In this paper, the properties of p-holomorphic maps are considered. An analog of Rolle’s theorem and a global p-diffeomorphism theorem for p-holomorphic functions are found. A modified Carathéodoryʼs theorem (conformal mapping) for p-holomorphic mappings is obtained. We prove the theorems of the mapping of rectangular domains and the compact exhaustion of bounded domains.

In this paper, we investigated a new linear integro-differential equation of arbitrary order given on the closed curve located on a complex plane. The coefficients of the equation are variables and have a special form. The characteristic feature is the presence of linear functions in the coefficients. The equation is reduced to the consecutive solution of a Riemann boundary value problem on an initial curve and two linear differential equations. Differential equations are solved for analytic functions in areas into which the initial curve separates a complex plane. The corresponding fundamental systems of solutions are found, after that the arbitrary-constant variation method is applied. To achieve the analyticity of the obtained solutions the restrictions are imposed. All the arising conditions of resolvability of the input equation are written down explicitly, and if they are carried out then the solution is written in an explicit form. We represent the example demonstrating the existence of the cases when all conditions of resolvability are satisfied.

In this paper, we consider the stable compact difference schemes of 4 + 4 approximation order for the multidimensional hyperbolic-parabolic equation with constant coefficients. A priori estimates for the stability and convergence of the difference solution in strong mesh norms are obtained. The theoretical results are confirmed by test numerical calculations.

In this paper, we consider the properties of uniformly convergent sequences of h-holomorphic functions on the set of h-complex numbers. Theorems on the global antiderivative and on the uniform approximation of h-holomorphic functions by polynomials are formulated and proven. The sufficient conditions for the h-holomorphism of the limit function are obtained. The compactness principle for functions of an h-complex variable and an analog of Vitaliʼs theorem for h-analytic functions are formulated and proven.

In this paper, we consider the class of functional integrals with respect to the conditional Wiener measure, which is important for applications. These integrals are written using the action functional containing terms corresponding to kinetic and potential energies. For the considered class of integrals an approach to semiclassical approximation is developed. This approach is based on the decomposition of the action with respect to the classical trajectory. In the expansion of the action, only terms with degrees zero and two are used. A numerical analysis of the accuracy of the semiclassical approximation for functional integrals containing the centrifugal potential is done.

In this paper, the system of equations describing a spin 1 particle is studied in cylindric coordinates with the use of tetrad formalism and the matrix 10-dimension formalism of Duffin – Kemmer – Petieau. After separating the variables, we apply the method proposed by Fedorov – Gronskiy and based on the use of projective operators to resolve the system of 10 equations in the r variable. In the presence of an external uniform magnetic field, we construct in an explicit form three independent classes of wave functions with corresponding energy spectra. Separately the massless field with spin 1 is studied; there are found four linearly independent solutions, two of which are gauge ones, and other two do not contain gauge degrees of freedom. Meanwhile, the method of Fedorov – Gronskiy is also used.

We analyze herein the higher orders contributions of expansions in the fine structure constant α to the anomalous magnetic moment of leptons coming from the diagrams of vacuum polarization by lepton loops in the case when the ratio of the mass of lepton in the loop to the mass of external lepton is less than unity. The dependence of the expansion coefficients a_n on the ratio of lepton masses is found and a comparison is made with the previously known analytical estimates. It is shown that for real values of lepton masses the new analytical expressions turn out to be more accurate than the known ones. Estimates are given for the order of expansion n*, starting from which one or another accuracy is guaranteed for the coefficients a_n.

Theoretical modeling within LDA, GGA, and PBE approximations was herein performed to determine the electronic band structures of MgGeN₂, MgSiN₂, ZnGeN₂, and ZnSiN₂  nitride compounds and their optical properties. It is established that the compounds with germanium are direct-gap semiconductors with the band gap values of 3.0 eV (MgGeN₂) and 1.7 eV (ZnGeN₂), while the silicon-based compounds are indirect-gap semiconductors with the band gap values of 4.6 eV (MgSiN₂) and 3.7 eV (ZnSiN₂). Optical properties analysis showed the prospects of using MgGeN₂  and ZnGeN₂  in optoelectronics.