In this paper we establish some results concerning the mixed-type reverse order laws for the Moore-Penrose inverse of various products of three elements in rings with involution.

This qualitative research study investigates the cultural symbolism and identity exhibited by ethnic minorities in Yunnan, China, through vocal music, as well as the ethnic minority music consequences. The study adopts a three-step coding analysis based on interviews with 18 respondents representing diverse ethnic minority groups to discover themes and patterns in the cultural symbolism buried within vocal music. For data analysis, the MAXQDA-2020 software is used. The findings show that ethnic minority groups in Yunnan rely heavily on vocal music to preserve cultural history, generate community pride, and promote intercultural understanding. The rich connection between vocal music and ethnic identity, the influence of nature and environment on musical expressions, the preservation of cultural heritage through rituals and celebrations, and the evolving collaborations and contemporary adaptations in vocal music are among the themes extracted from the data. This study has three implications: general, theoretical, and practical. In overall, this study adds to our understanding of vocal music as a potent medium for cultural expression and identity building. The identification of recurring themes and patterns has theoretical significance, offering a platform for further investigation and comparative investigations. In practice, the findings influence ethnic minority music practices by emphasizing the importance of introducing vocal music traditions into curricula, encouraging cultural awareness, and developing international discussion.

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According to the features of asynchronous interaction in systems, a new-type Pi calculus with buffers-Buffer-Pi calculus is proposed, the new labelled transition system based on buffers is introduced, the enhanced describing capability is shown to apply Buffer-Pi calculus to modeling with the concrete example of asynchronous interaction, and the new behavior equivalence relations are defined, several propositions and properties of Buffer-Pi calculus are given. The study shows that compared with Pi calculus, Buffer-Pi calculus can provide more powerful support for asynchronous behavior modeling in systems.