In comparison to the traditional use of glass substrates, the thin films onto metal substrates offer improved device cooling, economical large-scale roll-to-roll processing, and applicability in lightweight, as well as flexible products. However, unlike glass, metal foils tend to exhibit rough surfaces. This article studies the substrate-type (Mo/glass and Мо-foil) effect on the topographic characteristics of the Cu2ZnSnSe4 films by atomic force microscopy (AFM). Cu2ZnSnSe4 thin films were prepared by the electrodeposition of stack copper/tin/copper/zinc (Cu/Sn/Cu/Zn) precursors, followed by selenization. AFM was
used to study the topographic characteristics of thin films, including grain size, surface roughness, and maximum height of the profile. It is shown that the films obtained on Mo/glass and Mo-foil substrates have similar roughness and in the both cases the grain structure is formed. The Cu2ZnSnSe4 thin films show relatively high surface roughness and maximum roughness profile height compared to Cu-Zn-Sn precursors. The increase in the surface roughness of the films was caused by the growth of grains during annealing and selenization processes.

The quaternary compound Cu2ZnSnS4 was synthesized by the one-temperature method from the elementary components. Single crystals of Cu2ZnSnS4 were grown by chemical gas-transmission. Samples were prepared in the form of plane singlecrystal plates with a size of more than 2×5×0.5 mm. The electrical conductivity and dielectric properties of Cu2ZnSnS4 single crystals are investigated in the temperature range 100–300 K at the measuring field frequencies of 103–106 Hz. Non-irradiated samples and those irradiated by electrons with an energy of 4 MeV and doses of 1015 and 1016 cm-2are studied. It is shown that the absolute values of the studied characteristics increase with temperature. The curves σ = f(T) have the areas with different slopes, which indicates the presence of several types of conduction in the investigated semiconductors. The dispersion of the dielectric properties of the studied single crystals is revealed: as the frequency is increased, dielectric constant values decrease and, as electrical conductivity is increased, these values grow. Increasing the radiation decreases dielectric permittivity and causes a significant growth of electrical conductivity in the entire investigated temperature range.

An object of investigation is a tube having helical fins on the internal surface and different geometric sizes. Investigation methods: experiments to obtain quantitative results hydraulic resistance of tubes with internal helical finning and to verify computational algorithm; numerical simulation to visualize the flow structure in the tube. Studies of hydraulic resistance of tubes with internal helical finning over a wide range of operating and geometric parameters were made: ReD=2⋅103...2.5⋅105, under the variation of the angle of swirling a = 14–87°, the relative height of a protrusion h/d = (25–87,5)⋅10–3, the relative axial pitch p/d = 0.16–12.73. It is revealed that the hydraulic resistance of tubes with helical finning increases by a factor of 1.1 to 11.7. The numerical simulation results showed that, as the angle of helical swirling is increased, in the near-wall layers the share of the circumferential velocity component increases and the share of the longitudinal component decreases. And since the finning height exceeds the boundary layer thickness, the hydraulic resistance grows.

The band structure and optical properties of the bulk and one monolayer of SnS2, SnSe2, and SnTe2 were established by means of ab initio theoretical modeling. The first two bulk compounds were found to be indirect gap semiconductors, while SnTe2 behaves like a gapless semiconductor. As the atomic number of chalcogen is increased, the compounds considered show an increase in lattice constants and interatomic distances, as well as a decrease in the band gap from 2.4 to 0 eV. Upon transition from the bulk material to a single monolayer, the structural parameters remain practically unchanged. There is a proportional increase in the energy gap, whereby SnTe2 becomes a narrow-gap semiconductor with a band gap of 0.17 eV. The most interesting compound according to a practical use is tin diselenide SnSe2 due to the band gap (1.0–1.5 eV) and the absorption coefficient near the absorption edge more than 105 cm‑1 that are appropriate for photovoltaics. Ternary tin dichalcogenides are also of great interest as the variation of the chemical composition of the latter allows modifying the electronic structure and the optical properties in a wide range.

