Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences

MODELING AND ANALYSIS OF THE STABILITY OF A PULSE-CONTROLLED GENERATOR WITH A PENDULUM-INERTIAL CONICAL ROTOR AND MAGNETS

2026-06-17T13:35:00+06:00

This study examines the stability and efficiency of a generator with a pendulum-inertial cone-shaped rotor using permanent and variable magnets under pulse control and increased electrical load. The research was conducted to determine how pulsed energy input and different magnetic configurations influence rotor stability, angular velocity fluctuations, and energy conversion efficiency, especially near critical load resistance. A mathematical model was developed that incorporates rotational dynamics, magnetic interactions, pulsed disturbances, friction losses and electrical load. Numerical simulations using Python and NumPy evaluated the behaviour of angular velocity, induced EMF, current and accumulated energy across varying rotor masses, geometric parameters, pulse amplitudes and load resistances. Special attention was given to pulse-feeding modes activated when rotor speed decreases. The results show that adaptive pulsed pumping and variable magnets enhance stability and prolong efficient operation under high loads. These findings support the design of autonomous and energy-efficient generator systems with nonlinear mechanics and intelligent control.

COMPARATIVE ANALYSIS OF BEHAVIORAL VIDEO ANALYTICS APPROACHES BASED ON MACHINE LEARNING

2026-05-30T11:09:33+06:00

This review systematizes and analyzes modern approaches to the intelligent detection of anomalies in human behavior based on deep learning in video surveillance systems. The work explores key methods, including hybrid architectures, generative models, and multimodal approaches. The main purpose of the study is to identify the key limitations of existing solutions and propose ways to overcome them by developing a new conceptual architecture.

The analysis showed that modern models achieve high accuracy (F1-score in the range of 90-95%) on standard datasets, but face three fundamental problems: a lack of labeled anomaly data, high computational complexity that prevents real-time operation on edge devices, and low reliability with external interference.

To solve these problems, a hybrid multimodal architecture is proposed that uses compressed-domain analysis to optimize the speed of inference and a Gated Cross-Attention mechanism for intelligent merging of video and audio streams. The proposed architecture demonstrates the potential for creating a reliable, scalable and proactive monitoring system.

STRESS LEVEL MONITORING: A REVIEW OF IOT SOLUTIONS AND PHYSIOLOGICAL INDICATORS

2026-06-16T22:51:37+06:00

Stress can have a significant impact on a person's physical and psychological health. It may lead to serious and severe consequences such as anxiety, depression, cardiovascular diseases, cognitive impairments, and a weakened immune system. Therefore, early detection of stress, continuous monitoring of its levels, and the implementation of preventive measures are among the key areas of modern medicine, psychology, and healthcare systems. In recent years, there has been a growing interest in innovative platforms for stress assessment and management, including devices based on artificial intelligence, the Internet of Things (IoT), and biosensors. The role of these technologies in healthcare is becoming increasingly significant, as they enable continuous recording, analysis, and real-time data processing of an individual’s physiological parameters. This paper provides a comprehensive review of stress monitoring and assessment methods based on modern technologies. In particular, it explores the potential of using sensors for physiological parameter registration and IoT-based technologies. Sensors designed for measuring physiological indicators allow for the continuous monitoring of parameters such as heart rate, respiratory rate, skin conductivity, body temperature, blood pressure, blood oxygen level, and other vital metrics. These data are processed using various methods for determining and assessing stress levels. These technologies facilitate timely responses to an individual's condition, the implementation of effective measures for stress reduction, overall health improvement, and personalized patient care. Such innovative approaches open new perspectives in the healthcare system, contributing to an enhanced quality of life and improved well-being of the population.

GRAPH-MATRIX DESCRIPTION OF MULTI-CONTOUR INTERACTIONS IN A BIOTECHNOLOGICAL BEER FERMENTATION SYSTEM

2026-06-16T22:35:39+06:00

The paper presents a graph-matrix model of the biotechnological beer fermentation process based on a systematic approach to describing interactions between substrates, enzymes, products, and environmental regulators. The proposed method combines the structural and kinetic aspects of the process, which makes it possible to formalize the relationship between biochemical reactions, heat and mass transfer, and control actions. The model is implemented in the MATLAB environment and tested in a series of numerical experiments confirming its adequacy to experimental data. It is shown that the graph-matrix representation effectively describes multi-contour interactions and provides a convenient tool for stability analysis, parameter optimization, and the development of automatic fermentation control systems. The obtained results demonstrate the possibilities of integrating graph theory methods, biotechnological modeling and the digital twin of the process, which helps to increase the accuracy of forecasting and controllability of technological parameters at the stages of brewing.

