Modern problems of modeling

GEOMETRIC METHOD FOR CONSTRUCTING THE BASIS OF THE DISCRETE ELEMENT H32

Ihor Astionenko — 2025-07-09

This work examines serendipity finite elements (SFEs), which emerged in the late 1960s as an ingenious modification of Lagrangian two- and three-dimensional elements through the removal of internal interpolation nodes. The basis functions of serendipity finite elements found in the literature are referred to as standard functions. These functions are well-suited for the task of isoparametric mapping of a square into an arbitrary quadrilateral; however, the interpolation quality of standard bases is not always flawless. All standard bases of serendipity finite elements, except for the bilinear one, have disadvantages. The construction of SFE basis functions, in addition to traditional methods of matrix analysis, can also be performed using the Taylor series procedure. Unfortunately, polynomials created by both methods for higher-order elements exhibit certain drawbacks: an excessive number of multiple zeros at interpolation nodes (rigid model) and negative values when distributing uniform body forces among the nodes (unnatural distribution). A relevant issue in serendipity interpolation is the development of methods for constructing alternative higher-order bases that are free from these drawbacks. The existence of such bases was proven in the early 1980s. The specific nature of serendipity models requires abandoning the Taylor procedure, which is based on one-dimensional Lagrangian interpolation traditions. Only for first- and second-order elements do the results of both approaches coincide. Complex problems of higher-order SFEs are better addressed using methods of applied geometry. Using the example of a three-dimensional finite element with 32 nodes, the deficiencies of the Taylor interpolation procedure are analyzed. A geometric method for eliminating the shortcomings of the classical basis is proposed. The advantages of geometric construction of SFEs are undeniable – it enables the creation of alternative bases that are free from the limitations of standard bases. Moreover, successful geometric design makes it possible to optimize bases for improved interpolation and computational performance.

Key words: serendipity finite elements, interpolation, geometric construction, Taylor procedures, three-dimensional finite element, alternative higher-order basis.

NUMERICAL MODELING OF NONLINEAR POWER RECTIFIER CIRCUITS USING THE METHOD OF VARIABLE TRANSFORMATION

Oleg Bondar — 2025-07-09

The article presents a numerical simulation of a specific topology of a power rectifier circuit with a load that includes components exhibiting nonlinear voltage-current characteristics. The analyzed scheme comprises reactive elements (inductors and capacitors), as well as semiconductor devices whose dynamic behavior is determined not only by control parameters but also by their internal electrophysical properties. To simplify the mathematical description of the nonlinear model, the method of variable transformation was applied. This approach enables the reduction of the governing differential equations to a form suitable for numerical integration with a controlled level of accuracy.

The analysis of the simulation results made it possible to identify the characteristic features of electromagnetic processes arising in the circuit under near-resonant and boundary operating conditions. The study determined the conditions under which stationary oscillations occur, accompanied by hysteresis, magnetic saturation, and oscillatory instability. Phase portraits and time-domain graphs were obtained, illustrating transitions from regular operation to self-oscillatory or chaotic modes, depending on the parameters of the circuit. A comparison of the results before and after the variable transformation demonstrated improved accuracy, enhanced numerical stability, and reduced computational effort.

The proposed approach has proven effective for analyzing complex nonlinear processes in power electronic circuits. It can be applied in the design of adaptive electric drive systems, in the modeling of energy conversion devices, and in the educational process for courses related to electrical engineering, electromechanics, and automation.

Keywords: numerical modeling, nonlinear electrical circuits, power rectifier

ONE-DIMENSIONAL PARAMETRIC COMPOSITION MATRICES

Viktor Vereshchaha — 2025-07-09

The article proposes a systematic description of one-dimensional parametric composition matrices, which serve as a formal tool for parameterising spatial discretely defined curves and subsequently constructing continuous composition lines. It is shown that each element of a one-dimensional point composition matrix corresponds to a unique characteristic function, which, together with other functions, forms the functional basis of a point polynomial that continuously interpolates the initial base points.

General forms of parametric composition matrices for three parametric directions are presented, along with a universal symbolic notation that ensures invariance with respect to the choice of coordinate system and enables extension of the approach to composition surfaces. An algorithm is proposed for constructing characteristic functions for each base point of a discretely defined curve, and its operation is demonstrated on an example of a curve with four points, including special cases of point coincidence.

