A relationship between macroscopic parameters, such as the Young’s modulus in Hooke’s law, the speed of sound, and the Debye temperature, and the binding energy of an individual atom has been analyzed. A formula for the elastic deformation modulus is proposed. The speed of sound in isotropic solids is determined by the elastic properties of the substance. A relation between the speed of sound and the binding energy of an individual atom in a solid is obtained. A simple formula for the speed of sound in a metal rod is established. The relationship between the characteristic Debye temperature and the binding energy of an ion in the lattice of the solid is proposed. It is shown that the Young’s modulus in Hooke’s law, the speed of sound, and the Debye temperature are not independent, but are determined by the ion binding energy in the crystal lattice.

   A system of MHD equilibrium equations is considered. Approaches used to solve this system lead to an overdetermined system of nonlinear partial differential equations. The overdetermined system is solved with analytical approaches based on the reduction of systems of partial differential equations to systems of ordinary differential equations with the subsequent solution of such systems. With such reductions, various quantities can be selected as an independent variable in the system of ordinary differential equations. An algorithm is described to reduce the overdetermined system of equations to systems of ordinary differential equations in which the independent variable ψ is such that ψ(x, y, z) = const is the level surface for solutions of the system of MHD equilibrium equations. It is shown that there is more than one way to choose such an independent variable. Exact solutions of the original system are obtained in the cases ψ = x/(1 – my – nz) (m = const, n = const) and ψ = x + my + nz + l (m = const, n = const, l = const). The exact solutions obtained depend on arbitrary constants. It is shown that the solution in the ψ = x + my + nz + l (m = const, n = const, l = const) can be use to construct a magnetic field of a given direction. One particular solution of the system of MHD equilibrium equations is presented for a vortex-free solenoidal magnetic field. It is noted that these exact solutions can be used as tests in numerical calculations.

   The generalized third order nonlinear Schrödinger equation is considered. This equation is a non-linear partial differential equation, which can be used to describe pulses in optical fibers. The Cauchy problem for it is not solved by the inverse scattering transform; for this reason, the solution of the equation is sought in the travelling wave variables. A system of differential equations for the imaginary and real parts has been obtained in these variables. Conditions for the existence of a solution of an overdetermined system of differential equations are determined. An analytical solution is found in terms of the Jacobi elliptic function. The solutions are presented in the form of periodic and solitary waves of the nonlinear Schrrödinger equation under various conditions on the coefficients. The found solutions are plotted in the graphical form.

   Rational solutions of the Burgers hierarchy have been obtained using the linearization of the differential equation. Hierarchy equations allow a group of stretch transformations. Using self-similar variables, Burgers equations are converted after integration into an n-order differential equation where the function depends only on one variable. The linearization of the equation is done using the Cole–Hopf transformation. A solution of an ordinary linear differential equation is sought in the form of an (n + 1)-order polynomial. The substitution of the polynomial with indefinite coefficients into the equations transforms the linear differential equation into a system of (n + 1)-order algebraic equations with constant coefficients. The kind of a rational solution depends on the constant of integration, which has certain values related to the degree of polynomial. Rational solutions of the Burgers hierarchy have points of discontinuity.

   It is known that the excess of maximum one-time concentrations of harmful substances in the air of the working area is one of the main reasons for an increase in the level of occupational diseases in various industries (primarily chemical). In order to determine the cause of occupational diseases, assess their consequences, as well as develop a set of measures aimed at their prevention, it is necessary to establish a connection between such effects with the manifestation of occupational diseases, as well as to assess the incidence of diseases in occupational groups. Thus, the problem of assessing the impact of vapors, fine aerosols, and dust particles of hazardous chemicals used in technological processes that can be in the air of working rooms on personnel is extremely important, and studies aimed at quantifying the inhalation effects of hazardous chemical substances on personnel, are relevant. In this work, repeated short-term inhalation effects of hazardous chemicals on the working personnel of chemical enterprises are mathematically simulated, the developed software package including a database of hazardous chemicals and mathematical models that allow simulating toxicokinetics during inhalation effects on personnel in working rooms, as well as assessing risk to personnel, has been described.

   An essential part of the simulation of the behavior of a fuel rod for fast-neutron and thermal-neutron nuclear reactors is finding the stress–strain state of the fuel pellet and the fuel rod cladding in addition to the thermal problem. The system of equations that describes the fuel rod mechanical behavior includes not only the elastic strain, thermal strain, and radiation swelling strain but also thermal and radiation creep strain that makes the system of equations nonlinear with respect to the interlayer and axial pressure variables. The solution of that kind of equations demands effective iteration methods. In this work, the algorithm for search for the solution of the system of nonlinear equations of nuclear fuel rod mechanics has been advanced by replacing the Gauss elimination method with the sweep method adapted to the matrix type of the linearized system. Optimization gives substantial acceleration in finding the solution of the system of linear equations that is raised in the problem: the discretization computational time at 10 and 20 layers decreases by one and two orders of magnitude, respectively. As a result, the time of finding the solution of the nonlinear system of equations by the Newton method for 10 and 20 layers decreases by a factor of 1.3 and 2, respectively.

