Both temperature and air-velocity fields inside the instrumentation container have been calculated and results of these computational studies have been presented.

   The aim of this work is to determine the number of OverFrost 40 thermoelectric air conditioners and Schroff CR 027 air heaters needed to maintain the optimal air temperatures inside the container when the multi-frame recording system in the linear induction accelerator of electrons operates under the conditions of negative and positive ambient temperatures.

   Calculations have been carried out in several stages in the 3D finite element model, in the gas-dynamic module of the engineering program that implements the finite element method. The temperature and air-velocity fields inside the container for one air conditioner and one heater, both located at the rear wall inside the instrumentation container have been determined in the first stage. Calculations have been performed for operational conditions of positive and negative ambient temperatures. The number of conditioners needed to ensure the sufficient cooling of air inside the container down to the optimal temperature in the positive temperature operational environment has been determined in the second stage. Results of settlement studies have shown that six conditioners and one heater are enough to maintain an air temperature of (20 ± 5) °С in the office accommodation volume at an environment temperature of – 5 °С. At the operation with the maximum capacity, a temperature of (20 ± 5)°С will be reached in about 5 min. At the block operation, a temperature of (20 ± 5)°С is supported by means of six conditioners during 120 min.

   At present, a wide range of tests of materials used in nuclear power engineering is carried out in the BOR-60 reactor. The study of changes in the properties of materials requires the determination of temperature irradiation conditions with high accuracy. The materials that have been tested in the BOR-60 reactor from the start up to the present are summarized in this work. Each of the research areas requires an individual approach and the creation of a unique experimental device. Experimental devices differ in various criteria that affect the efficiency of the research. The geometrical parameters of the cells of the reactor and the characteristics of the pressure collector require external similarity of the devices used, and the internal design varies considerably depending on the type of research conducted. Various classifications of experimental devices of the BOR-60 reactor differing in the method of obtaining information, in the f irradiation temperature, and in the design and testing environment are presented. In addition, a methodical experiment conducted in the instrumented cell of the BOR-60 reactor is described. For each type of irradiation devices, the design features and test applications are discussed. An autonomous loop channel that allows testing in environments different from the reactor coolant is described separately.

   CuAlO₂ is an important material having diverse applications in thermoelectricity, photoelectricity, and optoelectronics. Most of the well-known and widely used transparent conducting oxides such as ZnO and SnO₂ and their doped versions are n-type material, but corresponding p-type transparent conducting oxides were surprisingly missing for a long time. Its unique crystallographic structure manifests an anisotropic environment for charge carriers and phonons, which is a reason for the enhancement of the thermopower. The scheme for the sol–gel synthesis of nanocrystalline powders CuAlO₂ has been presented in this work. It has been found that the energy released by the combustion of citric acid in the process of synthesis is “extra fuel,” which allows overcoming the energy barrier of the reaction. X-ray diffraction analysis and simultaneous thermal and electron-microscopic analyzes have been used to demonstrate the possibility of copper aluminate synthesis with a delafossite structure at a temperature of 1000 °C. It has been established that the heating mode significantly affects the phase composition of the final product: with the gradual heating, copper aluminate with a spinel structure is formed, and with the sharp heating, delafossite is formed. The scanning electron microscopy studies have also shown that the typical particle sizes in copper aluminate samples calcined at 1000 °C are 35 nm. The temperature dependence of conductivity has been measured and activation energy has been determined.

   Various classes of nonlinear mass and heat transfer equations with variable coefficients, c(x)u_t = ⌠a(x)f(u)u_xI_x = b(x)g(u)u_x, which admit exact solutions, are considered. The main attention is focused on nonlinear equations of a sufficiently general form, which contain several arbitrary functions that depend on the unknown function u and the spatial variable x. It is important to note that the exact solutions of nonlinear partial differential equations that contain arbitrary functions and are, therefore, sufficiently general, are of the greatest practical interest for testing various numerical and approximate analytical methods to solve corresponding initial-boundary value problems. The method used to find exact solutions is based on the representation of the solution in the implicit form⌠ h(u)du =  ξ(t) + η(x), where the functions h(u), ξ(t), η(x) are determined further by analyzing resulting functional-differential equations. Examples of specif-ic reaction–diffusion type equations and their exact solutions are given. Many new generalized traveling wave solutions and functional separable solutions are described.

   The classification of ordinary differential equations with exact solutions is a classical mathematical problem. In this work, the classification problem is considered for ordinary differential equations with solutions expressed in terms of the Weierstrass elliptic function. The algorithm of search for such equations is as follows. First, the order of the singularity of the solution is chosen. Then, the order of the sought nonlinear differential equation is set. Next, Newton polygons are used to write the general form of the nonlinear differential equation taking into account the singularity of the solution and the given order for the nonlinear differential equation. After that, limitations for the parameters are found so that the general form of the nonlinear differential equation has an exact solution expressed in terms of the Weierstrass elliptic function. Theorems used to look for parameter limitations are presented. The nonlinear autonomous ordinary differential equations of the third and fourth orders are constructed using the described algorithm. Moreover, nonlinear autonomous differential equations and their solutions expressed in terms of the Weierstrass elliptic function are presented.