An example is given of the formation of characteristic functions for the base points of a discretely defined curve, illustrating the rules of DDC (Discretely Defined Curve) point parameterisation and, directly, the generation of characteristic functions. It is stated that the set of characteristic functions for all base points of the DDC constitutes a functional basis of a point polynomial that continuously interpolates the initial base points. A general expression for this point polynomial is provided.

Variants of specific configurations of the initial base points are discussed, including those where points coincide. It is emphasised that the presence of such configurations does not require alternative algorithms for constructing characteristic functions. That is, characteristic functions are generated according to a unified method for both single and multiple points.

Furthermore, it is underlined that the formation of segments of composition surfaces requires the same number of points along each parametric edge, and this requirement is substantiated.

Keywords. Composite line, composite surface, composite parametric matrix, characteristic functions, point polynomial, functional basis from a point polynomial.

ERROR FINGERPRINT AND ENTROPY MAPS AS A WAY TO VISUALIZE THE EVALUATION OF THE QUALITY OF PERFORMANCE OF DEEP LEARNING MODELS FOR AERIAL IMAGE PROCESSING

Vitalii Vlasenko — 2025-07-09

The work is devoted to the research process and the development of our own approach to create a standardized error map of the quality of deep learning models and entropy in cases of analysis of aerial images obtained from drones. With the help of such maps, it is possible to understand in which areas of the images the models are most often mistaken in the test data and obtain the corresponding uncertainties. Building such maps helps to identify weaknesses in models and is especially useful for cases where the average standard metrics are almost the same. To conduct research and experiments, the most popular neural network architectures U-Net, DeepLabV3+ and Feature Pyramid Network, which are most often used in image segmentation problems in the field of computer vision, were used. Semantic segmentation plays a key role in the analysis of spatial data in the tasks of environmental monitoring, mapping and land management. The high accuracy of popular architectures does not eliminate the problem of analyzing errors and uncertainties. The paper presents an approach to building error heat maps, which show where models most incorrectly classify pixels on test data, and to estimate uncertainties, it is proposed to build special entropy maps. In addition to determining quantitative indicators of the quality of models (accuracy, Jaccard index, F-metrics), such visualizations allow for a qualitative analysis of weaknesses, for example, in cases where images have heterogeneous terrain or fuzzy contours. In the course of experiments on our own set of aerial photographs, it was determined that the effectiveness of the proposed imaging approaches can significantly increase the understanding of the results of the trained models. The results of such studies can have a significant impact on improving segmentation architectures, training data preparation, and error analysis in general. Future research may focus on expanding the aerial imagery dataset to evaluate model performance and reduce errors.

Keywords: deep learning, semantic segmentation, U-Net, DeepLabV3+, FPN, neural networks, error maps, entropy maps, visualization method, drones, aerial images.

THE INFLUENCE OF SUPERPOSITION COEFFICIENTS ON THE DISCRETE FORMATION OF ELEMENTARY FUNCTIONAL DEPENDENCIES

Oleg Vorontsov — 2025-07-09

The article presents a generalized approach to the discrete modeling of one-dimensional discrete geometric objects (DGOs) using templates, where the shape control of a discretely represented curve (DRC) being modeled is performed not only by the distribution function of the finite difference magnitude between adjacent nodes of the framework, but also by the distribution function of superposition coefficients.

Patterns of change in the values of superposition coefficients and the magnitude of the finite difference—which serves as a functional analog of load within the framework of the static-geometric modeling method—have been established during the modeling of one-dimensional point sets. This approach enables the effective solving of problems related to continuous discrete interpolation and extrapolation using numerical sequences for arbitrary one-dimensional functional dependencies defined by two selected nodal points.

One of the primary objectives of the study is to further develop the theoretical foundations for constructing discrete analogs of curvilinear objects based on the classical apparatus of finite differences, the static-geometric modeling method, and the geometric tools of superposition.

The research analyzes the process of forming discrete one-dimensional geometric objects using polynomial functional dependencies, with predefined values of superposition coefficients. It identifies patterns in the variation of superposition coefficients between adjacent nodal points of the polynomial function, as well as in the magnitude of the finite difference, illustrated through graphs of numerical sequences for a selected computational configuration.

The study establishes the dependencies of the finite difference magnitude on the ordinates of the modeled curve and the values of superposition coefficients between adjacent nodal points.

The obtained results make it possible to form one-dimensional geometric objects within a given computational scheme, based on known ordinates of two reference nodal points, superposition coefficients, and the corresponding finite difference.

