In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

This comprehensive study delves into the implementation and effectiveness of the flipped classroom model in engineering EMI (English as a Medium of Instruction) courses in Taiwan. Situated within the strategic framework of Taiwan's Bilingual Nation 2023 policy, this research aims to enhance the global competitiveness of Taiwan's semiconducting manufacturing engineering sector. The study’s primary focus is on understanding the perceptions and learning outcomes of EFL (English as a Foreign Language) students in engineering courses. Through a methodical approach using descriptive statistics and chi-square tests, the study analyzes data from 33 junior engineering students specializing in mechanical and mechatronic fields. The findings highlight a significant positive influence of the flipped classroom model on various aspects of student learning, including increased engagement, heightened creativity, and enhanced problem-solving skills. Additionally, the study emphasizes the critical role of timely L1 support in facilitating effective learning in EMI settings. These outcomes not only demonstrate the flipped classroom's viability as a sustainable educational approach but also provide crucial insights for educators and policymakers. By aligning cutting-edge teaching methods with the evolving needs of engineering education, this study contributes to the discourse on educational innovation. It ultimately guides the development of future engineering programs that cater to diverse student needs and fosters the cultivation of skilled engineering professionals.

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.