   The effect of monophenol on the activity of seston esterases has been simulated. The experiments have been carried out on natural water from the Don river in aquariums where phenol concentrations of 0.02, 0.010, 0.020, and 0.050 mg/l are added. One aquarium is a control tank. Changes in the activity of esterases under the influence of phenol concentrations 0.02, 0.010, 0.020 0.050 mg/l have been analyzed in model experiments and in studies performed on the site of the Seversky Donets river near the Lisichansk city. A canonical and regression analysis has been performed between the activity of extracellular seston esterases and hydrochemical parameters and phenolic compounds. It has been found that in model experiments and studies on drive aquatic ecosystems, the activity of extracellular seston esterases is an informative indicator for assessing the quality of water contaminated with phenolic compounds in the range from 2 to 50 μg/l, the effect of which lasts up to 6 d. +++The regression analysis has shown that the extracellular esterase activity in the section of the Seversky Donets river is predicted by a complex of chemical components consisting of phenols and biogenic substances, as well as by the temperature. The canonical analysis has shown that the state of the aquatic ecosystem can be mathematically described using activity of seston esterases`, as well as a complex of hydrochemical indicators. Accordingly, indicators of extracellular esterase activity can be used to assess the ecological state of aquatic ecosystems in environmental monitoring.

   Liquid level and temperature are employed in different industries such as nuclear power plant and power generator units. The water level and temperature have an essential role in nuclear and thermal power plants. A model predictive controller is a feedback control algorithm involving a model of a system to predict its future behavior, and it solves an online optimization algorithm to select the best control action that drives the predicted output to the reference. In this paper, a model predictive controller has been proposed to monitor the temperature and water level. The model predictive controller has two inputs from the temperature sensor and the water level. The system is based on a mathematical model for the task, and then a space model is developed. These systems with multiple inputs and multiple outputs are used as a testbed to find the optimal sequence of control inputs that drive the predicted plant output to track setpoint by changing the prediction horizon and control horizon.

   Sensitive elements of current sensors on surface acoustic waves based on magnetically sensitive FeNi films have been studied. It is shown that the use of such films provides a sufficiently high sensitivity to a magnetic field and allows refusing magnetic cores, which significantly reduces the weight and size parameters of the sensor and facilitates its installation on a conductive bus. The design of the current sensor with a FeNi film in the acoustic channel and the layout of the sensor on the conductive bus are shown. The influence of the location of the antennas on the sensor readings have been studied. It is shown that when the distance between the antennas changes, the resonant frequencies of the resonator surfactant do not shift. It is revealed that the ratio of the peak areas of primary and secondary reflections on the pulse responses of surfactant sensors does not depend on the distance between the sensor and reader antennas. This will eliminate the reference channel, which will simplify the design of the surfactant sensor.

   The multidimensional normal distribution of random variables is one of the main distributions in solving a large number of statistical problems. It is well known that the joint marginal distribution of a subset of random variables is also normal. In the literature, this fact is proved by finding the characteristic function for a given set of random variables and then finding the characteristic function for the subset of random variables and comparing the characteristic functions. Using this approach, it is relatively easy to prove that the subset of random variables belongs to the normal distribution. However, the density function of the multivariate normal distribution is not found explicitly, the vector of mathematical expectations and the covariance matrix of the distribution are not calculated. It would be interesting to obtain a rigorous proof that if the original set of random variables has a multidimensional normal distribution, then the subset of random variables of this set also has a multidimensional normal distribution with certain parameters. This work provides a rigorous derivation of the multivariate normal distribution density function for the subset of random variables. The vector of mathematical expectations and the covariance distribution matrix are calculated. The inference is based on block matrix operations. The presented formulas for calculating the inverse matrix, when the original matrix is presented in the form of blocks, are of independent interest.

   A modern method based on the wavelet transform has been proposed to clean diffractometric measurement data from noise. This method involves a multilevel one-dimensional discrete wavelet decomposition of the diffraction pattern and allows decomposing the source signal into approximating and detailing coefficients containing information with useful and noise components. The noise component of the diffraction pattern is mainly manifested in the detail coefficients obtained at the lowest decomposition level, to which threshold processing must be applied. This removes sufficiently small coefficients that are considered noise. The reconstruction of the diffraction pattern from the detail coefficients that have passed this processing will significantly reduce the noise level and, as a result, the localization error of the diffraction maxima. In order to obtain the best picture, it is necessary to use certain parameters for wavelet processing. In this work, the multilevel wavelet transform of the original diffraction pattern has been performed using real wavelets of various families with further analysis of the dependence of the processing quality on the choice of a basis. The efficiency of various algorithms for automatic threshold processing of decomposition coefficients in the MatLab software environment and the influence of the choice of thresholding parameters on the quality of cleaning are investigated. The results obtained are evaluated by comparing the standard deviation of the reconstructed and original diffraction patterns, as well as by comparing them visually. Examples of filtering diffraction patterns by the proposed method are given. In conclusion, the optimal parameters of thresholding for
processing diffraction patterns are given. The analysis of the displacement of the location of peaks on the processed diffraction pattern is performed. The detected peaks, which could not be localized on the original spectra, have been interpreted.

   The application features of the principal component analysis (PCA) in problems of electromechanical equipment diagnostics are studied. The sequence of analysis of diagnostic data presented in the form of time series is described by constructing a time series trajectory matrix followed by its singular value decomposition. The task is to determine the parameters of the PCA, which will provide the best depth of equipment diagnostics with the reduction of the probabilities of errors of the first and second kinds. To solve the problem, ergodic test signals of vibration of a rotating mechanism are synthesized: serviceable and with a kinematic pair defect. The test signals are processed using PCA and represented in the principal component basis. It is shown that the selection of the sampling characteristics and the parameters of the PCA should be carried out in such a way as to ensure the best separation of the serviceable and defective states of the examined mechanism in the principal component basis. The requirements for the required volume and sampling rate of the processed sample are justified. Recommendations on the choice of window length for the use of PCA have been developed. The effectiveness of the proposed approach is demonstrated by processing both synthetic and real signals. Using the example of vibration analysis of serviceable and defective bearings, it is shown that following the developed recommendations leads to a better separation of serviceable and defective states in the space of the principal components.