Thus, the research proposes a universal approach to identifying the patterns of variation in superposition coefficients and finite differences within given computational schemes, enabling the modeling of point ordinates for arbitrary one-dimensional functional dependencies and point sets.

Keywords: discrete modeling, static-geometric method, geometric apparatus of superposition, finite difference magnitude, superposition coefficients.

CALCULATION OF ELECTRIC POWER AND TEMPERATURE DISTRIBUTION IN AN ELECTRIC HEAT ACCUMULATING CONVERTER

Viktor Kovalenko — 2025-07-09

This study presents the development and implementation of a three-dimensional mathematical model for calculating the distribution of active electric power and temperature within an Electric Heat-Storage Converter (EHSC). The model is based on the method of secondary sources, which enables precise simulation of electromagnetic processes in inhomogeneous media with complex geometry and variable material properties. The proposed approach is particularly relevant for autonomous hot-water-supply systems, where energy efficiency, reliability, and the integration of renewable energy sources are critical.

The electric heat-storage converter operates by accumulating thermal energy during off-peak hours when electricity tariffs are low, and releasing this energy during peak demand periods. This mode of operation significantly reduces energy costs, enhances energy autonomy, and reduces the load on electrical grids. The converter’s design–compact, reliable, and compatible with standard three-phase grids–allows for easy integration into both new and existing systems, including solar thermal complexes, mini-boiler plants, and hybrid configurations.

A numerical method was applied using discretization of the converter volume into elementary geometric subdomains, with linear temperature distribution assumed within each. The Fredholm integral equations of the second kind, derived through the method of secondary sources, were solved numerically to simulate the electromagnetic field and calculate instantaneous power density. The results confirm the feasibility and accuracy of the proposed model for engineering applications.

Ultimately, the findings support the effectiveness of electric heat-storage converters as energy-efficient components in decentralized heating infrastructures, and demonstrate the practical value of advanced computational methods–especially the method of secondary sources–for analyzing and optimizing such systems.

Keywords: electric heat-storage converter, energy efficiency, secondary sources method, hot-water supply systems, electromagnetic field modeling, thermal storage, numerical simulation

RESEARCH OF ANALYTICAL CURVES IN GEOMETRIC MODELING OF PROJECTILE NOSE PROFILES

Dmytro Kotliar — 2025-07-09

This study investigates the feasibility of applying nine analytical curves for the geometric modeling of a ballistic projectile's nose section to achieve high aerodynamic efficiency. The research focuses on approximating the ogival profile while adhering to physical constraints, including the initial point, final height, horizontal tangent, and monotonicity. Various curve types were employed: polynomial (third-degree polynomial, parabola), sigmoidal (logistic, hyperbolic tangent), and nonlinear (elliptical arc, power, logarithmic, exponential, Nosek curve), enabling an evaluation of their ability to accurately reproduce the profile and meet aerodynamic requirements.

The methodology is based on numerical optimization using seven methods: Nelder-Mead, Powell, CG, BFGS, L-BFGS-B, TNC, and SLSQP. Penalty functions and direct constraints (for SLSQP) were applied to enforce physical constraints, ensuring compliance with boundary conditions and monotonicity. Penalty functions accounted for deviations from specified boundary points, tangent slope, and non-negative derivatives at key profile points. Optimization was performed by minimizing the root mean square error (RMSE) between reference points and modeled curves. The quality of approximation was assessed by analyzing deviations in the front, middle, and rear zones of the profile, alongside aerodynamic efficiency, which depends on profile smoothness and horizontal tangent compliance.

The study compared the curves based on fitting accuracy, numerical error stability, and physical correctness. Special attention was given to the impact of penalty functions on the convergence of optimization methods and their ability to compensate for limitations of unconstrained methods. Aerodynamic characteristics were analyzed, evaluating the effect of profile deviations on drag and stability. The findings provide recommendations for selecting optimal curves and optimization methods for geometric modeling of aerodynamic shapes.

The research highlights the importance of a comprehensive approach to modeling, combining analytical curves, numerical optimization, and physical constraints to achieve high accuracy and aerodynamic efficiency. The conclusions can be applied to improve ballistic projectile designs and related fields requiring precise geometric modeling of complex profiles.

Keywords: geometric modeling, ballistic projectile, analytical curves, aerodynamic efficiency, numerical optimization, penalty functions, polynomial curves, sigmoidal curves, nonlinear curves.

