Hypercubic billiard word: Difference between revisions
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== Determining whether a scale word is a billiard scale == | == Determining whether a scale word is a billiard scale == | ||
== Questions and conjectures == | == Questions and conjectures == | ||
1. A circular word ''s'' is ''d | 1. A circular word ''s'' is ''d-balanced'' if for any ''k'' ≥ 1 and for any pair of length-''k'' subwords ''w'' and <i>w'</i> of ''s'', | ||
<math> \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d,</math> | <math> \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d,</math> | ||