INTEGRATION OF MULTISENSORY DETECTION SYSTEMS AND ARTIFICIAL INTELLIGENCE ALGORITHMS IN A MOBILE ANTI-DRONE SYSTEM

2026-06-11T13:13:24+06:00

The authors of the article comprehensively investigate a promising direction in the field of security – the integration of multisensor detection systems and artificial intelligence (AI) algorithms into mobile anti-drone systems. The purpose of this study is to analyze modern technologies of multisensor detection of unmanned aerial vehicles (UAV), as well as to develop principles for building a mobile anti-drone system that integrates detection systems and artificial intelligence algorithms. It is argued that effective counteraction in a dynamic environment requires a mobile system with high autonomy and reaction speed. The principles of building a multisensor system combining radar, optoelectronic, and radio-frequency means are considered, which allows overcoming the limitations of each individual sensor through multimodal data fusion. The role of artificial intelligence algorithms in target detection, classification, data fusion, and real-time decision support is analyzed in detail. An architecture for integrating sensors and artificial intelligence into a single mobile platform is proposed, ensuring adaptability and resilience to challenging operating conditions. The paper summarizes key technological trends and forms a holistic vision for building highly effective next-generation mobile means of countering unmanned aerial vehicles (UAV).

AN ON-DEVICE KAZAKH LANGUAGE AGENT FOR HUMAN–ROBOT INTERACTION

2026-06-21T14:59:55+06:00

Service and assistive robots deployed in Kazakhstan must understand commands in Kazakh, an agglutinative, lower-resource language that general-purpose large language models handle poorly at the small scales that fit on a robot’s on-board computer. We present Farabi-0.6B, a 596-million-parameter Kazakh-centric (Kazakh/Russian/English) language agent obtained by continued pre-training and supervised fine-tuning of Qwen3-0.6B, and study its use as the language-understanding, retrieval, and action-selection core of an intelligent human–robot interface that is small enough to run locally on edge hardware. We make three contributions. First, we describe the interface architecture: a Kazakh command is mapped to an intent, the agent decides whether to invoke a robot skill, request a missing argument, abstain, or seek confirmation, and grounds informational answers in a retrieved manual. Second, we evaluate the language core on a purpose-built, by-construction simulated robot-command benchmark (68 Kazakh/Russian/English commands across five decision categories) and on standardized Kazakh benchmarks. The model maps commands to robot skills with 82% skill-selection and 79% slot-filling accuracy when it acts, and the agentic fine-tuning improves genuine clarification (+20 pp) and out-of-scope abstention (+20 pp) over a pre-agentic baseline; on standardized Kazakh tasks it far exceeds the same-size base (e.g. Belebele-kk 34.0 vs 25.5; FLORES en→kk chrF 37.4 vs 0.0). Third, we characterize edge feasibility: the 0.6B model needs 0.30–1.19 GB for weights and sustains 21 tok/s on a CPU and a projected 57–228 tok/s on named embedded accelerators — comfortably above conversational real time. We also report two honest weaknesses with direct safety implications: the model does not seek confirmation before irreversible actions (100% violation) and under-routes informational queries to retrieval (10%), which motivates an explicit safety-gating layer in the interface rather than reliance on the model alone.

A REVIEW OF DEEP LEARNING METHODS FOR DETECTING CORROSION AND CRACKS IN INDUSTRIAL PIPELINES

2026-06-18T23:33:47+06:00

Corrosion and cracks in industrial pipelines cause safety hazards, economic losses, and environmental damage. Timely detection is crucial for avoiding catastrophic failures. Traditional nondestructive testing (NDT) methods require human interpretation and often lack reliability in harsh environments. The purpose of this review is to analyze deep learning techniques used for pipeline defect detection and evaluation. The study focuses on convolutional neural networks (CNN), vision transformers (ViT), and image segmentation models applied to corrosion and crack identification. Different data sources are considered, including visual inspection, ultrasound imaging, radiography, and drone-based monitoring. The review compares available datasets, their limitations, and labeling difficulties. Inspection scenarios such as real-time monitoring, underwater pipelines, and high-temperature environments are highlighted. Challenges related to noise, occlusion, illumination changes, and generalization are discussed. The analysis also covers decision-making systems that support risk assessment and maintenance planning. The findings show that deep learning significantly improves defect detection accuracy compared to traditional approaches. However, model performance depends on data quality and domain adaptation. The review concludes that integrating multimodal sensing, real-time inference, and explainable artificial intelligence (XAI) is essential for practical deployment. Future research must focus on robust dataset development, uncertainty estimation, and autonomous inspection systems.