EVALUATION OF THE ADEQUACY OF A MATHEMATICAL MODEL OF AN ASYNCHRONOUS MOTOR UNDER POOR POWER SUPPLY CONDITIONS

Vitaliy Kuznetsov — 2025-07-09

The article presents a comprehensive approach to the mathematical modeling of an asynchronous electric motor operating under low-quality or unstable power supply conditions. The proposed model is based on analyzing the instantaneous values of voltage and current using space-time vector complexes (STVC), eliminating the need for signal transformation into harmonics or symmetrical components. This approach allows for more accurate and flexible simulation of electromagnetic and energy processes in the motor under real-world industrial conditions.

The study’s objective is to verify the adequacy of the developed model by comparing the calculated and experimental values of key performance parameters such as efficiency (η), power factor (cos φ), and total power losses. Field experiments were conducted in an industrial workshop where the power supply was distorted, with phase asymmetry and load fluctuations.

An asynchronous motor with a squirrel-cage rotor rated at 11 kW was used for testing, with mechanical loading ranging from 2.3 kW to 12.8 kW. Data collection was performed using a precision measurement system including Hall-effect sensors and a tachogenerator. Verification of the model was carried out through regression analysis and evaluation of root mean square error metrics.

The results confirm high predictive accuracy of the model: relative root mean square error was 2.72% for efficiency, 3.0% for power factor, and 3.99% for total losses. The correlation coefficient between simulated and measured values reached 0.99, affirming the validity of the modeling approach.

This model is both scientifically sound and practically applicable, offering opportunities for implementation in intelligent monitoring and control systems for electric drives. It supports the detection of operational anomalies, enhances energy efficiency, and is particularly valuable in settings with distorted or unbalanced power supply.

Keywords : asynchronous motor, mathematical model, poor power quality, space-time vectors, energy indicators, efficiency, power factor, total losses, experimental verification, digital diagnostics

MATHEMATICAL MODEL OF INFORMATION REPRESENTATION IN MEMORY FOR SOFTWARE PERFORMANCE ANALYSIS

Mykola Mitikov — 2025-07-09

In today's digital landscape, the efficacy of software is pivotal for the success of key economic sectors, including finance, commerce, healthcare, freight, and energy. Suboptimal software performance can incur serious repercussions, such as diminished computing system productivity and reputational and potential losses.

Software performance issue detection is multifaceted and challenging. Problems can stem from various causes: low algorithm efficiency, outdated technologies with poor performance, programming languages failing to fully leverage computing capabilities, and mismanagement of resources or software architecture. These factors may interact in complex, unpredictable ways, complicating effective diagnosis and resolution, leading to unforeseeable breakdowns, data loss, and erroneous system operations.

The gravity of solving these issues is accentuated by the prevalent use distributed systems, where minor defects can significantly impair productivity. Thus, the capability to identify, analyze, and resolve software issues is crucial in software system development and maintenance. Hence, the task of developing new mathematical models for detecting and analyzing software performance problems is of current interest. In the IT sector, software analysis aimed at optimization is essential for devising competitive software solutions. Software reliability becomes a decisive factor in developer selection and correlates with a firm's market sustainability. Quality software utilizing fewer computing resources reduces costs, vital for competitive viability. Reliability, security, and vulnerability resilience are key in preventing personal data leaks or other sensitive information exposure.

Considering these factors, examining and addressing software performance issues is not only academically significant but a strategic global IT industry priority. This study examines software performance problems, presenting approaches for identifying software inefficiencies and optimization. It conducts an analysis of methods for detecting software performance issues, offering recommendations and constraints on the application of these methodologies.

Key words: software, performance, memory snapshot, duplication, granularity.

ANALYSIS OF RECENT RESEARCH ON COMPLEX SPATIAL FORMS

Maryna Morozova — 2025-07-09

The work is devoted to the analysis of recent research on complex spatial forms in various fields of science and technology. Currently, in geometry, computer modeling, and engineering, special attention is focused on the study of complex geometric objects (forms), particularly spatial ones. The main role in such works is occupied by the study of polyhedral, lattice, and interconnected structures. In modern scientific works, these objects are considered from the point of view of both their theoretical utility and practical use. Today, geometric design, modeling, and application of complex spatial structures are relevant for various fields of human activity and constitute a fundamental problem that has both theoretical and applied significance. The growing complexity of the needs of science, education, and technology, caused by the rapid development of human society, forces researchers to look for new ways of using and methods of implementing complex spatial forms to solve a wide class of problems. Understanding such structures is fundamental for studying their characteristics and properties, which can be important for both technical and pedagogical sciences. Scientific works devoted to this topic serve as a basis for further research in the fields of higher mathematics, computer science, engineering, manufacturing industry, architecture, design, etc. To confirm the relevance of the topic, a review of recent literature related to the study of complex geometric objects in space, their characteristics, and recommendations for practical application is proposed. The article analyzes and attempts to systematize works on the topic. The article aims to reveal the main directions of research pursued by modern scientists. Particular attention is paid to the fields of geometric modeling, computer graphics, and other practical sciences aimed at the visualization of complex geometric structures.

