Eurasian Journal of Mathematical Theory and Computer Sciences

The following explains that if (a,b;c) is a Pythagorean pair, then the number a is a Pythagorean pair of the number b, and the Pythagorean triples … , then the Pythagorean pairs for the number a are given. It is also stated that if the number a is a Pythagorean pair of the number b, then the number b is also a Pythagorean pair of the number a. Pythagorean numbers, determining the number of Pythagorean pairs of a Pythagorean number are explained.

This paper considers the automorphisms of central extensions of Type I algebras. It is proved that any automorphism that fixes the center and is continuous in the local measure topology is inner.

This article examines the technical analysis of human speech in educational activities. During the study, speech signals were converted into digital format and analyzed in terms of formants, spectrograms, amplitude, and temporal parameters. The obtained results allow assessment of pronunciation, intonation, stress, and rhythm of speech. Moreover, automatic signal analysis algorithms enable the objectification of the pedagogical process and the organization of individual training sessions. The study demonstrates that technical analysis of speech is an effective tool for scientifically substantiating the pedagogical process and improving speech skills.

This article presents the development and validation of a numerical simulator for optical communication systems that enables comprehensive modeling of signal transmission, propagation, and reception in fiber-optic channels. The proposed approach combines physical accuracy with computational efficiency, allowing for both scientific analysis and practical system design. The simulator implements the nonlinear Schrödinger equation using the split-step Fourier method to account for chromatic dispersion, nonlinearity, attenuation, and amplifier noise. Simulation results demonstrate a strong correlation with experimental data, showing a deviation within 5–7%, thus confirming the reliability of the model.

This article analyzes the functional model of the class teacher in the pedagogical management system, its theoretical foundations, as well as problems in the process of practical activity and their solutions. During the study, the effectiveness of the educational, organizational, communicative and monitoring functions of the class teacher was studied, and scientific and theoretical approaches to improving these functions in a modern educational environment were developed. Also, the problems encountered in the activities of the class teacher - such factors as poor individual work with students, insufficient communication with parents, and low use of digital tools - were deeply analyzed. The article provides effective proposals for the implementation of innovative management technologies, digital monitoring systems, and educational work based on cooperation to solve these problems. The results of the study serve to optimize the pedagogical management system and increase the effectiveness of the activities of class teachers.

This article provides an in-depth analysis of the impact of Artificial Intelligence (AI) tools, specifically ChatGPT, Photomath, and WolframAlpha, on higher education, with a focus on mathematics instruction. The vulnerability of traditional assessment systems to modern technologies, the crisis of academic integrity, and the risk of declining critical thinking skills in students are highlighted. The primary objective of this article is to develop new pedagogical criteria, "AI-proof" assignments, and strategies for fair assessment of student knowledge in an environment where AI tools are freely accessible. As a practical case study, the topic "Differential of a function and main theorems (Rolle, Lagrange, Cauchy)" is selected, presenting contextual and analytical assignment examples and assessment rubrics for each theorem.