The problem of using burnable absorbers in VVER reactors to reduce the volume of liquid regulation of excess fuel burnup reactivity has been considered. The use of burnable absorbers such as Gd₂O₃ and Eu₂O₃ in nuclear fuel has a positive effect on the nuclear fuel cycle of light-water reactors PWR (VVER-1000) and BWR (RBMK) and leads to an increase in the duration of the campaign and some other positive properties. However, the presence of burnable absorbers in fuel simultaneously deteriorates the uniformity of the energy field and a number of other negative properties. Previous studies of the possibility of reducing these negative results have been presented. One of the proposed technical solutions is to change the structure of gadolinium placement in fuel rods. In this work, the influence of the placement of burnable absorbers (Gd₂O₃ and Eu₂O₃) on the neutron-physical characteristics of nuclear fuel of VVER-1000 reactors has been studied. These characteristics are the infinite neutron multiplication factor, the coefficient of uneven distribution of energy release, and the accumulation of isotopes depending on the burnup.

The mathematical model of instrumental errors of a three-accelerometer module has been analyzed in application to the determination of the apparent acceleration vector. To this end, the vector calibration method is used on a stationary testing bench with the gravitational acceleration as a reference. The instrumental errors of each accelerometer in the module are determined by a set of parameters, which are refined at calibration. This set consists of two angular errors of the sensitivity axis orientation, the deviation of the scaling coefficient, and bias. An approach has been proposed to determine the optimal set of angular calibration positions for the accelerometer module to achieve the best accuracy for estimating the listed parameters of the instrumental error model. The improvement is achieved due to a smaller condition number of an intermediate matrix arising when solving the calibration system of linear equations by the least squares method. The decrease in the condition number reduces the influence of unaccounted factors, such as instrumental noise and alignment error, on final parameter estimate. The approach has been tested through a computer simulation by adding noise to the array of measurements and comparison with a nonoptimal set of angular calibration positions.

   Various classes of nonlinear telegraph equations with variable coefficients c(x)u_{n +} d(x)u_{τ =} [a(x)u_x]_x + b(x)u_x + p(x)u_x + p(x)f(u), which allow exact solutions with a functional separation of variables of the form u = U(z), z = ϕ , (x, t ). , have been described. It has been shown that the source function f(u) and any four of the five coefficients a(x), b(x), c(x),  d(x), p(x) of these equations can be chosen arbitrarily, and the remaining coefficient is expressed in terms of them. The properties have been studied of the overdetermined system of differential equations for the function ф(x, t)  and  some its solutions have been obtained. Examples of particular equations and their exact solutions are given. Some exact generalized traveling-wave solutions of more complex nonlinear telegraph equations with delay of the form c(x)u_{n +} d(x)u_{τ =} [a(x)u_x]_x + b(x)u_x + p(x)u_x + p(x)f(u, w), w = u(x,t - τ), where τ > 0 is the delay time and f(u, w) is an arbitrary function of two arguments, are also obtained.

   The Fermi–Pasta–Ulam model including the fourth and fifth terms in the potential of interaction between neighboring particles has been considered. A passage to a continuum limit has been performed when the distance between the particles approaches zero and the number of particles tends to infinity. It has been shown that, a nonlinear partial differential equation of the sixth order is obtained instead of the well-known Korteweg–de Vries equation taking into account the quadratic interaction between particles. The fifth-order evolution partial differential equation has been obtained. The analytical properties of the resulting equations have been investigated. It has been shown that the general solution of the fifth-order differential equation obtained during the passage to traveling wave variables has four branches in the expansion into a Laurent series. In the second step of the Painlevé test, Fuchs indices two of which are complex have been found. It has been shown that the fifth-order nonlinear partial differential equations found from the Fermi–Pasta–Ulam model do not pass the Painlevé test. The exact solutions of the fifth-order evolution equations have been obtained using the simplest equation method. The chart of the solutions has been constructed.

