Morphing Parallel Graph Drawings

A pair of straight-line drawings of a graph is called parallel if, for every edge of the graph, the line segment that represents the edge in one drawing is parallel with the line segment that represents the edge in the other drawing. We study the problem of morphing between pairs of parallel planar drawings of a graph, keeping all intermediate drawings planar and parallel with the source and target drawings. We call such a morph a parallel morph. Parallel morphs have application to graph visualization. The problem of deciding whether two parallel drawings in the plane admit a parallel morph turns out to be NP-hard in general. However, for some restricted classes of graphs and drawings, we can efficiently decide parallel morphability. Our main positive result is that every pair of parallel simple orthogonal drawings in the plane admits a parallel morph. We give an efficient algorithm that computes such a morph. The number of steps required in a morph produced by our algorithm is linear in the complexity of the graph, where a step involves moving each vertex along a straight line at constant speed. We prove that this upper bound on the number of steps is within a constant factor of the worst-case lower bound. We explore the related problem of computing a parallel morph where edges are required to change length monotonically, i.e. to be either non-increasing or non-decreasing in length. Although parallel orthogonally-convex polygons always admit a monotone parallel morph, deciding morphability under these constraints is NP-hard, even for orthogonal polygons. We also begin a study of parallel morphing in higher dimensions. Parallel drawings of trees in any dimension always admit a parallel morph. This is not so for parallel drawings of cycles in 3-space, even if orthogonal. Similarly, not all pairs of parallel orthogonal polyhedra admit a parallel morph, even if they are topological spheres. In fact, deciding parallel morphability turns out to be PSPACE-hard for both parallel orthogonal polyhedra, and parallel orthogonal drawings in 3-space

The Astropy Problem

The Astropy Project (http://astropy.org) is, in its own words, "a community effort to develop a single core package for Astronomy in Python and foster interoperability between Python astronomy packages." For five years this project has been managed, written, and operated as a grassroots, self-organized, almost entirely volunteer effort while the software is used by the majority of the astronomical community. Despite this, the project has always been and remains to this day effectively unfunded. Further, contributors receive little or no formal recognition for creating and supporting what is now critical software. This paper explores the problem in detail, outlines possible solutions to correct this, and presents a few suggestions on how to address the sustainability of general purpose astronomical software

2-year outcomes after transcatheter versus surgical aortic valve replacement in low-risk patients

Background: The Evolut Low Risk Trial (Medtronic Evolut Transcatheter Aortic Valve Replacement in Low Risk Patients) showed that transcatheter aortic valve replacement (TAVR) with a supra-annular, self-expanding valve was noninferior to surgery for the primary endpoint of all-cause mortality or disabling stroke at 2 years. This finding was based on a Bayesian analysis performed after 850 patients had reached 1 year of follow-up. Objectives: The goal of this study was to report the full 2-year clinical and echocardiographic outcomes for patients enrolled in the Evolut Low Risk Trial. Methods: A total of 1,414 low-surgical risk patients with severe aortic stenosis were randomized to receive TAVR or surgical AVR. An independent clinical events committee adjudicated adverse events, and a central echocardiographic core laboratory assessed hemodynamic endpoints. Results: An attempted implant was performed in 730 TAVR and 684 surgical patients from March 2016 to May 2019. The Kaplan-Meier rates for the complete 2-year primary endpoint of death or disabling stroke were 4.3% in the TAVR group and 6.3% in the surgery group (P = 0.084). These rates were comparable to the interim Bayesian rates of 5.3% with TAVR and 6.7% with surgery (difference: -1.4%; 95% Bayesian credible interval: -4.9% to 2.1%). All-cause mortality rates were 3.5% vs 4.4% (P = 0.366), and disabling stroke rates were 1.5% vs 2.7% (P = 0.119), respectively. Between years 1 and 2, there was no convergence of the primary outcome curves. Conclusions: The complete 2-year follow-up from the Evolut Low Risk Trial found that TAVR is noninferior to surgery for the primary endpoint of all-cause mortality or disabling stroke, with event rates that were slightly better than those predicted by using the Bayesian analysis. (Medtronic Evolut Transcatheter Aortic Valve Replacement in Low Risk Patients [Evolut Low Risk Trial]; NCT02701283)