A Characterization of Mixed Unit Interval Graphs

We give a complete characterization of mixed unit interval graphs, the intersection graphs of closed, open, and half-open unit intervals of the real line. This is a proper superclass of the well known unit interval graphs. Our result solves a problem posed by Dourado, Le, Protti, Rautenbach and Szwarcfiter (Mixed unit interval graphs, Discrete Math. 312, 3357-3363 (2012)).Comment: 17 pages, referees' comments adde

Number Theory And Formal Languages

. I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorphisms, automaticity, and k-regular sequences. Key words. finite automata, automatic sequences, transcendence, automaticity AMS(MOS) subject classifications. Primary 11B85, Secondary 11A63 11A55 11J81 1. Introduction. In this paper, I survey some interesting connections between number theory and the theory of formal languages. This is a very large and rapidly growing area, and I focus on a few areas that interest me, rather than attempting to be comprehensive. (An earlier survey of this area, written in French, is [1].) I also give a number of open questions. Number theory deals with the properties of integers, and formal language theory deals with the properties of strings. At the interse..

Status of hormones and painkillers in wastewater effluents across several European states - considerations for the EU watch list concerning estradiols and diclofenac.

Present technologies for wastewater treatment do not sufficiently address the increasing pollution situation of receiving water bodies, especially with the growing use of personal care products and pharmaceuticals (PPCP) in the private household and health sector. The relevance of addressing this problem of organic pollutants was taken into account by the Directive 2013/39/EU that introduced (i) the quality evaluation of aquatic compartments, (ii) the polluter pays principle, (iii) the need for innovative and affordable wastewater treatment technologies, and (iv) the identification of pollution causes including a list of principal compounds to be monitored. In addition, a watch list of 10 other substances was recently defined by Decision 2015/495 on March 20, 2015. This list contains, among several recalcitrant chemicals, the painkiller diclofenac and the hormones 17β-estradiol and 17α-ethinylestradiol. Although some modern approaches for their removal exist, such as advanced oxidation processes (AOPs), retrofitting most wastewater treatment plants with AOPs will not be acceptable as consistent investment at reasonable operational cost. Additionally, by-product and transformation product formation has to be considered. The same is true for membrane-based technologies (nanofiltration, reversed osmosis) despite of the incredible progress that has been made during recent years, because these systems lead to higher operation costs (mainly due to higher energy consumption) so that the majority of communities will not easily accept them. Advanced technologies in wastewater treatment like membrane bioreactors (MBR) that integrate biological degradation of organic matter with membrane filtration have proven a more complete elimination of emerging pollutants in a rather cost- and labor-intensive technology. Still, most of the presently applied methods are incapable of removing critical compounds completely. In this opinion paper, the state of the art of European WWTPs is reflected, and capacities of single methods are described. Furthermore, the need for analytical standards, risk assessment, and economic planning is stressed. The survey results in the conclusion that combinations of different conventional and advanced technologies including biological and plant-based strategies seem to be most promising to solve the burning problem of polluting our environment with hazardous emerging xenobiotics