   Our previous studies were devoted to Bessel functions of the first kind J_ν(z) and modified Bessel functions of the first kind J_ν(z) (Infeld functions) with the parameter ν > –1. In this work, Bessel functions of the first kind of an arbitrary real order ν are considered. All the zeros of any such function are simple, and only a finite number of zeros (regulated by the Hurwitz theorem) can be located outside the real line. An auxiliary even entire function of the exponential type L(z, ν) constructed with respect to J_ν(z) and having the same nontrivial zeros is involved, allowing the application of the well-developed entire function method. The problem of expanding the function 1/L(z, ν) into a series of simple fractions with a special structure (Krein’s type series) has been studied. This general representation is used to derive formulas for calculating special series containing negative powers of zeros of the Bessel function J_ν(z). Particular attention is focused on integer and semi-integer orders ν. Examples of specific expansions of 1/J_ν(z) and the corresponding summation formulas for various parameters ν are given.

   It is shown that the direct method of functional separation of variables can sometimes provide a larger number of exact solutions of nonlinear partial differential equations than the method of differential constraints (with a single constraint) and the nonclassical method of symmetry reduction (based on the invariant surface condition). This fact is illustrated on nonlinear reaction–diffusion and convection–diffusion equations with variable coefficients, nonlinear Klein–Gordon type equations, and hydrodynamic boundary layer equations. Some new exact solutions are given.

   The following nonlinear telegraph equations with delay are considered: u_n + H(u)u_t = (G(u)u_x)_x + F (u, w); u_n = (G(u)u_x)_x + P(u)u_x)_x + F (u, w), where u = u(x, t), w = u(x, t -  τ), and τ – is the constant delay time. The equations contain the nonlinear transfer coefficient G(u) of the power-law or exponential type, as well as the coefficients H(u) and P(u) that either are constant or are nonlinear and have the form similar to the form of G(u) The kinetic functions F of all the equations consist of one or several arbitrary functions of one argument. For the equations under consideration, new exact travelling-wave solutions, as well as new exact solutions with generalized and functional separation of variables, have been obtained by means of the modified method of functional constraints. All the solutions are expressed in terms of elementary functions, contain free parameters, and can be used for the formulation of test problems to assess the accuracy of numerical methods for solving nonlinear partial differential equations with delay. Publications presenting exact solutions of equations with delay and describing methods for constructing exact solutions have been reviewed.

   The principle of the ultrasonic method for determining fuel assemblies forming in cooling pond of nuclear power plants is considered. It is shown that natural convection caused by residual heat generation affects the increase in the measurement error of the ultrasonic method. A method is proposed for calculating the convective heat transfer with the boundary condition of the second kind. The developed method has been tested by performing calculations under the conditions of experiments in which temperature change in water inside a free convective laminar and turbulent thermal boundary layer has been studied. Analysis of experimental and calculated data shows that the method used adequately describes natural-convection heat transfer. Using this method, an algorithm and a program are developed for calculating the speed of sound in water in a laminar and turbulent boundary layer near the surface of a vertical heated plate, simulating in the first approximation the heated surface of a WWER-1000 FA. The developed program also computes the parameters of the boundary layer, heating temperature, and temperature profile along the acoustic axis of the ultrasonic sensor. The program can be used to approximately estimate the influence of convection on measurement results and to verify CFD codes as applied to the calculation of the parameters of natural convection at the surface of the WWER-1000 FA, taking into account the decay heat.

   The aim of this work is to create a voxel phantom with the use of DICOM (Digital Imaging and Communications in Medicine) images for verification of dosimetric Monte Carlo calculations of the Gamma Knife Perfexion.

   The voxel phantom has been created using the Labview program, and the Monte Carlo simulation has been carried out using the Penelope / PenEasy program. To simulate the radiation transfer using Penelope, it is required to specify the number and size of voxels along the x, y, and z axes, as well as the material and density in each voxel. The data used are computed tomography files of a real patient in the DICOM format with information about the type of study, the size of pixels, and the distance between them, etc., as well as the value of pixels in Hounsfield Units (HU). To create the voxel phantom, data from DICOM images have been imported and converted into a file readable by penEasy step by step: – the first step involves importing information about the pixels to create the header of the file, – the second step is to assign materials and densities to the HU units. HU units are converted into the density, then are divided into four groups, and a material is assigned to each group for Monte Carlo simulation. To save the computational time, the image resolution of 512×512 pixels is reduced to 256 × 256 pixels by removing one HU unit of even rows and columns from the original matrix. As a first result, the voxel phantom has been created using images in the DICOM format.

   Approximation of a set of experimental points using a set of functions is actual for many engineering studies. To solve such problems, the concept of a linear normed space, whose elements are bounded real functions, is introduced, and the concept of a metric (norm), i.e., a measure of proximity between the elements of the space is used. In many cases, it is required to approximate a complex function by a polynomial of a given order and, at the same time, to ensure the maximum deviation of the polynomial from the function by no more than a certain specified error. In this case, it is reasonable to use the Chebyshev norm and look for a polynomial of the best uniform approximation. However, there are no universal effective algorithms for finding the best-dimensional approximation polynomial. In this work, a simple and efficient algorithm is proposed for constructing the best uniform approximation polynomial for continuously differentiable functions. The algorithm consists of three stages. At the first stage, a polynomial of the degree n is constructed using the least squares method. At the second stage, a special system of nonlinear equations is obtained. At the third stage, the coefficients of the best uniform approximation polynomial and the Chebyshev alternance point are found by solving a system of nonlinear equations by any iterative method. This algorithm is implemented in the SciLab 6.0 system and is experimentally tested.