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By Kaczynski , Mischaikow , Mrozek
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This publication constitutes the completely refereed post-proceedings of the second one foreign convention on Numerical research and Its purposes, NAA 2000, held in Rousse, Bulgaria in June 2000. The ninety revised papers offered have been conscientiously chosen for inclusion within the e-book throughout the rounds of inspection and reviewing.
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7 Let ;n := fx 2 Rn+1 j jjxjj = 1g: There is no deformation retraction of ;n to a point. We include this example at this point to try to indicate that this is a nontrivial problem. In particular, we encourage you to try to nd a proof of this fact. As motivation for the study of this subject we assure you that once you know homology theory, this example will become a triviality. 1 Prove that homotopy is an equivalence relation. 2 Let f g : X ! Y be continuous maps. Under the following assumptions on X and Y prove that f g.
Since T has no free vertices, there is an edge e2 with vertices v2 such that v1+ = v2;. Continuing in this manner we can label the edges by ei and the vertices by vi where vi; = vi+;1. Note since there are only a nite number of vertices, at some point vi+ = vj; for some i > j 1. Then fej ej+1 : : : eig forms a loop. This is a contradition. 11 Every edge is homotopic to a point. Proof: Let e be an edge with vertices v ; and v+. Since e is a line segment it is homeomorphic to 0 1]. Let h : 0 1] !
But what about f#1 ( 0 1]) where f#1(f0g) = f#1(f1g) = f0g? e. that f#1 ( 0 1]) = 0. 2. APPROXIMATION OF MAPS 65 rules to each of the intervals we obtain the following matrix 20 0 0 03 66 0 0 0 0 77 66 0 1 0 1 77 f#1 = 66 1 0 0 1 77 66 7 4 1 0 0 0 75 1 0 0 0 In guring out how to de ne f#1 we used the phrase \it seems reasonable to de ne" but this does not mean we should not de ne it a di erent way. Given our choice for f#0 are there any restrictions on the way we de ne f#1? The answer is an emphatic yes.