Stars and bunches in planar graphs. Part I: Triangulations

Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We prove a theorem on the structure of plane triangulations in terms of stars and bunches. The result states that a plane triangulation contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch. \u

The issue of the senate of the polish-lithuanian commonwealth in the late 16thЦ18th centuries in russian historiography

This article is focused on the Russian historiography of Polish parliamentarism in the late 16thЦ18th centuries. Particular attention is paid to the specifics of the vision of the political institutions of the Polish-Lithuanian Commonwealth of the late 16th - 18th centuries, taking into account the political realities of different stages of Russian historiography. The periodization of the study of the Polish political system is provided. The specifics of the perception of Polish history by Russian researchers is determined

Stars and bunches in planar graphs. Part II: General planar graphs and colourings

Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We first prove a theorem on the structure of plane graphs in terms of stars and bunches. The result states that a plane graph contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch. \u

The issue of the senate of the polish-lithuanian commonwealth in the late 16thЦ18th centuries in russian historiography

This article is focused on the Russian historiography of Polish parliamentarism in the late 16thЦ18th centuries. Particular attention is paid to the specifics of the vision of the political institutions of the Polish-Lithuanian Commonwealth of the late 16th - 18th centuries, taking into account the political realities of different stages of Russian historiography. The periodization of the study of the Polish political system is provided. The specifics of the perception of Polish history by Russian researchers is determined

Autoresonance in a Dissipative System

We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations.Comment: 22 pages, 3 figure

Quantum-chemical calculation of the adsorption energy of the Pb atom on the graphene

© 2018 Author(s). Graphene can be used as sensors to detect toxic heavy metals in peripheral blood. This requires a clear understanding of adsorption process on the graphene surface. The DFT-functional method is used to calculate adsorption energy. In the article we present the calculation results and analyse interaction of Pb atom with graphene surface. It is shown that the Pb atom in B-site position on a graphene plane has a minimum of energy