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We show that -tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the -tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of -tilting finite sincere simply connected algebras.
The compact simply connected Riemannian -symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form , where is a connected compact simple Lie group with an automorphism of order four on and is a fixed points subgroup of . According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian -symmetric spaces in the case where is of type . In the present article, we give the explicit form of automorphisms , and of order four on induced by the -linear transformations , and of the 248-dimensional -vector space , respectively. Further, we determine the structure of these fixed points subgroups , and of . These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces above corresponding to the Lie algebras , and , where . With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in .
Akiyama classified all cubic Pisot units with finite -expansions. In this paper, we show a generalization of Akiyama’s theorem by using some reduction theorem. Moreover we give another application of this reduction theorem.
We present new relationships between the work of H. Davenport and A. I. Popov. A new general formula involving the von Mangoldt function is presented, as well as expressions under the Riemann Hypothesis.
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