Chord complexity: Difference between revisions
000masa000 (talk | contribs) |
000masa000 (talk | contribs) |
||
| Line 195: | Line 195: | ||
We can use this to define the [[Tenney-Weil Height]] (or Benedetti-Weil height, if you prefer) for the chord, with a free parameter <math>k</math> interpolating between the two (and thus determining just how much we care about the span). We will define this as follows: | We can use this to define the [[Tenney-Weil Height]] (or Benedetti-Weil height, if you prefer) for the chord, with a free parameter <math>k</math> interpolating between the two (and thus determining just how much we care about the span). We will define this as follows: | ||
<math>\displaystyle TW_s(x_1, x_2, \ldots, x_N) = (1-k) \log B_s(x_1, x_2, \ldots, x_N) + (k) \log | <math>\displaystyle TW_s(x_1, x_2, \ldots, x_N) = (1-k) \log B_s(x_1, x_2, \ldots, x_N) + (k) \log W_s(x_1, x_2, \ldots, x_N)</math> | ||
so that for <math>k=0</math> we get the Tenney height, for <math>k=1</math> we get the Weil height (with even greater emphasis on smaller intervals), and for other values of <math>k</math> we can get values in between or beyond. | so that for <math>k=0</math> we get the Tenney height, for <math>k=1</math> we get the Weil height (with even greater emphasis on smaller intervals), and for other values of <math>k</math> we can get values in between or beyond. | ||