Rank-3 scale theorems: Difference between revisions

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# ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''.

==== ~~Definition~~: LQ ====

==== Definitions: LQ ====

A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k.

First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."

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

The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of LQ property.

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

* 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.

===== MV2 is equivalent to LQ in 2-step scales (WIP) =====

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 # ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''.
-==== Definition: LQ ====
+==== Definitions: LQ ====
-A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k.
+First attempt at a definition: "A scale word ''S'' with ''k'' step sizes X_1, ..., X_k (with a_1 X_1's, ..., and a_k X_k's) is ''line-quantizing'' (LQ) if ''S'', when viewed as a set of instructions tracing a path in Z^k from the origin (in which each X_i means "go 1 step in the positive x_i direction"), results in a path that is a closest approximation to the line [a_1 : a_2 : ... : a_k] intersecting the origin in R^k."
-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-minimax-LQ'' if the max distance max{|R(k)-L(k)| : k} is minimized. 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.
+The problem with this definition is that it is unclear what "closest approximation" means. We take several different definitions of that phrase and define different versions of LQ property.
+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-minimax-LQ'' if the max distance max{|R(k)-L(k)| : k} is minimized.
+* 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.
 ===== MV2 is equivalent to LQ in 2-step scales (WIP) =====