Rank-3 scale theorems: Difference between revisions
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Tags: Mobile edit Mobile web edit |
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M_b <= F: Prove that F(x) describes a mos. Two possibilities, for each k from 0 to a-1: (a) F(k + 1) = F(k) or (b) F(k + 1) = F(k) + 1. (b) happens b times, and (a) happens a-b times. For a given k, in case (a) (b/a)x hasn't crossed a y-gridline (i.e. one indicating a y-value such as y = m), during time k < x <= k+1, and in case (b), (b/a)x ''has'' crossed a y-gridline during time k < x <= k+1. | M_b <= F: Prove that F(x) describes a mos. Two possibilities, for each k from 0 to a-1: (a) F(k + 1) = F(k) or (b) F(k + 1) = F(k) + 1. (b) happens b times, and (a) happens a-b times. For a given k, in case (a) (b/a)x hasn't crossed a y-gridline (i.e. one indicating a y-value such as y = m), during time k < x <= k+1, and in case (b), (b/a)x ''has'' crossed a y-gridline during time k < x <= k+1. | ||
Suppose that some | Suppose that some r-step comes in 3 "sizes". | ||
M_b >= F: (bc it's a mos) Suppose there is an x-value x_0 where M_b(x_0) < F(x_0). Let m = min(n_0, n-n_0), n = scale size. Then find three different m-mossteps by taking one interval before n_0, one interval containing n_0 and one interval after n_0. | M_b >= F: (bc it's a mos) Suppose there is an x-value x_0 where M_b(x_0) < F(x_0). Let m = min(n_0, n-n_0), n = scale size. Then find three different m-mossteps by taking one interval before n_0, one interval containing n_0 and one interval after n_0. | ||