Keywords: complex spatial forms, polyhedron, polyhedral structure, lattice structure, interconnected structure, computer graphics, computer modeling, additive printing.

AERODYNAMICS AND AESTHETICS AS THE MAIN FACTORS IN THE SHAPE OF CAR BODY PARTS

Olga Nazarko — 2025-07-09

One of the defining indicators of body aerodynamics is its dimensions and shape. The interaction of car body elements during movement with the air flow, namely the ability to reduce air resistance, is characterized by the streamlining of its shape. In addition, the features of the body shape are a key factor in the aesthetic perception of the car as a finished product.

The capabilities of modern CAD, as well as the ability to integrate them, make it possible to carry out a preliminary analysis of the functionality of the future product using virtual tests.

An important aspect when designing body parts is compliance with the design concept of the future car. Assessment of the appearance of the finished product is also provided by CAD visualization tools. Computer prototyping allows, based on an existing object, to make modifications to the geometry of the digital model in order to improve its final characteristics. Currently, the development of objects of any type is impossible without creating their 3D model. Using the reverse engineering method allows you to quickly obtain three-dimensional computer prototypes for their further optimization. Thus, photogrammetry, as one of the reverse engineering tools, allows you to reproduce digital models of vehicle body parts according to the original or its fragment.

This work analyzes historical and technical factors that influenced the development of the shape of the car body. It is proposed to use the reverse engineering method to assess the streamline of the car based on a comparative analysis of the values of the aerodynamic drag coefficient of a surface simplified 3D model of the body and obtained using photogrammetry from a real object.

Key words: aerodynamic drag coefficient, body shape, shaping, design, three-dimensional modeling, photogrammetry, reverse engineering, Autodesk Inventor Professional, Autodesk ReCap Photo.

SOLVING THE RIDDLE OF SPHERICAL PERSPECTIVE OF THE PAINTERS OF THE RENAISSANCE IN NORTHERN EUROPE

Alexander Nitsyn — 2025-07-09

A method has been proposed for constructing spherical perspective using the simplest drawing tools, namely a compass and a ruler. The method is based on the notion that spherical perspective is nothing more than a reflection in a round convex mirror. The clue to the mystery of spherical perspective was the round convex mirror depicted in many portraits in the interior by Jan van Eyck, Robert Campin, Hieronymus Bosch, Pieter Bruegel and other outstanding painters of the Renaissance in Northern Europe. Moreover, it has been suggested that they knew a way to construct a reflection of a geometric figure in a round convex mirror using those drawing tools that were known to painters of the 15th-16th centuries, namely a compass and a ruler, and were able to apply the acquired knowledge to construct the spherical perspective.

Therefore, the proposed method of constructing spherical perspective can be considered as a reconstruction of the geometric constructions with which the artists of the Renaissance in Northern Europe reproduced the visually perceived space on the plane of the picture.

What’s more, thanks to the study of the geometry of paintings by the painters of the Renaissance in Northern Europe, it was given the definition of spherical perspective. Let’s call spherical perspective the parallel projection of a three-dimensional image of a geometric figure, which is its reflection relative to the sphere, onto a plane tangent to it.

Thus, in this work we have given an outline of the theory of spherical perspective that corresponds to the features of natural human visual perception. In addition to its theoretical value, our research also has practical significance, which consists in the fact that spherical perspective, represented as a reflection in a spherical mirror, can be applied in ‘virtual reality’ technology. This will allow the picture of a three-dimensional scene to be brought closer to natural visual perception and to convey objects in its foreground without the monstrous distortions inherent in linear perspective.

Key words: spherical perspective, geometric constructions using a compass and ruler, art of the Renaissance in Northern Europe.

