FractionalGeometry-Chap2

B find the minimum for which o is simply connected

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Unformatted text preview: act sets A, B , and C , we show h(A, B )+ h(B, C ) ≥ h(A, C ). Write h(A, B ) = δ1 and h(B, C ) = δ2 . A ⊆ Bδ1 (2.4) B ⊆ Cδ2 (2.5) and From Eq (2.5), we see Bδ1 ⊆ (Cδ2 )δ1 ⊆ Cδ1 +δ2 , where the second containment follows from Lemma 2.2.1. Together with Eq (2.4), this shows A ⊆ Cδ1 +δ2 . A similar argument gives C ⊆ Aδ1 +δ2 , from which we see h(A, C ) ≤ δ1 + δ2 = h(A, B ) + h(B, C ). In the next section, we use the Hausdorff metric to show that the deterministic IFS algorithm generates a sequence of compact sets converging to a unique limit, the attractor of the IFS. Practice problems Prxs 2.2.1 (a) Find the Hausdorff distance between Q, the set of rational numbers in [0, 1], and I , the set of irrational numbers in [0, 1]. (b) Interpret the answer to (a) in terms ofthe positive-defninteness of the Hausdorff metric. 41 2.2. THE HAUSDORFF METRIC Prxs 2.2.2 In Fig. 2.20 we see the Sierpinski carpet O, constructed from the filled-in unit square by removing the...
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