Vol. 55 No. 2 (255) (2021)

In this paper we generalize locally-balanced 2-partitions of graphs and introduce a new notion, the locally-balanced k-partitions of graphs, defined as  follows: a k-partition of a graph G is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$.  A k-partition (k$\geq $2) f of a graph G is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A k-partition  (k$\geq $2) $f^{\prime}$ of a graph G is a locally-balanced with a closed  neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=i\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=j\}\vert \right\vert\leq 1.$$ The minimum number k (k$\geq $2), for which a graph G has a locally-balanced k-partition with an open (a closed) neighborhood, is called an lb-open (lb-closed) chromatic number of G and denoted by $\chi_{(lb)}(G)$ ($\chi_{[lb]}(G)$). In this paper we determine or bound the lb-open and lb-closed chromatic numbers of several families of graphs. We also consider the connections of lb-open and lb-closed chromatic numbers of graphs with other chromatic numbers such as injective and 2-distance chromatic numbers.

An edge-coloring of a graph $G$ with consecutive integers $c_1,\ldots,c_t$ is called an interval t-coloring, if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if it has an interval t-coloring for some positive integer $t$. In this paper, we consider the case, where there are restrictions on the edges of the tree and provide a polynomial algorithm for checking interval colorability that satisfies those restrictions.

The concept of averaged controllability has been introduced relatively recently aiming to analyse the controllability of systems or processes containing some important parameters that may affect the controllability in usual sense. The averaged controllability of various specific and abstract equations has been studied so far. Relatively little attention has been paid to averaged controllability of coupled systems. The averaged state of a thermoelastic rectangular plate is studied in this paper using the well-known Green's function approach. The aim of the paper is to provide a theoretical background for further exact and approximate controllability analysis of fully coupled thermoelasticity equations which will appear elsewhere.

Simple harmonic motion was investigated of a rotational oscillating system. The effect of dumping and forcing on motion of the system was examined and measurements were taken. Resonance in a oscillating system was investigated and quality factor of the dumping system was measured at different damping forces using three different methods. Resonance curves were constructed at two different damping forces. A probabilistic model was built and system parameters were estimated from the resonance curves using Stan sampling platform. The quality factor of the oscillating system when the additional dumping was turned off was estimated to be $Q = \num{71 \pm 1}$ and natural frequency $\omega_0 = \num{3.105 \pm 0.008}\, \si{\per\second}$.

The use of the method developed in the CLAS collaboration (Jefferson Lab, USA) of improving leptons momentum for more correct studies of final state selection of quasi-real photoproduction resulting from the reaction to near threshold photoproduction of the $J/\psi$ meson is described. The radiation photons that were detected in electromagnetic calorimeter were studied with electrons and positrons accompanied them in very narrow angles. The method of radiated photon selection of $e^+ e^- p^{\prime}(e^{\prime})$ reaction is given, where $e^+ e^-$ lepton pair is formed during the decay of $J/\psi$ meson.

In the context of the Abraham–Minkowski controversy, the problem of the propagation of electromagnetic waves in a linear dielectric medium with a time-varying dielectric constant is considered. It is shown that the momentum of an electromagnetic wave in the form of Minkowski is preserved with an instantaneous change in the dielectric permittivity of the medium. At the same time, the Abraham momentum is not conserved, despite the spatial homogeneity of the problem. This circumstance is interpreted as a manifestation of the Abraham force.