Article ID Journal Published Year Pages File Type
840793 Nonlinear Analysis: Theory, Methods & Applications 2012 6 Pages PDF
Abstract
According to the Knaster conjecture, for any continuous function f:Sm+n−2→Rm and n distinct points u1,u2,…,un∈Sm+n−2, there exists a rotation r∈SO(m+n−1) such that f(ru1)=f(ru2)=⋯=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 ui⋅uj=ui+α⋅uj+α(1≤i,j≤n,un+1=u1). Furthermore, if n=p, where p is an odd prime number, this conjecture is proved for the case where ui⋅uj=ui+1⋅uj+1(1≤i,j≤p,up+1=u1) and the dimension of the sphere is not less than m+n−2.
Related Topics
Physical Sciences and Engineering Engineering Engineering (General)
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