THE NUMBER DETERMINATION OF A LIGHT BEAM REFLECTIONS INSIDE A LIGHT SHAFT IN THE FORM OF A REGULAR HEXAGONAL PRISM

Evgen Pugachev — 2025-07-09

In the article, an algorithm and corresponding software for determining the number of reflections in a vertical prismatic light shaft with horizontal upper and lower bases in the form of a regular hexagon are developed.

To determine the number of beam reflections from the surface of a prismatic shaft with horizontal upper and lower bases in the form of a regular hexagon, the method of straightening a billiard trajectory proposed by the German mathematician G.A. Schwartz was used. The essence of this method is that the shaft and the beam incident on its face are mirrored relative to the reflecting face. In this case, the incident and reflected beams form one straight line. By repeating this mapping for each face of the shaft, you can pave the space around the shaft with prisms with regular hexagons at the bases and straighten the beam trajectory.

To demonstrate the algorithm's operation, a prism in the form of a regular hexagon was given. The side of the base of the shaft was 2 m, and the height of the prism was 3 m. The calculated point was located in the plane of the lower base, through which the vectors of the output beam were set. The abscissa and ordinate coordinates of the vector remained constant, and the angle of its inclination to the horizontal plane was changed in steps of 5 degrees. The figures show the results of the algorithm at angles of inclination of the output beam to the horizontal plane of 30⁰, 25⁰, and 20⁰. At an angle of inclination of the incident beam to the horizontal plane of 30⁰, there were two reflections of the light beam, and at 25⁰ and 20⁰, there were 3 and 4 reflections, respectively. Therefore, it was concluded that when the angle of inclination of the output beam to the horizontal plane decreases, the number of reflections increases, and the lengths of the segments of the horizontal projection of the straightened beam remain constant within a separate hexagon.

The coordinates of the incoming beam can be obtained by specularly reflecting the light beam from the side plane of the prism on which the first reflecting point is located.

The developed algorithm for determining the number of reflections and the trajectory of the beam allows you to calculate the brightness of the output beam for modern models of brightness distribution across the firmaments approved by the CIE (International Commission on Illumination) and, accordingly, to model the illumination by light reflected from the shaft surfaces.

Keywords: light shaft, light beam tracing, hexagonal prism, beam brightness.

GEOMETRIC MODELING OF THE RECONSTRUCTION OF SOME OBJECTS BASED ON THEIR SHADOW PROJECTIONS

Elizaveta Sivak — 2025-07-09

There are many tools for constructing images and processing graphic information, which today are perfectly operated by a modern experienced engineer, who must be able to find new solutions to technical problems based on a combination of originality, mathematics, updated methods, applied scientific knowledge and professionally approach new implementations.

In the course of descriptive geometry, the direct problem is the study of methods for constructing images. In this case, a solution can be obtained regardless of the projection task and the type of object being displayed. The inverse problem can be considered the problem of recognizing the depicted images, finding their positional and metric characteristics. It is precisely such a problem that cannot always be solved. Restoring information about a geometric object is an important necessity.

The relevance of the topic is determined by the need for research in finding and indentifying general tecniques that often arise when applying various aspects of the reconstruction of geometric objects from their shadow proections.

The work defines the concept of «еllipsoid» as a common object of research. It is noted that in some implementations the problem of reconstructing an «ellipsoid» from its orthogonal shadow projections arises, which are used, for example, in nuclear technology to determine the energy parameters of microfuels, with stereological control to determine the parameters of inclusion domains. In this case, it is assumed that the surface of a granule or domain can be approximated with sufficient accuracy by an «ellipsoid».

There are many problems from various branches of science and technology, in the solution of which it is possible to apply the method of reconstructing geometring objects from their sufficiently informative shadow projections, for example, in aerial photography as a means of qualitatively improving X-ray images, in the field of plasma diagnostics, and especially in experiments on controlled thermonuclear fusion.

Key words: geometric objct; shadow projection; ellipsoid; reconstruction; stereology; granule; doman.