MODELING AND OPTIMIZATION OF A HYBRID HADOOP–SPARK ARCHITECTURE TO IMPROVE BIG DATA PROCESSING EFFICIENCY

2026-06-12T14:47:36+06:00

With the rapid growth of data volumes, heterogeneity, and intensity, the demands on architectures that provide not only high performance, but also robust scalability, efficient use of computing resources, and fault tolerance are increasing. This article examines a hybrid big data processing architecture that combines the Hadoop Distributed File System and Apache Spark operational processing mechanisms. The goal of the study is to develop a formalized approach to evaluating and optimizing the efficiency of a hybrid environment compared to the standalone use of Hadoop and Spark.

The paper proposes a system of analytical models describing the processing speed, scalability, resource utilization, overhead, and overall efficiency of the hybrid architecture. Unlike studies that limit platform comparisons to general characteristics or isolated benchmarks, this article focuses on the relationship between data storage, inter-node communication, computing load, and cluster configuration parameters. It is demonstrated that combining Hadoop's distributed storage mechanisms with Spark's in-memory processing reduces the impact of disk I/O, improves resilience to increasing load, and ensures a more balanced use of memory and CPU resources.

These results confirm that the hybrid architecture is a promising solution for building scalable analytics platforms designed to process heterogeneous data under variable and intensive workloads. The practical significance of this study lies in the potential use of the proposed models in the design and configuration of regional and enterprise big data analytics systems.

MATHEMATICAL MODEL OF A NEW POST-QUANTUM CRYPTOGRAPHIC ALGORITHM FOR DIGITAL SIGNATURE

2026-06-19T09:59:34+06:00

Abstract

This research is dedicated to formulating mathematical model for Falcon-M, an innovative post-quantum digital signature algorithm that operates without relying on trapdoor mechanisms. The primary intent is to mitigate the architectural complexity commonly associated with lattice-based signature schemes, while simultaneously ensuring their inherent structural integrity and computational efficacy. The findings delineate the convergence characteristics of discrete Gaussian sampling within the quotient polynomial ring establish an upper bound for the likelihood of hash collisions within the signature scheme, and deduce explicit analytical boundaries for the propagation of floating-point errors during both forward and inverse Number-Theoretic Transform computations. It is observed that Falcon-M streamlines the key generation process when juxtaposed with trapdoor-based designs, all while maintaining an optimal quasilinear computational complexity of O(n log n). These results advance the theoretical underpinnings of lattice-based post-quantum signatures and support the creation of cryptographic systems that are both numerically stable and structurally less intricate.

HYBRID RAG WITH KNOWLEDGE GRAPH FOR 5G/O-RAN CODE DOCUMENTATION

2026-06-09T14:05:51+06:00

This paper addresses a critical limitation of Retrieval-Augmented Generation (RAG) systems in domain-specific code question-answering: imprecise context retrieval actively misleads small language models, degrading accuracy below the baseline. We propose an entity-aware hybrid retrieval architecture that extracts the target code entity (class or function name) directly from each question and applies it as a hard filter on retrieved chunks. The system integrates semantic vector search (ChromaDB) with a kNN-based knowledge graph (NetworkX, 53,585 nodes) fused via Weighted Reciprocal Rank Fusion (WRRF). A two-stage quality-aware fallback mechanism rejects low-relevance context before generation. Evaluation on the srsRANBench dataset (1,502 multiple-choice questions over 500,000+ lines of srsRAN C++ code) shows the Graph-Enhanced system achieving 65.51% accuracy vs. 63.65% for Vector-only, with ROUGE-L of 0.1820-the best among all systems. Error analysis reveals that the knowledge graph rescues 87 vector-only failures. The "RAG hurts" phenomenon-where retrieved context misleads a 3B-parameter LLM-is systematically analyzed and shown to be mitigated by entity-aware filtering.

DEEP LEARNING-BASED AIR QUALITY FORECASTING AND ANOMALY DETECTION USING CNN AND DBSCAN CLUSTERING

2026-06-21T16:39:07+06:00

Air pollution poses a serious risk to environmental sustainability and public health, especially in urban agglomerations. Accurate air quality prediction and early anomaly detection are crucial for effective environmental management and preventive medicine. This paper proposes a deep learning-based method that integrates Convolutional Neural Networks (CNNs) for air quality level prediction and DBSCAN clustering for identifying anomalous pollution patterns. A dataset spanning five years (2018–2022) consisting of 2,797 samples and 13 attributes was utilized for the training and evaluation of the models. The dataset was divided into 70% for training, 15% for validation, and 15% for testing. The CNN model was trained for 50 epochs using the Adam optimizer with has a batch size of 32, employing Mean Squared Error (MSE) as the loss function. The model's performance was assessed using standard metrics, including Accuracy, Precision, Recall, and F1-score findings demonstrate a consistent and dependable performance, achieving an overall accuracy of 76.0%, precision of 77.5%, recall of 76.2%, and an F1-score of 76.3%. The confusion matrix also indicated areas of strength and weakness, particularly the impact of false negatives on public safety. With a view of improving anomaly sensitivity, this currently proposed technique in this study, has the potential to be used in real-time monitoring of air quality and policy support.

