Vol. 51 No. 3 (244) (2017)

In the paper, using a relationship between probability $P(L(\omega)\subset \mathbf {D}) $ that a random segment of length $l$ in $R^{n}$ having a common point with body $D$ entirely lying in $D$ and the covariogram of $D$, we obtain the explicit form of $P(L(\omega)\subset \mathbf {D}) $ for any triangle on the plane.

We prove that the free Burnside groups $B(m,3)$ of period 3 and rank $m\geq1$ have Magnus's property, that is if in $B(m,3)$ the normal closures of $r$ and $s$ coincide, then $r$ is conjugate to $s$ or $s^{-1}$. We also prove that any automorphism of $B(m,3)$ induced by a Nielsen automorphism of the free group $F_m$ of rank $m$. We show that the kernel of the natural homomorphism $\text{Aut}(B(2,3)) \rightarrow GL_2(\mathbb{Z}_3)$ is the group of inner automorphisms of $B(2,3)$.

In the paper $K$-groups of $C^*$-subalgebras of the Toeplitz algebra generated by inverse subsemigroups of the bicyclic semigroup are discussed. For these algebras inductive limit of inductive sequence of $K$-groups, which are generated by the corresponding inductive sequence of $C^*$-algebras is constructed.

Some aspects on strict topology in algebras of continuous and bounded functions on completely regular spaces are discussed in this work. Also some properties of their dual spaces are investigated.

In this paper we estimate the minimal number of systems of linear equations of $n+m$ variables over a finite field $F_q$ such that the union of all solutions of all the systems coincides exactly with all elements of $\overset{\ast}{\mathbb{F}_{q}^{n}} \times \overset{\ast}{\mathbb{F}_{q}^{m}}$.

It is proved, that if the square partial sums $\sigma_{q_n}(\textbf{x})$ of a multiple Franklin series converge in measure to a function $f$, the ratio $\dfrac{q_{n+1}}{q_n}$ is bounded and the majorant of partial sums satisfies to a necessary condition, then the coefficients of the series are restored by the function $f$.

The Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$ on the unit circle $T=\{z: |z|=1\}$ is investigated, where $\rho(t)={|t-t_{k}|}^{\alpha_{k}}$,~~$k=1,\dots,m$, \lb $t_{k}\in T$ and $\alpha_{k}$ are arbitrary real numbers. The problem is to determine a function $\Phi(z)$ analytic in unit disc such that: $ \lim_{r\rightarrow 1-0}\|Re\Phi(rt)-f(t)\|_{L^{1}(\rho_{r})}=0, $ where $f\in L^{1}(\rho)$. In the paper necessary and sufficient conditions for solvability of the problem are given and the general solution is written in the explicit form.

The equations of oscillations of a two-layer plate are obtained on the basis of the assumption of Kirchhoff's hypothesis concerning the packet as a whole when the contact surfaces of the plate can slide freely relative to each other. It is assumed that the tangential stresses in the boundary conditions on the contact surface of the plates are zero. The dependence of bending and planar vibrations is obtained. The conditions for the appearance of a resonance are obtained.

The present work considers the optimal stabilization problem in rotational motion of a rigid body around its center of gravity. The case of the Euler rotational motion of a rigid body around a fixed point is considered. The optimal stabilization problem of the considered motion is assumed and solved. Input controls are introduced in the direction of the generalized coordinates, full controllability of linear approximation of the system is checked. Besides the optimal stabilization problem of the system on classical sense is solved, optimal Lyapunov function, optimal controls and value of functional are obtained.

$In~vivo$ effect of non thermal millimeter electromagnetic waves (MEMWs) with $50.3~GHz$ (which coincides with resonant frequencies of oscillations of water molecular structures) and $48.3~GHz$ (which does not coincide with resonant frequencies) frequencies on DNA primary and secondary structures of liver of healthy rats as well as Sarcoma-45 tumor carrying rats has been studied. It was shown, that under the effect of MEMWs with $50.3~GHz$ during $60~min$, the content of 5-methylcitosine, the melting parameters of DNA of Sarcoma-45 tumors changed and approached to respective values of DNA extracted from the liver of healthy animals.

Electromagnetic field localization in a subwavelength metallic slit by a thermo-elastic optical indicator microscope (TEOIM) was investigated. As an indicator for the TEOIM system a slide glass with sizes of $20\times 20\times0.5~(mm)$ was used, on the surface of which an $\mathrm{Al}$ film of $20~nm$ thickness was vaporized using the vacuum evaporation technique, with various slit width $(10-50~\mu m).$  Strongly localized electromagnetic field has been exited in the slits by $50~ GHz$ generator and was visualized by a TEOIM. The waveguide properties of  the system were characterized by COMSOL Multiphysics$^{\circledR}$ additionally. The simulation results are in good agreement with the visualization data set.