Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets $\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points

The Minimum of the Maximum Rectilinear Crossing Numbers of Small Cubic Graphs

Here we consider the minimum of the maximum rectilin­ear crossing numbers for all d-regular graphs of order n. The case of connected graphs only is investigated also. For d = 3 exact values are determined for n are less than or equal to 12 and some estimations are given in general

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 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

Comprehensive analysis of NuMA variation in breast cancer

<p>Abstract</p> <p>Background</p> <p>A recent genome wide case-control association study identified <it>NuMA </it>region on 11q13 as a candidate locus for breast cancer susceptibility. Specifically, the variant Ala794Gly was suggested to be associated with increased risk of breast cancer.</p> <p>Methods</p> <p>In order to evaluate the <it>NuMa </it>gene for breast cancer susceptibility, we have here screened the entire coding region and exon-intron boundaries of <it>NuMa </it>in 92 familial breast cancer patients and constructed haplotypes of the identified variants. Five missense variants were further screened in 341 breast cancer cases with a positive family history and 368 controls. We examined the frequency of Ala794Gly in an extensive series of familial (n = 910) and unselected (n = 884) breast cancer cases and controls (n = 906), with a high power to detect the suggested breast cancer risk. We also tested if the variant is associated with histopathologic features of breast tumors.</p> <p>Results</p> <p>Screening of <it>NuMA </it>resulted in identification of 11 exonic variants and 12 variants in introns or untranslated regions. Five missense variants that were further screened in breast cancer cases with a positive family history and controls, were each carried on a unique haplotype. None of the variants, or the haplotypes represented by them, was associated with breast cancer risk although due to low power in this analysis, very low risk alleles may go unrecognized. The <it>NuMA </it>Ala794Gly showed no difference in frequency in the unselected breast cancer case series or familial case series compared to control cases. Furthermore, Ala794Gly did not show any significant association with histopathologic characteristics of the tumors, though Ala794Gly was slightly more frequent among unselected cases with lymph node involvement.</p> <p>Conclusion</p> <p>Our results do not support the role of <it>NuMA </it>variants as breast cancer susceptibility alleles.</p

String Matching and 1d Lattice Gases

We calculate the probability distributions for the number of occurrences $n$ of a given $l$ letter word in a random string of $k$ letters. Analytical expressions for the distribution are known for the asymptotic regimes (i) $k \gg r^l \gg 1$ (Gaussian) and $k,l \to \infty$ such that $k/r^l$ is finite (Compound Poisson). However, it is known that these distributions do now work well in the intermediate regime $k \gtrsim r^l \gtrsim 1$. We show that the problem of calculating the string matching probability can be cast into a determining the configurational partition function of a 1d lattice gas with interacting particles so that the matching probability becomes the grand-partition sum of the lattice gas, with the number of particles corresponding to the number of matches. We perform a virial expansion of the effective equation of state and obtain the probability distribution. Our result reproduces the behavior of the distribution in all regimes. We are also able to show analytically how the limiting distributions arise. Our analysis builds on the fact that the effective interactions between the particles consist of a relatively strong core of size $l$, the word length, followed by a weak, exponentially decaying tail. We find that the asymptotic regimes correspond to the case where the tail of the interactions can be neglected, while in the intermediate regime they need to be kept in the analysis. Our results are readily generalized to the case where the random strings are generated by more complicated stochastic processes such as a non-uniform letter probability distribution or Markov chains. We show that in these cases the tails of the effective interactions can be made even more dominant rendering thus the asymptotic approximations less accurate in such a regime.Comment: 44 pages and 8 figures. Major revision of previous version. The lattice gas analogy has been worked out in full, including virial expansion and equation of state. This constitutes the main part of the paper now. Connections with existing work is made and references should be up to date now. To be submitted for publicatio

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]

Emissions from mechanically-biologically treated waste landfills at field scale

Modern waste management tends towards greater sustainability in landfilling, with the implementation of strategies such as the pretreatment of solid waste. This work assesses the behaviour of rejects from a refining stage of mechanically-biologically treated municipal solid waste at the landfill. The main results of 18 months' monitoring of an experimental pilot cell with waste from a full-scale plant are presented. This first stages are expected to be the most problematic period for this type of waste. The evolution of the temperature and the composition of leachate and gas at various points within the cell are included. During the first weeks, pollutant concentrations in the leachate exceeded the reference ranges in the literature, coinciding with a rapid onset of methanogenic conditions. However, there was a quick wash, reducing concentrations to below one third of the initial values before the first year. pH values influenced concentrations of some pollutants such as copper. These results indicate that, right from the beginning of disposal, such facilities should be prepared to treat a high pollution load in the leachate and install the gas emissions control elements due to the rapid onset of methanogenesis.This work is funded by the Spanish Ministry of Economics and Competitiveness through the CTM2012-35055 project. The project is financed jointly by the European Regional Development Fund, FEDER (operational period 2007-2013). The authors wish to thank the Government of Cantabria, through the public company MARE, and TirCantabria, the landfill operator company, for their collaboration

An accurate and interpretable model for siRNA efficacy prediction

BACKGROUND: The use of exogenous small interfering RNAs (siRNAs) for gene silencing has quickly become a widespread molecular tool providing a powerful means for gene functional study and new drug target identification. Although considerable progress has been made recently in understanding how the RNAi pathway mediates gene silencing, the design of potent siRNAs remains challenging. RESULTS: We propose a simple linear model combining basic features of siRNA sequences for siRNA efficacy prediction. Trained and tested on a large dataset of siRNA sequences made recently available, it performs as well as more complex state-of-the-art models in terms of potency prediction accuracy, with the advantage of being directly interpretable. The analysis of this linear model allows us to detect and quantify the effect of nucleotide preferences at particular positions, including previously known and new observations. We also detect and quantify a strong propensity of potent siRNAs to contain short asymmetric motifs in their sequence, and show that, surprisingly, these motifs alone contain at least as much relevant information for potency prediction as the nucleotide preferences for particular positions. CONCLUSION: The model proposed for prediction of siRNA potency is as accurate as a state-of-the-art nonlinear model and is easily interpretable in terms of biological features. It is freely available on the web a

NuMA Overexpression in Epithelial Ovarian Cancer

Highly aneuploid tumours are common in epithelial ovarian cancers (EOC). We investigated whether NuMA expression was associated with this phenomenon