Recursive structure of MOS scales: Difference between revisions

Line 458:

Assume ''c'' = ''b'' and ''e'' = −''a''. Let χ = L − s > 0; then χ is ''p''-equivalent to ''+ng''. Now by generatedness, binarity, and linear independence of ''p'' and ''g'', the class of ''k''-steps must have two sizes separated by ''ng'', and changing the k-steps outside these two sizes leaves the range −(''n'' − 1)''g'', ..., 0, ..., (''n'' − 1)''g''. Thus the class of ''k''-steps has two sizes for 1 ≤ ''k'' ≤ (''n'' − 1).

To show Myhillness for ~~rational step ratios~~, note that every ''k''-step (which is a specific linear combination of L and s) in the scale with rational step ratio is a limit point of the same linear combination of L and s in versions of the binary scale with ~~irrational step ratios~~, and thus there must be ''at most'' 2 sizes for each generic interval. Since χ, which separates the two sizes, is ''p''-equivalent to ''ng'', which remains ''p''-inequivalent to 0 since L/s ≠ 1/1, each generic interval has ''exactly'' 2 sizes.

To show Myhillness for non-linearly-independent ''p'' and ''g'', note that every ''k''-step (which is a specific linear combination of L and s) in the scale with rational step ratio is a limit point of the same linear combination of L and s in versions of the binary scale with linearly independent ''p'' and ''g'', and thus there must be ''at most'' 2 sizes for each generic interval. Since χ, which separates the two sizes, is ''p''-equivalent to ''ng'', which remains ''p''-inequivalent to 0 since L/s ≠ 1/1, each generic interval has ''exactly'' 2 sizes.

== See also ==

@@ Line 458: / Line 458: @@
 Assume ''c'' = ''b'' and ''e'' = &minus;''a''. Let χ = L &minus; s > 0; then χ is ''p''-equivalent to ''+ng''. Now by generatedness, binarity, and linear independence of ''p'' and ''g'', the class of ''k''-steps must have two sizes separated by ''ng'', and changing the k-steps outside these two sizes leaves the range &minus;(''n'' &minus; 1)''g'', ..., 0, ..., (''n'' &minus; 1)''g''. Thus the class of ''k''-steps has two sizes for 1 ≤ ''k'' ≤ (''n'' &minus; 1).
-To show Myhillness for rational step ratios, note that every ''k''-step (which is a specific linear combination of L and s) in the scale with rational step ratio is a limit point of the same linear combination of L and s in versions of the binary scale with irrational step ratios, and thus there must be ''at most'' 2 sizes for each generic interval. Since χ, which separates the two sizes, is ''p''-equivalent to ''ng'', which remains ''p''-inequivalent to 0 since L/s ≠ 1/1, each generic interval has ''exactly'' 2 sizes.
+To show Myhillness for non-linearly-independent ''p'' and ''g'', note that every ''k''-step (which is a specific linear combination of L and s) in the scale with rational step ratio is a limit point of the same linear combination of L and s in versions of the binary scale with linearly independent ''p'' and ''g'', and thus there must be ''at most'' 2 sizes for each generic interval. Since χ, which separates the two sizes, is ''p''-equivalent to ''ng'', which remains ''p''-inequivalent to 0 since L/s ≠ 1/1, each generic interval has ''exactly'' 2 sizes.
 == See also ==