Rank-3 scale theorems: Difference between revisions
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Take the graph of the brightest mode of the mos, M_b(x) (right = L, up = s). We claim that this is the required graph of F(x) = floor(b/a*x). | Take the graph of the brightest mode of the mos, M_b(x) (right = L, up = s). We claim that this is the required graph of F(x) = floor(b/a*x). | ||
M_b <= F: (bc it's the brightest mode) Prve that F(x) describes a mos. | M_b <= F: (bc it's the brightest mode) Prve that F(x) describes a mos. This is equivalent to having the line y = b/a*x start at the origin and cross grid lines in x and y coordinates. Every time this line crosses a gridline y = (integer), we write down an s step. Every time it crosses a gridline x = integer, we write down an L. We want to show that this sequence of gridline crossings forms a MOS. | ||
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. | ||