Vol. 53 No. 3 (250) (2019)

We prove that each automorphism of order 2 of any non-abelian periodic group of odd period has a non-trivial fixed point.

We consider equations of associativity and transitivity. We obtain necessary conditions for the solution of these equations for right quasigroup operations, generalizing the classical quasigroup case.

In this work we construct a weighted space $ L_{\mu}^{p},\ p \geq 1 $, in which functions with the norm of that space are presented by Walsh double series, which coefficients are monotone in all ways.

We consider boundary value problem for degenerate first order differentialoperator equation $Lu \mathclose{\equiv} t^{\alpha} u^{\prime} \mathclose{-} P u \mathclose{=} f $, $ u(0) \mathclose{-} \mu u(b) \mathclose{=} 0 $, where $ t \mathclose{\in} (0,b) $,  $ a \mathclose{\geq} 0 $, $ P: H \mathclose{\rightarrow} H $ is linear operator in separable Hilbert space $ H $, $ f \mathclose{\in} L_{2, \beta} ((0,b),H) $,  $ \mu \mathclose{\in} \mathbb{C} $. We prove that under some conditions on the operator $ P $ and number $ \mu $ the boundary value problem has unique generalized solution $ u \mathclose{\in} L_{2, \beta} ((0,b),H) $ when $ 2 \alpha \mathclose{+} \beta \mathclose{<} 1 $,  $ \beta \mathclose{\geq} 0 $ and for any $ f \mathclose{\in} L_{2, \beta} ((0,b),H) $.

In this paper we provide a necessity condition for embedding of the binary algebra into the quasi-boolean power of a rectangular algebra. It is also proved that every idempotent and hyperassociative algebra via the weak bihomomorphism maps in an idempotent and commutative algebra.

The problem of optimal control of the spatial motion of a dynamic object in order to find a moving object performing a simple motion in a rectangular domain on the plane is considered. A method for controlling the motion of the searching object, as well as the corresponding law of variation of the electric current in the light source circuit, ensuring the detection of the sought object with minimal light energy consumption for a guaranteed search time, are proposed.

thickness, the equations of the problem of bending of an elastically clamped beam in the case of simultaneous action of its own weight and axial compressive forces are obtained. The effects of transverse shear and the effect of reducing the compressive force of the support are taken into account. Turning to dimensionless quantities, the specific problem for a beam of linearly variable thickness is solved by the collocation method. The question of the stability of the beam is discussed.

The built-in functions of programming languages are functions with indeterminate values of arguments. The built-in McCarthy functions $ car $, $ cdr $, $ cons $, $ null $, $ atom $, $ if $, $ eq $, $ not $, $ and $, $ or $, are used in all functional programming languages. In this paper we show the $ \lambda $-definability of the built-in McCarthy functions as functions with indeterminate values of arguments. This result is necessary when translating typed functional programming languages into untyped functional programming languages.

The noncompetitive DNA hybridization and conductivity in the presence of non-charged ligands have been investigated and the comparison is made with the ligand-free case. It has been shown that the intercalating ligands enhance the sensitivity of the DNA chips as compared with the ligand-free case at the high enough concentration of ligands.

The effect of zinc oxide nanoparticles ($ \mathrm{ZnO} $) on the stability and conductivity of the bilayer lipid membrane (BLM) in solution was studied. It has been shown that $ \mathrm{ZnO} $ nanoparticles increase the stability of BLM in an electric field, and BLM becomes more stable with increasing their concentration. The increase in the stability of BLM in an electrical field is mainly due to the increase in the coefficient of linear tension of the pore edge, which is forming in BLM. It is also shown, that the presence of $ \mathrm{ZnO} $ nanoparticles in the solution surrounding BLM leads to a decrease in the BLM conductivity.