Algebras of acyclic cluster type: tree type and type A~\widetilde{A}

In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type \A_n for each possible orientation of \A_n. We give an explicit way to read off in which derived equivalence class such an algebra lies, and describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.Comment: v2: 37 pages. Title is changed. A mistake in the previous version is now corrected (see Remark 3.14). Other changes making the paper coherent with the version 2 of 1003.491

Mother of God, Mother of Christianity: The Development of the Marian Tradition in Early Modern Japan

The Christian figure of the Virgin Mary, first introduced as Jesus’ mother in the Bible, has since been repeatedly reinterpreted in various roles and imagery through her incorporation into different cultures. This project analyses the historical adoption and adaptation of Mary among Christian converts in Japan, from the arrival of Jesuit missionaries in 1549 to the end of the Tokugawa era in the nineteenth century. An examination of doctrinal prayers, the rosary, and Marian iconography within Japan illustrates Mary’s role as the Mother of God and compassionate intercessor for early Japanese Christians. Moreover, their affinity for Mary enabled Christianity to endure centuries of religious oppression under the Tokugawa shogunate as the hidden believers accommodated Mary within preexisting Buddhist traditions and Japanese customs. In studying the primacy of Mary and her roles within Japanese Christianity, this research explains a main reason behind the religion’s survival in early modern Japan