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

Математична модель маршрутизації в ТКМ

In this paper, drawing on the work of Michel Foucault, I argue that academics are enmeshed in power relations in which confession operates, both on and through academics. Drawing on Foucault’s genealogy of confession, I illustrate how academics are not only invited to reflect on performance, faults, temptations and desires in their work and private life, but as teachers they mobilise the same kind of technology in relation to students. These power relations are connected to wider changes in society, where discourses on New Public Management have become all pervasive in organising and governing public institutions. The examples of the use of appraisal interviews and logbooks as governing techniques illustrate how government currently operates through the freedom of the individual. The paper ends with a discussion on how books of life could introduce a different relation of the self to the self in academia, and thus provide opportunities to live the present otherwise.

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

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

Landscape of transcription in human cells

Eukaryotic cells make many types of primary and processed RNAs that are found either in specific subcellular compartments or throughout the cells. A complete catalogue of these RNAs is not yet available and their characteristic subcellular localizations are also poorly understood. Because RNA represents the direct output of the genetic information encoded by genomes and a significant proportion of a cell’s regulatory capabilities are focused on its synthesis, processing, transport, modification and translation, the generation of such a catalogue is crucial for understanding genome function. Here we report evidence that three-quarters of the human genome is capable of being transcribed, as well as observations about the range and levels of expression, localization, processing fates, regulatory regions and modifications of almost all currently annotated and thousands of previously unannotated RNAs. These observations, taken together, prompt a redefinition of the concept of a gene