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Some global error bounds with undetermined parameters, which are not always valid, for the extended vertical linear complementarity problems (LCP) of CKV-type matrices and CKV-type B -matrices, are presented by Yan and Wang (Jpn. J. Ind. Appl. Math. 41:129–150, 2024). In this paper, new global error bounds for the extended vertical LCP of CKV-type matrices and CKV-type B -matrices are given, which depend only on the entries of the involved matrices. Numerical examples show that the new bounds are better than those provided in Zhang et al. (Comput. Optim. Appl. 42(3):335–352, 2009) and Wang et al. (Comput. Appl. Math. 40:148, 2021) in some cases. [ABSTRACT FROM AUTHOR]

Lesson study (LS), a teacher-oriented, student-focused professional development (PD) approach that originated in Japan, has spread globally. However, existing literature on the implementation of LS and its effectiveness provides inconsistent results, suggesting a need to review current research on LS. With a focus on LS in mathematics education, we examined 75 recent LS studies using Lewis's (ZDM Mathematics Education 48:571–580, 2016. https://doi.org/10.1007/s11858-016-0792-x) framework to understand how LS is implemented and the pathways by which LS can impact teaching practice and student learning. We found that new developments have taken place in LS implementation, although challenges persist throughout its process. Regarding the implementation of LS, even though the examined LS studies generally contained four phases (study, plan, teach, and reflect), we found many LS lacked a research question, and the study of teaching materials was sometimes invisible. Across phases, studies shared a consensus that it is most critical to focus on students' thinking and learning, yet reported challenges in maintaining this focus. In addition, the role of knowledgeable others (KOs) was recognized but inconsistently understood. Collaboration was also widely reported as a challenge. Finally, there were large variations in LS duration, with some LS implementing overly brief cycles. Regarding LS impact, the literature has more frequently reported changes in teachers' knowledge and beliefs and less frequently in curriculum, learning community/professional norms, teaching practice, and student learning outcomes. The above challenges reflect a need for culturally relevant systemic support for developing sustainable and large-scale LS. We suggest future directions for continued research and practice improvement. [ABSTRACT FROM AUTHOR]

Pedagogical content knowledge (PCK) is one form of teachers' professional knowledge in subject teaching, and teachers' rich PCK enables effective instruction and improves students' academic performance. However, there has been limited research on the relationships of individual difference characteristics of teachers to PCK among in-service elementary school teachers. Therefore, in addition to the demographic variables (gender and years of teaching experience) and psychological variables (beliefs about teaching and learning and teacher efficacy) examined in previous studies, this study attempted to clarify whether motivation for teaching is related to PCK. We conducted a web survey of inservice elementary school teachers in Japan (n D 267). The results showed that the traditional beliefs that students are to be controlled by their teachers and indifference, which describes a state of lack of motivation to prepare for class, were negatively associated with two elements of mathematical PCK (knowledge of learners and knowledge of instruction). Furthermore, multiple regression analysis revealed that traditional beliefs about teaching and learning were negatively associated with the knowledge of learners and indifference to subject instruction with knowledge of instruction. This suggests that teachers' motivation for teaching is related to PCK, in addition to the variables that have been previously examined. [ABSTRACT FROM AUTHOR]

Let G be a group. An automorphism α of G is called a commuting automorphism if [α(g), g] = 1 for all g ∈ G. Let A(G) denote the set of all commuting automorphisms of G. A group G is said to be an A(G)-group if A(G) forms a subgroup of Aut(G), where Aut(G) denotes the group of all automorphisms of G. In [Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 5, 57-60] Rai proved that a finite p-group G of co-class 2 for an odd prime p is an A(G)-group. We prove that a finite p-group G of co-class 3 for an odd prime p, under some conditions, is an A(G)-group. [ABSTRACT FROM AUTHOR]