koz"Вестник Северо-Казахстанского университета имени Манаша Козыбаева"Bulletin of Manash Kozybayev North Kazakhstan University2958-003X2958-0048М. Қозыбаев атындағы СҚУ10.54596/2958-0048-2025-3-193-205koz-2101Research ArticleИНФОРМАЦИОННО-КОММУНИКАЦИОННЫЕ ТЕХНОЛОГИИINFORMATION AND COMMUNICATION TECHNOLOGIESМатематическая сингулярность: как модели поведения роя превращаются в роевой интеллект и позволяют строить образовательную траекторию обученияMathematical singularity: how swarm behaviour models turn into swarm intelligence and allow building educational trajectory of learninghttps://orcid.org/0000-0002-6299-3728КопноваО. Л.KopnovaO. L.

Оксана Леонидовна Копнова - кафедра Математики и физики, ст. преподаватель,

Петропавловск

Senior Lecturer, Department of Mathematics and Physics, PhD, Academician of the International Academy of Informatization, Department of Mathematics and Physics,

Petropavlovsk

oksanakopnova@gmail.comТаджигитовА. А.TadzhigitovA. A.

Петропавловск

Head of the Department of Mathematics and Physics, Candidate of Physical and Mathematical Sciences, Department of Mathematics and Physics,

Petrolpavlovsk

askartadzhigitov@mail.ruГригоренкоО. В.GrigorenkoO. V.

Новосибирск

Head of the Department of Postgraduate and Doctoral Studies, Candidate of Physical and Mathematical Sciences, AssociateProfessor,

Novosibirsk

ogrigorenko2311@mail.ruНАО «Северо-Казахстанский университет имени Манаша Козыбаева»КазахстанManash Kozybayev North Kazakhstan University NPLCKazakhstanСибирский государственный университета геосистем и технологийРоссияSiberian State University o f Geosystems and TechnologyRussian Federation20252410202503 (67)193205Copyright © Копнова О.Л., Таджигитов А.А., Григоренко О.В., 20252025Копнова О.Л., Таджигитов А.А., Григоренко О.В.Kopnova O.L., Tadzhigitov A.A., Grigorenko O.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.ku.edu.kz/jour/article/view/2101

Статья исследует феномен математической сингулярности в контексте роевого интеллекта и его применение для построения адаптивных образовательных траекторий. На примере модели движения муравьиной колонии (Ant Colony Optimization) демонстрируется, как локальные взаимодействия агентов приводят к глобально оптимальным решениям, преодолевая ограничения индивидуального поведения. Анализируется аналогия между поиском кратчайшего пути в природе и выбором персонализированной траектории обучения. Рассматриваются технологические и этические аспекты внедрения роевых алгоритмов в образовательные системы, а также перспективы создания самоорганизующихся платформ, способных адаптироваться к потребностям учащихся. Статья подчеркивает, что сингулярность в роевых системах открывает путь к созданию "умных" образовательных сред, где траектории формируются не сверху вниз, а через взаимодействие учащихся, данных и адаптивных алгоритмов.

The article investigates the phenomenon of mathematical singularity in the context of swarm intelligence and its application to the construction of adaptive educational trajectories. Using the example of the ant colony movement model (Ant Colony Optimisation), it is demonstrated how local interactions between agents lead to globally optimal solutions, overcoming the limitations of individual behaviour. An analogy between finding the shortest path in nature and choosing a personalised learning trajectory is analysed. The technological and ethical aspects of introducing swarm algorithms into educational systems are discussed, as well as the prospects of creating self-organising platforms capable of adapting to learners' needs. The article emphasises that singularity in swarm systems opens the way to the creation of smart educational environments where trajectories are not shaped from the top down, but through the interaction of learners, data and adaptive algorithms

роевой интеллектматематическая сингулярностьобразовательные траекторииинтеллектуальные системы обученияперсонализированное обучениеswarm intelligencemathematical singularityeducational trajectoriesintelligent learning systemspersonalised learningReferencesAltshuler, Y. (2023). Recent Developments in the Theory and Applicability of Swarm Search. Entropy, 25(5), 710.Altshuler, Y. (2023). Recent Developments in the Theory and Applicability of Swarm Search. Entropy, 25(5), 710.Abraham, A., Guo, H., & Liu, H. (2006). Swarm Intelligence: Foundations, Perspectives and Applications. In Swarm Intelligence: Foundations, Perspectives and Applications (pp. 23-42). Springer Berlin Heidelberg.Abraham, A., Guo, H., & Liu, H. (2006). Swarm Intelligence: Foundations, Perspectives and Applications. In Swarm Intelligence: Foundations, Perspectives and Applications (pp. 23-42). Springer Berlin Heidelberg.Ahmed, H.R., & Glasgow, J.I. (2012). 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Coordinating massive robot swarms. International Journal of Robotics Applications and Technologies, 2(2), 1-19; MacLennan, B.J. (2015). The morphogenetic path to programmable matter. Proceedings of the IEEE, 103(7), 1226-1232; MacLennan, B.J. (2018). Coordinating swarms of microscopic agents to assemble complex structures. In Swarm Intelligence, Vol. 1: Principles, Current Algorithms and Methods (pp. 583-612). Institution of Engineering and Technology.).Excerpts from "08.pdf' (This excerpt references several works by Bruce J. MacLennan and others, making it difficult to cite as a single entity. Some referenced works include: MacLennan, B.J. (2014). Coordinating massive robot swarms. International Journal of Robotics Applications and Technologies, 2(2), 1-19; MacLennan, B.J. (2015). The morphogenetic path to programmable matter. Proceedings of the IEEE, 103(7), 1226-1232; MacLennan, B.J. (2018). Coordinating swarms of microscopic agents to assemble complex structures. In Swarm Intelligence, Vol. 1: Principles, Current Algorithms and Methods (pp. 583-612). Institution of Engineering and Technology.).Excerpts from "09.pdf" (This appears to be an internal report supported by NASA, the specific citation details are not fully present in the excerpt).Excerpts from "09.pdf" (This appears to be an internal report supported by NASA, the specific citation details are not fully present in the excerpt).Excerpts from "15.pdf" (This excerpt contains references to various works in swarm intelligence and related fields, making it challenging to cite as a single source. Examples include: Kennedy J., Eberhardt R. (1995) Particle swarm optimization, Proc. IEEE Int. Conf. Neural Network, 1942-1948; Weber H. (1974) GrundriMer Insektenkunde, Gutav Fischer Verlag, Stuttgart.)Excerpts from "15.pdf" (This excerpt contains references to various works in swarm intelligence and related fields, making it challenging to cite as a single source. Examples include: Kennedy J., Eberhardt R. (1995) Particle swarm optimization, Proc. IEEE Int. Conf. Neural Network, 1942-1948; Weber H. (1974) GrundriMer Insektenkunde, Gutav Fischer Verlag, Stuttgart.)

The authors declare that there are no conflicts of interest present.