FORECASTING DEMAND FOR COMPUTER CENTER SERVICES MACHINE LEARNING METHODS

2026-06-15T11:08:57+06:00

This paper presents an applied study of the problem of short-term forecasting of the daily number of requests to a computer service center. For a two-year time series covering 2023–2024, calendar, lag-based, and meteorological features were constructed, and a hybrid ensemble of base models ˗ Ridge, ElasticNet, SVR, Random Forest, XGBoost, and LightGBM ˗ was implemented. To prevent data leakage, strict temporal feature shifting, logarithmic transformation of the target variable, and a TimeSeriesSplit scheme with five splits were employed. Ensemble weights were optimized as a constrained optimization problem using SLSQP, Particle Swarm Optimization (PSO), Differential Evolution (DE), and Simulated Annealing (SA). The best performance was achieved on the baseline feature set: the hybrid ensemble with PSO-based weight optimization reached RMSE = 2.0847, MAE = 1.5988, MAPE = 20.62%, and R² = 0.6315. Compared to the best single model in terms of MAPE (SVR, 21.29%), the improvement amounted to 0.67 percentage points. It is shown that increasing feature space complexity through trigonometric time encoding and additional derived features does not improve generalization performance. The obtained results confirm the practical applicability of compact ensembles for forecasting noisy operational demand affected by chaotic factors.

IMPROVING CLASSIFICATION ACCURACY ON IMBALANCED DATA USING A HYBRID MODEL

2026-06-17T10:51:47+06:00

In the context of the rapid growth of data volumes, the problem of class imbalance has become one of the key challenges in classification tasks, significantly reducing the accuracy and generalization ability of machine learning models. The aim of this study is to improve classification accuracy on imbalanced datasets through the development and application of a hybrid model that combines data preprocessing techniques and ensemble learning methods. To achieve this goal, existing approaches to handling class imbalance were analyzed, including resampling techniques (oversampling and undersampling), cost-sensitive learning, and modern ensemble strategies.

The research methodology is based on the integration of synthetic data generation with gradient boosting and random forest algorithms. This approach enhances sensitivity to the minority class while maintaining model robustness against overfitting. The proposed hybrid model was evaluated on several open-source and applied datasets with varying degrees of class imbalance. The performance assessment was conducted using metrics suitable for imbalanced data, including F1-score, balanced accuracy and other.

The obtained results demonstrate a statistically significant improvement in classification performance compared to baseline models, especially in detecting the minority class. The scientific significance of the study lies in the development of a reproducible approach to improving classification effectiveness under class imbalance conditions, thereby expanding the applicability of machine learning methods in domains such as healthcare, finance, and risk analysis.

A REVIEW OF AI-BASED INTELLIGENT TEACHING SYSTEMS FOR PERSONALIZING EDUCATION

2026-06-29T17:12:55+06:00

The integration of artificial intelligence (AI) technologies into educational processes is significantly impacting the development of personalized learning systems. An analysis of the key functions of intelligent tutoring systems demonstrates that these AI technologies provide educators with opportunities to continuously improve their teaching methods while meeting students' educational needs. The aim of this study is to determine the current state of development of intelligent tutoring systems.

This article presents a comprehensive overview of AI technologies used to implement personalized learning in the education system, proposes a classification, and analyzes the specific features of their practical use. Specific systems are used to examine data on the goals, technologies, and content generation in intelligent tutoring systems. This systematic literature review utilizes the PRISMA method to select relevant studies, allowing for an analysis of various approaches to the use of intelligent tutoring systems, the tools used, their results, and methods for evaluating their effectiveness. The paper highlights the important role of intelligent tutoring systems in the educational process and their positive impact on students' academic performance: their use in both school and university settings has contributed to a significant increase in knowledge levels. The basic architecture of the developed intelligent learning system, aimed at supporting gifted children, is considered.

The basic architecture of the developed intelligent learning system, aimed at supporting gifted children, is considered.

