describe an algorithm for computing the separating common tangents of two simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies to the same side of the line. A separating common tangent of two polygons is a tangent of both polygons where the polygons are lying on different sides of the tangent. Each polygon is given as a read-only array of its corners. If a separating common tangent does not exist, the algorithm reports that. Otherwise, two corners defining a separating common tangent are returned. The algorithm is simple and implies an optimal algorithm for deciding if the convex hulls of two polygons are disjoint or not. This was not known to be possible in linear time and constant workspace prior to this paper. An outer common tangent is a tangent of both polygons where the polygons are on the same side of the tangent. In the case where the convex hulls of the polygons are disjoint, we give an algorithm for computing the outer common tangents in linear time using constant workspace.Comment: 12 pages, 6 figures. A preliminary version of this paper appeared at SoCG 201

Diagrams as Vehicles for Scientific Reasoning

We argue that diagrams are not just a communicative tool but play important roles in the reasoning of biologists: in characterizing the phenomenon to be explained, identifying explanatory relations, and developing an account of the responsible mechanism. In the first two tasks diagrams facilitate applying visual processing to the detection of patterns that constitute phenomena or explanatory relations. Diagrams of a mechanism serve to guide reasoning about what parts and operations are needed and how potential parts of the mechanism are related to each other. Further they guide the development of computational models used to determine how the mechanism will behave. We illustrate each of these uses of diagrams with examples from research on circadian rhythm

Convergence analysis of block Gibbs samplers for Bayesian linear mixed models with p>Np>N

Exploration of the intractable posterior distributions associated with Bayesian versions of the general linear mixed model is often performed using Markov chain Monte Carlo. In particular, if a conditionally conjugate prior is used, then there is a simple two-block Gibbs sampler available. Rom\'{a}n and Hobert [Linear Algebra Appl. 473 (2015) 54-77] showed that, when the priors are proper and the $X$ matrix has full column rank, the Markov chains underlying these Gibbs samplers are nearly always geometrically ergodic. In this paper, Rom\'{a}n and Hobert's (2015) result is extended by allowing improper priors on the variance components, and, more importantly, by removing all assumptions on the $X$ matrix. So, not only is $X$ allowed to be (column) rank deficient, which provides additional flexibility in parameterizing the fixed effects, it is also allowed to have more columns than rows, which is necessary in the increasingly important situation where $p>N$. The full rank assumption on $X$ is at the heart of Rom\'{a}n and Hobert's (2015) proof. Consequently, the extension to unrestricted $X$ requires a substantially different analysis.Comment: Published at http://dx.doi.org/10.3150/15-BEJ749 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Diameter two properties, convexity and smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.Comment: Removed Proposition 2.7 from version [v1] because of a gap in the proof. arXiv admin note: text overlap with arXiv:1506.0523