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The crossing number of composite knots
It is a very old conjecture that the crossing number of knots is additive
under connected sum. In other words, if K#K' is the connected sum of knots K
and K', then does the equality c(K#K') = c(K) + c(K') hold? We prove that
c(K#K') is at most c(K) + c(K') and at least (c(K) + c(K'))/152.Comment: 28 pages, 20 figures; final version, to appear in the Journal of
Topolog
Smooth norms and approximation in Banach spaces of the type C(K)
We prove two theorems about differentiable functions on the Banach space
C(K), where K is compact.
(i) If C(K) admits a non-trivial function of class C^m and of bounded
support, then all continuous real-valued functions on C(K) may be uniformly
approximated by functions of class C^m.
(ii) If C(K) admits an equivalent norm with locally uniformly convex dual
norm, then C(K) admits an equivalent norm which is of class C^infty (except at
0)
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