Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series

Generalizations of the Rudin - Keisler preorder and their model-theoretic applications

Nikolai Polyakov, Denis Saveliev — Mon, 30 Jun 2025 00:00:00 +0000

Generalizing of the Rudin--Keisler preorder, we introduce relations $R_\alpha$ (and $R_{<\alpha}$) on the set $\beta\omega$ of ultrafilters on~$\omega$. They form an ordinal sequence of length~$\omega_1$ which is strictly increasing by inclusion and lies between the Rudin--Keisler preorder and the Comfort preorder. We show that the composition of these relations is expressed via a~multiplication-like operation on ordinals. Explicit calculations of this operation show that $R_{<\alpha}$ is transitive (and so, a preorder) if the ordinal~$\alpha$ is multiplicatively indecomposable. The proposed constructions have several model-theoretic consequences. Generalizing significantly results of Garc{\'\i}a-Ferreira, Hindman, and Strauss concerning an interplay between ultrafilter extensions of semigroups and the Comfort preorder, we prove that for every model~$\mathfrak A$, ultrafilter~$\mathfrak u$, and ordinal~$\alpha$, the set $\{\mathfrak u:\mathfrak u\,R_{<\alpha}\,\mathfrak v\}$ forms a~submodel of the ultrafilter extension~$\beta\mathfrak A$ of~$\mathfrak A$ if the ordinal~$\alpha$ is additively indecomposable. Furthermore, generalizing Blass' characterization of the Rudin--Keisler preorder via ultrapowers, we characterize the relations $R_\alpha$, and in particular, the Comfort preorder, via a~specific version of limit ultrapowers.

Multilevel Feature Fusion and Spatial Attention-Based Deep Learning Models for Personal Protective Equipment Detection

Nurzada Amangeldy, Alibek Baktybayevich, Nazira Tursynova — Mon, 30 Jun 2025 00:00:00 +0000

The proposed research work examines the effectiveness of deep learning models for the automatic recognition of personal protective equipment (PPE) in complex environments encountered in various industrial workplaces. Since PPE is designed to protect workers from multiple injuries and hazards. Monitoring their use using state-of-the-art deep learning models is an important solution. Given the diversity of these hazards, a state-of-the-art but unexplored multi-class dataset containing diverse scenes and fully annotated was trained on the Yolov11 deep learning network to evaluate their robustness and potential in real-world applications. The results show that the architecturally improved models can effectively handle complex and imbalanced data. In addition, the study conducted a comparative analysis with YOLOv8 and YOLOv10 models and performed an analysis for each architecture variant. This research contributes to the understanding of the topic by providing new insights and perspectives on the development of intelligent systems in the field of industrial safety, offering a practical solution to monitor workers' compliance with safety rules in the workplace and thereby saving lives.