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Unformatted text preview: } , then d 1 ( ~x, ~ y ) is just the Hamming distance between the two vectors (i.e., the number of places in which the two strings diﬀer). We can also deﬁne  ~x  ∞ = lim p →∞ (  ~x  p ) . Going through the math, we have that  ~x  ∞ = max i (  x i  ) . We then have that C ∞ is a square: 5 This means that for a mathematician, a circle is a circle, is a diamond, is a square (which may explain why I always had trouble with those “shape matching” tests . . . :) In general, we have the following sort of picture of various circles: 6 Homework exercises: What happens for 0 < p < 1? What happens if we take the limit as p goes to 0? Show that in the limit as p goes to 0, the corresponding distance d ( ~x, ~ y ) =  ~x~ y  is a generalized Hamming distance that counts the number of coordinates that are diﬀerent from each other . . . 7...
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 Fall '09
 Carter
 Differential Equations, Equations, Vectors, Vector Space, Santa Fe Institute, Complex Systems Summer, generalized Hamming distance

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