Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
840793 | Nonlinear Analysis: Theory, Methods & Applications | 2012 | 6 Pages |
Abstract
According to the Knaster conjecture, for any continuous function f:Sm+nâ2âRm and n distinct points u1,u2,â¦,unâSm+nâ2, there exists a rotation râSO(m+nâ1) such that f(ru1)=f(ru2)=â¯=f(run). In this paper, we focus on the study of the properties of a continuous mapping from a sphere to a Euclidean space by using the theory of Smith periodic transformation and the Smith special index of the Stiefel manifold under periodic transformation. We obtain some mapping theorems for the case where n=αβ,β is an odd prime number and uiâ
uj=ui+뱉
uj+α(1â¤i,jâ¤n,un+1=u1). Furthermore, if n=p, where p is an odd prime number, this conjecture is proved for the case where uiâ
uj=ui+1â
uj+1(1â¤i,jâ¤p,up+1=u1) and the dimension of the sphere is not less than m+nâ2.
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Authors
Yuhong Liu,