PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number
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Ivano-Frankivsk Department of Shevchenko Scientific Society
INFLUENCE OF BOTTOMHOLE ZONE TREATMENTS ON THE DURATION OF WATERED GAS WELL FLOWING
The problems of gas wells operation, in particular, the impact of waterflooding on their productivity, are characterized. The main analytical dependencies for determining the effect of bottomhole zone treatments on the duration of waterflooding of a watered gas well are presented. For the conditions of a hypothetical model well, the influence of the water factor, tubing diameter, and the degree of reduction of the coefficients of filtration resistances of the bottomhole formation zone (BFZ) on the conditions of waterflooding of a watered gas well was studied using analytical dependencies. The studies were carried out at different tubing diameters (0.0503, 0.062, 0.0759 m) and different degrees of reduction of the filtration resistance coefficients of the BFZ by 3 and 6 times. The research results are presented in the form of graphical dependences of the pressure at the wellhead on the water factor for different values of the tubing diameter and the degree of reduction of the coefficients of the filtration resistances of the BFZ. The generalized research results are summarized in the table. The duration of the fountain period of operation of a watered gas well was estimated by the maximum value of the water factor achieved when the pressure at the wellhead decreases to a given value (0.4 MPa), at which the natural fountain of the well stops. According to the research results, there is a clear tendency to increase the water factor and wellhead pressure with an increase in the degree of decrease in the coefficients of filtration resistances of the BFZ and a decrease in the diameter of the tubing. Depending on the degree of reduction of the coefficients of filtration resistances of the BFZ and the value of the internal diameter of the tubing, the duration of the fountain period of well operation increases by 1.09-1.45 times. The results of the studies performed on the example of a model well indicate the technological efficiency of BFZ treatment to extend the flowing period of a watered gas well. Keywords: well, operation, waterflooding, water factor, bottomhole zone, filtration resistance coefficients, fountaining.
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ГІДРОДИНАМІЧНИЙ АНАЛІЗ ТА ІННОВАЦІЇ СЕРВІСНОГО ОБСЛУГОВУВАННЯ НАФТОПРОВОДУ З ДЕФЕКТАМИ
The article presents an analysis of the hydrodynamic parameters of a pipeline with curvilinear shape defects caused by accidents, using modern methods of mathematical modeling. The Navier-Stokes equations, mass equations, and pressure equations were used to describe the oil flow through the pipeline. The objective of the study is to determine the impact of defects on the key hydrodynamic parameters of the pipeline, including flow velocity, pressure drop, hydraulic losses, and energy costs. A numerical solution algorithm has been developed, particularly for crack-like curvilinear defects, which affect the oil flow distribution in the pipeline. The modeling was conducted for a pipeline made of 12GSB material, with a diameter of 530 mm and a wall thickness of 7 mm, and a length of 40 km. The impact of different defect types (ranging from 0% to 10% of the cross-sectional area) on the hydrodynamic characteristics of the system was evaluated. The results show that defects significantly reduce the efficiency of pipeline operation, increasing hydraulic losses and altering flow velocity, which affects energy consumption. The article also proposes new scientific approaches to mathematical models of defects and methods for optimizing energy costs. Additionally, a concept for a monitoring system for the pipeline using unmanned aerial vehicles (UAVs) and innovative technologies is presented. UAVs are equipped with muon sensors to detect anomalies, as well as gas analyzers and lidars for precise data collection. Information from the sensors is transmitted to a computational system where machine learning algorithms detect potential leaks. Interaction with the user is provided through a chatbot that delivers notifications about anomalies. Innovative pipeline repair methods, including CIPP technology for restoring pipes without disassembly, laser-acoustic cleaning, and automated welding to enhance repair efficiency, are also discussed.
MATHEMATICAL MODELLING OF THE PHYSICAL PROPERTIES OF ENERGY CARRIERS IN THE BIOGAS PRODUCTION SYSTEM
The article discusses the topical issue of mathematical modelling of the physical properties of energy carriers in grain distiller's grains biogas production systems. The efficiency of biogas technologies largely depends on the accurate design and operation of systems for transporting raw materials and the biogas produced. The main objective of the work is to develop mathematical models of the key physical properties (density, dynamic viscosity, coefficient of volume expansion, specific isobaric heat capacity and thermal conductivity coefficient) of grain distiller's grains, industrial effluents and biogas. For grain distiller's grains, the models are developed based on laboratory studies and take into account the dependence of properties on temperature and mass concentration of dry substances. For industrial wastewater, a 4% caustic soda solution is taken as the base liquid. Modelling of biogas properties is based on its component composition. The developed models are the basis for performing thermohydraulic calculations, optimising the design and operation of pipeline systems in biogas plants, which contributes to increasing their reliability, safety and economic feasibility.
RESEARCH ON ENERGY CONSUMPTION IN GAS STORAGE IN UNDERGROUND STORAGE FACILITIES
The problem of estimating energy costs during gas storage in underground gas storage facilities is considered. For this purpose, it is proposed to use the concept of the efficiency coefficient of an underground gas storage facility, which is determined by the ratio of useful work performed to the total amount of energy consumed. This approach will allow analyzing the ways of using energy resources in the process of gas storage, assessing the efficiency of energy resources use and outlining areas for their savings. The useful energy of gas, which must be stored in the productive horizon of an underground gas storage facility during its injection and long-term storage, should be considered the potential energy of gas in the formation, while the energy losses of gas during injection and withdrawal from the storage facility should be attributed to unprofitable energy losses. Using mathematical models of energy efficiency of gas in the reservoir during storage and wasteful energy consumption in the processes of its injection from the storage and subsequent withdrawal, an expression was obtained that links the value of the efficiency coefficient of the underground gas storage with the technological storage parameters, which are considered as the volume of active and buffer gas in the storage, maximum and minimum reservoir pressures, as well as the operating parameters of the booster compressor station, including the efficiency coefficient of gas compression. The result makes it possible to forecast energy consumption for gas storage in a specific gas storage facility if there is information about the real state of the equipment and communications and the parameters of the operation mode of the booster compressor station during injection. However, of greater interest is the analysis of the obtained dependence, thanks to which it is possible to outline the ways of operating the underground gas storage facility with the lowest energy costs. Based on the analysis, it was found that an increase in the relative volume of buffer gas leads to an increase in the storage efficiency, and an increase in the ratio of maximum and minimum reservoir pressures leads to its decrease, since with an increase in reservoir pressure, the duration of the compressor period for gas injection into the storage will increase, which will cause an increase in the volume of fuel gas for the compression process.
ANALYSIS OF OPERATING CONDITIONS OF MAIN PIPELINES
Pipelines are the most economical way of transporting oil, oil products and gas from the places of production and primary preparation over long distances by the shortest route with the least losses to the areas of their use and processing. Significant service life and operational factors (vibration, static and dynamic load, damage to the insulating coating, subsidence of the soil, the occurrence of washed-out areas) cause fatigue and corrosion-fatigue damage to pipeline elements, which, in turn, leads to their destruction, causes significant economic losses and is a dangerous phenomenon from the point of view of safety and ecology. Ukraine has a developed network of pipelines for the transportation of natural gas and oil and product pipelines. Main pipelines are the largest underground structures, the operation of which depends on the characteristics of interaction with the environment. The location of pipelines in hard-to-reach places complicates their diagnosis, which increases the probability of failure [1]. Even slight deviations of the actual operating conditions from those assumed to be initial can bring the entire pipeline system to the limit state. Routes of main oil and gas pipelines pass through natural (lakes, swamps, rivers, etc.) and artificial (railway tracks, highways, etc.) obstacles. Routes of main oil and gas pipelines pass through natural (lakes, swamps, rivers, etc.) and artificial (railway tracks, highways, etc.) obstacles. Economically, pipelines are a reliable and environmentally safe way of transporting hydrocarbons. Their advantages are achieved thanks to good projects, high-quality equipment, materials and compliance with design parameters during construction and operation. However, these components do not always withstand, as a result of which accidents and failures occur on pipelines. Also, emergency situations are associated with their service life.
OBTAINING THE STIFFNESS MATRIX OF THE CARRYING CABLE OF A SINGLE-SPAN SUSPENSION PIPELINE
The paper considers the development of a discrete model of the carrying cable of a suspension pipeline, in which the suspension attachment points are simultaneously serve as the points for discretizing the cable parameters. To obtain the stiffness matrix of the carrying cable, the behavior of two processes was modeled: the sag of the supporting cable under its own weight and additional load. The discrete model was constructed in two stages. First, the values of vertical coordinates were obtained in the case of symmetrical loading. The second stage involved verification of the model and its extension to asymmetric loading. The application of numerical calculation methods based on a discrete model made it possible to obtain the coordinates of the cable's nodal points in two extreme positions - the initial one and under the action of five equal forces of 20 KN each, which were applied to the nodes. The linearization of the dependence “force-displacement of a knot” was carried out over the interval of change in the additional force load [0...20 KN]. This made it possible to determine the shares of each force in the total vector of node displacements. On the basis of these data the linearized matrix of pliability of flexible carrying cable with nonlinear characteristics of the system “pipeline - carrying cable” was finally obtained. An example of using the ductility matrix to determine the displacements of the cable nodes under asymmetric additional load is shown.
RESEARCH OF THE STRESS-STRAIN STATE OF PIPELINES EXPOSED TO THE INFLUENCE OF LANDSLIDE MASSIF ON TRANSVERSE SLOPES
Engineering preparation of the route in mountainous conditions includes measures, the implementation of which makes it possible to carry out the main types of linear works strictly according to the schedule with optimal labor and material costs and the use of the necessary machines and mechanisms. These include the implementation of anti-landfall and anti-lanslide measures, clearing the route from forest and bushes, arranging of shelves and notches, temporary exits from the main roads and entrances to the route, repairing and strengthening existing road surfaces, bridges and crossings, arranging of culverts and drainage structures, intermediate platforms for storage and installation of pipes, as well as construction of crossings over small rivers, streams and ravines. In order to correctly solve the issues that are part of the engineering preparation of the route, the construction organization project, which is issued by the design institute as part of the technical project, must necessarily take into account the materials of the route survey in kind. The survey must be carried out before the release of working drawings in conjunction with construction organizations. The survey materials should determine the optimal transport schemes for the delivery of pipes, materials and heavy construction equipment to the route, the volumes and methods of work performance to prevent landfalls and landslides, the volumes and costs associated with the reconstruction of existing roads and bridges, as well as the construction of access roads and culverts on them, the volumes and costs associated with the construction of assembly and storage platforms specifically for each difficult section of the route with ascent and descent, as well as the methods of work performance on these sections of the route.
IMPROVEMENT OF METHODS AND TOOLS FOR MONITORING THE CONDITION OF OIL AND GAS EQUIPMENT
This paper analyses current methods of non-destructive testing (NDT) for evaluating the technical condition of oil and gas equipment, focusing on strain gauge and ultrasonic techniques. Their advantages and limitations under complex geological conditions are reviewed. Improvements are proposed via temperature compensation in strain gauges and new piezoelectric transducer designs. The influence of thermal variations on transducer frequency characteristics is studied, and methods are proposed to stabilize acoustic coupling. Resonant frequencies of novel piezo plates and their ability to detect metal anisotropy are determined. A newly developed thickness gauge enables continuous monitoring of pipe walls, including oval cross-sections, which is essential for long-distance gas pipelines.
THE LATEST TECHNOLOGIES OF AUTOMATIC WELDING OF THE COMPANY CRC
Currently, welding is one of the most common technologies for joining metal parts in many branches of modern industry. Its wide application is due to a number of advantages, such as reducing the weight of structures, reducing the time for manufacturing products of complex shape, great opportunities for mechanization and automation of the process. Automation of the welding process is a promising direction of welding production. Automatic welding, today, has the best combination of such indicators as: the quality of finished welded joints, the productivity of the welding process, as well as the economic efficiency of application. In addition, the undoubted advantage of using automatic welding technology in shielding gases in the construction of main pipelines is the full automation of the welding process, which provides high productivity for a wide range of pipe sizes being welded in combination with high mechanical properties of welded joints. The introduction of an automatic welding method minimizes the amount of deposited metal, which, in addition to saving welding materials, reduces the time for manufacturing welding materials. The use of new technology minimizes the impact of the human factor on the welding process, and also facilitates the training of welders, which ultimately increases the mechanical properties of welded joints and ensures the stability of their production.
ANALYSIS OF ENVIRONMENTAL NOISE POLLUTION CAUSED BY GAS PUMPING UNITS IN BLOCK-CONTAINER DESIGN
Compressor stations play a crucial role in the technological processes of hydrocarbon energy carrier preparation and transportation. However, they are also significant sources of environmental pollution. The expansion of gas pipeline networks, particularly those located near residential areas, has created a serious ecological concern, as the noise generated by compressor stations contributes to environmental degradation. Exceeding permissible noise levels established by current sanitary standards adversely affects both the operational personnel of compressor stations and residents of nearby communities. Therefore, mitigating various types of noise produced by gas transport equipment is a critical social issue. This study aims to investigate environmental noise pollution caused by gas pumping units in block-container configurations. To achieve this goal, the research focuses on identifying the primary noise sources of such units and evaluating their environmental impact. Gas pumping units in block-container design are widely used for the transportation of natural gas, and it should be noted that they exhibit high levels of sound power, which negatively affect the environment. Consequently, noise reduction and improved working conditions during the operation of these units are particularly relevant. Effective implementation of noise mitigation measures requires a comprehensive understanding of the physical nature of key noise sources, as well as the mechanisms of sound generation and propagation throughout the gas pumping unit. Research findings indicate that intake and exhaust ducts, the walls of engine and compressor compartments, pipeline systems, and fans are the main sources of noise. An analysis of the acoustic characteristics of block-container units was conducted. The primary contributors to noise levels within industrial and residential areas are gas-air pathways, the container itself, and the associated pipeline systems. Enhancing the sound insulation of containers and pipeline systems represents a key strategy for mitigating noise emissions from gas pumping units.
FEATURES OF THERMAL-HYDRAULIC CALCULATIONS OF PIPELINES FOR BIOGAS PRODUCTION SYSTEMS
Hydraulic and thermal processes of pipeline transport in biogas production systems have been investigated. An algorithm for thermal-hydraulic calculation for slurry pipelines and sewer pipelines has been developed, which takes into account variable physical and chemical properties, heat exchange with the ground (determination of heat transfer coefficients, similarity criteria) and hydraulic resistance (based on Bernoulli's equation and Darcy-Weisbach's dependence). This allows predicting the temperature of the transported medium along the length of the pipeline, as well as the total head and pressure losses in the system at a given capacity. For biogas pipelines, where the flow is considered isothermal, the algorithm provides for the calculation of the energy parameter and iterative selection of the optimal internal pipe diameter to ensure the specified nominal productivity and the minimum permissible final pressure. Based on the calculations for manure pipelines and sewer pipelines, a reasonable selection of pumping equipment was made according to the criteria of productivity and required head. It was established that the hydraulic characteristics of the manure pipeline significantly depend on the mass content of dry substances, the average pumping temperature and the energy carrier flow regime.
STUDY OF THE STRESS-STRAIN STATE OF FOUNDATION ELEMENTS
The foundation is the basis of any construction. Currently, various types of foundations and methods of their construction have been developed. Modern construction is faced with complex geological conditions and limited building space, which requires the use of the latest foundation construction technologies. To replace the existing deep-buried concrete foundations, the Pin Pile foundation is becoming widespread - it is a promising alternative, especially in cases where the use of heavy equipment for the delivery of materials and the construction of traditional foundations is impossible or economically inexpedient. However, it is necessary to study and research promising Pin Pile foundations in order to adapt them to specific operating conditions. For this, it is advisable to use simulation modeling based on the finite element method. To establish the characteristics of the foundation constructed using the Pin Pile technology, its three-dimensional model was built. To simplify the simulation modeling, the calculation scheme and the relationships of the structural elements were adjusted. As a result of simulation modeling, the distributions of equivalent stresses and displacements in the studied foundation model were obtained. According to the distribution of equivalent stresses, their maximum values ​​are concentrated in the piles at the exit from the concrete head pipes. This is explained by the boundary conditions and the principle of operation of the foundation. The maximum value of equivalent stresses is 53 MPa, which is not critical for the selected material of the studied piles. As for the displacements observed in the studied foundation model, their maximum value is 0.4 mm. The value of the contact pressure on the “pile-pipe” surfaces was also determined. The maximum value of this pressure is 6.5 MPa, which, given the properties of the materials from which the pipes and piles are made, is acceptable. In further studies, it is planned to establish the influence of soil properties on the stress-strain state of the foundation and to improve it in order to increase its operational characteristics. The coefficient of friction between the foundation elements and the soil will also be taken into account.
ON THE BOUNDEDNESS OF THE L-INDEX IN DIRECTION OF ENTIRE SOLUTIONS OF THE GOLDBERG-STRELITZ EQUATION
For some linear first-order partial differential equation there is proved the boundedness of the $L$-index in the direction $\mathbf{b}=(b_1,b_2)$ of all its entire bivariate solutions with bounded multiplicity of zero points. The function $L$ is defined as the modulus of the product of the derivative in the direction $\mathbf{b}$ of the function $zw$ on the function $l(zw)$, and an entire solution is represented by the product of the exponential function $e^z$ and an arbitrary entire function $f(zw)$ such that $f: \mathbb{C}\to\mathbb{C}$ has bounded $l$-index.
ON THE STABILITY OF THE MAXIMUM TERM OF FUNCTIONAL SERIES IN A SYSTEM OF FUNCTIONS
By $L_+$ we denote the class of positive continuous on $\mathbb{R}_+:=[0,+\infty)$ functions $l(t)$ such that $l(t) \uparrow +\infty$ $(t\to +\infty)$, and by $\mathcal{W}$ the class of functions $w\in L_+$ such that $ \int\nolimits_{1}^{+\infty}{x^{-2}{w(x)}dx}<+\infty. $ The article deals with the functional series of the form $F(x)=\sum\nolimits_{k=0}^{+\infty}a_kf(x\lambda_k),$ where $\Lambda=(\lambda_k)$ is some sequence of non-negative numbers, $a_k\geq 0$ $(k\geq 0)$, and a positive increase to $+\infty$\ on $[0;+\infty)$\ function $f$ is such that $f(0)=1$\ and $\ln f(x)$\ is a convex on the interval $[0;+\infty)$\ function. Let us denote $F_w(x)=\sum\nolimits_{k=0}^{+\infty}a_ke^{w(\lambda_k)}f(x\lambda_k),$ $$ \nu_0(t)=\nu\{u\geq 0\colon \ln f(u)\leq t\},\quad \nu(G)=\sum\nolimits_{\lambda_{n}\in G}{e^{w(\lambda_{n})}} $$ for every bounded set $G\in\mathbb{R}_+$, where $w\in L_+$. The main result the paper is the following statement: If there exists a function $w\in L_+$ such that $a_ne^{w(\lambda_n)}f(\lambda_nx)\to 0,$ for every $x>0$,\ $\ln\nu_0\in \mathcal{W}$, then there exists a set $E\subset \mathbb{R}_{+}$ of finite Lebesque measure such that the asymptotic relation $\ln \mu(x,F)=(1+o(1))\ln \mu(x,F_w)$ holds as $x\to +\infty$ outside the set $E$, where $\mu(x,F)=\max\{a_kf(x\lambda_k)\colon k\geq 0\}$.
SYMMETRIC POLYNOMIALS ON CARTESIAN PRODUCTS OF REAL BANACH SPACES OF ABSOLUTELY SUMMABLE SEQUENCES
Let $\ell_p^{(\mathbb{R})}$ be the real Banach space of all absolutely summable in the power $p\in [1,+\infty)$ sequences of real numbers. For $n\in\N,$ let $\ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ be the Cartesian product of Banach spaces $\ell_{p_1}^{(\mathbb{R})},\ldots, \ell_{p_n}^{(\mathbb{R})},$ where $p_1,\ldots,p_n\in [1,+\infty).$ We consider the space $\ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ as the space of all sequences $x= (x_1,x_2,\ldots)$ of $n$-dimensional vectors $x_j = \bigl( x_j^{(1)},\ldots,x_j^{(n)} \bigr),$ where $j\in\mathbb{N},$ such that the sequence $\bigl( x_1^{(s)}, x_2^{(s)},\ldots \bigr)$ belongs to the space $\ell_{p_s}^{(\mathbb{R})}$ for every $s\in \{1,\ldots,n\}.$ A function $f$ defined on the space $\ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ is called symmetric if $\displaystyle f\bigl( \bigl( x_{\sigma(1)},x_{\sigma(2)},\ldots \bigr) \bigr) = f((x_1,x_2,\ldots)) $ for every $(x_1,x_2,\ldots) \in \ell_{p_1}^{(\mathbb{R})}\times\ldots\times \ell_{p_n}^{(\mathbb{R})}$ and for every bijection $\sigma:\mathbb{N}\to\mathbb{N}.$ We show that the set $\displaystyle \left\{ H_{\boldsymbol{p},\boldsymbol{k}}^{(\R)}:\; \k\in\mathbb{Z}_+^n\;\textrm{such that}\;k_1/p_1+\ldots+k_n/p_n \geq 1 \right\} $
THE WARING-GIRARD FORMULAS FOR SYMMETRIC AND SUPERSYMMETRIC POLYNOMIALS AND THEIR APPLICATION
У статті досліджується застосування формул Варінга-Жирара до симетричних та суперсиметричних поліномів у контексті банахових просторів. Окреслено ключові алгебраїчні базиси симетричних функцій, включаючи степеневі, елементарні та повні симетричні поліноми. Розглянуто комбінаторні тотожності та розширення класичних формул для нескінченновимірних структур. Запропоновано аналог формули Варінга для суперсиметричних поліномів, що спирається на ізоморфізми між відповідними алгебрами. Дослідження має потенційне застосування в теорії інваріантів, комбінаториці та математичній фізиці.
MULTIPLICATIVE CONVOLUTION FOR THE BLOCK-SYMMETRIC POLYNOMIALS
This paper investigates the concept of multiplicative convolution for block-symmetric polynomials defined on infinite-dimensional spaces of the form $\ell_p(C^s)$. A multiplicative shift operation and the corresponding convolution operator are developed for the algebra of block-symmetric analytic functions of bounded type. It is shown that the defined convolution is both commutative and associative, forming a commutative semiring with unity on the spectrum of the algebra. Estimates for the spectral radius function of homomorphisms are provided, and essential properties such as linearity, continuity, and compatibility with the algebraic structure are established. The construction of spectral elements with prescribed values on elementary polynomials is proposed. This work generalizes earlier results for the $\ell_1(C^s)$ case to the broader setting of arbitrary $1 < p < \infty$, contributing to the theory of symmetric analytic functions in infinite-dimensional analysis.
The article considers a degenerate impulse boundary value problem for systems of ordinary differential equations with a matrix degenerate on the entire segment where the problem is considered, with respect to the derivative, for the case when the number of unknowns does not coincide with the number of boundary conditions. A criterion for the existence of solutions to a degenerate impulse boundary value problem is obtained, assuming that the degenerate differential system reduces to the central canonical form. For the problem under consideration, the conditions for the existence of solutions to a degenerate impulse boundary value problem are reduced to the solution of the algebraic system.
Investigation of differential pursuit games under perturbation of the motion equation
The article investigates differential pursuit games in conditions where the movement of the pursuer and the escaper is affected by external small disturbances. Similar problems but without taking into account perturbations were considered in the works of A.O. Chikrii. In this article the problem is considered in the case where the control matrix changes slowly over time and is affected by multi-frequency disturbances. The averaging method is justified in the case of slowly changing frequencies. Estimates of the deviation were constructed between the solutions of the exact and averaged systems under the same initial conditions on a finite time interval. An example of a modified differential game "Crocodile and Boy" with imposed disturbances is given. The pursuit completion time for the averaged model is found. The influence of disturbances with different strength on the game completion time is analyzed. The obtained results can be applied in various control problems, where it is important to take into account the influence of external factors on the behavior of participants in dynamic processes.
LEMMA ON THE DETERMINANTAL RELATION OF SEQUENCES
In The On-Line Encyclopedia of Integer Sequences (OEIS), relationships between numerical sequences are frequently cited, although these connections mostly arise spontaneously. Often, the source of such relationships lies in second-order, and less frequently third-order, linear recurrence sequences. This article considers numerical sequences associated with infinite linear recurrence relations and proves a lemma on their determinantal relation. To this end, matrices are introduced that generalize the Hessenberg–Toeplitz matrices. These matrices are associated with unordered partitions of a natural number into natural summands. This general, matrix-based approach to the study of sequences naturally establishes a bijection between mutually conjugate partitions of a natural number into natural summands and provides a framework for identifying connections between sequences associated with linear recurrence relations. The article also presents important corollaries of the lemma and illustrates them by examining the connection between the $n-$th term of the Pell number sequence and the Fibonacci sequence, as well as their principal determinants.
PAR-FUNCTIONS OF TOEPLITZ MATRICES AND PARTITION POLYNOMIALS
This paper explores certain properties of the par and par^{+} functions of Toeplitz matrices. These functions are studied in tandem due to their many shared properties. Since the combinatorial foundation of these functions lies in ordered partitions of a natural number into non-negative integers, it becomes possible to represent them as partition polynomials and to construct recursive algorithms for their computation. In addition to a brief introduction to these functions, the paper presents a recurrence relation for computing the par-functions of Toeplitz matrices, which enables the unification of a broad class of linear recurrence relations. As linear recurrence relations are often related to partition polynomials, the representation of these functions as partition polynomials is also studied. The article includes an example that utilizes the fact that the multilinear polynomials of par^+ and par-functions of square matrices contain 2^(n-1) terms, with the par-function comprising half positive and half negative terms. Two combinatorial identities are derived using a Toeplitz matrix whose entries are all equal to one.
PERTURBATION OF AN α-STABLE STOCHASTIC PROCESS AND THE FUNDAMENTAL SOLUTION OF A CERTAIN PARABOLIC PSEUDODIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS
Using the method of perturbation of a rotationally invariant alpha-stable stochastic process, an explicit representation of the fundamental solution of the pseudodifferential equation \(\frac{\partial}{\partial t}u(t,x)=-c(-\Delta)^{\alpha/2}u(t,\cdot)(x)+ (a,\nabla_\beta u(t,\cdot)(x)), t>0, x\in\mathbb{R}^d\), where \(\Delta\) is the Laplace operator, \(\nabla_\beta\) is the "pseudogradient" operator of order \(0<\beta<\alpha\) (given by the symbol \(i|\lambda|^{\beta-1}\lambda\)). The case of fixed \(0<\alpha<2,\) \(c>0\) and constant \(a\in\mathbb{R}^d\) is considered.
METHODOLOGY FOR PROCESSING THE RESULTS OF CHANGES IN TECHNOLOGICAL PARAMETERS OF DRILLING COMPLETED OIL AND GAS WELLS
Modern drilling rigs, such as the Ultra Single 150 drilling complex developed by the Canadian company Sense EDM, are equipped with control and measuring instruments. The monitors are located in the drilling master's control room. Their use allows for recording discrete values of all technological parameters of the drilling process with a sampling step of 1 m of well depth. These values are stored for completed oil and gas wells in the appropriate data bank on electronic media. Fixing changes in drilling technological parameters allows for analyzing the impact of using modern technologies on the rock destruction process. To conduct such an analysis, it is necessary to select homogeneous areas of well deepening according to the accepted drilling criterion, for example, according to the change in mechanical drilling speed. The authors have developed a method for processing the results of changes in technological drilling parameters of completed wells. The program App.2 is proposed for determining homogeneous borehole sections of the well, which is developed in the Visual Studio software environment in the C# programming language (C Sharp) using the Windows Form Application framework using Windows operating systems. The peculiarity of this technique is that for homogeneous drillability sections of the deepening, the distribution law and statistical characteristics are determined for the main technological parameters, which include the mechanical drilling speed (drillability criterion), axial static load on the bit, the torque of the installation rotor, and the rotation speeds of the rotor and bit. Belonging homogeneous well-depth sections to one general population is determined by the testing results statistical parametric hypotheses for all the above technological parameters. If at least one parametric hypothesis about the equality of variances and mean values of the compared samples of discrete values of these parameters is rejected, the hypothesis about the belonging homogeneous linear sections to the well interval with the same physical and mechanical characteristics of the rock is rejected.
FORCE LOAD ON MINE TRAWL ASSEMBLIES DURING EXPLOSION
Heavy armoured vehicles used as a mine trawl vehicle are subjected to high loads from the shock wave during demining. The study of mechanisms for reducing the force load on mine trawl components during an explosion is relevant, of scientific and practical value. Therefore, the aim of the work is to reduce the force load on a mine trawl vehicle in the event of an explosion, and to make it possible to use existing motor-tractor equipment of less mass as vehicles for mine trawls. In the actuating mechanism, the mine trawl roller is rigidly connected through a rocker arm to the comb, which, when exploded, enters the ground with spikes, creates a rigid pinching resistance, while reducing the force load on the vehicle. The force load on the comb spike was determined, depending on the mechanical characteristics of the soil, which made it possible to determine the number of spikes to obtain the required load reduction on the mine trawl components. To reduce the maximum peak impact load on the vehicle at the time of explosion, the installation of an elastic shock absorber between the actuator and the vehicle is justified. The obtained results make it possible to reduce the power load on the mine trawl components in order to use existing motor-tractor equipment of lower mass as vehicles.
On Bending of Transversely Inhomogeneous Plate with Periodic System of Collinear Contact Cracks
The bending problem of functionally graded plate with periodic system of collinear through cracks with contact interaction of their faces, which arises due to non-uniform stress distribution through the plate thickness, is considered. Based on the classical Kirchhoff plate bending theory and special contact model along a line, which allows to eliminate the kinematic contradiction of mutual penetration of crack faces, the boundary value problem for a pair of biharmonic equations with interrelated conditions on cuts is formulated. Using the method of singular integral equations with periodic Hilbert kernel, an exact analytical solution of the problem in closed form is constructed. The influence of material inhomogeneity profile and mutual arrangement of cracks on stress and moment intensity factors, as well as on limit load value determined by energy fracture criterion, is studied in detail. Numerical analysis is performed for the specific case of inhomogeneity with quadratic distribution of elastic modulus through the plate thickness, described by the second-order Legendre polynomial, which models both plates with stiffer periphery and with rigid core.