On the density of sets of the Euclidean plane avoiding distance 1

A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25646$ and $\chi_f(\mathbb R^2) \geq 3.8992$.Comment: 11 pages, 5 figure

Broken-Symmetry Ground States of Halogen-Bridged Binuclear Metal Complexes

Based on a symmetry argument, we study ground states of what we call MMX-chain compounds, which are the new class of halogen-bridged metal complexes. Commensurate density-wave solutions of a relevant multi-band Peierls-Hubbard model are systematically revealed within the Hartree-Fock approximation. We numerically draw ground-state phase diagrams, where various novel density-wave states appear.Comment: 5 pages, 4 figures embedded, to appear in Phys. Lett.

On the density of sets avoiding parallelohedron distance 1

The maximal density of a measurable subset of R^n avoiding Euclidean distance1 is unknown except in the trivial case of dimension 1. In this paper, we consider thecase of a distance associated to a polytope that tiles space, where it is likely that the setsavoiding distance 1 are of maximal density 2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n = 2, and for the Vorono\"i regions of the lattices An, n >= 2

Next Generation Cluster Editing

This work aims at improving the quality of structural variant prediction from the mapped reads of a sequenced genome. We suggest a new model based on cluster editing in weighted graphs and introduce a new heuristic algorithm that allows to solve this problem quickly and with a good approximation on the huge graphs that arise from biological datasets

On DP-Coloring of Digraphs

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure

Human progress and its socioeconomic effects in society

Abstract. The goal of this paper is to suggest a definition of human progress given by: an inexhaustible process driven by an ideal of maximum wellbeing of purposeful people which, on attainment of any of its objectives for increasing wellbeing, then seek another consequential objective. The human progress improves the fundamental life-interests of people represented by health, wealth, expansion of knowledge, technology and freedom directed to increase wellbeing throughout the society. These factors support the acquisition by humanity of better and more complex forms of life. However, this study also shows the inconsistency of the equation economic growth= progress because human progress also generates negative effects for human being, environment and society, such as increasing incidence of cancer in advanced countries. Keywords. Human Progress, Economic Growth, Wellbeing, Social Progress, Environmental Degradation, Cancer.JEL. B50, B59, I00, I10, I30, O10, O30, O33, O40

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete