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With a decrease in the electric field strength, the values of the specific electric charge on negatively and positively charged aluminum oxide particles increase. At the same time, the absolute values of the specific charge on negatively charged particles are 2 times higher than the values of the specific charge on positively charged particles. This is due to the peculiarities of the formation of positive and negative charges on the reagent particles.
Additionally, experiments conducted to determine the specific yield of ice-forming nuclei showed that in the temperature range of -10...-12 °C, negatively charged particles give a specific yield of ice-forming nuclei of the order of 1012 particles per 1 gram of reagent, which is 2 times higher than positively charged ones. Conclusion
A complex of laboratory equipment and a method of conducting experiments for the study of aluminum oxide nanostructures have been developed. It is found that the electric charge on the particles affects their specific yield of ice-forming nuclei.
With a decrease in the electric field strength, the values of the specific electric charge on negatively and positively charged aluminum oxide particles increase.
The specific yield of ice-forming nuclei showed that in the temperature range of -10...-12 °C, negatively charged particles give a specific yield of ice-forming nuclei of the order of 1012 particles per 1 gram of reagent. References
1. Kachurin L.G., Bekryaev V.I. Investigation of the electrification process of crystallizing water. - Reports of the Academy of Sciences USSR, 1960. vol. 130, №1, pp. 57-60.
2. Kachurin L.G. Physical bases of influence on atmospheric processes: An experiment. atmospheric physics: [Study for universities on spec. «Meteorology»] / L.G. Kachurin. - L.: Hydrometeoizdat. 1990. - 462 p.
3. Imyanitov I.M., Chuvaev A.P. On the question of the main processes leading to electrification in thunderclouds. Proceedings of the MGO, 1957. issue 67 (129).
4. Mason. B.J., Maybank J. The fragmentation and electrification of freezing water drops. Quarterly Journal of the Royal Meteorological Society, 1960, 86(371), 176-185.
5. Khuchunaev B.M., Gekkieva S.O., Budaev A.Kh. Laboratory studies of the effect of electric field strength on the specific charge on reagent particles formed during the sublimation of pyrotechnic compositions / «Science. Innovation. Technologies», 2021 - №4. pp. 209-226.
© Khuchunaev B.M., Budaev A.Kh., Daov I.S., 2024
УДК 53
Owezov B.,
student. Pudakov B.,
teacher.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
APPLICATIONS OF MATHEMATICAL METHODS FOR ANALYZING BIOLOGICAL DATA
Abstract
The increasing complexity of biological data has made mathematical methods indispensable for
meaningful analysis. From modeling population dynamics to analyzing genetic information, mathematical tools provide robust frameworks for extracting insights from biological phenomena. This paper explores key applications of mathematical methods in biology, including statistical techniques, differential equations, and machine learning approaches. These tools not only enhance the understanding of biological systems but also facilitate predictions and decision-making in research and healthcare.
Biological data is inherently complex, characterized by high dimensionality and variability. To decipher patterns and relationships, researchers employ mathematical methods as foundational tools. These methods provide structured approaches to analyze, interpret, and predict biological phenomena. This paper highlights the critical applications of mathematical techniques in analyzing biological data, emphasizing their importance in modern biology and biotechnology.
Role of Mathematics in Biology Mathematics bridges the gap between raw biological data and meaningful insights: Data Modeling: Mathematical models simulate biological processes like disease spread and population growth. Statistical Analysis: Statistical methods quantify variability and identify patterns in experimental data. Computational Biology: Algorithms based on mathematics analyze large datasets, such as genomic sequences.
Applications of Mathematical Methods Population Dynamics Population dynamics use differential equations to model changes in population size over time:
where N is the population size, r is the growth rate, and K is the carrying capacity.
This logistic growth model predicts population stabilization when resources become limited. Statistical analysis in Genetics s tatistical methods are crucial for identifying associations between genes and traits. For example:
• Chi-Square Tests: Analyze genetic linkage in inheritance studies.
• Regression Analysis: Predict phenotypic traits based on genetic markers.
Machine Learning for Biological Data
Machine learning algorithms process large-scale biological datasets, such as gene expression profiles. Key applications include:
• Predicting protein structures using deep learning.
• Classifying diseases based on medical imaging data.
Challenges in Applying Mathematical Methods: Data Complexity: Biological data is often noisy and incomplete, complicating analysis. Computational Cost: Advanced methods like machine learning require substantial computational resources. Interdisciplinary Expertise: Effective application demands knowledge of both biology and mathematics. Recommendations for Effective Use Integrating Tools: Combine multiple mathematical methods to enhance analysis accuracy. Collaboration: Foster interdisciplinary collaboration between biologists and mathematicians. Training Programs: Develop educational programs to equip researchers with mathematical skills.
Mathematical methods have revolutionized the analysis of biological data, offering powerful tools for understanding complex systems. From statistical analysis to predictive modeling, these methods enable researchers to make informed decisions and advance biological knowledge. Continued innovation in mathematical techniques will further expand their role in addressing challenges in biology and healthcare.
Keywords:
mathematical methods, biological data, population dynamics, statistical analysis, machine learning, data modeling.
References
1. Murray, J. D. (2002). Mathematical Biology: I. An Introduction.
2. Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning.
3. Alon, U. (2006). An Introduction to Systems Biology: Design Principles of Biological Circuits.
© Owezov B., Pudakov B., 2024
УДК 53
Rahimjanov A.,
student.
Yagmyrova M.,
teacher.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
OPTIMIZATION TECHNIQUES IN MACHINE LEARNING Abstract
Optimization plays a central role in machine learning by enabling models to minimize errors and improve predictive accuracy. This paper explores key optimization techniques used in machine learning, including gradient descent, stochastic optimization, and second-order methods. By examining their mathematical foundations and practical applications, the study highlights the advantages, limitations, and selection criteria for different optimization strategies. Furthermore, the paper emphasizes the critical role of hyperparameter tuning and regularization in achieving robust and generalizable models.
Keywords:
machine learning, optimization, gradient descent, stochastic methods, regularization, hyperparameter tuning.
Optimization is a fundamental aspect of machine learning, as it directly affects the performance and efficiency of models. At its core, optimization involves minimizing or maximizing an objective function, such as loss functions in supervised learning or reward functions in reinforcement learning. This paper provides a comprehensive overview of key optimization techniques, focusing on their mathematical principles and practical implications in machine learning applications.
Mathematical Foundations of Optimization
Optimization aims to find the minimum (or maximum) of an objective function f(x)defined in a given domain:
where x* represents the optimal solution. Machine learning models typically involve non-convex functions, making the optimization process challenging.
Gradient-based methods rely on the gradient Vf(x) of the objective function:
where n is the learning rate. Variants include:
