This paper explores the role of invariants in the analysis of special points within various classes of differential equations. By leveraging invariants, the study provides a framework for simplifying the identification and characterization of equilibrium points, singularities, and other critical features. The results demonstrate that invariants offer powerful tools for analyzing the structure and solutions of differential equations, especially in complex systems.

The efficiency of cotton cleaning machines plays a critical role in the textile industry, particularly in removing fine impurities. This paper focuses on enhancing the design of cotton cleaning machines to improve their performance. By analyzing the shortcomings of existing models, this research proposes modifications to the design of spiked drums and air suction systems to enhance cleaning efficiency. The results demonstrate that the proposed enhancements lead to significant improvements in impurity removal, reducing energy consumption and increasing cotton quality.

This study investigates the influence of nano-clay on the properties of alkali-activated water-cooled slag geopolymer, aiming to enhance its mechanical performance and durability. Alkali-activated materials have garnered significant attention as sustainable alternatives to conventional cement due to their lower carbon footprint and superior resistance to environmental degradation. In this research, varying proportions of nano-clay were incorporated into the geopolymer matrix, and a series of experiments were conducted to assess the resulting mechanical properties, including compressive strength, flexural strength, and workability.

Additionally, the microstructural characteristics were analyzed using techniques such as scanning electron microscopy (SEM) and X-ray diffraction (XRD) to elucidate the effects of nano-clay on the geopolymer formation process and bonding mechanisms. The results indicate a marked improvement in mechanical properties with optimal nano-clay incorporation, alongside enhanced durability against chemical attacks and freeze-thaw cycles. This study contributes to the growing body of knowledge on the use of nano-materials in geopolymer technology, highlighting the potential for developing high-performance, eco-friendly construction materials.

This article contains the following skills that students of architecture, design, engineering graphics and visual arts need to acquire: prognostic, reflexive, analytical, mobilization, perspective, orientation, informational, artistic skill, projective, communicative, engineer-technician, the ideas and contents of all approaches are fully covered.

This article examines the role of modern technologies in improving irrigation and soil moisture management in Uzbekistan’s agricultural sector. With 9.7% of the country’s land under irrigation, efficient water use is critical. Traditional soil moisture measurement methods are labor-intensive, while IoT-based sensors and automated systems offer real-time monitoring and optimization. Research findings show that these technologies enhance water conservation, increase crop productivity, and prevent soil degradation, making them essential for sustainable agriculture in water-scarce regions like Uzbekistan.

This study explores the role of local structural characteristics in influencing tracer diffusion mechanisms within amorphous Fe-based alloys. Amorphous materials, characterized by their lack of long-range order, exhibit unique properties that differ significantly from their crystalline counterparts. The diffusion behavior of tracer atoms within these alloys is critical for understanding their thermal stability, mechanical properties, and overall performance in various applications. Utilizing advanced characterization techniques, including nuclear magnetic resonance (NMR), X-ray diffraction (XRD), and atomic pair distribution function (PDF) analysis, we investigate the local structural arrangements and their correlation with diffusion pathways.

Our findings reveal that variations in the local environment, such as atomic coordination and clustering, significantly affect the mobility of tracer atoms. The results indicate that regions of increased atomic density facilitate higher diffusion rates, while disordered environments impede tracer movement. Additionally, we assess the impact of alloying elements on the local structure and diffusion behavior, providing insights into how compositional changes can be leveraged to optimize the properties of Fe-based alloys for specific applications.

This research contributes to a deeper understanding of the interplay between local structural characteristics and diffusion mechanisms in amorphous Fe-based alloys, offering valuable guidance for the design and development of advanced materials with tailored properties for industrial applications. Ultimately, the insights gained from this study may pave the way for enhancing the performance of amorphous alloys in a range of engineering fields.

Filtration processes of liquid solutions are fundamental in many natural and industrial applications, such as environmental protection, chemical engineering, water purification, and petroleum extraction. This article develops a mathematical model for describing filtration processes and explores various numerical methods for solving the governing equations. The model is based on Darcy’s law, continuity equation, and constitutive relations of liquid solutions in porous media. Numerical methods, including finite difference, finite element, and finite volume approaches, are discussed with applications to various filtration scenarios. We also provide analysis of the stability, convergence, and efficiency of these methods.