Vol. 55 No. 3 (256) (2021)

We construct systems of independent defining relations for free Burnside groups $B(m,3)$ of ranks $m=2, 3$. The proof for the case $m=2$ is established using the matrix representation of $B(2,3)$. For the case $m=3$ the approach is based on the natural embedding of $B(2,3)$ into $B(3,3)$.

An interval vertex-coloring of a graph $G$ is a coloring of the vertices of the graph with intervals of integers such that the intervals of any two adjacent vertices do not intersect. In this paper we consider the case, where for each vertex $v$ there is a length $l(v)$ and a set of colors $S(v),$ from which the colors should be and it is required to find an interval vertex-coloring $\gamma$ such that for each vertex $v$ the restrictions are met, i.e. $|\gamma(v)|=l(v),\gamma(v) \subseteq S(v) $. We will provide a pseudo-polynomial algorithm for cactus graphs. If it is impossible to have an interval vertex-coloring that satisfies all the restrictions, then the algorithm will tell that as well.

In the problem of electron diffraction by a standing light wave (the Kapitza–Dirac effect), an electronic refractive index can be defined as the ratio of electron momenta in the wave field and outside it. Moreover, both kinetic and canonical electron momenta can be used for this purpose, which corresponds to the Abraham–Minkowski controversy in photonic optics. It is shown that in both cases the same expression for the electronic refractive index is obtained. This is consistent with Barnett's resolution of the Abraham–Minkowski dilemma.

In the article, we present a report on the observation of microwave  absorption of sodium chloride (NaCl) and glucose aqueous solution of various concentrations using a thermoelastic optical indicator microscope (TEOIM). The non-invasive sensing technology of TEOIM offers a great advantage over other detection technologies. With the help of TECIM, the change in the distribution of the microwave field was visualized depending on the concentration of NaCl and glucose in the range of $8-14~GHz.$ This article shows the investigation results for various concentrations of NaCl $(0-2.5\%)$ and presents the behavior of average microwave near-field intensity for glucose (varied in the range of $0-10\%)$ and NaCl (fixed at $0.9\%$) complex aqueous solutions.