Spectral properties of nickel phthalocyanine (NiPc) and silver (Ag) thin films, as well as of planar hybrid nanostructures composed of organic semiconductor nanometer films contacting with silver island structures were studied. All nanostructures were fabricated by thermal vacuum evaporation on glass and quartz substrates (S). Two configurations of planar hybrid nanostructures were investigated, in which the silver nanoparticle monolayer was placed under the NiPc film (S/Ag/NiPc) and over the NiPc film (S/NiPc/Ag). The NiPc film thickness was changed from 10 to 30 nm. The silver surface density was about 2⋅10-6 g/cm2. The surface structure of films was studied with the use of a scanning probe microscope “Solver P47 - PRO” in the semi-contact regime. Optical spectra were recorded by a spectrophotomer “Cary 500”. The most significant increase
in the organic film absorption in a presence of Ag nanoparticles was observed for the NiPc film thickness of 10 nm over the spectral range of electronic absorption bands λ ~ 600–700 nm. The effect is due to the local field strengthening near the plasmonic nanoparticles surface for distances compared with nanoparticle sizes. Quantitative regards showed that for the nanostructures of S/Ag/NiPc and S/NiPc/Ag the existence of Ag nanoparticles leads to an increase in the optical density at the wavelength λ = 625 nm at 25 and 33 %, respectively. We suppose that the dependence of the NiPc film effective absorption on the hybrid nanostructure configuration may be related to the features of the nanostructure formation in the process of thermal evaporation.

For autonomous systems with smooth right sides the problem of precise non-local estimation of the limit cycles number is considered in a simply-connected domain of a real phase plane containing three equilibrium points with a total Poincaré index +1. To solve this problem, we are constructing successively two Dulac-Cherkas functions which provide the closed transversal curves decomposing the simply-connected domain in simply-connected subdomains, doubly-connected subdomains, and possibly a three-connected subdomain. The efficiency of the developed approach is demonstrated by the examples of
the polynomial Liènard systems, for which it is proved that there exist a limit cycle in each of the doubly-connected subdomains and two limit cycles in the three-connected subdomain. We determine the configurations of these limit cycles. The obtained results can be applied in the qualitative theory and in the theory of bifurcations of ordinary differential equations, as well as in the theory of nonlinear oscillations.

All groups considered are finite. The symbol Fp(G) denotes the p-nilpotent radical of a group G; p(G) is the set of primes dividing the order of G. Let f be a function of the form f : ℙ → {formations of groups}. (Ú) We consider the class of groups LF ( f ) = (G | G / Fp (G) ∈ f (p) for all p ∈ p(G)). If F is a formation such that F = LF ( f ) for
a function f of the form (Ú), then F is said to be saturated and f is said to be a local satellite of F. Let F be a saturated formation. We write F /l F ∩ N to denote the lattice of all saturated formations lying between F and F ∩ N, where N is the class of all
nilpotent groups. Let Q be a complete lattice of formations. Then we denote by Ql the set of all formations having a local Q-valued satellite. A satellite f is called Q-valued if all values of f belong to Q. Let X ⊆ F ∈ Q be a collection of group. We write QformX
to denote the intersection of all formations of Θ containing all groups of X. Let {Fi | i ∈ I} be an arbitrary collection of formations in Θ. We denote ∨Q (Fi | i ∈ I) = Qform i
i∈I        F . Let { fi | i ∈ I } be a collection of Θ-valued satellites. Then ∨Q ( fi | i ∈ I ) denotes the satellite f such that ( ) form i ( ) i I f p f p ∈   = Q      for every p ∈ ℙ. A complete lattice Ql is called inductive (see Skiba A. N. Algebra formacij [Algebra of Formations]. Minsk, Belaruskaja navuka Publ., 1997) if for any collection {Fi = LF ( fi ) | i ∈ I }, where fi is an integrated satellite of Fi ∈ Ql, the following equality holds: ( | ) ( ( | )) l i i i I LF f i I Q Q ∨∈ = ∨ ∈ F . In this paper, we prove the following
T h e o r e m. Let F be a saturated formation. Then the lattice F /l F ∩ N is inductive.

