Vol. 52 No. 3 (247) (2018)

Three-dimensional submanifolds in affine space $ A^6 $ have been studied by the method of exterior forms. It is proved that the structure of total space induces a special type of affine connection on this submanifold. The structure equations of this submanifold have been found.

In this work the question of algebraic closeness of $ \beta $-uniform algebra $ A (\Omega) $ defined on locally compact space $ \Omega $ is investigated.

Using the Fourier transforms method for meromorphic functions we characterize the behavior of the integral logarithmic mean of arbitrary order of Blaschke products for the half-plane.

In the paper Taylor–Maclaurin type formulas for some classes of functions are obtained. The main result of this study introduces an idea of the generalized classes of $ \langle \rho_j \rangle $ completely monotone function. Under the various conditions $ \left( \sum\limits_{j=1}^{\infty} \dfrac{1}{\rho_j} < + \infty ,\ \sum\limits_{j=1}^{\infty} \dfrac{1}{\rho_j} = + \infty \right) $ the terms of their representation are obtained and some related theorems are proved.

We obtain upper bounds of the complexity of linearized coverings for some special solutions of the equation $$ x_{1}x_{2}x_{3} \mathclose{+} x_{2}x_{3}x_{4} \mathclose{+} \cdots \mathclose{+} x_{3n}x_{1}x_{2} \mathclose{+} x_{1}x_{3}x_{5} \mathclose{+} x_{4}x_{6}x_{8} \mathclose{+} \cdots \mathclose{+} x_{3n-2}x_{3n}x_{2} \mathclose{=} b $$ over an arbitrary finite field.

In this paper the main canonical notion of $ \delta $-reduction is considered. Typed $ \lambda $-terms use variables of any order and constants of order $ \leq 1 $, where constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The canonical notion of $ \delta $-reduction is the notion of $ \delta $-reduction that is used in the implementation of functional programming languages. For main canonical notion of d-reduction the uniqueness of $ \beta\delta $-normal form of typed $ \lambda $-terms is shown.

The detection and recognition of objects in images has found a wide application in many spheres. It is used in the surveillance system, cartography, robotics, medicine, etc. In many surveillance systems the localization of the detected object has a great significance. After localizing the object, it will be possible to carry out the calculation of its coordinates. The formulas for calculating the object-camera distance are rather theoretical. Conducting experiments and verifying formulas in practice would have a high significance, as the distance of the object from camera is an essential precondition for object’s coordinate calculation.

In this paper the effect of donor impurity on the Aharonov–Bohm oscillations of the electronic states in a double quantum ring with Gaussian confinement has been studied. Three different impurity positions: namely in the inner ring, in the outer ring and in the barrier between the rings, have been considered. It is shown that the energies of the two lowest states are almost constant, while for the higher levels the Aharonov–Bohm oscillations are observed. The obtained results indicate on the possibility of the effective control of the quantum states by means of donor impurity and external magnetic field.

In this work we study the influence of hydrodynamic flows on the optical properties of hybrid aligned nematic liquid crystals (NLC) structure caused by direct volume expansion mechanism for the case, when the direction of flow velocity is perpendicular to the angular distribution of the molecules in the cell. It has been shown that the hydrodynamic flow leads to reorientation of NLC molecules. The behavior of the polarized light passing through the NLC layer for two opposite directions of the flow was observed.

Using computer simulations and simultaneous X-ray diffraction at large and small angles, the dependence of the swelling coefficient of the phospholipid bilayer on the angle between the inclination of dipole fragments and the plane of the lamella was investigated. It was shown that in this case the balance between electrostatic, repulsive and van der Waals attraction forces is a particular importance. As the dipole angle increases, the repulsive forces of the electrostatic dipole-dipole interaction increase, resulting the increase of the bilayer thickness as a function of the number of –CH-groups in the phospholipid. At a certain angle the bilayer is destroyed. It is also shown that at certain small angles the bilayer takes a minimum thickness. In this case a phase transformation takes place and the system enters the “gel”-phase.