Vol. 54 No. 3 (253) (2020)

We investigate the relations between the proof lines of non-minimal tautologies and its minimal tautologies for the Frege systems, the sequent systems with cut rule and the systems of natural deductions of classical and nonclassical logics. We show that for these systems there are sequences of tautologies ψ_n, every one of which has unique minimal tautologies φ_n such that for each n the minimal proof lines of φ_n are an order more than the minimal proof lines of ψ_n.

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.

The paper considers the problem of vibration of a piezoelectric layer of the class $6~mm$ with initial conditions in the form of impact of an external electric field or displacement, when one edge is rigidly grounded and the other is free. The layer displacement and internal electric field are determined.

The article considers the problem for an elastic infinite sheet (plate), which is strengthened on two parallel finite parts of its upper surface by two parallel finite stringers with different elastic properties. The parallel stringers are located asymmetrically with respect to the horizontal axis of the sheet and deform under the action of horizontal forces. The interaction between the infinite sheet and stringers takes place through thin elastic adhesive layers. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on two parallel finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations in Banach space can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated.

The manufacturing technique of a millimetric sizes cantilever from photo-driven azobenzene polymer is described. The cantilever oscillations under the influence of laser radiation are studied. The possibility of making a micron-sized cantilever by a femtosecond laser initiated two-photon polymerization technique is shown. Such cantilever can become the basis for a high sensitive sensor, controlled directly by light.

This paper shows a non-destructive visualization of the absorption of microwave filed by a graphite periodic structure. The visualization system was a thermo-elastic optical indicator microscope. The article presents the interaction of the electromagnetic field with graphite cylindrical cells of periodicity and shows the distribution of the electromagnetic field over the graphite cells. Depending on the distance between the periodic structure of graphite and the microwave source, the electromagnetic field distribution and absorption rate were different. The visualization was performed using a microwave signal with a frequency of $11~GHz$ and a maximum power of $35~dBm$.