Eurasian Journal of Mathematical Theory and Computer Sciences

This article explores the nature of sound vibrations and their wave properties. Sound is defined as a mechanical wave generated by a vibrating object, such as a tuning fork or human vocal cords, which causes the surrounding air molecules to oscillate. These oscillations are transmitted through the medium, eventually reaching the eardrum and converting into nerve impulses perceived by the brain. The study discusses the frequency range of sound waves audible to the human ear (approximately 20 to 20,000 Hz) and distinguishes between audible sound, infrasound (below 20 Hz), and ultrasound (above 20,000 Hz), which are inaudible to humans. It also examines the propagation characteristics of sound in different media: longitudinal waves in gases and liquids, and both longitudinal and transverse waves in solids. Furthermore, the paper addresses the subjective perception of sound (pitch, timbre, and loudness) and their corresponding physical parameters. It is noted that any real sound consists of a combination of harmonic oscillations with a specific frequency spectrum, referred to as the acoustic spectrum.

This article analyzes the possibilities of using geometric methods in solving quadratic equations. These approaches are important not only for strengthening theoretical knowledge, but also for solving practical problems.

The article presents a method for calculating the oil filter of a diesel locomotive taking into account the increase in its filtering capacity, reliability and performance, mathematical modeling was performed, numerical studies were carried out in the programming environment MATHCAD 15.

Ushbu maqola semantik qidiruv tizimlarining ilmiy asoslari, texnologik arxitekturasi va amaliy qo‘llanilish sohalari haqida tahliliy ma’lumot taqdim etadi. Semantik qidiruvning an’anaviy kalit so‘zga asoslangan tizimlardan farqi, tabiiy tilni ishlab chiqarish (NLP), ontologiyalar va graf bilimlari kabi texnologiyalarning roli yoritiladi. Maqolada tibbiyot, ta’lim, elektron tijorat va huquq sohalari misolida semantik qidiruvning amaliy qo‘llanilishi tahlil qilinadi. Shuningdek, tizimning texnik cheklovlari (hisoblash resurslari, ma’lumot sifati, til cheklovlari) va kelajakdagi rivojlanish yo‘nalishlari (ko‘p modal qidiruv, personalizatsiya, kvantsal hisoblashlar) ko‘rib chiqiladi. O‘zbekiston kontekstida semantik qidiruvni mahalliy tillarga moslashtirish va ilmiy tadqiqotlarni qo‘llab-quvvatlash bo‘yicha tavsiyalar beriladi. Maqola ilmiy jurnallar va amaliyot uchun qimmatli manba sifatida xizmat qiladi.

This article discusses the conditions under which a topological space is metrizable and how this relates to compactness. Key metrization theorems, such as those by Urysohn and Nagata-Smirnov, are examined. The article also explores the role of compactness in metrizable spaces, supported by examples like the Sorgenfrey line and Cantor set. Applications in analysis, computer science, and control theory demonstrate the practical importance of these concepts.

The article proposes a method and algorithm for constructing an informative phase, which is one of the most important issues of symbol recognition. All the concepts related to the problems of symbol recognition have been attempted to be expressed in a logical perfection, mathematical clarity and simplicity. The main issue of classification of symbols⁇, the formation of the phase of informative signs, the determination of the levels of importance of the objects and symbols of the study sample were thoroughly analyzed. The Fisher functional elements used to determine the values of the proximity function in the informative signs phase Upper and lower bounds of the similarity function between objects are identified and supporting data is provided.

This paper explores how artificial intelligence (AI) can be applied to enhance user interface (UI) and user experience (UX) design in web applications. By analyzing real-time user interaction data such as mouse movements, click patterns, and session time, machine learning models identify usability issues and recommend interface improvements. The study compares traditional heuristic evaluation with AI-driven approaches and demonstrates how data-informed UX redesigns significantly improve engagement metrics. Experimental results suggest that AI integration leads to reduced bounce rates and more efficient user navigation, validating its role in next-generation UX design workflows.

This article explores the integration of large language models (LLMs), such as ChatGPT, into modern web applications. It discusses the practical benefits—enhanced user interaction, automation, and intelligent response generation—alongside potential security risks including data exposure, bias propagation, and misuse. A technical prototype is implemented to assess the impact of LLMs on performance and user experience metrics. Empirical analysis shows measurable gains in task efficiency but highlights challenges in privacy and latency. The study provides a framework for ethical, efficient, and secure deployment of LLMs in dynamic web environments