We consider a three-state solid-on-solid (SOS) model in the presence of a nonzero external field on a Cayley tree. A system of functional equations corresponding to this model is derived, where each solution defines a quasi Gibbs measure. Based on this system, we investigate the translation-invariant p -adic quasi Gibbs measures (TIpQGMs) of the model. In the case of a binary Cayley tree, we explicitly determine the TIpQGMs under the same conditions. Furthermore, by analyzing the boundedness of these measures, we establish the existence of a phase transition.
In contemporary education, immersive technologies (VR/AR) and artificial intelligence (AI) are opening up new opportunities for preserving and teaching the national cultural heritage. This article proposes an innovative pedagogical model developed for music clubs in Uzbekistan, in which virtual reality (VR), augmented reality (AR) and AI integrate the teaching of national musical instruments (Kashgar rubab, doira, gijjak, dutor) with the 4C skills (critical thinking, creativity, collaboration, communication). The model was formed on the basis of the electronic programs, the "Innovative Music Notebook", the game "Musical Wonders" and the VR mock-up ("Centre for the Creative Development of Music Clubs with the Help of 4C Technologies") created within the framework of the project "Development of Modernized Innovative Systems of Contemporary '4C' Music Teaching Technologies". The novelty lies in the fact that the model combines traditional pedagogical approaches (Orff, Kodály, gamification) with an immersive, AI-based environment, thereby strengthening pupils' cultural identity and developing their life skills. Experimental results show an increase in pupils' motivation and musical literacy in an immersive environment.
The paper investigates a Dirichlet-type boundary value problem for a three-dimensional elliptic equation with three singular coefficients in a domain bounded by two parallel planes in the first octant. The unique solvability of the stated problem in the class of regular solutions is established. The uniqueness of a solution is proved using the method of energy integrals. To prove the existence of a solution, the Hankel integral transform method is employed.
Speech etiquette is a dynamic system of communicative norms that reflects social structures, cultural values, and personal identity. It functions not only as a set of polite formulas but as a mirror of society, encoding collective traditions and individual strategies of interaction. The choice of etiquette forms is shaped by social identity, which regulates how speakers perceive themselves and others in communicative contexts. Key factors such as age, gender, profession, national culture, and virtual identity influence the expression of respect, status, equality, and social distance. Professional groups often rely on formal address to maintain hierarchy, while youth communities prefer informal greetings and slang to emphasize solidarity. Thus, speech etiquette emerges as a sociolinguistic phenomenon that evolves with social and cultural transformations, serving as a vital tool for expressing belonging, respect, and identity.
The article examines the influence of social networks and information media on the formation of a health-saving culture of future teachers. The concepts of health, culture, health-saving culture, its formation are analyzed, indicating the importance of personal health. Human health represents not only an individual value but also a strategic resource of society. In the context of social transformation and modernization processes, the preservation and strengthening of public health becomes a priority direction of state policy. The level of health of citizens directly influences the socio-economic development, competitiveness, and sustainable progress of the country. After gaining independence, the Republic of Kazakhstan has placed significant emphasis on strengthening the health of the population. In modern conditions of spiritual and social renewal, personal responsibility for health is becoming a key component of civic consciousness. Public health is viewed as a fundamental condition for entering the group of the thirty most competitive countries in the world.
