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

Effect of different compositions on anaerobic co-digestion of cattle manure and agro-industrial by-products

ArticleThe present research is dedicated to the study of anaerobic co-digestion process of different biomass materials. Anaerobic co-digestion of digested sludge, grass silage, haylage and cattle manure was evaluated in mesophilic tank reactors in the lab-scale experiment. Twelve laboratory scale tank reactors (1.5 L) were used during the incubation period of 45 days. First triplet of reactors was fed with pure digested sludge and the other three with different mixtures having the volumetric ratios of 30/35/25/10, 40/30/20/10 and 50/25/15/10 for digested sludge/corn silage/grass haylage/cattle manure. Methane production was analyzed for all labscale reactors individually. The resulting specific methane production of above-mentioned batches was 336.34, 238.1 and 233.23 LSTP[CH4] kg-1 [TVS], respectively. Other results such as cumulative biogas and methane yield, volumetric biogas and methane yield, volumetric biogas and methane yield per day were also assessed. These results had the highest meaning when complex substrate had no more than 30% of inoculum

On colouring point visibility graphs

In this paper we show that it can be decided in polynomial time whether or not the visibility graph of a given point set is 4-colourable, and such a 4-colouring, if it exists, can also be constructed in polynomial time. We show that the problem of deciding whether the visibility graph of a point set is 5-colourable, is NP-complete. We give an example of a point visibility graph that has chromatic number 6 while its clique number is only 4

First apsidal motion and light curve analysis of 162 eccentric eclipsing binaries from LMC

We present an extensive study of 162 early-type binary systems located in the LMC galaxy that show apsidal motion and have never been studied before. For the ample systems, we performed light curve and apsidal motion modelling for the first time. These systems have a median orbital period of 2.2 days and typical periods of the apsidal motion were derived to be of the order of decades. We identified two record-breaking systems. The first, OGLE LMC-ECL-22613, shows the shortest known apsidal motion period among systems with main sequence components (6.6 years); it contains a third component with an orbital period of 23 years. The second, OGLE LMC-ECL-17226, is an eccentric system with the shortest known orbital period (0.9879 days) and with quite fast apsidal motion period (11 years). Among the studied systems, 36 new triple-star candidates were identified based on the additional period variations. This represents more than 20% of all studied systems, which is in agreement with the statistics of multiples in our Galaxy. However, the fraction should only be considered as a lower limit of these early-type stars in the LMC because of our method of detection, data coverage, and limited precision of individual times of eclipses.Comment: 24 pages, 18 figures, 5 tables, published in 2020A&A...640A..33

The period-gap cataclysmic variable CzeV404 Her: A link between SW Sex and SU UMa systems

Context: We present a new study of the eclipsing cataclysmic variable CzeV404 Her (Porb = 0.098 d) that is located in the period gap. Aims: This report determines the origin of the object and the system parameters and probes the accretion flow structure of the system. Methods: We conducted simultaneous time-resolved photometric and spectroscopic observations of CzeV404 Her. We applied our light-curve modelling techniques and the Doppler tomography method to determine the system parameters and analyse the structure of the accretion disk. Results: We found that the system has a massive white dwarf M_WD = 1.00(2) M_sun a mass ratio of q = 0.16, and a relatively hot secondary with an effective temperature T_2 = 4100(50) K. The system inclination is i = 78.8{\deg}. The accretion disk spreads out to the tidal limitation radius and has an extended hot spot or line region. The hot spot or line is hotter than the remaining outer part of the disk in quiescence or in intermediate state, but does not stand out completely from the disk flux in (super)outbursts. Conclusions: We claim that this object represents a link between two distinct classes of SU UMa-type and SW Sex-type cataclysmic variables. The accretion flow structure in the disk corresponds to the SW Sex systems, but the physical conditions inside the disk fit the behaviour of SU UMa-type objects

Comprehensive analysis of a symbiotic candidate V503 Her

V503 Her was previously proposed as an eclipsing symbiotic candidate based on photometric behavior and spectroscopic appearance indicating the composite optical spectrum. To investigate its nature, we analyzed long-term photometric observations covering one hundred years of its photometric history and new low-resolution optical spectroscopic data, supplemented with the multifrequency measurements collected from several surveys and satellites. Based on the analysis presented in this paper, we claim that V503 Her is not an eclipsing binary star. The optical and infrared wavelengths are dominated by a K-type bright giant with an effective temperature of 4 500 K, luminosity of 1 900 L$_\odot$, and sub-solar metallicity on the asymptotic giant branch showing semiregular complex multi-periodic pulsation behavior. V503 Her does not show the characteristics of strongly interacting symbiotic variables, but some pieces of evidence suggest that it could still be one of the 'hidden' accreting-only symbiotic systems. However, the currently available data do not allow us to fully confirm or constrain the parameters of a possible companion.Comment: 16 pages, 10 figures, 4 tables; accepted in the Astronomical Journa

Science with a small two-band UV-photometry mission II: Observations of stars and stellar systems

We outline the impact of a small two-band UV-photometry satellite mission on the field of stellar physics, magnetospheres of stars, binaries, stellar clusters, interstellar matter, and exoplanets. On specific examples of different types of stars and stellar systems, we discuss particular requirements for such satellite missions in terms of specific mission parameters such as bandpass, precision, cadence, and mission duration. We show that such a mission may provide crucial data not only for hot stars that emit most of their light in UV, but also for cool stars, where UV traces their activity. This is important, for instance, for exoplanetary studies, because the level of stellar activity influences habitability. While the main asset of the two-band UV mission rests in time-domain astronomy, an example of open clusters proves that such a mission would be important also for the study of stellar populations. Properties of the interstellar dust are best explored when combining optical and IR information with observations in UV. It is well known that dust absorbs UV radiation efficiently. Consequently, we outline how such a UV mission can be used to detect eclipses of sufficiently hot stars by various dusty objects and study disks, rings, clouds, disintegrating exoplanets or exoasteroids. Furthermore, UV radiation can be used to study the cooling of neutron stars providing information about the extreme states of matter in the interiors of neutron stars and used for mapping heated spots on their surfaces.Comment: Submitted to Space Science Review

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]