The strong thirteen spheres problem

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

Piwi induces piRNA-guided transcriptional silencing and establishment of a repressive chromatin state

In the metazoan germline, piwi proteins and associated piwi-interacting RNAs (piRNAs) provide a defense system against the expression of transposable elements. In the cytoplasm, piRNA sequences guide piwi complexes to destroy complementary transposon transcripts by endonucleolytic cleavage. However, some piwi family members are nuclear, raising the possibility of alternative pathways for piRNA-mediated regulation of gene expression. We found that Drosophila Piwi is recruited to chromatin, colocalizing with RNA polymerase II (Pol II) on polytene chromosomes. Knockdown of Piwi in the germline increases expression of transposable elements that are targeted by piRNAs, whereas protein-coding genes remain largely unaffected. Derepression of transposons upon Piwi depletion correlates with increased occupancy of Pol II on their promoters. Expression of piRNAs that target a reporter construct results in a decrease in Pol II occupancy and an increase in repressive H3K9me3 marks and heterochromatin protein 1 (HP1) on the reporter locus. Our results indicate that Piwi identifies targets complementary to the associated piRNA and induces transcriptional repression by establishing a repressive chromatin state when correct targets are found

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

Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.Comment: 26 pages, 10 figure

Su(var)2-10 and the SUMO Pathway Link piRNA-Guided Target Recognition to Chromatin Silencing

Regulation of transcription is the main mechanism responsible for precise control of gene expression. Whereas the majority of transcriptional regulation is mediated by DNA-binding transcription factors that bind to regulatory gene regions, an elegant alternative strategy employs small RNA guides, Piwi-interacting RNAs (piRNAs) to identify targets of transcriptional repression. Here, we show that in Drosophila the small ubiquitin-like protein SUMO and the SUMO E3 ligase Su(var)2-10 are required for piRNA-guided deposition of repressive chromatin marks and transcriptional silencing of piRNA targets. Su(var)2-10 links the piRNA-guided target recognition complex to the silencing effector by binding the piRNA/Piwi complex and inducing SUMO-dependent recruitment of the SetDB1/Wde histone methyltransferase effector. We propose that in Drosophila, the nuclear piRNA pathway has co-opted a conserved mechanism of SUMO-dependent recruitment of the SetDB1/Wde chromatin modifier to confer repression of genomic parasites

Su(var)2-10 and the SUMO Pathway Link piRNA-Guided Target Recognition to Chromatin Silencing

Regulation of transcription is the main mechanism responsible for precise control of gene expression. Whereas the majority of transcriptional regulation is mediated by DNA-binding transcription factors that bind to regulatory gene regions, an elegant alternative strategy employs small RNA guides, Piwi-interacting RNAs (piRNAs) to identify targets of transcriptional repression. Here, we show that in Drosophila the small ubiquitin-like protein SUMO and the SUMO E3 ligase Su(var)2-10 are required for piRNA-guided deposition of repressive chromatin marks and transcriptional silencing of piRNA targets. Su(var)2-10 links the piRNA-guided target recognition complex to the silencing effector by binding the piRNA/Piwi complex and inducing SUMO-dependent recruitment of the SetDB1/Wde histone methyltransferase effector. We propose that in Drosophila, the nuclear piRNA pathway has co-opted a conserved mechanism of SUMO-dependent recruitment of the SetDB1/Wde chromatin modifier to confer repression of genomic parasites

Optimal Packings of Superballs

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a versatile family of convex particles (p >= 0.5) with both cubic- and octahedral-like shapes as well as concave particles (0 < p < 0.5) with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings. The maximal packing density as a function of p is nonanalytic at the sphere-point (p = 1) and increases dramatically as p moves away from unity. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional "superdisk" counterparts, and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.Comment: 28 pages, 16 figure