Hypercubic billiard word: Difference between revisions
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** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact. | ** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact. | ||
* Not all billiard words of arity higher than 2 are [[balance|balanced]]. (In binary scales, the term "(1-)balanced" becomes one characterization of the [[MOS]] property.) Ternary billiard scales that are 1-balanced are [[MV3]].<ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> | * Not all billiard words of arity higher than 2 are [[balance|balanced]]. (In binary scales, the term "(1-)balanced" becomes one characterization of the [[MOS]] property.) Ternary billiard scales that are 1-balanced are [[MV3]].<ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> | ||
* There exist billiard scales of length up to 29 that do not have a well-formed [[generating sequence]]. | |||
* A ''d''-ary billiard scale has [[balance]] at most (''d'' − 1).<ref name="vuillon"/> However, scales of arity ≥ 3 satisfying this bound need not be billiard scales. | * A ''d''-ary billiard scale has [[balance]] at most (''d'' − 1).<ref name="vuillon"/> However, scales of arity ≥ 3 satisfying this bound need not be billiard scales. | ||
* An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Com- | * An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Com- | ||
binatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> Hence this bound must hold for periodic ''d''-ary billiard scales as well. | binatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> Hence this bound must hold for periodic ''d''-ary billiard scales as well. | ||
=== Determining whether a scale word is a billiard scale === | === Determining whether a scale word is a billiard scale === | ||
The following discussion documents a naive algorithm for answering whether a circular word ''s'' of arity ''d'' with velocity '''a''' = ∑<sub>''i''</sub> ''a''<sub>''i''</sub> '''e'''<sub>''i''</sub> ∈ ℤ<sup>''d''</sup> (representing the signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub>) is a billiard word: | The following discussion documents a naive algorithm for answering whether a circular word ''s'' of arity ''d'' with velocity '''a''' = ∑<sub>''i''</sub> ''a''<sub>''i''</sub> '''e'''<sub>''i''</sub> ∈ ℤ<sup>''d''</sup> (representing the signature ''a''<sub>1</sub>''x''<sub>1</sub>...''a''<sub>''d''</sub>''x''<sub>''d''</sub>) is a billiard word: | ||