Rank-3 scale theorems: Difference between revisions

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Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let R be the corresponding path following the word w (R(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.

* Say ''S'' is ''pointwise-least-squares-LQ'' if the sum of (R(k) - L(k))^2 over k ∈ {0, ..., n} is minimized.

* Say ''S'' is ''pointwise-least-squares-LQ'' if the sum of (R(k) - L(k))^2 over k ∈ {0, ..., n} is minimized for *all* modes of ''S''.

* Say ''S'' is ''pointwise-minimax-LQ'' if the max distance max{|R(k)-L(k)| : k} is minimized.

* Say ''S'' is ''pointwise-minimax-LQ'' if the max distance max{|R(k)-L(k)| : k} is minimized for *all* modes of ''S''.

* Say that a [[mos]] (MV2) scale ''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.

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 Let n = a+b+c be the scale size, w = aX bY cZ be the scale word, let R be the corresponding path following the word w (R(k) = your location after taking k steps according to w), and put n+1 equally spaced points p_n on the line segment L = {(a,b,c)t : 0 <= t <= n}, i.e. the points {L(k) = (a,b,c) k : k ∈ {0, ..., n}}.
-* Say ''S'' is ''pointwise-least-squares-LQ'' if the sum of (R(k) - L(k))^2 over k ∈ {0, ..., n} is minimized.
+* Say ''S'' is ''pointwise-least-squares-LQ'' if the sum of (R(k) - L(k))^2 over k ∈ {0, ..., n} is minimized for *all* modes of ''S''.
-* Say ''S'' is ''pointwise-minimax-LQ'' if the max distance max{|R(k)-L(k)| : k} is minimized.
+* Say ''S'' is ''pointwise-minimax-LQ'' if the max distance max{|R(k)-L(k)| : k} is minimized for *all* modes of ''S''.
 * Say that a [[mos]] (MV2) scale ''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.