Vol 74, No 1 (2019)

Explicit relations for harmonic potentials are obtained when the density of the potentials is defined at two segments of parallel straight lines. These results can be used for testing numerical algorithms of calculating the harmonic potentials in boundary value problems outside open arcs in a plane. Moreover, explicit solutions to two systems of singular integral equations are obtained, and these solutions can be used for testing numerical methods for solving singular integral equations. Futhermore, explicit solutions to Neumann problem for the Laplace equation are obtained for specific functions in the boudary conditions.

The evolution function of the differential operator −(−Δ)^v in a d-dimensional Euclidean space is calculated. We obtain its analytic expression through the Fox-Wright psi-functions, which is well-defined for both integer and noninteger values of v and d. Possible applications of the obtained functions in quantum field theory and the connection with fractional calculus are discussed.

This article presents the results of studies of the associated production of the Higgs boson and a heavy fermion pair in arbitrarily polarized electron-positron collisions: \({e^ - }{e^ + } \to {H_{{\rm{SM}}}}f \bar{f}\), \({e^ - }{e^ + } \to Hf \bar{f}\), \({e^ - }{e^ + } \to hf \bar{f}\), and \({e^ - }{e^ + } \to Af \bar{f}\). Analytical expressions for the differential cross sections of the processes are obtained, the specific features of the behavior of the cross sections, angular and spin asymmetries are investigated depending on the energies and angles of emission of particles.

This paper proposes the use of surfaces with a preformed ordered nanotopography to study the mechanisms of the evolution of surface topography under ion beam irradiation. The proposed approach is used for silicon surface bombardment with an oblique beam of accelerated cluster ions. Samples with an ordered topography were formed using electron lithography. The surface was studied using the SEM and AFM techniques. It is shown that the resulting topography is formed as a result of the competition between processes of sputtering and redistribution of atoms. The effectiveness of these processes is determined by the local incidence angles of the ions and the surface curvature. The possibility of obtaining an asymmetric surface profile with the specified parameters is shown by selecting the incidence angle of the ion beam, the irradiation dose, and the initial surface topography.

This paper describes investigation of the possibilities of low-frequency ultrasonic tomography in medicine. The main application is the development of tomographic methods for differential diagnosis of breast cancer. To solve the inverse problem, a mathematical model is used that describes both the phenomena of diffraction and refraction and the absorption of ultrasound in an inhomogeneous medium. The algorithms developed for reconstructing the velocity structure were evaluated using the test bench for experimental studies. Broadband ultrasonic sources in the 50–600 kHz range were used for sounding. A Teledyne Reson TC4038 hydrophone with a sensitivity of approximately 4 µV/Pa and a frequency range of 10–800 kHz was used as a receiver. A Teledyne Reson VP1000 preamplifier was used to amplify the signals. The studies were carried out on phantoms with acoustic parameters close to the properties of human soft tissues. A layerwise model was used to reconstruct the three-dimensional velocity structure. The sound speed cross-sections reconstructed from the experimental data have a resolution of approximately 2 mm, while the central wavelength is approximately 4 mm. An important result of the work is an experimental confirmation of the correspondence of the model to physical processes.

The possibilities and prospects of laser generation in active media based on organic dyes that are cooled to cryogenic temperatures are considered. The results demonstrate the possibility of creating highly effective active laser media (based on the analog of the Shpol’sky matrix) on the basis of cooled carbon-containing matrices.

An analogue of the Heisenberg uncertainty principle (complexity) is introduced for complex biological systems in the framework of the new chaos-self-organization theory. The requirement for such an analysis is determined by the absence of stationary regimes of biosystems (dx/dt ≠ 0 is continuous for the state vector complexity x(t)), uninterrupted chaotic change of the statistical functions f (x) and other parameters. At present, this is defined as type 2 uncertainty. At the same time, type 1 uncertainty is introduced for the complexity when f (x) do not change, and the quasi-attractor parameters can change. A neuron network simulator finds the differences between the samples in the absence of statistical differences.

In this study, the dependence between the spring thermal bar (TB) propagation velocity in a fresh water reservoir and the morphometric parameters of its bottom is obtained via mathematical modeling. Characteristic intervals of TB propagation velocities in reservoirs for different depths and bottom inclinations are found. It is demonstrated that the main factor that affects the TB lifetime is the reservoir depth. The influence of the relative air humidity and solar radiation flux on the spring TB propagation velocity is estimated. Isolines of space—time variability of the TB location in the water area of the reservoir are constructed using the simulations and the quantitative parameters of Lake Ladoga bottom relief. Satisfactory agreement between the simulated data and the field observations is achieved.

The interaction of a short cylindrical vortex elongated along the transverse coordinate with a shear flow near a solid surface is considered. A cylindrical vortex with its ends lowered to the bottom of the channel is shown to turn into a vortex ring. The obtained qualitative model is verified by experiment.