S-expression: Difference between revisions

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5. If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because:

: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} > {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} < {{sfrac|''k'' + 2|''k'' + 1}}}}.

: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}}}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} > {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} < {{sfrac|''k'' + 2|''k'' + 1}}}}.

: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*…*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].)

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 . If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because:
-: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &gt; {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &lt; {{sfrac|''k'' + 2|''k'' + 1}}}}.
+: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}}}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &gt; {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &lt; {{sfrac|''k'' + 2|''k'' + 1}}}}.
 : (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*…*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].)