Virtual Polytopes

Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications

A Characterization of Visibility Graphs for Pseudo-Polygons

In this paper, we give a characterization of the visibility graphs of pseudo-polygons. We first identify some key combinatorial properties of pseudo-polygons, and we then give a set of five necessary conditions based off our identified properties. We then prove that these necessary conditions are also sufficient via a reduction to a characterization of vertex-edge visibility graphs given by O'Rourke and Streinu

Locked and Unlocked Polygonal Chains in 3D

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D with a polynomial number of moves.Comment: To appear in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan. 199

Maximizing Maximal Angles for Plane Straight-Line Graphs

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

Auxetic regions in large deformations of periodic frameworks

In materials science, auxetic behavior refers to lateral widening upon stretching. We investigate the problem of finding domains of auxeticity in global deformation spaces of periodic frameworks. Case studies include planar periodic mechanisms constructed from quadrilaterals with diagonals as periods and other frameworks with two vertex orbits. We relate several geometric and kinematic descriptions.Comment: Presented at the International Conference on "Interdisciplinary Applications of Kinematics" (IAK18), Lima, Peru, March 201

Age of HIV Acquisition Affects the Risk of Multi-Morbidity after 25 Years of Infection Exposure

Introduction: Understanding the intersection of HIV, aging and health is crucial due to the increasing number of people aging with HIV. Objective: The objective of the study was to assess the prevalence of, and risk factors for individual comorbidities and multi-morbidity in people living with HIV with similar duration of HIV infection, notwithstanding a 25-year difference at the time of HIV acquisition. Methods: In a cross-sectional multicentre retrospective study, we compared three match-control age groups. The "Young" were selected from Romania and included HIV-positive patients prenatally infected and assessed at the age of 25-30 years. The "Old" and the "Geriatric" were selected from Italy. These respectively included subjects infected with HIV at the age of 25 years and assessed at the age of 50-55 years, and those infected at the age of 50 years and assessed at the age of 75-80 years. Each group was sex and age matched in a 1: 5 ratio with controls selected from the CINECA ARNO database from Italy. We described non-infectious comorbidities (NICM), including cardiovascular disease, hypertension, dyslipidaemia, diabetes, chronic kidney disease, and multi-morbidity (MM >= 3 NICM). Results: MM prevalence in the "Young" group compared to controls was 6.2% vs 0%, while in the "Geriatric" was "68.2% vs 3.6%. Using "Young" as a reference, in multivariate analyses, predictors for MM were as follows: HIV serostatus (OR=47.75, IQR 14.78-154.25, p<0.01) and "Geriatric" vs "Young" (OR=30.32, IQR 5.89-155.98, p<0.01). Conclusion: These data suggest that age at acquisition of HIV should be considered as a risk factor for NICM and MM

Locked and Unlocked Polygonal Chains in Three Dimensions

This paper studies movements of polygonal chains in three dimensions whose links are not allowed to cross or change length. Our main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions. Other results include an algorithm for straightening open chains having a simple orthogonal projection onto some plane, and an algorithm for making convex any open chain initially configured on the surface of a polytope. All our algorithms require only O (n) basic moves.

Safety and on-treatment efficacy of telaprevir: the early access programme for patients with advanced hepatitis C

Background and aim Severe adverse events (AEs) compromise the outcome of direct antiviral agent-based treatment in patients with advanced liver fibrosis due to HCV infection. HEP3002 is an ongoing multinational programme to evaluate safety and efficacy of telaprevir (TVR) plus pegylated-interferon-alpha (PEG-IFN alpha) and ribavirin (RBV) in patients with advanced liver fibrosis caused by HCV genotype 1 (HCV-1).Methods 1782 patients with HCV-1 and bridging fibrosis or compensated cirrhosis were prospectively recruited from 16 countries worldwide, and treated with 12 weeks of TVR plus PEG-IFN/RBV, followed by 12 or 36 weeks of PEG-IFN and RBV (PR) alone dependent on virological response to treatment and previous response type.Results 1587 patients completed 12 weeks of triple therapy and 4 weeks of PR tail (53% cirrhosis, 22% HCV-1a). By week 12, HCV RNA was undetectable in 85% of naives, 88% of relapsers, 80% of partial responders and 72% of null responders. Overall, 931 patients (59%) developed grade 1-4 anaemia (grade 3/4 in 31%), 630 (40%) dose reduced RBV, 332 (21%) received erythropoietin and 157 (10%) were transfused. Age and female gender were the strongest predictors of anaemia. 64 patients (4%) developed a grade 3/4 rash. Discontinuation of TVR due to AEs was necessary in 193 patients (12%). Seven patients died (0.4%, six had cirrhosis).Conclusions in compensated patients with advanced fibrosis due to HCV-1, triple therapy with TVR led to satisfactory rates of safety, tolerability and on-treatment virological response with adequate managements of AEs.Janssen PharmaceuticsUniv Milan, Div Gastroenterol, Dept Med, Fdn IRCCS Ca Granda Osped Maggiore Policlin, Milan, ItalyHosp Univ 12 Octubre, Secc Aparato Digest, Madrid, SpainIM Sechenov First Moscow State Med Univ, EM Tareev Clin Nephrol Internal & Occupat Med, Moscow, RussiaUniversidade Federal de São Paulo, Viral Hepatitis Div Infect Dis, Outpatient Clin HIV, São Paulo, BrazilUniv Sydney, Royal Prince Alfred Hosp, AW Morrow Gastroenterol & Liver Ctr, Sydney, NSW 2006, AustraliaCharles Univ Prague, Fac Med 1, Dept Internal Med, Prague, Czech RepublicCent Mil Hosp Prague, Prague, Czech RepublicUniv Libre Brussels, Dept Gastroenterol Hepatopancreatol & Digest Onco, Erasme Univ Hosp, Liver Unit, Brussels, BelgiumCarol Davila Univ Med & Pharm, Natl Inst Infect Dis, Bucharest, RomaniaJanssen Pharmaceut, B-2340 Beerse, BelgiumJanssen Pharmaceut, Paris, FranceJanssen Res & Dev, Titusville, NJ USAJanssen Res & Dev, High Wycombe, Bucks, EnglandJanssen Cilag AG, Zug, SwitzerlandHannover Med Sch, D-30623 Hannover, GermanyUniversidade Federal de São Paulo, Viral Hepatitis Div Infect Dis, Outpatient Clin HIV, São Paulo, BrazilWeb of Scienc

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016