Rank-3 scale theorems: Difference between revisions

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* Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.

* Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode of ''S'' gives the graph of floor(b/a*x).

* Say that a k-step scale ''S'' is ''LQ'' if

* Say that a k-step scale ''S'' is ''LQ'' if ...

===== MV2 is equivalent to floor-LQ in 2-step scales =====

Assume wlog there are more L's than s's.

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Thus b/a#L <= b/a(#L-t), a contradiction.

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/average slopes by taking one interval before n_0, one interval containing n_0 and one interval after n_0. (We already know that mosses are "slope-LQ".)

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/average slopes by taking one interval before n_0, one interval containing n_0 and one interval after n_0. (We already know that mosses are slope-LQ.)

==== MV3 Theorem 1 (WIP) ====

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 * Assume ''S'' is a 2-step scale. Then ''S'' is ''slope-LQ'' if the slope between any two pair of points (representing a ''k''-mosstep) is one of the two nearest possible slopes (in the set {k/0,...,0/k}) to b/a.
 * Say that a 2-step scale ''S'' is ''floor-LQ'' if some mode of ''S'' gives the graph of floor(b/a*x).
-* Say that a k-step scale ''S'' is ''LQ'' if
+* Say that a k-step scale ''S'' is ''LQ'' if ...
 ===== MV2 is equivalent to floor-LQ in 2-step scales =====
 Assume wlog there are more L's than s's.
@@ Line 51: / Line 51: @@
 Thus b/a#L <= b/a(#L-t), a contradiction.
-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/average slopes by taking one interval before n_0, one interval containing n_0 and one interval after n_0. (We already know that mosses are "slope-LQ".)
+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/average slopes by taking one interval before n_0, one interval containing n_0 and one interval after n_0. (We already know that mosses are slope-LQ.)
 ==== MV3 Theorem 1 (WIP) ====