Interleaving: Difference between revisions

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If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w''(''x'', ''y'') where |''w''| > 2, and ''k'' is an odd number greater than 1 and less than |''S''| - 1, then the class of ''k''-steps has more than 3 abstract intervals.

Proof: Denote by |w| the length of ''w'' in letters and by ||''w''|| the interval subtended by ''w'' in its scale word.

Proof: Denote by |w| the length of ''w'' in letters and by ‖''w''‖ the interval subtended by ''w'' in its scale word.

Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| ≥ 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>.

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(i), (1) => ''b'' = ''b'x'', ''p'' = ''xb' '', have ''xbx'' = ''xb'xx'', continue scooting to the left until you find a ''y'' to the left, then same case as (ii), (2).

(ii), (1) => ''b'' = ''b'y'', ''p'' = ''yb' '', we have ||''p''('''XZ''', '''YZ''')'''Y'''|| as a fourth interval size.

(ii), (1) => ''b'' = ''b'y'', ''p'' = ''yb' '', we have ‖''p''('''XZ''', '''YZ''')'''Y'''‖ as a fourth interval size.

(i), (2) => ''b'' = ''b'y'', ''p'' = ''xb' '', we have ||''p''('''XZ''', '''YZ''')'''X'''|| as a fourth interval size.

(i), (2) => ''b'' = ''b'y'', ''p'' = ''xb' '', we have ‖''p''('''XZ''', '''YZ''')'''X'''‖ as a fourth interval size.

(ii), (2) => ''b'' =''b'x'', ''p'' = ''yb' '', we have ||''p''('''XZ''', '''YZ''')'''X'''|| as a fourth interval size.

(ii), (2) => ''b'' =''b'x'', ''p'' = ''yb' '', we have ‖''p''('''XZ''', '''YZ''')'''X'''‖ as a fourth interval size.

(c) is similar to (b).

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 If ''S'' consists of the subwords '''XZ''' and '''YZ''' arranged in the pattern of a single-period binary circular word ''w''(''x'', ''y'') where |''w''| > 2, and ''k'' is an odd number greater than 1 and less than |''S''| - 1, then the class of ''k''-steps has more than 3 abstract intervals.
-Proof: Denote by |w| the length of ''w'' in letters and by ||''w''|| the interval subtended by ''w'' in its scale word.
+Proof: Denote by |w| the length of ''w'' in letters and by ‖''w''‖ the interval subtended by ''w'' in its scale word.
 Let ''A'' be the set of all (''k'' - 1)/2-step intervals of ''w''. |''A''| = 1 implies that ''k'' = 1 mod |''w''|, so |''A''| &ge; 2 and contains at least two intervals '''w'''<sub>1</sub> and '''w'''<sub>2</sub>.
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 (i), (1) => ''b'' = ''b'x'', ''p'' = ''xb' '', have ''xbx'' = ''xb'xx'', continue scooting to the left until you find a ''y'' to the left, then same case as (ii), (2).
-(ii), (1) => ''b'' = ''b'y'', ''p'' = ''yb' '', we have ||''p''('''XZ''', '''YZ''')'''Y'''|| as a fourth interval size.
+(ii), (1) => ''b'' = ''b'y'', ''p'' = ''yb' '', we have ‖''p''('''XZ''', '''YZ''')'''Y'''‖ as a fourth interval size.
-(i), (2) => ''b'' = ''b'y'', ''p'' = ''xb' '', we have ||''p''('''XZ''', '''YZ''')'''X'''|| as a fourth interval size.
+(i), (2) => ''b'' = ''b'y'', ''p'' = ''xb' '', we have ‖''p''('''XZ''', '''YZ''')'''X'''‖ as a fourth interval size.
-(ii), (2) => ''b'' =''b'x'', ''p'' = ''yb' '', we have ||''p''('''XZ''', '''YZ''')'''X'''|| as a fourth interval size.
+(ii), (2) => ''b'' =''b'x'', ''p'' = ''yb' '', we have ‖''p''('''XZ''', '''YZ''')'''X'''‖ as a fourth interval size.
 (c) is similar to (b).