   Qualitative features of numerical integration of initial-boundary value problems for partial differential equations with delay by the method of lines have been described. The method of lines is based on the approximation of spatial derivatives by corresponding finite differences, which allows reducing the initial equation to an approximate system of ordinary differential equations with delay. The system is then solved by the Runge–Kutta methods of the second and fourth orders and by the BDF method, which are built into Wolfram Mathematica. Test problems for nonlinear Klein–Gordon type equations with a constant delay τ whose solutions are expressed in terms of elementary functions have been formulated. The extensive comparison of numerical and exact solutions of the test problems on a significant time interval from 0 to 50 τ has been made. It has been found that the numerical method under consideration with moderate delay times ensures high accuracy of the results obtained.

   Systems of equations of two-dimensional shallow water over an uneven bottom have been considered in both the Eulerian and Lagrangian variables. An intermediate system of equations has been introduced. Its solutions are simultaneously solutions of the system of equations of two-dimensional shallow water in the Eulerian variables and implicit solutions of the system of equations of two-dimensional shallow water in the Lagrangian variables. All basic hydrodynamic conservation laws of the intermediate system of equations have been found without using symmetries. A relationship has been obtained between the conservation laws of the intermediate system of equations and the system of equations of two-dimensional shallow water in the Lagrangian variables. The basic hydrodynamic conservation laws of the intermediate system of equations have been used to construct the basic conservation laws of the first order for the system of equations of two-dimensional shallow water in the Lagrangian variables.

   The Steklov type spectral problem for the Laplace operator and the corresponding boundary value problem in a bounded domain with a smooth boundary has been considered. It is assumed that the homogeneous Dirichlet condition is set on a small part of the boundary, and the Steklov condition (or the corresponding Neumann condition) is imposed on the rest of the boundary. It is known that the Steklov problem perturbed on a small part of the boundary by the Dirichlet condition has a countable set of eigenvalues with finite multiplicity. Moreover, the limit problem is a problem for the Laplace operator with the Steklov condition on the entire boundary. It is also known that the problem for the Laplace operator with the Steklov condition on the entire boundary has a countable set of eigenvalues with finite multiplicity. A two-term asymptotic expressions have been constructed for the eigenvalues and the corresponding eigenfunctions of the original problem as the small parameter characterizing the size of the boundary part with the Dirichlet condition tends to zero. It has been shown that the asymptotic expression for the eigenvalue has the second term inversely proportional to the logarithm of the small parameter. Moreover, the asymptotic expression is strictly justified with the estimate of the rest term inversely proportional to the square of the logarithm of the small parameter.

   First integrals for one fourth-order ordinary differential equation that is a special case of one of the higher analogues of the Painlevé equation have been found. The equation has been studied for the presence of the Painlevé property using the Kovalevskaya algorithm. It has been shown that the method used does not allow to accurately determining whether the general solution of the equation has critical moving singular points or not. Two particular cases of the equation under study with certain values of its parameters contained have been considered in detail. To find the first integrals of the resulting equations, their linear dependence on the highest derivative is assumed. The found first integrals are used to reduce the order of the equations under study. This equation does not have any first integral of a similar form. The found first integral of this equation is used to reduce the equation to a second order equation. It has been shown that the resulting equation does not have a first integral of the form similar to the cases of the third and fourth order equations.