DEVELOPMENT OF IT MICRO-QUALIFICATIONS AND THEIR INTEGRATION INTO THE TRAINING OF TEACHING PERSONNEL

2026-06-09T18:24:15+06:00

The ongoing transformation of higher education requires not only the enhancement of teachers’ professional competencies but also the development of new approaches to increasing their professional engagement and motivation. The digitalization of educational processes, the widespread adoption of artificial intelligence technologies, and changes in the academic environment have a direct impact on teachers’ emotional resilience, their ability to manage professional stress, and their overall work performance. In this context, the integration of emotional intelligence and digital technologies is considered an important factor in supporting the professional development of higher education teachers.

This article analyzes the pedagogical potential of applying emotional intelligence-based digital technologies to enhance the professional engagement of higher education teachers. The study examines mechanisms for monitoring teachers’ emotional well-being and fostering professional engagement through digital platforms, artificial intelligence tools, online assessment systems, digital coaching, and analytical services. The paper presents the pedagogical opportunities offered by emotional intelligence-based digital technologies in higher education institutions and highlights their role in creating supportive and sustainable professional environments for educators.

DEVELOPING STUDENTS’ DIGITAL COMPETENCE THROUGH THE USE OF ARTIFICIAL INTELLIGENCE TOOLS IN TEACHING THEORETICAL FOUNDATIONS OF INFORMATICS

2026-06-18T00:00:36+06:00

In the context of global technological transformation and the transition to the Industry 4.0 concept, the integration of artificial intelligence tools into higher education has become increasingly relevant. Alongside unprecedented opportunities for the personalization of learning, this process also generates significant cognitive, ethical, and pedagogical challenges. The purpose of this study is to develop, implement, and comprehensively evaluate the effectiveness of innovative pedagogical strategies aimed at enhancing students’ digital competencies through the use of artificial intelligence and interactive visualization tools in teaching the theoretical foundations of informatics.

The methodological framework of the study was based on the Action Research approach, which enables the gradual improvement of educational practices within a natural academic environment. During two research cycles, the process of students’ mastery of algorithms with hidden logic was analyzed. For this purpose, the VisuAlgo algorithm visualization platform, as well as the neural network models ChatGPT, Gemini, Claude, and the Gamma presentation generation platform, were employed. The findings statistically and qualitatively confirm that the integration of artificial intelligence tools into the educational process requires a highly structured pedagogical design. The scientific significance of the study lies in extending the DigComp 2.2 conceptual framework and substantiating the need to develop AI literacy as an essential component of the future specialist’s digital competence. The results demonstrate that artificial intelligence tools should function not as a means of replacing students’ cognitive efforts but as a powerful catalyst for developing computational thinking and analytical skills.

THE EFFECTIVENESS OF TEACHING DIGITAL LITERACY TO PRIMARY SCHOOL STUDENTS BASED ON ELECTRONIC HIERARCHICAL STRUCTURES

2026-06-13T14:52:29+06:00

This article examines the use of electronic hierarchical structures in teaching digital literacy to primary school students. The study compared the learning outcomes of control and experimental groups in order to evaluate the effectiveness of the proposed approach. Within the framework of a pedagogical experiment, students’ academic achievement, learning activity, learning motivation, and the development of digital skills were investigated. Assessment was carried out according to motivational, content-based, and technological criteria. Statistical analysis of the collected data showed that the performance indicators of students in the experimental group were higher than those of the control group. The findings indicate that a structured presentation of learning materials contributes to a more systematic understanding of digital literacy content, improves students’ ability to identify relationships between concepts, and supports the development of digital skills.

STEM ТЕХНОЛОГИЯСЫ АРҚЫЛЫ ИНЖЕНЕРЛІК ОЙЛАУДЫ ДАМЫТУ: ЭМПИРИКАЛЫҚ ДЕРЕКТЕРГЕ НЕГІЗДЕЛГЕН ЗЕРТТЕУ

2026-06-11T21:19:25+06:00

This article examines the development of engineering and computational thinking within the framework of STEM education. The purpose of the study is to empirically assess the level of engineering thinking among school learners and university students engaged in STEM-oriented learning and to analyze its relationship with academic achievement in computer science. The study involved school learners from Kazakhstan and 50 undergraduate students from the Department of Informatics at M. Auezov South Kazakhstan University. Research instruments included the Computational Thinking Scale (CTS) and author-designed test tasks, with data analyzed using Item Response Theory (IRT). The relationship between engineering thinking and academic performance was examined through regression analysis. The results indicate that STEM-based instruction significantly enhances algorithmic thinking, modeling skills, and problem structuring abilities. The findings confirm the effectiveness of STEM education in fostering engineering thinking and provide practical implications for educational practice.

