Ternary scale theorems: Difference between revisions

Line 265:

Note: The xen term "brightest mos word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multimos word" is equivalent to "powers of a Christoffel word". Also see [[Glossary for combinatorics on words]] for more equivalents between xen community terms and standard academic terminology.

To show that balanced scales of type (3) are ''not'' SV3, first observe that (''a''+''b'')-steps come in only 2 sizes: floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''b'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''b'''''Z''', since the underlying MOS 2''a'''''X''' 2''b'''''Y''' only has the (''a''+''b'')-step ''a'''''X''' + ''b'''''Z'''. The construction replaces ~~these~~ '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either ~~letter~~ may occur first, since ''a'' is odd and thus the (''a''+''b'')-step subword ''w''[''k'' : ''k''+''a''+''b''] becomes the subword ''w''[''k''+''a''+''b'' : ''k''+2''a''+2''b''] via interchanging '''X''' and '''Y'''.

To show that balanced scales of type (3) are ''not'' SV3, first observe that (''a''+''b'')-steps come in only 2 sizes in such a scale ''s'': floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''b'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''b'''''Z''', since the underlying MOS 2''a'''''X''' 2''b'''''Y''' only has the (''a''+''b'')-step ''a'''''X''' + ''b'''''Z'''. The construction replaces the '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either of '''X''' or '''Y''' may occur first, since ''a'' is odd and thus the (''a''+''b'')-step subword ''s''[''k'' : ''k''+''a''+''b''] becomes the subword ''s''[''k''+''a''+''b'' : ''k''+2''a''+2''b''] via interchanging '''X''' and '''Y'''.

Claim 6.1.4 can be verified by noting that such scales are PWF and using Theorem 4. {{Qed}}

@@ Line 265: / Line 265: @@
 Note: The xen term "brightest mos word" is equivalent to "Christoffel word" in the paper, and similarly "brightest multimos word" is equivalent to "powers of a Christoffel word". Also see [[Glossary for combinatorics on words]] for more equivalents between xen community terms and standard academic terminology.
-To show that balanced scales of type (3) are ''not'' SV3, first observe that (''a''+''b'')-steps come in only 2 sizes: floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''b'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''b'''''Z''', since the underlying MOS 2''a'''''X''' 2''b'''''Y''' only has the (''a''+''b'')-step ''a'''''X''' + ''b'''''Z'''. The construction replaces these '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either letter may occur first, since ''a'' is odd and thus the (''a''+''b'')-step subword ''w''[''k'' : ''k''+''a''+''b''] becomes the subword ''w''[''k''+''a''+''b'' : ''k''+2''a''+2''b''] via interchanging '''X''' and '''Y'''.
+To show that balanced scales of type (3) are ''not'' SV3, first observe that (''a''+''b'')-steps come in only 2 sizes in such a scale ''s'': floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''b'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''b'''''Z''', since the underlying MOS 2''a'''''X''' 2''b'''''Y''' only has the (''a''+''b'')-step ''a'''''X''' + ''b'''''Z'''. The construction replaces the '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either of '''X''' or '''Y''' may occur first, since ''a'' is odd and thus the (''a''+''b'')-step subword ''s''[''k'' : ''k''+''a''+''b''] becomes the subword ''s''[''k''+''a''+''b'' : ''k''+2''a''+2''b''] via interchanging '''X''' and '''Y'''.
 Claim 6.1.4 can be verified by noting that such scales are PWF and using Theorem 4. {{Qed}}