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Unformatted text preview: S , what is the shape resulting if T2 and T3 are never applied? Prob 2.4.9 Define T : K(R2 ) → K(R2 ) by T (A) = {(2x, 2y ) : (x, y ) ∈ A}. Is T continuous in the Hausdorff metric? That is, for every A ∈ K(R2 ) and every ǫ > 0, find a δ > 0 for which h(A, B ) < δ implies h(T (A), T (B )) < ǫ. If T is not continuous, find a specific C ∈ K(R2 ) at which continuity fails. Prob 2.4.10 For T1 , T3 and T4 of (2.10), find T2 so the attractor of the IFS {T1 , T2 , T3 , T4 } is the unit square, and the region with address 21 is as shown in Fig. 2.31 Prob 2.4.11 Recall Λ(A) denotes the set of limit points of A. For all sets A, B ⊆ R, prove Λ(A ∩ B ) = Λ(A) ∩ Λ(B ), or find a counterexample. 2.5 Faster than random: de Bruijn sequences We have seen that the random IFS algorithm eventually will generate points in all regions of length n addresses for any given n, but how much time is needed for “eventually”? For example, if the image is contained in 1024 × 1024 pixels, and is generated by transformations (2.10), then a single pixel has address about 64 CH...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

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