DEVELOPMENT OF A ROBOTICS-BASED LEARNING SYSTEM FOR CHILDREN WITH SPEECH IMPAIRMENTS

2026-06-13T15:26:09+06:00

The article provides a comprehensive examination of the methodology for using robotics to develop the linguistic, cognitive, and communicative skills of children with speech impairments. The main aim of the study is to contribute to the expansion of children’s language abilities, improvement of their academic achievements, and enhancement of their social adaptation levels through the effective integration of robotics into special and inclusive education systems. The study employed theoretical analysis and software modeling methods, and also considered pedagogical approaches aimed at organizing and optimizing the learning process. As a result, a comprehensive teaching model for children with speech impairments was developed, along with the structure and functional capabilities of the EduRobo learning platform. The findings of the study may serve as a basis for advancing inclusive education in Kazakhstan and providing quality support to children with special educational needs.

BOUNDARY VALUE PROBLEMS ON A TWO-LINK THERMAL GRAPH AND THEIR SOLUTIONS

2026-06-09T11:08:11+06:00

The paper investigates stationary and nonstationary boundary value problems on a two-link thermal graph with local and coupled boundary conditions. A linear graph consisting of two sequentially connected rods with different thermal parameters and a single junction point is considered. Based on the generalized functions method and the Fourier transform with respect to time, a unified technique for solving boundary value problems on a two-link thermal graph is developed. Resolving systems of equations for the Dirichlet and Neumann–Dirichlet problems are constructed, and analytical integral representations of the solutions are obtained. Numerical experiments were carried out to study temperature distribution and the influence of oscillation frequency on the thermal state of the graph. The obtained results can be applied to modeling thermal processes in multilink rod structures used in construction, mechanical engineering, and thermal networks.

ANALYSIS OF A PARABOLIC APPROXIMATION FOR A LINEAR UNSTEADY NAVIER–STOKES PROBLEM

2026-06-05T12:16:30+06:00

The paper investigates a parabolic approximation of a linear unsteady Navier–Stokes problem for an incompressible viscous fluid. The proposed approach replaces the incompressibility constraint with an evolutionary pressure equation that includes a small regularization parameter. Such a transformation makes it possible to apply methods from the theory of parabolic differential equations and simplify the analysis of the mathematical model.

The study employs techniques of functional analysis, energy estimates, and the Galerkin method to examine the properties of the approximating system. Conditions ensuring the existence, uniqueness, and regularity of generalized and strong solutions are established. Particular attention is paid to the convergence of the approximating solution to the solution of the original Navier–Stokes system as the approximation parameter tends to zero. Quantitative estimates characterizing the convergence rate in various functional spaces are obtained.

To demonstrate the practical relevance of the proposed approach, a model problem related to transient flow in a pipeline system is considered. The results indicate that the parabolic approximation provides a satisfactory balance between computational efficiency and solution accuracy. The estimates obtained justify the method's applicability to the mathematical modeling of unsteady hydrodynamic processes and may serve as a basis for developing efficient numerical algorithms in computational fluid dynamics.

SYNTHESIS OF SELF-ORGANIZING CONTROL SYSTEMS IN THE CLASS OF TWO-PARAMETER STRUCTURALLY STABLE MAPPINGS

2026-06-09T11:24:15+06:00

This paper presents a methodology for the synthesis of a self-organizing control system for a single-input single-output object within the class of two-parameter structurally stable mappings. The proposed approach is based on a Lyapunov framework using a gradient-type method applied to Lyapunov vector functions. Conditions defining aperiodic robust stability of stationary states are derived for both the control system and its corresponding model with prescribed transient characteristics. These conditions are formulated as parameter inequalities that define admissible regions ensuring non-oscillatory convergence of system trajectories. The obtained stability conditions are used to compute the controller coefficients, establishing a relationship between system parameters and admissible stability regions. Simulation results show that the state trajectories converge to the equilibrium point, while the control signal remains bounded. The transient response is well damped and does not exhibit oscillatory behavior. The results demonstrate that the proposed approach can be applied to the design of control systems operating under parametric uncertainty.

APPLICATION OF THE ADJOINT EQUATION METHOD IN THE NUMERICAL SOLUTION OF THE INVERSE PROBLEM OF MAGNETOTELLURIC SOUNDING

2026-07-01T01:59:28+06:00

The article discusses the theoretical foundations and numerical methods for solving the direct and inverse problems of magnetotelluric sounding (MTS) based on Maxwell’s equations. A mathematical description of the geoelectrical model consisting of horizontally layered isotropic media with different electrical resistivities is presented. The inverse problem is solved using the optimal control approach and the adjoint equation method. A gradient iterative algorithm is developed to reconstruct the distribution of electrical conductivity from observational MTS data. Numerical experiments confirm the stability and convergence of the proposed method: as the number of iterations increases, the objective functional decreases significantly, and the reconstructed parameters closely approach the true values. The results demonstrate the effectiveness of the MTS technique in interpreting geoelectrical structures and its practical importance for assessing the deep-layer electrophysical properties of the Earth’s crust in geophysical exploration.

