Note on a Conjecture of Wegner

The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner's conjecture.Comment: 5 pages; Contributions to Algebra and Geometry, Vol.52 No1 April 201

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

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

Contact numbers for congruent sphere packings in Euclidean 3-space

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.Comment: 18 page

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

OligoWalk: an online siRNA design tool utilizing hybridization thermodynamics

Given an mRNA sequence as input, the OligoWalk web server generates a list of small interfering RNA (siRNA) candidate sequences, ranked by the probability of being efficient siRNA (silencing efficacy greater than 70%). To accomplish this, the server predicts the free energy changes of the hybridization of an siRNA to a target mRNA, considering both siRNA and mRNA self-structure. The free energy changes of the structures are rigorously calculated using a partition function calculation. By changing advanced options, the free energy changes can also be calculated using less rigorous lowest free energy structure or suboptimal structure prediction methods for the purpose of comparison. Considering the predicted free energy changes and local siRNA sequence features, the server selects efficient siRNA with high accuracy using a support vector machine. On average, the fraction of efficient siRNAs selected by the server that will be efficient at silencing is 78.6%. The OligoWalk web server is freely accessible through internet at http://rna.urmc.rochester.edu/servers/oligowalk

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

Fundamental differences in the equilibrium considerations for siRNA and antisense oligodeoxynucleotide design

Both siRNA and antisense oligodeoxynucleotides (ODNs) inhibit the expression of a complementary gene. In this study, fundamental differences in the considerations for RNA interference and antisense ODNs are reported. In siRNA and antisense ODN databases, positive correlations are observed between the cost to open the mRNA target self-structure and the stability of the duplex to be formed, meaning the sites along the mRNA target with highest potential to form strong duplexes with antisense strands also have the greatest tendency to be involved in pre-existing structure. Efficient siRNA have less stable siRNA–target duplex stability than inefficient siRNA, but the opposite is true for antisense ODNs. It is, therefore, more difficult to avoid target self-structure in antisense ODN design. Self-structure stabilities of oligonucleotide and target correlate to the silencing efficacy of siRNA. Oligonucleotide self-structure correlations to efficacy of antisense ODNs, conversely, are insignificant. Furthermore, self-structure in the target appears to correlate with antisense ODN efficacy, but such that more effective antisense ODNs appear to target mRNA regions with greater self-structure. Therefore, different criteria are suggested for the design of efficient siRNA and antisense ODNs and the design of antisense ODNs is more challenging