Aim. Purpose is to find exact solutions of the boundary value problem for the biharmonic equation in an infinite 𝑛𝑛-dimensional layer with Navier boundary conditions.

   Methodology. The paper considers a boundary value problem for a biharmonic equation in an infinite n-dimensional layer. The paper considers a boundary value problem for a biharmonic equation in an infinite n-dimensional layer 𝑥 ∈ Rⁿ, 0 < y < a with Navier boundary conditions. This problem reduces to the sequential solution of two Dirichlet problems for the Poisson equation, the explicit solutions of which were obtained earlier by the authors using the Fourier transform of generalized functions of slow growth.

   Results. Exact solutions of the Navier boundary value problem are obtained for a biharmonic equation whose right-hand side is a polyharmonic function in 𝑥𝑥, in particular a polynomial. In this case, the solution is also a polyharmonic function in 𝑥, in particular a polynomial.

   Research implications. They consist in obtaining exact solutions of the Navier boundary value problem for a biharmonic equation in an infinite 𝑛𝑛-dimensional layer.

   Aim of this work is to theoretically investigate the temporal evolution of Bose-condensed atoms in a quadrupole trap.

   Methodology. Theoretical studies of the interaction Hamiltonian describing the time evolution of Bose-condensed atoms in a quadrupole trap under linear tunneling conditions have been carried out.

   Results. Analytical solutions are obtained for a system of differential equations describing the time evolution of Bose-condensed atoms in a quadrupole trap.

   Research implications. The time evolution of Bose-condensed atoms in a quadrupole trap is determined by the initial phase difference, which makes it possible to phase-control the process of Bose-atom tunneling in traps.

   Aim. Review of destructive and non-destructive methods for controlling the thickness of the epitaxial silicon layer (Si). The determination of thin film parameters is an important problem for condensed matter physics. Modern methods of control, such as spherical slot, ellipsometry and IR-Fourier spectrometry, which are poorly represented in the scientific literature, are given.

   Methodology. Practical experience is analyzed and the main results are presented.

   Results. The main approaches to determining the thickness of epitaxial layers are summarized.

   Research implications. A method for determining the thickness of the Si epitaxial film and the depth of the p–n junction with a spherical strip is considered in depth.

   Aim. Investigation of the physical mechanism of the selective effect of laser radiation on the micropore system in the surface layer of an opaque material with high thermal conductivity.

   Methodology. Theoretical study of the specifics of heating the surface for a material with a system of micropores using a nonlinear differential equation of thermal conductivity by the finite element method.

   Results. A physical mechanism for the selective effect of short-pulse laser radiation on micropores in an opaque material is proposed. It is established that under conditions of impulsed laser heating of the material, the specificity of the propagation of isotherms significantly depends on the configuration of the micropore system. For the upper micropore is realized speeded heating of the material above the micropore and slowed heating under the micropore. The specific heating pattern of the material, supplemented by the impact of the shock wave, will stimulate the movement of the material in the direction of the micropore and its partial/complete healing.

   Research implications. The results obtained expand the understanding of the physics of selective laser healing of micropores.

Aim of this work is a theoretical study of the time evolution of Bose-condensed atoms in a four-well trap

Methodology. Theoretical studies of the interaction Hamiltonian describing the time evolution of Bose-condensed atoms in a four-well trap under linear tunneling conditions have been carried out

Results. Analytical solutions are obtained for a system of differential equations describing the time evolution of Bose-condensed atoms in a four-well trap.

Research implications. The time evolution of Bose-condensed atoms in a four-well trap is determined by the initial phase difference, which makes it possible to phase-control the process of Bose-atom tunneling in traps

   Aim. Demonstration of the relationship between the linear-fractional function, analyzed by students of technical universities in the course of complex function theory, and the Lorentz group, which students study in the course of theoretical physics.

   Methodology. Demonstration of the relationship between the fractional linear function, which is analyzed by students of technical universities in the course "Theory of Function of Complex Variable (TFCV)", and the Lorentz group, which students study in the course of theoretical physics.

   Results. The one-to-one correspondence between the classes of fractional-linear transformations of the extended complex plane and the corresponding Lorentz transformations of inertial frames of reference is described in an explicit form. Physically significant examples of light aberration and Wigner rotation are described.

   Research implications. The necessity of taking into account the interdisciplinary connections of theoretical physics and "Theory of Function of Complex Variable (TFCV)" in the study of the foundations of the special theory of relativity is demonstrated.