In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R³ is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited simply-connected region with a smooth boundary in R³. Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem. The obtained condition permits to prove that the set of regularizable Riemann – Hilbert boundary value problems for the considered class of systems has two components of homotopic connectedness, and the index of an arbitrary regularizable problem equals to minus one.

In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.

As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.

The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.

In this paper we consider mixed-type stochastic differential equations driven by standard and fractional Brownian motions with Hurst indices greater than 1/3. There are proved theorems on the existence, uniqueness, and continuous dependence of solutions on the initial data. We provide an analog of the Ito formula to change variables. Asymptotic expansions of functionals on the solutions of mixed-type stochastic differential equations for small times are obtained. We receive analogs of the Kolmogorov equations for mathematical expectations and probability densities in the commutative case. Finally, we consider an application of mixed-type stochastic differential equations to solving the problem of macroeconomic variables extrapolation in credit risks models.

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

An approximate evaluation of matrix-valued functional integrals generated by the relativistic Hamiltonian is considered. The method of evaluation of functional integrals is based on the expansion in the eigenfunctions of Hamiltonian generating the functional integral. To find the eigenfunctions and the eigenvalues the initial Hamiltonian is considered as a sum of the unperturbed operator and a small correction to it, and the perturbation theory is used. The eigenvalues and the eigenfunctions of the unperturbed operator are found using the Sturm sequence method and the reverse iteration method. This approach allows one to significantly reduce the computation time and the used computer memory compared to the other known methods.

Quark gluon plasma (QGP) is a special state of nuclear matter where quarks and gluons behave like free particles. Recently, a number of investigations of this state with high temperature and/or density have been conducted using collisions of relativistic and ultra-relativistic heavy nuclei. It is accepted that depending on the temperature and density, 1^st or the 2^nd order phase transitions take place in hadron matter during the formation of QGP. Herein, we have modeled heavy ion collisions using a HIJING Monte-Carlo generator, taking into account the description of the 1^st order phase transition as a probabilistic process. We analyzed the behavior of the fluctuations of the total (N = N₊ – N_–) and resultant (Q = N₊ – N_–) electric charges of the system. Different phases were introduced using the BDMPS (Baier – Dokshitzer – Mueller – Piegne – Schiff) model of parton energy loss during crossing through a dense nuclear medium.

In the diffusion-drift approximation, we have constructed a phenomenological theory of the coexisting migration of v-band holes and holes by means of hopping from hydrogen-like acceptors in the charge state (0) to acceptors in the charge state (−1). A p-type crystalline semiconductor is considered at a constant temperature, to which an external stationary electric field is applied. In the linear approximation, analytical expressions for the screening length of the static electric field and the length of the diffusion of v-band holes and the holes quasilocalized on acceptors are obtained for the first time. The presented relations, as special cases, contain well-known expressions. It is shown that the hopping migration of holes via acceptors leads to a decrease in the screening length and in the diffusion length.

Herein, the influence of water vapor adsorption and desorption processes on the surface of SnO_2−δ nanocrystalline films with different concentrations of oxygen vacancies on their electrical conductivity at room temperature was studied. SnO_2−δ films were synthesized by means of reactive magnetron sputtering of tin in an argon-oxygen plasma followed by 2-stage oxidative annealing. The concentration of oxygen vacancies in the films was varied by changing the 2nd stage annealing temperature within the range 350–400 °C. It was found that in the films with the highest concentration of oxygen vacancies (~10²⁰ cm⁻³) in the region of low relative humidity (less than ~30 %), an increase in electrical conductivity was observed due to the dissociative adsorption of water molecules with the formation of hydroxyl groups. The adsorption of water vapor on the surface of SnO_2−δ films at room temperature at relative humidity values higher than ~30 % was found to induce a decrease in the electrical conductivity of the samples. The generation of positive and negative EMF pulses between the open surface of SnO_2−δ nanocrystalline films and the one covered by waterproof materials under the adsorption and desorption of water vapor, respectively, was detected. The change of resistance and the generated EMF value under the adsorption-desorption processes was found to increase with the concentration of free charge carriers in the films.

Tiling is a widely used technique to solve the problems of the efficient use of multilevel memory and optimize data exchanges when developing both sequential and parallel programs. This paper investigates the problem of obtaining global dependencies, i.e. informational dependencies between tiles. The problem is solved in the context of parametrized hexagonal tiling in application to algorithms with a two-dimensional computational domain. The paper includes a formalized definition of the hexagonal tile and the criteria for dense coverage of the computational domain with hexagonal tiles. Herein, we have formulated a statement that permits to obtain all global dependencies between tiles. Formulas are constructed for the determination of sets of iterations of hexagonal tiles generating these dependencies. The sets of iterations that generate global dependencies are obtained in the form of polyhedra with an explicit expression of their boundaries.