СРАВНИТЕЛЬНЫЙ АНАЛИЗ МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ ДИНАМИКИ pH В ПРОЦЕССЕ ФЕРМЕНТАЦИИ АЙРАНА

2026-06-24T13:19:24+06:00

This paper presents a comparative analysis of four mathematical models (logarithmic, modified Gompertz, logistic, and Baranyi) for describing pH dynamics during ayran fermentation. Experimental data were obtained from laboratory fermentation runs with two functional additives at varying dosages (0–4% w/w). pH measurements were performed using a calibrated potentiometric pH meter at 2-hour intervals (at 2, 4, 6, 8, and 10 hours) over a 10-hour fermentation period. The logarithmic model pH(t) = a – b·ln(t+1) – c·Dose showed highest accuracy: R²=0.9985, MAE=0.015, RMSE=0.018. The Baranyi model showed moderate performance, with R²=0.9856 for the control, 0.9717 for Additive 1, and 0.9928 for Additive 2. The Gompertz and logistic models showed poor performance, with highly negative R² values, including −180.5 (Gompertz) and −55.8 (logistic) for the control group, indicating a severe mismatch with the experimental data. The inverse problem for target pH=4.30±0.05 is solved analytically. One‑way ANOVA confirmed a statistically significant effect of additive type on final pH (F(2,3)=25.00, p=0.013) and fermentation time (F(2,3)=13.69, p=0.031). Post‑hoc Tukey HSD analysis revealed that both additives significantly altered acidification kinetics compared to the control (p<0.05). The inverse problem of predicting the time required to reach the target pH = 4.30±0.05 is solved analytically using the logarithmic model. The proposed dose‑dependent logarithmic model provides a robust, computationally efficient tool for quantitative analysis of ayran fermentation kinetics and can be applied to other low‑viscosity fermented dairy systems, enabling formulation optimization and process control without specialized hardware.

ON THE DISCRETIZATION OF THE HEAT EQUATION SOLUTION FROM NIKOL’SKII - BESOV CLASS

2026-06-05T11:34:41+06:00

The aim of this study is to construct a finite object (discretization operator) that approximates with any accuracy the solution of the heat equation with an initial condition from the multidimensional

Nikol’skii - Besov

functional class in a certain metric. The research methodology is based on well-known statements of approximation theory. The relevance of the study carried out here is explained by the following circumstances: firstly, in this article, for a solution of the heat equation representable in the form of an absolutely convergent multiple functional series and belonging to the periodic

Nikol’skii - Besov

class, a discretization operator approximating it with any accuracy is proposed, constructed from the values of the initial condition at the points of a uniform grid of the unit cube; secondly, it is proven that any discretization operator constructed from a given finite set of values of linear functionals defined on the functional class under consideration does not improve the order of the error obtained when approximating the solution by the proposed discretization operator.

CURRENT ADVANCES IN THE STUDY OF WAVE PROCESSES IN DEFORMABLE FLUID-SATURATED POROUS MEDIA

2026-06-22T00:10:17+06:00

The study of wave processes in fluid-saturated porous media represents a fundamental problem in continuum mechanics, holding key significance for geophysics, seismic exploration, hydrogeology, and acoustics. Over the past seven decades, significant research efforts have been directed toward developing mathematical models describing coupled deformation and filtration processes in such media. This review systematizes and analyzes the main stages in the development of the theory of wave processes in deformable fluid-saturated porous media, starting from the foundational works of Maurice Biot and extending to modern generalizations incorporating anisotropy, viscoelasticity, relaxation effects, and stochastic parameter variability. Special attention is given to the comparative analysis of constitutive relations, wave types (fast and slow compressional, shear), interface boundary conditions, and numerical methods for implementing the considered models. Current trends in the development of the theory are identified, including the application of fractional calculus to account for memory effects in porous media and stochastic approaches for quantifying parameter uncertainties.

