L-Visibility Drawings of IC-planar Graphs

An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either a horizontal or vertical segment. As a byproduct of our drawing technique, we prove that an IC-plane graph has a RAC drawing in quadratic area with at most two bends per edge

Genomic dynamics of transposable elements in the western clawed frog (Silurana tropicalis)

Transposable elements (TEs) are repetitive DNA sequences that can make new copies of themselves that are inserted elsewhere in a host genome. The abundance and distributions of TEs vary considerably among phylogenetically diverse hosts. With the aim of exploring the basis of this variation, we evaluated correlations between several genomic variables and the presence of TEs and non-TE repeats in the complete genome sequence of the Western clawed frog (Silurana tropicalis). This analysis reveals patterns of TE insertion consistent with gene disruption but not with the insertional preference model. Analysis of non-TE repeats recovered unique features of their genome-wide distribution when compared with TE repeats, including no strong correlation with exons and a particularly strong negative correlation with GC content. We also collected polymorphism data from 25 TE insertion sites in 19 wild-caught S. tropicalis individuals. DNA transposon insertions were fixed at eight of nine sites and at a high frequency at one of nine, whereas insertions of long terminal repeat (LTR) and non-LTR retrotransposons were fixed at only 4 of 16 sites and at low frequency at 12 of 16. A maximum likelihood model failed to attribute these differences in insertion frequencies to variation in selection pressure on different classes of TE, opening the possibility that other phenomena such as variation in rates of replication or duration of residence in the genome could play a role. Taken together, these results identify factors that sculpt heterogeneity in TE distribution in S. tropicalis and illustrate that genomic dynamics differ markedly among TE classes and between TE and non-TE repeats.published_or_final_versio

3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Acquired heart block: A possible complication of patent ductus arteriosus in a preterm infant

A large patent ductus arteriosus (PDA) is a frequently encountered clinical problem in extremely low birth weight (ELBW) infants. It leads to an increased pulmonary blood flow and in a decreased or reversed diastolic flow in the systemic circulation, resulting in complications. Here we report a possible complication of PDA not previously published. On day 8 of life, a male ELBW infant (birth weight 650 g) born at a gestational age of 23 weeks and 3 days developed an atrioventricular block (AV block). The heart rate dropped from 168/min to 90/min, and the ECG showed a Wenckebach second-degree AV block and intraventricular conduction disturbances. Echocardiography demonstrated a PDA with a large left-to-right shunt and large left atrium and left ventricle with high contractility. Within several minutes after surgical closure of the PDA, the heart rate increased, and after 30 min the AV block had improved to a 1: 1 conduction ratio. Echocardiography after 2 h revealed a significant decrease of the left ventricular and atrial dimensions. Within 12 h, the AV block completely reversed together with the intraventricular conduction disturbances. We suggest that PDA with a large left-to-right shunt and left ventricular volume overload may lead to an AV block in an ELBW infant. Surgical closure of the PDA may be indicated. Copyright (C) 2007 S. Karger AG, Basel

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

Shedding Light Onto the Nature of Iron Decorated Graphene and Graphite Oxide Nanohybrids for CO₂ Conversion at Atmospheric Pressure

We report on the design and testing of new graphite and graphene oxide‐based extended π‐conjugated synthetic scaffolds for applications in sustainable chemistry transformations. Nanoparticle‐functionalised carbonaceous catalysts for new Fischer Tropsch and Reverse GasWater Shift (RGWS) transformations were prepared: functional graphene oxides emerged from graphite powders via an adapted Hummer's method and subsequently impregnated with uniform‐sized nanoparticles. Then the resulting nanomaterials were imaged by TEM, SEM, EDX, AFM and characterised by IR, XPS and Raman spectroscopies prior to incorporation of Pd(II) promoters and further microscopic and spectroscopic analysis. Newly synthesised 2D and 3D layered nanostructures incorporating carbon‐supported iron oxide nanoparticulate pre‐catalysts were tested, upon hydrogen reduction in situ, for the conversion of CO2 to CO as well as for the selective formation of CH4 and longer chain hydrocarbons. The reduction reaction was also carried out and the catalytic species isolated and fully characterised. The catalytic activity of a graphene oxide‐supported iron oxide pre‐catalyst converted CO2 into hydrocarbons at different temperatures (305, 335, 370 and 405 °C), and its activity compared well with that of the analogues supported on graphite oxide, the 3‐dimensional material precursor to the graphene oxide. Investigation into the use of graphene oxide as a framework for catalysis showed that it has promising activity with respect to reverse gas water shift (RWGS) reaction of CO2 to CO, even at the low levels of catalyst used and under the rather mild conditions employed at atmospheric pressure. Whilst the γ‐Fe2O3 decorated graphene oxide‐based pre‐catalyst displays fairly constant activity up to 405 °C, it was found by GC‐MS analysis to be unstable with respect to decomposition at higher temperatures. The addition of palladium as a promoter increased the activity of the iron functionalised graphite oxide in the RWGS. The activity of graphene oxide supported catalysts was found to be enhanced with respect to that of iron‐functionalised graphite oxide with, or without palladium as a promoter, and comparable to that of Fe@carbon nanotube‐based systems tested under analogous conditions. These results display a significant step forward for the catalytic activity estimations for the iron functionalised and rapidly processable and scalable graphene oxide. The hereby investigated phenomena are of particular relevance for the understanding of the intimate surface morphologies and the potential role of non‐covalent interactions in the iron oxide‐graphene oxide networks, which could inform the design of nano‐materials with performance in future sustainable catalysis applications

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

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure