Applied Questions of Mathematical Modelling
AN AGENT-BASED MODEL FOR PERSONALIZED LEARNING IN NETLOGO USING Q-LEARNING
This study investigates the effectiveness of using the Q-learning reinforcement learning algorithm in the context of personalized student learning by constructing an agent-based model in the NetLogo simulation environment. The proposed model assumes the existence of a population of student agents, each possessing individual characteristics, especially a distinct learning style (visual, auditory, or kinesthetic). The learning environment is modeled as a grid of patches, each containing educational content of a particular type (textual, video, or audio) and a specified difficulty level.Agents interact with the environment by making decisions to either stay on their current location or move to another patch, based on an ε-greedy action selection strategy. The Q-learning algorithm enables each agent to form and continuously update its own Q-table, which stores the utility estimates of various actions in different states. The reward is determined by the degree of alignment between the agent’s learning style and the content type. If the style matches the content format (e.g., auditory style and audio material), the agent receives a positive reward; otherwise, a negative one.The results of a series of simulations showed that agents gradually learn to avoid mismatched content and begin to prefer patches that correspond to their perception style. An increase in average reward, a growing proportion of agents with positive outcomes, and a decrease in the frequency of random movements over time were observed. This indicates the formation of a stable learning policy and effective adaptation to the environmental conditions. Moreover, the influence of Q-learning parameters (ε, α, γ) on learning success was analyzed, allowing for the identification of optimal configurations for achieving the best performance. The obtained results confirm the relevance of using reinforcement learning approaches to model personalized educational strategies. The proposed model can be used as a foundation for the development of intelligent learning support systems capable of adapting the educational process to individual learners’ needs and traits.
RESEARCH OF TEMPERATURE CONTROL SYSTEMS IN THE HEAT SUPPLY SYSTEM OF INDIVIDUAL HEAT CONSUMERS
The article considers the problem of efficient heating control in individual residential buildings using mathematical modelling. Heating systems in modern conditions should take into account not only the efficiency of thermal devices, but also energy saving, which is especially important against the background of rising energy prices and the need to improve energy efficiency. The main goal of this work is to rationalise the distribution of heat load among several consumers, including heating devices, underfloor heating and hot water supply, which minimises energy costs and improves living comfort. The main objective of this work is to develop a mathematical model that allows for efficient control of the heating system of an individual house, taking into account the interaction of different heat sources and energy distribution among consumers.The mathematical model proposed in this paper describes the heat balance of a room through the models of two heat exchangers that take into account the conditions of heat exchange, hot water consumption and various methods of temperature control. The applied differential equations allow us to describe the heating and cooling processes in the room, as well as the water temperature in the heating devices. The simulation model allows us to test the effectiveness of different approaches to temperature control and draw conclusions about the optimal methods of heat load management. This makes it possible to simulate the temperature dynamics in a building with different types of control – quantitative and qualitative. The two approaches to temperature control are considered and the transient processes for each method are compared using a simulation model. Evaluation of transient processes using the simulation model has shown that the qualitative control system allows for greater temperature stability with less deviation compared to the quantitative control, which makes it more efficient for automated control systems.The proposed model makes it possible to increase the energy efficiency of heated buildings by rationally distributing heat across the premises, and can be used to develop intelligent heating systems in conditions of high energy costs.
METHODS OF ESTIMATING THE SOFTWARE DEVELOPMENT BUDGET
Software is currently used in all areas of human activity and is critical for many industries. Software development is a complex and resource-intensive process, the development team can have several dozen employees. There are also many risks during program development, such as inconvenience between the customer and developers, unnecessary expenditure of resources on fixing errors and exceeding the available time to meet the project requirements and the development budget, and as a result – the development failure and even the closure of the project.To avoid the worst-case scenarios in software development, various methods for estimating the development budget are continuously improved. These methods are based on different concepts and ideas, taking into account various aspects and parameters of the project. The paper analyzes existing estimating methods and examines the advantages and disadvantages of their practical use.After the analysis, a software product was developed for calculating the cost of program development, which used the parametric evaluation method with the heuristic coefficients. The parametric method takes into account each project parameter, such as development stages, development team, time for development of each stage of the task, availability of a development budget margin, to prevent budget exhaustion in case of unforeseen situations, such as equipment failure or involvement of additional developers. The use of the heuristic coefficients adds realism to the calculations due to the introduced coefficients of developer productivity and the complexity of the stages of software development.As a result of using this method and development experience from previous projects to determine the complexity of the stages and the productivity of developers depending on their experience qualitative calculations of the possible cost of software development were obtained. The results are displayed in the form of a graph and saved in a text file for a more detailed analysis of the cost of each specified stage of the software development project.
Games in today's world have evolved beyond mere entertainment to become a powerful medium of cultural expression, social interaction, and even personal development. With the rapid advancement of digital technologies, the gaming industry has transformed into a multi-billion-dollar global phenomenon with immense influence across various domains of society. From casual mobile apps to complex, immersive open-world experiences on PCs and consoles, games now encompass a vast range of genres, formats, and narratives. They are not only a source of fun but also a tool for developing creativity, strategic and critical thinking, teamwork, and problem-solving skills. Moreover, they are increasingly integrated into education, therapy, and professional training. Developing a game application involves a complex, multi-phase process, including pre-production, production, quality assurance, release, and post-production support. Each phase requires detailed planning and the use of modern tools, programming languages, game engines, and collaborative systems to ensure the reliability and appeal of the final product. This paper presents the development of a 2D adventure RPG game titled “Little Adventures in Balaburs”. The application immerses the player in a fictional virtual world with its own narrative, characters, dialogue systems, and quest structure. The project was developed using the Unity game engine and the C# programming language, which enabled the implementation of interactive gameplay mechanics, a user-friendly interface, and dynamic in-game events. Throughout the development process, key game elements were implemented, including the creation of a Game Design Document (GDD), the visual and narrative design, dialogue trees, quest logic, and audio components. The application serves as an example of a complete indie game development cycle – from concept to a playable prototype. This experience provides valuable insight into the structure of game software, the practical use of programming within game engines, and the multidisciplinary nature of game design and development.
AUTONOMOUS UAV NAVIGATION: TECHNOLOGIES FOR ORIENTATION AND LOCALIZATION
Navigation of autonomous vehicles and particularly unmanned aerial vehicles (UAV) remains one of the core challenges in achieving full autonomy. Autonomous vehicles must independently analyze an environment, react on the dynamic change of surroundings, create optimal trajectories, and perform mission-critical tasks independently. That is why drones need to precisely understand their pose (position and orientation). While in case of error, ground and marine drones might stop the performance, the error of UAV has significant consequences due to the high working speed and heights. It may lead both to losing UAV and harming infrastructure, nature, and civilian houses in accordance to the area, where the UAV is operating.Localization and orientation issues require information from both the state of the drone and information from the environment, as the vehicle’s position in relativeness with other objects is necessary for path planning tasks. This article analyzes the sensors and navigation systems that might give convenient data about both inertial parameters and surroundings. Solutions for both known, partially known, and unknown environments are described. Special attention is paid to Visual SLAM (Simultaneous Localization and Mapping) and Visual-Inertial Odometry, which are key for map creation and relative movement estimation in unknown environments. These approaches enable drones to navigate in GPS-denied or dynamic environments by fusing visual data with inertial measurements.This article also gives a comprehensive review of the map-based sensors, which greatly simplify the UAV performance in pre-known environments. Map-based sensors are applicable for both indoor and outdoor scenarios.The final section of the article introduces the sensor fusion technics, which are necessary for all kinds of UAVs to reduce drifting error and improve navigation accuracy. It enables the comparison of measurements from different sensors or navigation systems with two levels of filtering. Selecting an observer must be done with consideration of the nonlinear nature of autonomous vehicles.
CYBERSECURITY IN THE MARITIME INDUSTRY: THREATS, RESPONSES, AND INCIDENT MANAGEMENT
This article addresses the issue of cybersecurity in the maritime sector amidst the increasing digitalization of mar- itime infrastructure. Recent research and publications on cybersecurity in the shipping industry indicate an increase in threats due to the digitalization of the maritime industry. It emphasizes the importance of implementing effective cyber incident response plans to ensure business continuity and reduce risks associated with cyber threats. The implementation of international standards, scenario testing, and agile process management all form the foundation for cyber resilience in the maritime sector.The purpose of the study is to develop a model for building an effective action plan and response plan for cyber incidents using modern project management technologies and based on data protection standards. As authors, we see the following main tasks: building an action plan to ensure the cybersecurity of the ship based on governing documents, the experience of leading specialists, and an analysis of achievements in the field of cybersecurity on seagoing vessels; implementing the project into the education system of the relevant direction and testing it during the cadets’ swimming practices; further using it for operational training of crew members on ships. Currently, an agreement has been reached to include the built plan in the program of the educational component “Cybersecurity of ship computer networks and systems” of specific educational institutions, such as the Kherson State Maritime Academy.Existing approaches to incident response were analysed, and the possibilities of adapting project management to increase the effectiveness of such measures were investigated. Another goal of this study is to improve the plan for practical training of future shipowners in ensuring cybersecurity on a seagoing vessel by modernizing and changing the content of the educational component. A structured approach to creating a cyber incident response plan focused on the specifics of threats in shipping is proposed. The study’s practical significance lies in the possibility of using the results obtained by shipping companies and port authorities to train personnel by the requirements of international standards.
MODELIZATION OF MARITIME TRANSPORT SCHEMES OF UKRAINE IN MODERN CONDITIONS BASED ON MATHEMATICAL ANALYSIS
Based on the analysis of the current state of maritime transport in Ukraine, new options for cargo delivery schemes have been developed, knowing the starting and ending points. Transportation routes using several modes of transport have been described. Ukraine’s proposal to amend the existing traffic distribution systems “Approaches to the ports of Odesa and Illichivsk” and “Between the ports of Odesa and Illichivsk” by combining them into a single traffic distribution system “Approaches to the ports of Chornomorsk, Odesa and Pivdenny” has been considered. The updated traffic distribution system establishes the safest routes in the deepest waters for vessels heading to and from the Ukrainian ports of Chornomorsk, Odesa, Pivdenny, as well as towards the ports of Mykolaiv and Kherson. Traffic distribution systems are designed in such a way as to simplify the traffic flow patterns in the convergence/divergence zones and to organize the movement of vessels entering/exiting three ports – Chornomorsk, Odesa and Pivdennyi in the most optimal way.Analysis of the existing theory of mathematical analysis suggested that in our case the most optimal is the transport gravity model. It describes the interaction between spatial objects in the form of a mathematical analysis of transport flows, it is based on an analogy with Newton’s law of universal gravitation, which connects the force of attraction between two bodies of different masses and the distance between them.The developed transport gravity model connects the intensity of the transport flow with its volume of shipments, or movements from one area of a certain volume of shipments that have reached the destination area with the costs of movement.The conducted research will allow to use the gravity model and the correspondence calculation apparatus to develop a simulation model of cargo turnover in the maritime transport system of Ukraine. The methodology used opens up new opportunities for solving a group of forecasting tasks – such as determining the optimal number of vessels, determining the busiest port on the line, the number of transported cargoes, and delivery time.
MATHEMATICAL MODELS FOR DETECTING OUTLIERS IN TWO-DIMENSIONAL CBO AND RFC METRICS’ DATA OF OPEN-SOURCE KOTLIN-BASED APPLICATIONS
Modern software development requires robust methods for assessing the quality and complexity of object-oriented design. Such evaluation is typically performed using software metrics. However, this process usually requires metric data processing, including outlier detection. In real-world scenarios, metric distributions deviate from normality in the vast majority of cases. This reality necessitates the development of specialized mathematical models that account for such non-standard behavior of software metric data.The study proposes mathematical models for more reliable detection of outliers in two-dimensional CBO and RFC metric data specifically for Kotlin projects. Analysis of 102 open-source projects revealed substantial differences com- pared to Java: the average CBO value for Kotlin was only 2,20 (versus 11,88 for Java), and RFC – 7,46 (versus 18,36).Mardia’s test confirmed that the distribution of these metrics for Kotlin significantly deviates from normal (β₁ = 70,867, β₂ = 14,066 with critical values of 9,3 and 9,48).To solve this problem, we developed a model that includes: application of Box-Cox normalization transformation with parameters and implementation of two complementary approaches for outlier detection (based on χ²-distribution quantile and based on Fisher’s F-distribution quantile).The results demonstrate that our models overcome the key limitation of Java-oriented approaches, where up to 25% of Kotlin projects were misclassified as outliers. The refined models show greater reliability in analyzing Kotlin CBO and RFC metrics while preserving all advantages of the prediction ellipsoid approach. Notably, the χ²-criterion model proved more stringent, enabling more accurate outlier detection. This research provides practical value for Kotlin developers by offering specialized tools for CBO and RFC metric analysis. Our findings confirm the necessity of developing language-specific analytical models that account for unique metric distribution characteristics.
SOLVING APPLIED PROBLEMS USING BASIC ALGORITHMIC PRIMITIVES OF GRAPH THEORY
The possibility of formulating applied problems in terms of graph theory is considered. In particular, the problems of the optimal structure of the enterprise, control over financial operations, and determination of the optimal sequence of works are considered. An approach to solving such problems using algorithmic primitives on graphs is implemented. The use of graphs as models of many applied problems is due to both the clarity of the concept of a graph and the availability of exact algorithms on graphs with polynomial complexity.The process of solving the problem of constructing an optimal structure sequentially includes the calculation of strongly connected components on the graph of the problem, the construction of a graph of components, and topological sorting of the graph of components in order to transition to a linear structure. The optimal structure of the enterprise is understood as a linear ordering of its subdivisions. The method of solving the problem of control over financial operations is based on the properties of depth-first search for an acyclic graph. The classification of edges of a depth-first search forest allows us to make certain assumptions about hidden connections in the presence of reverse edges. It is worth noting that the proposed method does not give an answer about the quantitative nature of the connection, fixing only its fact. A partial case of the general linear programming problem is the problem of determining the sequence of work execution, given by a system of difference constraints. Solving a system of difference constraints is reduced to calculating the weight of the shortest path in the corresponding constraint graph. To calculate the shortest path from one vertex, the Bellman–Ford Algorithm was used, the application of which allows obtaining a result even in the presence of edges of negative weight.The correctness of theoretical considerations is confirmed by practical calculations using specially developed software. The calculations were performed in the Mathcad v. 13 environment. The corresponding procedures STRONG_ COMP, DFS_CTRL, BELL_FORD_CONST were developed.
APPLICATION OF THE INTEGRAL EQUATION IN THE SOLUTION OF ONE NONLINEAR BOUNDARY VALUE PROBLEM
This paper considers a mathematical model of the temperature field of an isotropic spherical area, rotating around one of the symmetry axes at a constant speed in the form of an initial-boundary-value problem for a parabolic-type equation with nonlinear boundary conditions. The purpose of the work is to apply the method of integral equations to the solution of a nonlinear boundary value problem in the spherical domain, to determine the conditions of thermodynamic equilibrium. A periodic problem is considered in one of the spatial coordinates, which is reduced to solving a nonlinear integral equation. The obtained integral equation is reduced to a Fredholm-type equation along the spatial coordinate using the second Green’s formula, and to a Volterra-type equation along the time coordinate. The kernel of the integral equation is found in the form of a Green’s function. The Green’s function is constructed using the properties of the Laplace operator in a spherical coordinate system, as well as the Legendre and Bessel functions of half-integer order. The linear part of the solution function of the problem both inside the sphere and on its surface was found using the properties of the Gamma function. To determine the conditions of thermodynamic equilibrium of a sphere with an unlimited number of its rotations around its axis, a limit transition was applied, as a result of which the conditions of thermodynamic equilibrium were determined. The solution of the nonlinear part of the problem for determining the periodic quas-stationary temperature field was obtained by solving the corresponding nonlinear integral equation of the Hammerstein type. It was also proved that the solution of the original problem, periodic both in time and in the spatial coordinate, does not depend on the initial condition. This condition does not affect the quasi-stationary temperature field of the sphere with an unlimited number of its rotations.
MODELING PLANAR ELECTRON BEAM TRAJECTORIES IN THE MAGNETIC FIELD OF MAGNETRON GUN
The paper considers the motion of electrons in the cylindrical magnetic field with variable strength along the main axis of the field. A numerical model for the distribution of the magnetic field of the magnetron gun, as well as an algorith- mic transformation of the experimental field data by analytical functions are used. The object of the study was to construct the model of the motion of an electron beam with an energy of tens of keV in gradient magnetic field, to study the formation of electron trajectories, the dependence of particle dynamics on the initial conditions and the distribution of the magnetic field along the axis of the system. The purpose of the work is to obtain precision characteristics of the electron beam with energy of 20–30 keV in the longitudinal and radial directions during its transportation in the disturbed magnetic field of the solenoid of the magnetron gun. An algorithm was synthesized, on the basis of which the software tool was built, the use of which made it possible to simulate the main dependencies of the electron beam motion in the given solenoidal magnetic field. The paper presents the results of numerical modeling of electron trajectories in the main magnetic field of the gra- dient-type magnetron gun with secondary-emission cathode. The formation of the beam with an energy of 20–30 keV in the working volume of the gun during its transportation in the solenoidal magnetic field with large gradient is considered.The main characteristics of the electron beam were obtained when performing modeling calculatio1ns. The modes of the particle beam during their movement along the transportation axis were determined. The dependence of the formation of the resulting distribution of particles in the magnetic field along the system axis was studied. The root-mean-square spread of electrons along the vertical was chosen as the quality criterion. The dependence of the longitudinal spread of particles on the distance traveled along the system axis was obtained.
CORRECTION OF AXIAL DISPLACEMENT ERRORS OF A DIGITAL MAGNETOMETER USING LABVIEW
When using a three-axis MEMS (Micro-Electro Mechanical Systems), digital magnetometer, in a working electro- magnetic environment, it is necessary to calibrate this device. In our case, the GY-273 module on the QMC5883L chip is used. For this, you can use several programs that allow you to calculate correction factors that take into account two main types of interference – hard-iron and soft-iron distortions. The most common method of calibrating a magnetometer is ellipsoid fitting – rotating the sensor in all directions to obtain a sufficient amount of data that is placed on the surface of the spheroid. Correction factors allow you to transform the spheroid into a regular sphere with the center at the origin.But using these calibration methods, for example, using the Magneto program, does not always give the desired result.The measurement error can remain quite significant and requires additional correction. When performing calibration measurements under conditions of external magnetic influences, situations arise when the coordinate axes in the space of measurement data are not mutually perpendicular and do not have a common intersection point. At the same time, the measured values of the components coming from the sensor are separate vectors that have their own deviations from the axes, both relative to the origin of coordinates and relative to the angles of inclination. In this work, dependencies are proposed for calculating correction matrices that take into account the features of the location of the measuring axes in space. Unlike existing correction methods, the shifts of the centers of the segments of each axis along each coordinate are taken into account, i.e. the shift matrix has a dimension of 3 × 3. This allows you to influence the measurement error. A set of virtual devices has also been developed for visualizing the calibration process, obtaining correction matrices based on the obtained analytical dependencies and processing magnetometer data in real time according to the proposed algorithm. The developed complex can be used in the educational process for laboratory practice for calibrating three-axis sensors and visualizing the three-dimensional measuring space.
THE IMPACT OF AUTOMATION AND ARTIFICIAL INTELLIGENCE ON EMPLOYMENT, WAGES, AND PRODUCTIVITY
The modern labor market is undergoing significant transformations under the influence of automation and artificial intelligence (AI) technologies. These changes impact employment structure, wage levels, and overall labor productivity.While automation creates new opportunities for increasing production efficiency, it also poses challenges related to job displacement. One of the key issues is balancing technological progress with socio-economic stability, which requires comprehensive analysis and effective adaptation mechanisms.This study proposes a mathematical model that describes the interrelationship between automation levels, workforce retraining opportunities, government policies, and education levels. The proposed system of equations is presented in a matrix form, enabling the analysis of employment dynamics in response to technological shifts. The model allows for assessing the impact of automation on the labor market under different levels of government support and investment in educational initiatives.To evaluate the effects of technological innovations, three key scenarios are considered: Low automation and high reskilling levels – this scenario helps maintain stable employment and mitigate the negative impact of automation. It demonstrates that the active implementation of retraining programs facilitates an effective transition of workers to new job roles. High automation and low reskilling levels – this leads to a significant decline in employment levels and a decrease in average wages. It may cause increased social tensions and economic instability due to rising unemployment.Effective government policy and educational support – this scenario reduces the adverse effects of automation by expanding workforce adaptation programs, thereby stabilizing the labor market.Numerical modeling results indicate that retraining, government support, and investments in educational initiatives are the key factors enabling labor market adaptation to automation and AI integration. The findings can be utilized to develop government regulations, shape policies for educational program development, and forecast the socio-economic consequences of technological advancements.This research may be useful for academics, labor market analysts, and government agencies involved in employment policies and workforce adaptation strategies. The applied approaches and modeling results provide insights into assessing automation risks and developing optimal mechanisms for effectively integrating technological changes into the socio-economic environment.
AUTOMATED RECOGNITION OF AUTOCLAVE SURFACE DEFECTS USING POINT CLOUD ANALYSIS
The problem of detecting defects on the surface of autoclaves is extremely relevant for ensuring safety, quality, and economic efficiency of production in critical industries such as the food industry. Defects, including cracks, corrosion, and irregularities, create risks for sterility, product quality, and can lead to emergency situations. The article analyzes modern methods of scanning the internal surface of autoclaves, including laser scanning and photogrammetry, and identifies their advantages and disadvantages in terms of accuracy, speed, and cost. The aim of the research is to develop an effective automated algorithm for processing the results of scanning the internal surface of an autoclave to detect defects. The input data is a point cloud obtained by photogrammetry. The stages of building a dense point cloud based on Structure from Motion (SfM) and Multi-View Stereo (MVS) algorithms are described. The developed point cloud processing algorithm is presented, which includes sequential filtering of points based on the analysis of normal coordinates calculated using the Principal Component Analysis (PCA) method with the k-nearest neighbors algorithm. Five filtering stages aimed at extracting points that may correspond to defects are described: removal of isolated points and points with degenerate local geometry (zero normals), removal of points belonging to large planar surfaces and symmetrical elements (identical values of normal coordinates), filtering of points with a nearly horizontal orientation (z-coordinate of the normal close to one), selection of points located below the main surface (negative z-coordinate), and selection of points with normal coordinates deviating from the horizontal orientation by a threshold value. The results of applying the developed algorithm to a point cloud representing the internal surface of an autoclave with traces of corrosion are presented. Visualization of the point cloud after each filtering stage is demonstrated. The importance of comparing the results of the current scan with previous ones for monitoring the dynamics of defect development and evaluating the effectiveness of repair work is emphasized. The developed automated approach contributes to increasing the efficiency and objectivity of the process of detecting defects on the surface of autoclaves.
ADAPTIVE CONTROL METHOD AND CRITERIA FOR VERIFYING THE RELIABILITY OF COMPUTATIONAL IMPLEMENTATION OF MODELS OF DYNAMIC SYSTEMS
The issues of studying dynamic systems are most acute in the development, operation and modernization of control systems for various objects and processes. A characteristic feature of modern control systems is the widespread use of various types of computer systems that implement methods and algorithms for modeling and control. Therefore, the quality of functioning of modeling and control systems is largely determined by the characteristics of the computing resources used.The problem of ensuring the quality of functioning, in particular, of control systems with a computer in the control loop is very complex from both a technical and mathematical point of view. In solving this problem, practice has put forward one of the first places a scientific direction related to the development of methods and tools that ensure the reliability and validity of computing and information processing processes. In a number of applied problems, in particular when assessing the reliability of the obtained solution, the dominant value is the forecast time (i.e. the speed of its receipt) and the simplicity of the computational procedures involved. In particular, such requirements may arise when solving control problems of fast-moving dynamic processes (objects), especially in conditions of multivariate formulation (such problems may include: optimization problems with iterative procedures for finding the optimal control law; dispatch control problems in abnormal and emergency situations, when the best solution according to a certain criterion is sought from possible alternative ones; problems that are solved in an “accelerated” time scale, when the laws of program control are sought, provided that there is no possibility to organize closed-loop control systems, etc.). It is obvious that to check the reliability of calculations regarding the dynamic state of the studied objects and processes, it is rational to use control algorithms. Moreover, in the case of limited time for calculations, it is advisable, as a control (verification) algorithm, to use a simplified algorithm for solving the problem of assessing the reliability of the obtained solution to the initial modeling or control problem.
APPLICATION OF THE CONICAL APPROXIMATION OF NAVIER-STOKES EQUATIONS TO COMPUTER MODELLING OF THE STRUCTURE OF SUPERSONIC SHEAR FLOWS
The study of supersonic turbulent separation flows is one of the main problems of modern aerodynamics, which has both fundamental and practical significance. From a practical point of view, the spatial distributions of gas-dynamic parameters affect the aerodynamic characteristics of supersonic aircraft, compressors, and turbines. The theoretical interest in this class of flows is due to the fact that all the main phenomena characteristic of viscous-inviscid interactions occur here: significant parameter gradients, flow separation, turbulence, intensive heat exchange with the flowing surface. Investigation of the physical features of the shock waves/turbulent boundary layers interactions (SWBLI) acquires special importance.Known results of studies of the interaction of shock waves with boundary layers were obtained mainly by experi- mental methods. However, the conducting of the blowing in wind tunnels or full-scale flight tests is associated with both large financial costs and a limited amount of measured data.The Navier – Stokes equations, Reynolds or Favre averages (RANS), are currently the basis of computational aero- dynamics. Complete information about the flow provides the researcher with more opportunities to identify the character- istic physical features of the interacting flows. The use of modern numerical methods, differential turbulence models, and detailed computational grids allows the flow under study to be reproduced with a high degree of reliability.The conical approximation of the compressible gas Navier – Stokes equations makes it possible to use a more detailed calculation grid for modeling supersonic flows with the appropriate symmetry of the flow separation, which in turn leads to the possibility of computer reconstruction of small details of the interactions. The implicit finite-volume technique in present paper is based on the second order Roe scheme. The implementation of the Spalart – Allmaras differential turbulence model adapted to supersonic flows is considered. Verification of the developed algorithms and the program complex was carried out on the problem of supersonic turbulent flow around a sharp cone at a supercritical angle of attack.
MATHEMATICAL MODELS FOR DETERMINING THE COMPLEXITY OF OBJECT-ORIENTED DESIGN DUE TO THE RELATIONSHIPS BETWEEN CLASSES OF OPEN-SOURCE WEB APPLICATIONS CREATED USING PHP FRAMEWORKS
The problem of determining the object-oriented design complexity of the open-source software, including Web applications created using the PHP frameworks, is important because nowadays open-source software is growing in popularity and using the PHP frameworks making application development faster. The object of the study is the process of determining the object-oriented design complexity due to the relationships between classes of the open-source web applications created using the PHP frameworks. The subject of the study is the mathematical models to determine the object-oriented design complexity due to the relationships between classes of the open-source web applications created using the PHP frameworks. The goal of the work is the build two mathematical models for determining the object-oriented design complexity due to the relationships between classes of the open-source web applications created using the PHP frameworks based on the two-variate Box-Cox normalizing transformation to increase confidence in evaluating the object-oriented design complexity of these apps.To determine the object-oriented design complexity due to the relationships between classes of open-source web applications created using PHP frameworks, two mathematical models in the form of a prediction ellipse were constructed based on the two-variate Box-Cox transformation for the normalized metrics RFC (the Response for Class) and CBO (Coupling Between Objects). The maximum likelihood method is used to compute the parameter estimates of the two-variate Box-Cox transformation. The above models are built using two test statistics based on the quantiles of the Chi-square distribution and the F-distribution, respectively. To build these mathematical models using the PhpMetrics tool (https://phpmetrics.org/), we obtained data from metrics for 125 open-source web applications created using five well-known PHP frameworks: CakePHP, Laravel, Symfony, CodeIgniter, and Yii. These applications are hosted on GitHub. The results of determining the complexity of object-oriented design due to the relationships between classes of open-source web applications created using PHP frameworks were compared.
REGRESSION MODELS FOR EARLY ESTIMATING THE LINES OF CODE COUNT OF WEB APPLICATIONS CREATED USING THE CODEIGNITER FRAMEWORK
The problem of early estimation of lines of code count in software projects holds significant importance, as it directly influences the prediction of software development effort, including Web applications created using such the well-known PHP framework, as the Codeigniter. The object of the study is the process of early estimating the lines of code count of web applications created using the Codeigniter framework. The subject of the study is the regression models for early estimating the lines of code count of web applications created using the Codeigniter framework.The goal of the work is to build some regression models with three factors for early estimating the lines of code count of web applications created using the Codeigniter framework depending on the factors that can be found in the class diagram.In the work, we built two linear regression models for early estimating the lines of code count of web applications created using the Codeigniter framework depending on three factors: the number of classes, the average number of methods per class, metric DIT (Depth of Inheritance Tree) on the application level. These factors were chosen for two reasons. First, the values of these factors can be found from the class diagram, and, second, there is no problem of multicollinearity for them. The parameter estimates of the obtained models were found by the method of least squares. The above models are constructed based on splitting the four-dimensional data set (the actual size in thousands of lines of code, the number of classes, the average number of methods per class, the DIT metric on the application level) into two clusters. This data of metrics for 50 open-source web applications created using the Codeigniter framework was obtained using the PhpMetrics tool (https://phpmetrics.org/). These applications are hosted on the GitHub platform (https://github. com/). The comparison of the constructed linear regression models with the non-linear regression models is performed.These two linear models, in comparison with existing non-linear ones, allow us to describe all the four-dimensional data on which they were built.
THE NEED FOR LARGE AND HIGH-QUALITY DATA VOLUMES FOR AI TRAINING: A REVIEW OF MARINE SCADA SYSTEMS AS PRIMARY DATA SOURCES FOR PREDICTIVE MAINTENANCE
This article explores the significance and necessity of large, high-quality datasets for the effective application of artificial intelligence (AI) in the maritime sector, particularly for predicting the technical condition of shipboard electrical equipment. The emphasis is placed on analyzing Supervisory Control and Data Acquisition (SCADA) systems to review it as the main data sources for training AI algorithms. The use of AI is increasing its criticality for enhancing operational efficiency, environmental sustainability, and safety in maritime shipping. The paper addresses the primary challenges associated with collection and processing of large data volumes, including the problem of data imbalance, where instances of equipment failure occur much less frequently than normal operational data. Special attention is given to the integration of machine learning algorithms with existing SCADA systems, the challenges of data standardization and quality. Examples of success- ful implementations of predictive maintenance leveraging AI algorithms are demonstrating the potential to significantly reduce operational costs and the risks of unexpected equipment failures. There are financial aspects highlighting substantial economic benefits from implementing advanced predictive maintenance systems. Additionally, specific cases of AI algorithm applications for predicting failures and detecting anomalies in shipboard equipment are analyzed, significantly enhancing the reliability and safety of vessels. Particular attention is given to cybersecurity issues essential for protecting data used in the training and operational phases of AI algorithms. Furthermore, promising directions for future research are outlined, including the development of new algorithms, adaptation of existing data analysis methods to maritime-specific conditions, and a systematic approach to ensuring cybersecurity in shipboard information systems. The conclusions emphasize the necessity of addressing these key tasks for effective and sustainable advancement in contemporary maritime shipping.
SYSTEMS OF KINETIC POPULATION EQUATIONS AND THEIR APPLICATIONS
A systematic analysis of systems of kinetic population equations is presented, including Volterra and Lotka-Vol- terra. Population problems that had to be solved are investigated, and their brief analysis is provided. These problems include demographic, ecological, etc. problems. From a conceptual point of view, these problems are divided into two types: the problem of two species eating the same food (Volterra equation) and the predator-prey problem (Lotka – Vol- terra equation). The first problem arose from the problem of rabbit reproduction in Australia. In addition, in the same population biology, the problem arose when one species eats another (predator and prey). This problem was solved by many researchers in the field of biology and medicine, in particular virology. Its partial solution is given in the book of A. Lotka, and a more general one in the lectures of V. Volterra. Because of this, these equations are sometimes called the Lotka-Volterra equations. As in the first and second problems, it is necessary that there is enough resource (food) for the stationary stable existence and development of the dynamical system. We have analyzed the problems that are solved or that are expedient to be solved using these methods. Problems with a non-uniform temporal hierarchy of processes have also been analyzed. It has been shown that for solving such problems it is expedient to use the method of adiabatic elim- ination of variables. This method was used to solve kinetic problems in relaxation optics. These equations are expedient to use when there are several competing in-phase processes. Based on the general analysis of the systems of Volterra equations, it is possible to construct system criteria for controlling and predicting the corresponding processes and phenomena. To move to spatial problems, it is necessary to introduce the corresponding transport and diffusion coefficients into the systems of equations of Volterra and Lotka – Volterra. In this case, these equations can also be considered as systems of nonlinear diffusion equations. A list of problems for which it is expedient to use such a formalism is given.
STRUCTURAL RELIABILITY ANALYSIS OF A TECHNICAL SYSTEM USING SPANNING TREES AND CYCLES
In this work, a study of the structural reliability of a technical system based on graph theory was conducted by analyzing spanning trees, cyclic subgraphs, and the total number of connected subgraphs modeling its topology. The focus was on examining the quantity of spanning trees, cyclic subgraphs, and the overall number of connected subgraphs passing through each section of the system. These indicators provide a quantitative assessment of the importance of individual structural elements and determine their impact on the overall system performance.The system’s reliability was evaluated based on the number of redundant paths that ensure functionality even in the event of individual element failures. Graph theory methods were employed for the analysis, including the depth-first search (DFS) algorithm for detecting cyclic subgraphs and Kirchhoff’s theorem for calculating the number of spanning trees.The study results revealed that the most critical sections of the structure are those through which the largest number of spanning trees and cyclic subgraphs pass. The edges providing the highest number of redundant connections are key to maintaining system functionality. Their failure significantly reduces reliability, while the failure of less significant sections, through which the fewest connected subgraphs pass, has minimal impact due to alternative paths. It was concluded that the reliability of a technical system depends on the number of connected subgraphs passing through each element.The practical application of the obtained results allows for optimizing the design of technical systems, enhancing their fault tolerance, and reducing failure risks by properly distributing resources and reserving critical sections. These findings can be applied to improve the reliability of engineering networks, computer systems, and infrastructure facilities.
MODELING THE IMPACT OF DEFECTS IN FERROMAGNETIC PRODUCTS ON THE QUALITY OF PROCESSING IN FINISHING OPERATIONS
To ensure the quality of processed surfaces, it is necessary, based on the functional relationships between the physical and mechanical properties of materials and parameters during finishing operations, to select machining conditions and tool characteristics such that the current grinding temperature values T (x, y, τ), thermal flux q (y, τ), maximum stresses σp max, grinding forces PY, PZ, and intensity coefficient K1 (S, α, σmax) do not exceed their specific values. These values should consider defects of certain geometric dimensions located in the surface layer, having an inherited nature, thus ensuring the required quality of the product’s working surfaces. This study addresses the mathematical formulation of the problem of detecting defects in ferromagnetic components resulting from previous operations during their magnetization. Mathematical expressions were derived to assess the geometric shape and depth of defect placement within the surface of a component based on measurements of magnetic induction distribution on its surface. The mechanism of technological crack formation during finishing operations on surfaces of ferro ceramic materials is considered from the viewpoint of the “weakest link” hypothesis, defined as an inherited defect whose size serves as a criterion for defect-free processing. Analytical conditions for preventing structural defects from developing into main cracks were established. These conditions depend on fracture toughness, physical and mechanical characteristics, and the contact temperature TK during processing, determined by operational parameters. As a result of the research informational support was developed for technological capabilities to achieve defect-free processing of products made from materials prone to cracking.This support involves establishing calculation dependencies to determine the influence of inherited defects formed during previous operations on the crack resistance of the surface layer during finishing operations. It includes technological processing conditions that account for accumulated damages and inhomogeneities in the surface layer of parts made from materials and alloys particularly susceptible to crack formation. This development holds significant economic importance by reducing defects in finishing operations and enhancing the operational properties of machine parts.
OPTIMIZATION OF THERMOMECHANICAL PHENOMENA IN FINISHING OPERATIONS OF PARTS MADE OF MATERIALS PRONE TO DEFECT FORMATION
The study is devoted to the development and application of mathematical models for optimizing thermomechanical processes occurring during finishing operations of machining parts made of materials prone to defect formation. The article analyzes the modern technological requirements associated with the processing of magnetically hard alloys, ferroceramic materials, and high-strength steels, where traditional methods do not always provide the necessary balance between productivity and processing quality. The main goal of the work is to create an integrated mathematical model that takes into account the influence of temperature, force, and geometric parameters on the formation of defects, including cracks, during finishing. The article presents a theoretical framework for formalizing the problem of multicriteria optimization, which allows unifying various partial criteria into a single system for assessing the quality of a working surface. The use of the principles of utility theory contributes to the development of a model that describes the nonlinear distribution of the temperature field, diffusion processes, and the occurrence of force stresses in the contact zone between the tool and the workpiece. Thanks to the use of specialized cooling and lubricating media, adhesion phenomena were reduced, which minimized local temperature overheating and reduced the risk of microcracks. Experimental studies confirmed the effectiveness of the modeling, as a clear dependence of the stress intensity factor on the size of defects and finishing modes was established. The results obtained allow us to formulate practical recommendations for optimizing technological parameters, including the processing temperature, the duration of exposure to the heat flow, and the geometry of the abrasive wheel. This contributes not only to improving the quality of the treated surfaces, but also to reducing economic costs by reducing the rejection rate and increasing production productivity. Thus, the study has significant practical potential for the implementation of innovative solutions in the field of thermomechanical processing of hard-to-machine materials. The developed methodology of mathematical modeling creates the basis for further experimental testing and optimization of production processes, which will help to increase the competitiveness of enterprises in modern conditions of high-tech production.
AUTONOMOUS IOT-BASED CLASSROOM MICROCLIMATE MONITORING SYSTEM USING AN OPEN DIY ARCHITECTURE
The article presents the architectural model of an autonomous system for monitoring microclimate parameters in educational environments, implemented using Internet of Things (IoT) technologies based on an open Do It Yourself (DIY) architecture. The proposed solution is aimed at creating a flexible, scalable, and cost-effective system that enables local collection, processing, and real-time visualization of environmental indicators without relying on external cloud services. The system follows a three-tier architecture: the edge level includes ESP32 microcontrollers with connected sensors for temperature, humidity, and CO₂; the server level is implemented using PostgreSQL extended with TimescaleDB and a RESTful API for data storage, verification, and processing; and the client level includes a web interface developed using React and Chart.js, providing interactive access to data visualization in accordance with standard thresholds (ISO 7730, DBN V.2.5-67:2013). The relevance of this research is driven by the need to implement affordable, open, and secure tools for environmental monitoring in the context of limited funding for educational institutions. Commercial IoT systems often exhibit significant drawbacks such as high cost, closed architecture, dependence on cloud infrastructure, and limited support for non-standard sensor configurations. The proposed solution addresses these limitations by offering complete autonomy, local data management, a high level of confidentiality, and adaptability to the specific requirements of individual classrooms. The system also includes a mechanism for automatic alerts in cases of regulatory parameter violations, allowing timely identification of potential risks to health and educational performance. In addition to its practical value, the proposed development has strong educational potential, serving as a platform for teaching students in the fields of microelectronics, programming, data analysis, and IoT security. The results confirm the feasibility of using open hardware and software solutions in education and highlight the prospects for scaling the system to other types of spaces, such as offices, laboratories, or public facilities.
AN OVERVIEW OF USING OF FRACTAL ANALYSIS FOR DETECTING DDOS NETWORK ATTACKS
Distributed Denial-of-Service (DDoS) attacks are among the most severe cybersecurity threats, continuously evolv- ing and causing extensive financial losses worldwide. Unlike traditional Denial-of-Service (DoS) attacks, DDoS attacks leverage multiple compromised systems, creating a coordinated effort to overwhelm network resources and disrupt ser- vice availability. The growing complexity of these attacks, often indistinguishable from legitimate traffic, presents sig- nificant challenges for detection and mitigation. This article examines various machine learning techniques for DDoS detection, including Support Vector Machine (SVM), K-Nearest Neighbors (KNN), Decision Trees, and Naïve Bayes, all of which demonstrate high accuracy in classifying attack patterns on synthetic datasets. While effective in controlled environments, these methods often struggle with the nuances of real-world network traffic, where hybrid and novel attack types obscure detection efforts. To address these challenges, we explore the application of fractal analysis, a promising approach for identifying self-similarity in network traffic.Fractal analysis, which captures the self-similarity in network traffic, has shown potential for identifying abnormal patterns indicative of DDoS activity. Although it has limitations, fractal analysis can improve the detection process when combined with statistical features and machine learning algorithms.Fractal characteristics, such as those quantified by the Hurst Exponent, reveal long-term dependencies and auto- correlation structures in traffic, making them suitable for detecting irregularities commonly associated with DDoS attacks.Our analysis demonstrates that fractal-based methods, when combined with statistical and machine learning approaches, can enhance detection accuracy and improve adaptability to real-world scenarios. Although no single method offers a universal solution, this study underscores the importance of using diverse techniques to effectively monitor and protect against DDoS threats. Further research should focus on integrating multifaceted detection models to better address the evolving landscape of cybersecurity threats posed by DDoS attacks.
STRUCTURING INFORMATION FOR SYSTEM ANALYSIS OF TRANSPORT LOGISTICS
The article presents a system approach to integrating analysis, synthesis and consideration of various aspects of the logistics transport system with the possibility of restructuring to change the functioning. It is established that the system approach in transport logistics is a methodology of scientific knowledge based on the consideration of moving objects as systems with their own properties and relationships. The use of the system approach to transport logistics involves interaction with systems consisting of structured and functionally organized elements. The formalization of the logistics transport system, its structure and properties is performed. Factors, parameters and sets of indicators characterizing logistics changes in transport systems are identified. The structuring of information for the system analysis of transport logistics has been developed, the peculiarity of which is the simultaneous assignment of states and situations determined by the external conditions of the functioning of transport systems. The prospects of using system analysis tools for operational access to information related to any element of transport logistics and the dynamics of transportation processes are shown. The use of systems analysis to study the problems of transport logistics is due to the interdependence of logistic and mathematical models and connections between goals, alternative means of achieving them, the environment and resources. At first, only visible structures are recorded, and then, by analyzing and transforming the logistic system, hidden structures are revealed, followed by a new quality of the system. The problems of logistics are considered as a set of unified ideas about structural changes in the management process under uncertainty. System calculation methods with elements of discrete mathematics allow us to describe the state of the system in dynamics and find rational management solutions. It is shown that transport management based on the principles of systems analysis in the formation of complex systems management can reveal a number of qualitatively new ideas that can significantly improve the efficiency of transportation.
ENHANCING DANGEROUS OBJECT DETECTION THROUGH INTERMEDIATE MULTIMODAL DATA FUSION
Reliable detection of dangerous objects is a critical component in numerous domains, including autonomous transportation, industrial safety, surveillance systems, and robotics. In real-world scenarios – such as poor lighting, adverse weather, occluded views, or noisy backgrounds – single-sensor detection systems often struggle to maintain accuracy and stability. To address these limitations, this paper proposes an approach based on multimodal data fusion, leveraging attention mechanisms and object criticality assessment.The proposed system integrates information from an RGB camera, LiDAR, thermal imager, and optionally, tactile sensors. Each sensor stream is processed by a dedicated feature encoder. Attention weights are dynamically calculated based on the environmental context to determine the relevance of each modality. The resulting fused feature vector is passed through a detection module that produces a base confidence score, while a parallel criticality module evaluates spatial distance, relative velocity, and orientation of the detected object.The final decision is formed by combining both the detection score and the criticality assessment using a weighted factor. If the resulting Sfinal value exceeds a predefined threshold, the system triggers a response – issuing a warning, initiating emergency braking, or notifying the operator. A hypothetical case study involving pedestrian detection in limited visibility illustrates the practical utility of the proposed architecture.The system exhibits high accuracy, robustness, and contextual adaptability. Its main advantages include preservation of spatial and semantic features, the ability to process inputs across multiple sensing domains, and risk-based prioritization of threats. This architecture presents a strong foundation for the development of reliable, safety-oriented autonomous solutions in transportation, industrial monitoring, and urban environments.
HYBRID MODEL FOR EVALUATING THE EFFECTIVENESS OF PROXIMITY METRICS FOR GENE EXPRESSION DATA
This paper presents the development and application of a hybrid model for evaluating the effectiveness of prox- imity metrics in high-dimensional gene expression data, integrating data mining and machine learning methods within a unified comprehensive framework. The study focuses on a comparative analysis of correlation distance, mutual infor- mation-based metrics, and the Wasserstein metric to assess their efficiency for clustering and classification tasks. The obtained results demonstrate that correlation distance and the Wasserstein metric provide high accuracy and stability, making them suitable for integration into diagnostic systems. To enhance classification reliability, a stacking model has been implemented to compensate for potential clustering errors and ensure consistent performance regardless of the applied metric and cluster structure. The proposed data processing pipeline enables automated, standardized, and scala- ble analysis of large-scale gene expression datasets, aligning with the principles of personalized medicine. By facilitating early disease diagnosis and supporting the development of individualized treatment strategies, the findings contribute sig- nificantly to improving modern diagnostic systems within the framework of precision medicine. The application of hybrid approaches allows for an effective combination of the advantages of different similarity evaluation and classification methods, enhancing prediction accuracy based on gene expression data. The proposed methodological approach can be utilized in bioinformatics for analyzing complex biological systems, optimizing genetic data processing workflows, and developing new algorithmic solutions in medical diagnostics. This approach promotes the adaptation of modern information technologies for identifying disease biomarkers, ensuring the integration of obtained results into clinical practice, and supporting the advancement of personalized treatment strategies.