DFAs and PFAs with Long Shortest Synchronizing Word Length

It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n =4,5,6$. For arbitrary n we give a construction of a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.Comment: 16 pages, 2 figures source code adde

Evolutional dynamics of 45S and 5S ribosomal DNA in ancient allohexaploid Atropa belladonna

Background: Polyploid hybrids represent a rich natural resource to study molecular evolution of plant genes and genomes. Here, we applied a combination of karyological and molecular methods to investigate chromosomal structure, molecular organization and evolution of ribosomal DNA (rDNA) in nightshade, Atropa belladonna (fam. Solanaceae), one of the oldest known allohexaploids among flowering plants. Because of their abundance and specific molecular organization (evolutionarily conserved coding regions linked to variable intergenic spacers, IGS), 45S and 5S rDNA are widely used in plant taxonomic and evolutionary studies. Results: Molecular cloning and nucleotide sequencing of A. belladonna 45S rDNA repeats revealed a general structure characteristic of other Solanaceae species, and a very high sequence similarity of two length variants, with the only difference in number of short IGS subrepeats. These results combined with the detection of three pairs of 45S rDNA loci on separate chromosomes, presumably inherited from both tetraploid and diploid ancestor species, example intensive sequence homogenization that led to substitution/elimination of rDNA repeats of one parent. Chromosome silver-staining revealed that only four out of six 45S rDNA sites are frequently transcriptionally active, demonstrating nucleolar dominance. For 5S rDNA, three size variants of repeats were detected, with the major class represented by repeats containing all functional IGS elements required for transcription, the intermediate size repeats containing partially deleted IGS sequences, and the short 5S repeats containing severe defects both in the IGS and coding sequences. While shorter variants demonstrate increased rate of based substitution, probably in their transition into pseudogenes, the functional 5S rDNA variants are nearly identical at the sequence level, pointing to their origin from a single parental species. Localization of the 5S rDNA genes on two chromosome pairs further supports uniparental inheritance from the tetraploid progenitor. Conclusions: The obtained molecular, cytogenetic and phylogenetic data demonstrate complex evolutionary dynamics of rDNA loci in allohexaploid species of Atropa belladonna. The high level of sequence unification revealed in 45S and 5S rDNA loci of this ancient hybrid species have been seemingly achieved by different molecular mechanisms

Water Droplet With Carbon Particles Moving Through High-Temperature Gases

An experimental investigation was carried out on the influence of solid inclusions (nonmetallic particles with sizes from a few tens to hundreds of micrometers) on water droplet evaporation during motion through high-temperature gases (more than 1000 K). Optical methods for diagnostics of two-phase (gas and vapor-liquid) flows (particle image velocimetry (PIV) and interferometric particle imaging (IPI)) were used. It was established that introducing foreign solid particles into the water droplets intensifies evaporation rate in high-temperature gas severalfold. Dependence of liquid evaporation on sizes and concentration of solid inclusion were obtained

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

MULTIFUNCTIONAL HEAVY-DUTY ROBOTIC FIRE EXTINGUISHING SYSTEM

In this paper there is considered the creation of a multifunctional heavy-duty robotic fire extinguishing complex with improved tactical and technical characteristics on a tracked chassis. It is designed to eliminate man-made accidents and fires associated with the risks of fatalities and injury of personnel as well as to conduct reconnaissance in fire seats and to deliver fire extinguishing agents to them. With appropriate retrofitting, it is planned to use this system during elimination of the consequences of accidents with chemical and radiation pollution and work with explosive objects

van der Waals materials for overcoming fundamental limitations in photonic integrated circuitry

With the advance of on-chip nanophotonics, there is a high demand for high-refractive-index and low-loss materials. Currently, this technology is dominated by silicon, but van der Waals (vdW) materials with a high refractive index can offer a very advanced alternative. Still, up to now, it was not clear if the optical anisotropy perpendicular to the layers might be a hindering factor for the development of vdW nanophotonics. Here, we studied WS2-based waveguides in terms of their optical properties and, particularly, in terms of possible crosstalk distance. Surprisingly, we discovered that the low refractive index in the direction perpendicular to the atomic layers improves the characteristics of such devices, mainly due to expanding the range of parameters at which single-mode propagation can be achieved. Thus, using anisotropic materials offers new opportunities and novel control knobs when designing nanophotonic devices.L.M.M. acknowledges Project PID2020-115221GB-C41, financed by MCIN/AEI/10.13039/501100011033, and the Aragon Government through Project Q-MAD. A.A.V., I.K., and D.I.Y. gratefully acknowledge the financial support from the Ministry of Science and Higher Education (Agreement No. 075-15-2021-606). I.A.K. gratefully acknowledges the financial support from the RSF (No. 22-19-00738) for first-principle calculations. K.S.N. is grateful to the Ministry of Education, Singapore (Research Centre of Excellence award to the Institute for Functional Intelligent Materials, I-FIM, project No. EDUNC-33-18-279-V12) and to the Royal Society (UK, grant number RSRP\R\190000) for support.Peer reviewe