A REVIEW AND CLASSIFICATION OF ONLINE TOOLS FOR SUPPORTING MATHEMATICAL MODELING IN EDUCATION

Tetiana Tabler — 2025-07-09

The article provides a review and classification of modern online services used to support mathematical modeling processes in educational settings. The relevance of using online calculators is substantiated in the context of the digital transformation of education and the implementation of a competency-based approach to teaching mathematics. More than twenty online services have been systematized according to their functional capabilities, user orientation (students, teachers, university students, researchers), and modeling potential. Four main groups of services are identified: for engineering and scientific modeling; for use in general secondary and higher education institutions; mobile assistants (including photo calculators); and self-learning services with explanations. For each group, a specific tool is highlighted and an example of its use in education is provided: Maple – as an environment for creating mathematical models and 3D visualizations of complex objects; Desmos – as an interactive function graphing tool and instrument for exploratory learning; Photomath – as a mobile assistant capable of recognizing problems and providing step-by-step explanations; MathDF – as a step-by-step calculator for studying matrices, complex numbers, integrals, and equations. Examples of tasks, illustrations, and usage scenarios are provided for classroom application, considering modern teaching approaches such as project-based learning and STEM orientation. The article concludes with the rationale for the gradual integration of digital services into school and higher education as a means of developing mathematical competence, critical thinking, and digital literacy among students. It outlines the prospects for developing methodological recommendations for educators with sample exercises adapted to curricula of various levels, and identifies opportunities for integrating digital tools into teacher professional development systems. The importance of aligning technological infrastructure with pedagogical appropriateness and a sustainable digital education strategy in Ukraine is emphasized.

Keywords: mathematical modeling, online calculators, digital educational tools, computer-based learning, MathDF, Photomath, Maple, Desmos, function graphing, mobile applications, step-by-step solutions, STEM, project-based learning, mathematical competencies, digital transformation in education, mathematical visualization, analytical tools.

INFLUENCE OF SURFACE INCLINATION ON THE STABILITY OF CONCRETE MIXTURES IN MONOLITHIC STRUCTURES

Valery Usenko — 2025-07-09

The article addresses a pressing issue in modern construction — ensuring the uniform distribution of concrete mix on inclined and curved surfaces of monolithic spatial structures. Given the widespread adoption of innovative architectural solutions and the use of shell forms in contemporary construction, there is a growing need for scientific justification of concrete placement processes on complex surfaces. An analytical model is proposed, which takes into account the influence of surface inclination, shear yield stress, dynamic viscosity of the concrete mix, and velocity gradients on the nature of its movement and distribution. Special attention is given to identifying critical zones where local deformations are likely to occur, potentially compromising the strength and durability of the concrete layer. As the inclination angle increases, the gravitational force component along the formwork surface grows, increasing the risk of the mix sliding downwards. Maintaining a balance between the physical properties of the mix (particularly its viscoplasticity) and the surface geometry is crucial. The paper presents the equation of motion for concrete mix on an inclined base, with key parameters including layer thickness, inclination angle, depth of a point relative to the top edge, and the shear yield stress. Solving this equation yielded an analytical expression for calculating the critical inclination angle at which the concrete mix transitions into a flowing state. The article provides a comparative analysis of traditional and innovative methods for forming monolithic structures. It is shown that the efficiency of concreting depends not only on the properties of the mix but also on the precise consideration of gravitational effects, surface geometry parameters, and formwork characteristics. Pneumatic technology demonstrates promising potential for rapid assembly of geometrically complex concrete structures. The results of this study lay the scientific foundation for future work in modeling the behavior of concrete mixes on non-standard surfaces, optimizing technical solutions during the concreting of complex architectural forms, and enhancing the reliability and longevity of buildings.

Keywords: critical deformations, monolithic spatial coatings, defect prevention.

WEB APPLICATION DEVELOPMENT FOR 3D MODELING OF INTERIORS WITH APPLICATIONS ALGORITHMS OF SPATIAL ANALYSIS

Hanna Fedchenko — 2025-07-09

The work explores a current topic, which consists in the constant development of web design and web development technologies, which open up new opportunities for creating interactive and visually attractive websites. The development of 3D technologies allows creating detailed virtual models of interiors, which is especially important for design studios, as well as individual customers who seek to visualize the future arrangement of their home or office. The object of the work is the process of developing websites, while the subject is methods of visualization of 3D models of interiors within the framework of web design. The scientific novelty of the work consists in the adaptation and optimization of modern web development technologies for the integration of complex 3D models, which previously required significant computer resources and high computing power. The development of easy and affordable visualization methods can greatly simplify the interior design process and make it more accessible to a wide range of users.