The present work deals with two directions of the theory of functional integration. The first is the representation of physical quantities, in particular the evolution operator kernel in the form of functional integrals. The second is concerned with the methods for calculation of functional integrals. A new method for approximate evaluation of functional integrals with respect the conditional Wiener measure is proposed in this work. This method is based both on the use of the Feynman – Kac formula giving the integral representation of the evolution operator kernel and on the representation of the kernel using eigenvalues
and eigenvectors of operator. The proposed method is effective for calculation of functional integrals over a space of functions defined on the intervals of large length.

In this article the canonical form of the vector-difference schemes is constructed. The definition of the monotonicity of difference schemes is given. This definition is related to the positivity property of the difference solution. Based on this definition, the monotone difference schemes for the Schnakenberg model with the Dirichlet and Neumann boundary
conditions are constructed. This model is a semi-nonlinear reaction-diffusion system, and it plays an important role in mathematical modeling in the fields of physical chemistry and biology. In constructing a monotone difference scheme for this model with the Neumann boundary condition, the idea of half-integral nodes at the boundary points under the secondkind boundary conditions is used. The results of numerical experiments have confirmed the effectiveness of the suggested methods. the numerical solution without nonphysical oscillation is obtained.

In this article we consider the initial boundary-value problem for the so-called Gamma equation, which can be derived by transforming the nonlinear Black – Scholes equation for option price into a quasi-linear parabolic equation for the second derivative
of option price, and for its exact solution the two-side estimates are obtained. By means of the regularization principle, the previous results are generalized to construct an unconditionally monotone finite-difference scheme (the maximum principle is satisfied without limitations on the relations between the coefficients and the grid parameters) of second-order approximation on uniform grids for this equation. With the help of the difference maximum principle, the two-side estimates for a difference solution are obtained using the arbitrary non-sign-constant input data of the problem. The a priori estimate in the maximum norm C is proved. It is interesting to note that the proven two-side estimates for the difference solution are fully consistent with the differential problem, and the maximal and minimal values of the difference solution do not depend on
the diffusion and convection coefficients. Computational experiments confirming the theoretical conclusions are given.

In the article [1], we have proved the existence of solutions for a model of plug flow at each time step tm = mτ, m = 0,1,…, M. In this article, a priori estimates of these solutions have been obtained, which do not depend on τ and allow passing to the limit as τ→0.

A problem of optimal design of processes of sequential machining of multiple parts on rotary table machines is considered. Batches are processed in a given sequence. Parts of the same batch are located at the working positions of rotary table and are machined simultaneously. Operations are divided into groups which are performed by spindle heads or by turrets. Constraints on the design of spindle heads, turrets, and working positions, as well as on the order of operations are given. The problem is to minimize the estimated cost of machine equipment while reaching a given output. The proposed method to solve the problem is based on its formulation in terms of mixed integer linear programming. Computational results are reported.

If the inhomogeneous equation of semi-bounded string vibration is a classical solution in the first quadrant, then the right-hand side of this equation is obviously continuous. We prove that in this case, a special integral of this right-hand side, which is a generalized solution of the inhomogeneous equation for semi-bounded string vibration, has continuous second derivatives and it is therefore a classical solution. This generalized solution differs from the known generalized solution of this equation in the presence of the upper half of the module of the spatial variable in the integrand, which is a continuous right-hand side of
the equation. This assertion can be used to identify the corresponding necessary smoothness requirements on the right-hand side of the equation for string vibration for the existence of classical solutions of different mixed problems in the quarter and the half-plane.