Hypercubic billiard word: Difference between revisions
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== Properties == | == Properties == | ||
Proofs to be added | Proofs to be added | ||
* A (circular) scale word is a rank-2 billiard scale | * A (circular) scale word is a rank-2 billiard scale iff it is a MOS scale. | ||
* All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position <math>\left(1, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{3}}\right)</math>, but it is not a Fokker block. | * All [[distributionally even]] scales on any finite number of letters are billiard scales<ref name="sano"/>. The converse fails, because not all billiard scales are Fokker blocks (DE implies the scale is a Fokker block); [[blackdye]] can be checked to be a billiard scale by using the initial position <math>\left(1, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{3}}\right)</math>, but it is not a Fokker block. | ||
* A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false. | * A billiard scale becomes a billiard scale over fewer letters when one removes all instances of some subset of its step sizes. In particular, ternary billiard scales are ''deletion-MOS'' (DMOS): deleting any step size results in a MOS. However, the converse is false. | ||