Chord complexity: Difference between revisions
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<math>\displaystyle D_s(x_1, x_2, \ldots, x_N)^{1/s} \cdot N^{1/s} = M_{-s}(x_1, x_2, \ldots, x_N)</math> | <math>\displaystyle D_s(x_1, x_2, \ldots, x_N)^{1/s} \cdot N^{1/s} = M_{-s}(x_1, x_2, \ldots, x_N)</math> | ||
And now we need only use the proof that it is well known that the power mean tends to the geometric mean as <math>p \to 0</math>, the minimum as <math>p \to -infty</math> and the maximum as <math>p \to \infty</math>. Thus, since we have flipped the sign so that <math>D_s = M_{-s}</math>, we have the aforementioned result, but with <math>s \to \infty</math> being the minimum and <math>s \to -\infty</math> being the maximum. | And now we need only use the proof that it is well known that the power mean tends to the geometric mean as <math>p \to 0</math>, the minimum as <math>p \to -\infty</math> and the maximum as <math>p \to \infty</math>. Thus, since we have flipped the sign so that <math>D_s = M_{-s}</math>, we have the aforementioned result, but with <math>s \to \infty</math> being the minimum and <math>s \to -\infty</math> being the maximum. | ||
Since for dyads, at least in terms of relative rankings, the geometric mean is equivalent to the Tenney Height, and the maximum the Weil Height, we have our result. | Since for dyads, at least in terms of relative rankings, the geometric mean is equivalent to the Tenney Height, and the maximum the Weil Height, we have our result. | ||