We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V 0 = {p 1 , p 2 , p 3 } be the set of vertices of SG and u i (x) = 1 2 (x + p i ) the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations u w = u w 1 u w 2 · · · u w n for any sequence w = (w 1 , w 2 , . . . , w n ) ∈ {1, 2, 3} n . The union of the images of V 0 under these iterations is the set of nth stage vertices V n of SG. Let F : V n → R be any function. Given any numbers α w (w ∈ {1, 2, 3} n ) with 0 < |α w | < 1, there exists a unique continuous extension f : SG → R of F , such thatfor x ∈ SG, where h w are harmonic functions on SG for w ∈ {1, 2, 3} n . Interpreting the harmonic functions as the "degree 1 polynomials" on SG is thus a self-similar interpolation obtained for any start function F : V n → R.
We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit nontrivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case. 0
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