The paper considers the pricing problem of complementary products in a fuzzy dual-channel supply chain environment where there are two manufacturers and one retailer. Four different decision models are established to study this problem: the centralized decision model, MS-Bertrand model, RS-Bertrand model and Nash game model, where the consumer demand and manufacturing cost for each product are characterized as fuzzy variables. A closed form solution has been obtained for each model by using game theory and fuzzy theory. Numerical examples are presented to compare the maximal expected profits and optimal pricing decisions, and to provide additional managerial insights. The finding shows that the decision makers are more likely to choose industries with higher self-price elastic coefficient and lower complementarity in the retail channel to cooperate. We can obtain that consumers can benefit from the cooperation of the two manufacturers because of lower prices. We can also find that it might not be bad for retailer because it can expand demand and obtain more maximal expected profits.

Introduction: The ability to solve complex mathematical problems has become a key indicator of students' mathematical literacy and innovative capacity.Methods: Based on the TIMSS 2023 data and focused on eighth-grade students in Taiwan, China, the United States, and Turkey, the study investigates the interaction mechanisms among students' learning interest (LI), classroom disruptive behavior (CDB), self-efficacy (SE), and complex mathematical problem-solving ability (CMPSA) through constructing a structural equation model.Results: LI, CDB, and SE are significantly correlated with CMPSA and can predict CMPSA. In addition, LI not only influences CMPSA directly, but also has an indirect effect through CDB and SE, including both parallel and chain mediation effects. Cross-cultural analysis further reveals significant regional differences in the impact mechanisms of CMPSA. To be specific, Taiwan, China is mainly characterized by the direct effect of LI, while the United States and Turkey rely on the indirect path of SE.Discussion: These explorations systematically reveal the formation mechanism of CMPSA from the perspective of multidimensional interaction of cognition, emotion, and environment, enrich the cross-cultural theoretical framework of CMPSA, and provide empirical basis for optimizing mathematics teaching practices and formulating regionally adaptive intervention strategies under different education systems.

Introduction: Learning engagement (LE) is a core determinant of academic achievement and educational quality. Methods: Based on PISA 2022 data and focusing on a sample of adolescents in Korea ( N = 6,308), the current study attempts to investigate how intellectual curiosity (IC) is associated with LE through multiple indirect pathways involving teacher support, encompassing universal, targeted, accompanying, and ongoing teacher support (UTS, TTS, ATS, OTS), and school belonging (SB). Results: The significant correlations were demonstrated of varying strengths among IC, UTS, TTS, ATS, OTS, SB, and LE. Further analysis indicated that IC not only influences LE directly, but also exerts indirect effects through UTS, TTS, ATS, OTS, and SB. Specifically, TS and SB exhibited both multiple parallel and multiple chain mediating effects in the relationship between IC and LE, thereby extending the self-determination theory. Additionally, subgroup analyses revealed that the mechanism exhibited significant and distinct divergence along the gender dimension. For instance, males reported a lower perception of TTS, whereas females' LE is more strongly shaped by the indirect effects of TS and SB. Discussion: These findings elucidate the complex mechanisms how IC affects LE, validate the four-dimensional structure of TS to extend SDT, offer a gender-sensitive intervention perspective for differentiated teaching, as well as provide practical insights for the holistic development of adolescents.

Mathematical problem-solving ability (MPSA), as a core indicator for measuring students’ higher-order thinking development, is crucial for fostering