THE EFFECTIVENESS OF USING STEM APPROACHES IN TEACHING GEOMETRY

2026-07-01T02:16:36+06:00

This article examines the pedagogical effectiveness of applying STEM approaches in teaching geometry. STEM-based education (Science, Technology, Engineering, and Mathematics) is considered a modern educational approach aimed at developing students’ spatial reasoning, analytical thinking, creativity, and research skills. The study analyzes the possibilities of using modeling, 3D visualization, GeoGebra software, project-based learning, and practical tasks in geometry instruction. A review of the scientific literature demonstrates that STEM approaches contribute to a deeper and more meaningful understanding of geometric concepts. In addition, these approaches increase students’ motivation and help connect geometry with real-life situations, engineering practices, and technological processes. The article emphasizes the importance of teachers’ professional preparedness, digital resources, and sufficient material and technical support for the successful implementation of STEM education. The findings indicate that integrating STEM technologies into geometry lessons positively influences students’ learning motivation and improves their analytical and spatial thinking skills. Thus, STEM-based instruction enhances both conceptual understanding and practical application of geometry in the educational process.

THE ROLE AND PLACE OF OLYMPIAD AND NON-STANDARD PROBLEMS IN THE SCHOOL MATHEMATICS CURRICULUM

2026-06-29T10:53:38+06:00

The main goal of teaching students to solve Olympiad and non-standard mathematical problems is to enhance their mathematical abilities and overall level of preparedness. This article examines the essence, classification, and theoretical and methodological foundations for the use of Olympiad and non-standard problems in the school mathematics curriculum. Based on an analysis of scientific and methodological literature, the conceptual differences between “standard problems,” “non-standard problems,” and “Olympiad problems” are identified, and their role in developing students’ logical, analytical, and creative thinking is substantiated. The views of domestic and international scholars are compared, and an author’s classification model of Olympiad problems according to content and solution methods is proposed. In addition, the main methods and heuristic techniques used in solving non-standard and Olympiad problems are systematized, and effective ways of integrating them into the instructional process are demonstrated. The research findings confirm that the systematic use of such problems promotes the development of students’ mathematical culture, cognitive activity, independent thinking, and research skills. The proposed methodological conclusions are intended for use in school mathematics teaching practice, elective courses, and Olympiad training programs.

Key words: school mathematics course, Olympiad problems, non-standard problems, problem-solving methods, heuristic techniques, mathematical thinking, mathematical culture, Olympiad training, teaching problem solving.

METHODOLOGICAL ASPECTS OF TEACHING STUDENTS TO SOLVE MATHEMATICAL PROBLEMS USING NON-STANDARD METHODS

2026-06-22T02:10:35+06:00

The article examines the theoretical and methodological aspects of teaching school students to solve mathematical problems using non-standard methods. The relevance of the study is determined by the need for advanced mathematics instruction in lyceums and gymnasiums, the development of students’ logical, analytical, and creative thinking, as well as preparation for effective performance in mathematical olympiads and final assessment examinations (BBJM, ENT).

The study analyzes the works of foreign and domestic scholars and methodologists devoted to non-standard methods of solving mathematical problems and systematizes their classification. The stages and specific features of applying non-standard methods are illustrated through concrete examples, and their influence on students’ cognitive activity is analyzed. In addition, the article describes the organization and results of a pedagogical experiment conducted with students of grades 10-11 in a general secondary school. The experimental data confirm that the purposeful use of non-standard methods has a positive effect on the dynamics of students’ problem-solving skills, logical thinking, and cognitive independence. The effectiveness of the proposed methodological approaches is substantiated, and the possibility of using the research materials in classes with advanced mathematics instruction, elective courses, and olympiad preparation is justified.

DIGITALIZATION OF AL-FARABI'S PLANIMETRIC LEGACY AND ITS ADAPTATION TO THE SCHOOL EDUCATION SYSTEM

2026-05-30T11:07:55+06:00

Al-Farabi's scientific heritage is one of the important but insufficiently studied areas in the modern Kazakhstani education system. Today, few compatriots know that in the Middle Ages his works formed the basis of world science and contributed to the formation of "quadrivium" disciplines in European universities. In order to fill this gap, the presented article is aimed at integrating Farabi's geometric heritage into the national educational process. The main goal of the study is to informatize teaching by supplementing the school curriculum with Farabi's geometric problems, their solution methods and substantive terminology. To this end, teachers’ continuous professional development and the flexible integration of the proposed content within the scope of their existing pedagogical workload are envisaged. The project is intended to be implemented through the initiative and commitment of teachers, without disrupting the existing curriculum framework. The primary challenge lies in the effective incorporation of theoretical materials that are not included in current textbooks into the educational process. To support the informatization of schools through this heritage, digital technologies and elements of the GeoGebra-based online approach are employed.

The principal outcome of the study is the provision of educational resources that enable young people in Kazakhstan to become acquainted from an early age with the scholarly contributions of Al-Farabi, a national figure of pride who made a substantial contribution to the development of mathematics. As such, the study constitutes a noteworthy informational and educational achievement, offering an innovative contribution to an issue of national importance through both its substantive content and its application of a digital GeoGebra-based approach