A hydrogen recombiner based on high-porosity cellular materials is currently studied at the Russian Federal Nuclear Center VNIITF. The hydrogen recombiner is used for flameless hydrogen combustion. Hydrogen flowing through a porous catalyst in the recombiner induces various physical and chemical processes essentially including the exothermal chemical reaction of hydrogen oxidation on the catalyzing surfaces of the recombiner and a convective flow of a gas mixture through the catalyst unit. These processes determine the technical characteristics of the recombiner including the most important efficiency measured at a given input gas composition. The BM-LR and BM-P chambers are used to conduct experiments to estimate the hydrogen concentration range within which the recombiner operates in a flameless mode at atmospheric pressure, high temperatures, and humidity, which are characteristic of severe nuclear power plant accidents [1, 2]. The BM-LR chamber includes an internal chamber (5-t railway container) and a thermally insulated external shell consisting of a mineral wool layer and a glass–magnesium plate. The internal chamber is airtight, one of the walls being covered with a polycarbonate sheet to provide video recording of experimental results. The sheet is pushed away when pressure increases during hydrogen ignition to preserve the integrity of the metal part of the chamber. Heat guns maintain the temperatures required for the experiments. The BM-P chamber is a 4-m-high steel cylinder 2 m in diameter. The walls of the chamber are equipped with windows for high-speed video recording of condensation and flame propagation processes. The chamber is designed to keep air-tightness at high pressures and to withstand high explosive loads produced by an ignited vapor–water mixture. The temperature fields in the BMLR and BM-P chambers for different recombiner operation modes have been computationally analyzed. The maximum temperatures and velocity fields are determined for the gas mixture flowing within the chambers when the recombiner is in operation.

   Charge collection from the tracks of single ionizing particles by two-phase CMOS inverters with a design norm of 65 nm on two mutually connected channels (phases) forming a chain has been simulated. The occurrence of error pulses caused by charge collection from tracks directed along the normal to the surface of the device part of the crystal has been analyzed. The input points of particle tracks are at the drain areas of transistors or at a distance of 0.3–0.7 μm from them. The durations of error pulses on the nodes of the elements which collecting the charge from the track are from 120 to 300 ps. The amplitudes of pulses relative to the voltage on the power bus or common bus range from 0.05 to 1.0 V. The propagation of noise pulses along the chain of two-phase CMOS inverters occurs only for tracks with entry points into the drain area of transistors. If the linear energy transfer to the track is 60 MeV cm² /mg, an error pulse can be transferred only to the next inverter if error pulses are formed at two outputs of the two-phase CMOS inverter and the sum of their amplitudes exceeds the supply voltage of the element.

   The ACNP (automatic construction of Newton polygons) program designed to automatically construct Newton polygons corresponding to polynomial differential equations has been described. A Newton polygon of an ordinary differential equation is a convex polygon whose vertices are the outer points of the carrier of this equation (the carrier is the set of points on the plane corresponding to the monomials of the differential equation according to a certain rule). Newton polygons for polynomial ordinary differential equations are useful for studying the integrability of nonlinear equations using the Kovalevskaya algorithm, for constructing asymptotic solutions, and for finding exact solutions of nonlinear differential equations. Automatic construction of Newton polygons in some cases allows finding the order of the pole of the equation, select the leading terms of the equation, speed up the process of finding the power asymptotic behavior of solutions of differential equations, and simplify the choice of the simplest equation when finding exact solutions of nonlinear differential equations. The ACNP program is written in the Maple computer algebra environment. The algorithm of the program and examples of its application have been presented.

   Approximation by standard functions is one of the methods to reconstruct the orientation distribution function of grains in polycrystalline materials from a set of experimentally measured pole figures. In practice, the central and canonical normal distributions are often used to solve this problem. The central distribution has a circular scattering character, whereas the canonical one is anisotropic. The orientation distribution function can have peak and axial components. The peak component is bell-shaped and has a single maximum in the orientational space. The axial component is the average of the peak component over rotations around the selected axis. The distribution functions for the orientations of the axial components of the central and canonical normal distributions have been calculated. Pole figures for the canonical normal distribution with different parameters are constructed. The exact and approximating expressions for the axial component of the central normal distribution have been compared quantitatively and qualitatively. It is appropriate to use an approximating function to simplify the calculation of the axial component for a normal distribution with circular and noncircular scattering patterns of the texture of polycrystals. Since real textures usually include several axial components with different parameters and weights, calculations of the orientation distribution function are greatly simplified when using the approximating expression.