Balanced word: Difference between revisions
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Created page with "Let ''d'' ≥ 0. A scale word ''s'' is ''d-'''balanced''' if its '''balance''' satisfies the following: <math> \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|..." |
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Let ''d'' ≥ 0. A scale word ''s'' is ''d-'''balanced''' if its '''balance''' satisfies the following: | Let ''d'' ≥ 0. A scale word ''s'' is ''d-'''balanced''' if its '''balance''' satisfies the following: | ||
<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> where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''. A scale is '''balanced''' if it is 1-balanced. | |||
<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> | |||
where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''. A scale is '''balanced''' if it is 1-balanced. | |||
[[Category:Scale]][[Category:Terms]] | [[Category:Scale]][[Category:Terms]] | ||
Revision as of 05:31, 8 December 2023
Let d ≥ 0. A scale word s is d-balanced if its balance satisfies the following:
[math]\displaystyle{ \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]
where |u|xi is the number of occurrences of the letter xi in the word u. A scale is balanced if it is 1-balanced.