The practical significance of the study is determined by the possibility of using the developed site for online presentations of interior solutions, which significantly increases the quality of customer service and ensures more effective interaction between designers and customers. When choosing a program for creating models for visualization, the Blender environment was preferred. Also, Verge3D technology was chosen for the implementation of the project. Vеrge3D is a powerful tool for creating interactive 3D web applications that allows you to directly interact with 3D models through a web browser. OpenServerPanel was chosen for the local server environment. This tool provides all the necessary tools for setting up a local web server, including support for Apache, MySQL, PHP and other necessary components. OpenServerPanel provides ease of installation and configuration, allowing you to quickly deploy web applications and test them before publishing. It also supports different server configurations, which allows you to adapt it to the specific requirements of the project. The result of the performed work is the analysis of the existing methods of computer technologies of 3D modeling and ray tracing, their possibilities and problems in solving interior visualization tasks, as well as the creation of a web application that provides interactive visualization of three-dimensional interior models

Keywords: 3D modeling, spatial analysis, ray tracing, Blender, Verge3D

DETERMINING THE MASS OF A SEMI-INFINITE STRING

Volodymyr Fomenko — 2025-07-09

A semi-infinite (singular) string with an unknown density distribution is considered. The mass of the string is also considered to be unknown and possibly infinite. The left end of the string, which can slide according to the known law f(t) in the direction perpendicular to its equilibrium state, generates a propagating wave. The interaction of this wave with the left end over a time of 2T produces a response R(f), which is measured.

As is known, the information f → R(f) (input – output) is sufficient to solve the inverse problem: find the density of the string at each point of the interval (0; x(T)) captured by the waves before the time T.

One of the effective methods for solving this problem is the BC method (Boundary Control method, M. Belishev, 1986), which is an approach to inverse problems based on their connection with the theory of boundary control. The ВС method uses waves that propagate inward, scatter on the inhomogeneities of the string and bring information to the boundary.

The ВС method allows one to find the parameters of the string in an optimal way in time based on the known response at the boundary: the reconstruction depth is proportional to the observation time at the boundary. This property is most relevant in geophysical studies, since it allows one to reconstruct the parameters of the medium in real time. Let us assume that an external observer has the ability to make measurements only at the boundary (left end) of the string. The question arises: what information about the qualitative nature of the string spectrum can he extract from these measurements?

As is known, the finiteness of the total mass of an string is a necessary condition for the discreteness of its spectrum [2], [3]. Therefore, the finiteness or infinity of the string mass is important information for the qualitative analysis of the string spectrum.

In this paper, in terms of the inverse problem data on the semiaxis (based on the known response operators R(f) for all T), a formula for the total mass of an inhomogeneous string with an unknown density distribution is obtained. The obtained formula makes it possible to effectively test the necessary condition for discreteness of the spectrum of a singular string.

Keywords: hyperbolic equations, singular boundary value problem, inverse problem, semi-infinite string, boundary control method, responsе operator.

ON THE DYNAMIC PROBLEM OF OPTIMAL SET PARTITIONING WITH FIXED CENTERS

Sergiy Yakovlev — 2025-07-09

The article formulates and investigates a dynamic optimal set partitioning (OSP) problem with fixed centers, which belongs to the class of spatial distribution problems with dynamically changing parameters. A formal problem statement, solution algorithm, and results of a numerical experiment confirming the efficiency and accuracy of the proposed approach are presented. The relevance of the selected topic is driven by the wide applicability of optimal partitioning problems in dynamic formulations for solving applied tasks in logistics, resource management, service area planning, and other domains where cost depends on time. Unlike static model problems, in real-world conditions, the transportation cost between elements may vary over time, significantly affecting both the search for the optimal solution and the value of the objective functional. The study considers several transportation cost models, including linear, exponential, and random (stochastic), enabling improved adaptation of models to specific applied contexts.

To conduct the numerical experiment, specialized software was developed in Python, integrating modern numerical methods such as step-by-step optimization, modified gradient approaches, and graphical result analysis tools. The model is based on the classical formulation of the optimal partitioning problem with fixed centers but introduces temporal dependencies into cost coefficients, allowing the simulation of complex distribution scenarios with time as a factor. A comparative analysis of modeling results for both dynamic and static approaches was carried out, demonstrating the advantages of the dynamic formulation: higher accuracy, greater flexibility, and adaptability to changing problem conditions.

The computations were performed on a personal computer with the following configuration: Intel Core i7-12700K, 8 cores, 3.6 GHz; 32 GB DDR4 RAM; 2 TB SSD. The obtained results confirmed not only the effectiveness of the mathematical model but also the correctness of the implemented algorithms, additionally illustrated through graphical phase trajectories and visualizations of the resulting partitions.

Keywords: dynamic problem, optimal set partitioning, objective functional, phase trajectory, numerical methods, time dependence, Python.