This article considers an initial-boundary value problem for a system of parabolic equations, which arises when studying the flow of a binary mixture in a horizontal channel with walls heated non-uniformly. The problem was reduced to a sequence of initial-boundary value problems with Dirichlet or Neumann conditions. Among them, an inverse problem with a non-local overdetermination condition was distinguished. The solution was constructed using the Fourier method and validated as classical. The behavior of the non-stationary solution at large times was discussed. It was shown that certain functions within the solution tend to their stationary analogs exponentially at large times. For some functions, only boundedness was proved. The problem and its solution are relevant for modeling the thermal modes associated with the separation of liquid mixtures.

   This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.

   The one-dimensional problem of the linear stability of dynamic states of local thermodynamic equilibria with respect to small perturbations was studied for the case when the Vlasov–Poisson electron gas contains electrons with a stationary distribution function that is constant in physical space and variable in a continuum of velocities. The absolute instability of all considered one-dimensional dynamic states of any local thermodynamic equilibrium was proved using the direct Lyapunov method. The scope of applicability of the Newcomb–Gardner–Rosenbluth sufficient condition for linear stability was outlined. An a priori exponential estimation was obtained for the rise of small one-dimensional perturbations from below. Analytic counterexamples to the spectral Newсomb–Gardner theorem and the Penrose criterion were constructed. Earnshaw’s theorem was extended from classical mechanics to
statistical one.

   A key exchange protocol over a special class of formal matrices B_n(R, P) was proposed. The potential of this design for constructing key exchange protocols using suitable associative rings and ideals over them was shown.

   This article considers how the shape of the inner channel in the anode assembly affects plasma flow velocity in a plasma torch. Three different shapes of the anode assembly were analyzed, all with a conical confusor part of 50 mm in length: with a diameter transition from 12 to 6 mm, from 12 to 8 mm, and from 12 to 10 mm. A computer experiment was performed using the finite element method and then validated by the subsequent full-scale experiment on a laboratory plasma unit. The obtained results were verified. The verification outcomes showed a satisfactory convergence and were consistent with the published data. A review of the existing plasma unit designs for powder production, application of functional coatings, and surface modification was carried out. The software packages implementing the finite element method to solve these problems were examined. The study yielded practical recommendations for consumers and developers of plasma equipment and identified the shapes of the anode assembly enabling both supersonic and subsonic plasma flow regimes.

   Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.

   The possibilities of combining known techniques for optimizing the securities portfolio (SP) structure were studied. A method was introduced that enables the simultaneous use of both passive and active approaches to managing the SP structure. The combined application of these methods is based on techniques for SP diversification and searching for an SP structure that mirrors the SP structure of an index fund. The objective function was modified in order to optimize the SP structure according to the traditional “return–risk” approach. The proposed objective function, along with the security risk, describes the degree to which the desired distribution of SP shares coincides with the distribution generated using an index fund. It was established that the main properties of optimal SPs obtained with the “return–risk” approach also occur in the case under consideration.

   The problem of the non-stationary flow of a viscous liquid with an external free boundary around a moving solid cylindrical body was formulated and solved. The liquid is subject to periodic impacts with or without the predominant direction in space. To formulate the problem, the Navier—Stokes equation, the continuity equation, and the equation of conditions at both the solid and free boundaries of the liquid were used. New hydro-mechanical effects were discovered.

   This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.