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Vol.20 (2007) >

 
Title :Inversions and Möbius invariants
Authors :Maehara, Hiroshi
Authors alternative :前原, 濶
Issue Date :30-Dec-2007
Abstract :Two $n$-point-sets in Euclidean space are said to be inversion-equivalent if one set can be transformed into the other set by applying inversions of the space. All 3-point-sets are inversion-equivalent to each other. For each four points $x,y,z,w$ in an $n$-point-set, $n\ge 4$, the ratio $\left( xy \cdot zw \right)$/$\left( xw \cdot yz \right)$ is invariant under inversions, which is called a Möbius invariant of the $n$-point-set. We prove that for $4\le n\le d+2$, the minimum number of Möbius invariants necessary to detetmine all Möbius invariants for every $n$-point-set in Euclidean $d$-space is equal to $n(n-3)/2$, and discuss the case of planar $n$-point-sets in some detail. We also characterize those fractional functions that are invariant under inversions.
Type Local :紀要論文
ISSN :1344-008X
Publisher :Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus
URI :http://hdl.handle.net/20.500.12000/4807
Citation :Ryukyu mathematical journal Vol.20 p.9 -23
Appears in Collections:Vol.20 (2007)

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