Ternary scale theorems: Difference between revisions
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# Single-period MV3 scales not of type (4) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size. | # Single-period MV3 scales not of type (4) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size. | ||
=== Proof === | === Proof === | ||
Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that PWF scales are S<sub>3</sub>-action images of φ(aX bY), where a is even, b | Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that PWF scales are S<sub>3</sub>-action images of φ(aX bY), where a is even, gcd(a,b) = 1, and φ is the operation of replacing every other X with Z. | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Ternary scale]] | [[Category:Ternary scale]] | ||
[[Category:Scale]] | [[Category:Scale]] | ||