Ternary scale theorems: Difference between revisions

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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.

For 6.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown that balanced scales of type (2) are SV3. To show that balanced scales of type (3) are ''not'' SV3, 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'''.

For 6.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown in Theorem 1 that balanced scales of type (2) are SV3. To show that balanced scales of type (3) are ''not'' SV3, 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}}

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 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.
-For 6.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown that balanced scales of type (2) are SV3. To show that balanced scales of type (3) are ''not'' SV3, 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'''.
+For 6.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown in Theorem 1 that balanced scales of type (2) are SV3. To show that balanced scales of type (3) are ''not'' SV3, 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}}