The paper considers the second-order differential equations, whose right-hand sides are polynomials with periodic coefficients, on the compactification of the cylindrical phase space. We describe bifurcations of an infinitely far triple limit cycle.

We consider integral estimates of Hardy-Littlewood type of the derivative of an analytic function in terms of the norm of the function, and similar estimates for the gradient of a function in Bergman type multifunctional spaces of analytic and harmonic functions. The method of proof uses the classical Whitney decomposition of a connected open set and allows us to extend these estimates for all domains of the complex plane in Lp-spaces with a weight representing a distance to the domain boundary for all 0 < p < +∞.

We report an analysis of nonlinear interaction of electromagnetic field with the classical Maxwellian collisionless plasma. Two electromagnetic waves propagate in plasma. Moreover, the wave vector of the first electromagnetic wave is directed along the x axis. The wave vector of the second electromagnetic wave is in the (x, z) plane and its direction makes an angle of ϕ, (ϕ ≠ 0) with the x axis. For this case, formulae for calculating the electric current are derived. It is found that the nonlinear analysis makes it possible to identify, apart from the known transverse current, the longitudinal current, which is proportional to the square of the electric field. The case of small values of the wave number is considered.

Using the methods of optical polarization microscopy, spectrophotometry and refractometry, we study the microstructure and optical properties of liquid crystal composites (LC composites) based on borosiloxane and LCD-1282 nematic liquid crystal that have been obtained for the first time. In the uniaxial stretching of composites, a new phenomenon in the formation of borosiloxane of microfilaments of liquid crystals in a matrix up to several centimeters in length is observed. The formation of microfilaments and their spontaneous destruction with the formation of microdroplets of liquid crystals are experimental and theoretically characterized. The model of observed processes and an analytical expression relating the length and radius of the formed filament with the size of the droplets formed during its destruction are proposed and experimentally confirmed. Materials based on the investigated LC composites can be used in display technology and optoelectronics, for example, as microresonators and electron-optical sensors.

We consider Stokes’s second problem in relation to the behavior of a rarefied gas filling a half-space, when the plane limiting the half-space performs harmonic oscillations in its own plane. Equations of continuum mechanics in the slip regime are used. It is shown that the approximation that is quadratic in the wall velocity is characterized by temperature effects in a gas due to the influence of viscous dissipation. In this case, there is a temperature drop between the body surface and the gas away from the surface.

А method for the formation of twist structures in objects of micron size using the phenomenon of the formation of capillary fluid bridges in composite materials based on polycaprolactone (PCL) and a nematic liquid crystal 4-cyano-4’-N-heptylbiphenyl (7CB) is proposed. Theoretical modeling of optical characteristics in micro-twist objects is carried out.

We analyze and discuss the limits of application of Mises-Hill plasticity conditions [2] to describe the plastic anisotropy of sheet-rolled aluminum alloys. This condition is often used in the design of processes of sheet metal processing at a pressure. We present in tabular form the results of computation of compatibility indicators and experimentally determine mechanical properties for aluminum alloys. Indicators of consistency are shown to have deviations exceeding the accuracy of the experiment due to non-submission of the material condition of the Mises plasticity [3] and the associated law of plastic deformation.

The properties of a dry flat acoustic contact are studied theoretically and experimentally. It is shown that the contact can be a generator of higher harmonics. It is found that resonance phenomena contribute to their experimental observation. In the case of subharmonic excitation it is possible to obtain the signals of the second, third, fifth, seventh and ninth harmonics of the basic signal.

We have performed a comparative analysis of the concept ‘velocity’, used in various areas of Natural Sciences. The feasibility of using computer technologies, which provide quick and easy information that helps to improve the cognitive activity of a student, is demonstrated.

We discuss the special role of students’ term papers in the formation of creative and research skills of future specialists and development of their ability of self-organization and self-development. We determine the features of the term paper in Functional Analysis. The most important feature consists in applying general abstract principles to specific cases and applications. The paper provides examples of topics of tasks of term papers on Functional Analysis. The role of the integrated interdisciplinary course in the educational process is specified. The topics and the list of tasks for interdisciplinary term projects on methods of calculation, functional analysis and integral equations are presented.