Symmetry

The incorporation of digital music into the reform of education has become one of the primary methods to improve educational outcomes, increase creativity, and innovate the practices in the classroom. This combination, together with the close industry–academia cooperation, presents the possibilities to integrate educational strategies in accordance with the technological and creative demands of the contemporary world. Nevertheless, uncertainty, reluctance, symmetry, and subjectivity in expert ratings are significant problems to cope with when considering multi-criteria decision-making (MCDM). To resolve them, this paper suggests a Picture Fuzzy Faire Un Choix Adequat (PF-FUCA) decision support model, where fifteen options will be rated by seven criteria, depending on the contribution of four professional decision-makers. These findings indicate that the PF-FUCA framework is effective and superior to the current PF-MCDM models, as illustrated by sensitivity and comparison analysis. The identified best strategies based on the framework are blockchain-based music copyright education and integrated music–STEM platforms that, when combined, offer a viable policy instrument to policymakers, educators, and industry stakeholders.

The main subject of this paper is the theme of differential operators defined for symmetric tensors on a Riemannian manifold and introduced in several new contexts. Some examples for 2-tensors are given and then the grad div operator for symmetric vector forms is defined. A few original operators in ${\mathbb{R}}^n$ related with grad div operator are discussed. Finally, important notions such as the Kenmotsu manifold, with some interesting examples, are also presented.

This paper tackles the challenge of disassembly sequence planning (DSP) in energy-efficient remanufacturing by introducing an innovative hybrid optimization framework. The proposed model integrates a Dynamic Time-Varying CRITIC–Entropy (DTVCE) decision-making framework with an Improved Honey Badger Algorithm (IHBA) to optimize disassembly sequences under key operational criteria, including idle rate, line smoothness, and energy consumption. The DTVCE framework constructs a dynamic composite score by normalizing evaluation criteria across time slices and incorporating temporal discounting to capture the evolving importance of each factor. Meanwhile, by establishing a symmetric disassembly constraint matrix to restrict the disassembly sequence and integrating exploration and exploitation mechanisms to enhance the IHBA, the solution process is empowered to efficiently generate feasible disassembly sequences and fulfill task allocation across workstations while satisfying takt time constraints. Experimental validation demonstrates that the proposed framework significantly outperforms traditional disassembly optimization approaches in both energy efficiency and line balance performance. In a case study involving an automotive drive axle, the method achieved a near-optimal configuration using only eight workstations, leading to a marked reduction in both energy consumption and idle times. Sensitivity analysis further verifies the model’s robustness, showing stable convergence and consistent performance under varying takt times and energy parameters. Overall, this study contributes to the advancement of green remanufacturing by offering a scalable, data-driven, and adaptive solution to disassembly optimization—paving the way toward sustainable and energy-aware production environments.