Patent val/Properties: Difference between revisions

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\mathrm {N} = \bigcap_{i = 1}^{\pi (p)} \mathrm {N}_i </math>

Then V is a GPV of every edo in N if N is non-empty and not a single point; otherwise it is not a GPV.

Then V is a GPV of every edo in N if N is non-empty and not a singleton (i.e. a single point); otherwise it is not a GPV.

== Cardinality ==

Given a finite prime limit, the set of all GPVs are [[Wikipedia: Countably infinite|countably infinite]].

== ~~Adjacent GPVs~~ property ==

== Sorting property ==

Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV V''k'' and its next GPV V''k'' + 1 are the same, and for the different entry, the latter increments the former by 1.

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This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7, and even ''some'' rationally dependent subgroups, such as 2.3.9.7. It does not hold, however, for other rationally dependent subgroups, such as 2.3.27.7, where at certain points of edo number ''n'', both the mappings for 3 and 27 increment.

~~===~~ Proof ~~===~~

{{Databox|Proof|

By definition, the ''p''-limit GPV of ''n''-edo is V (''n'') = round (''n'' log2 (Q)), where Q is the prime basis {{val| 2, 3, 5, …, ''p'' }}.

By definition, the ''p''-limit GPV of ''n''-edo is V (''n'') = round (''n'' log2 (Q)), where Q is the prime basis {{val| 2 3 5 … ''p'' }}.

The ~~adjacent GPVs~~ property is equivalent to

The sorting property is equivalent to

# for any prime ''q''''i'' in Q, there is a point of ''n'' to cause ''v''''i'' to increment to ''v''''i'' + 1; and

# for any distinct primes ''q''''i'', ''q''''j'' in Q, there is ''not'' a point of ''n'' to cause both ''v''''i'' and ''v''''j'' to increment to ''v''''i'' + 1 and ''v''''j'' + 1, respectively.

# for any distinct primes ''q''''i'', ''q''''j'' in Q, there is ''not'' a point of ''n'' to cause both ''v''''i'' and ''v''''j'' to increment to ''v''''i'' + 1 and ''v''''j'' + 1, respectively;

<~~nowiki~~>#1</~~nowiki~~> ~~holds because the point~~ is ''n'' ~~= (~~''v''~~''i''~~ + 1/2)~~/log~~2 (''q''~~~~''i''~~~~).

where an increment of ''f'' (''x'') at ''x''0 is defined as lim ''x''→''x''0+ ''f'' (''x'') = lim ''x''→ ''x''0- ''f'' (''x'') + 1.

To prove <nowiki>#2</nowiki>, let us assume there exists such an ''n''. By the ~~property~~ of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''''i'', ''q''''j'' in Q, 2''n'' log2 (''q''''i'') ∈ '''Z''', and 2''n'' log2 (''q''''j'') ∈ '''Z'''. If that is the case, then their quotient (2''n'' log2 (''q''''i''))/(2''n'' log2 (''q''''j'')) = log''q''''j'' (''q''''i'') ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''n'' does not exist.

<nowiki>#1</nowiki> holds immediately following the definition of the round function, and the point is ''n'' = (''v''''i'' + 1/2)/log2 (''q''''i'').

To prove <nowiki>#2</nowiki>, let us assume there exists such an ''n''. By the definition of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''''i'', ''q''''j'' in Q, 2''n'' log2 (''q''''i'') ∈ '''Z''', and 2''n'' log2 (''q''''j'') ∈ '''Z'''. If that is the case, then their quotient (2''n'' log2 (''q''''i''))/(2''n'' log2 (''q''''j'')) = log''q''''j'' (''q''''i'') ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''n'' does not exist.

}}

== Application ==

@@ Line 19: / Line 19: @@
 \mathrm {N} = \bigcap_{i = 1}^{\pi (p)} \mathrm {N}_i </math>
-Then V is a GPV of every edo in N if N is non-empty and not a single point; otherwise it is not a GPV.
+Then V is a GPV of every edo in N if N is non-empty and not a singleton (i.e. a single point); otherwise it is not a GPV.
 == Cardinality ==
 Given a finite prime limit, the set of all GPVs are [[Wikipedia: Countably infinite|countably infinite]].
-== Adjacent GPVs property ==
+== Sorting property ==
 Given a finite prime limit, the set of all GPVs can be ordered in this way such that all but one entry in the GPV V<sub>''k''</sub> and its next GPV V<sub>''k'' + 1</sub> are the same, and for the different entry, the latter increments the former by 1.
@@ Line 31: / Line 31: @@
 This also holds for any rationally independent subgroups, such as 2.3.7 and 2.9.7, and even ''some'' rationally dependent subgroups, such as 2.3.9.7. It does not hold, however, for other rationally dependent subgroups, such as 2.3.27.7, where at certain points of edo number ''n'', both the mappings for 3 and 27 increment.
-=== Proof ===
+{{Databox|Proof|
-By definition, the ''p''-limit GPV of ''n''-edo is V (''n'') = round (''n'' log<sub>2</sub> (Q)), where Q is the prime basis {{val| 2, 3, 5, …, ''p'' }}.
+By definition, the ''p''-limit GPV of ''n''-edo is V (''n'') &#61; round (''n'' log<sub>2</sub> (Q)), where Q is the prime basis {{val| 2 3 5 … ''p'' }}.
-The adjacent GPVs property is equivalent to
+The sorting property is equivalent to
 # for any prime ''q''<sub>''i''</sub> in Q, there is a point of ''n'' to cause ''v''<sub>''i''</sub> to increment to ''v''<sub>''i''</sub> + 1; and
-# for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in Q, there is ''not'' a point of ''n'' to cause both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> to increment to ''v''<sub>''i''</sub> + 1 and ''v''<sub>''j''</sub> + 1, respectively.
+# for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in Q, there is ''not'' a point of ''n'' to cause both ''v''<sub>''i''</sub> and ''v''<sub>''j''</sub> to increment to ''v''<sub>''i''</sub> + 1 and ''v''<sub>''j''</sub> + 1, respectively;
-<nowiki>#1</nowiki> holds because the point is ''n'' = (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub> (''q''<sub>''i''</sub>).
+where an increment of ''f'' (''x'') at ''x''<sub>0</sub> is defined as lim ''x''→''x''<sub>0</sub><sup>+</sup> ''f'' (''x'') &#61; lim ''x''→ ''x''<sub>0</sub><sup>-</sup> ''f'' (''x'') + 1.
-To prove <nowiki>#2</nowiki>, let us assume there exists such an ''n''. By the property of the round function, an increment of ''y'' = round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in Q, 2''n'' log<sub>2</sub> (''q''<sub>''i''</sub>) ∈ '''Z''', and 2''n'' log<sub>2</sub> (''q''<sub>''j''</sub>) ∈ '''Z'''. If that is the case, then their quotient (2''n'' log<sub>2</sub> (''q''<sub>''i''</sub>))/(2''n'' log<sub>2</sub> (''q''<sub>''j''</sub>)) = log<sub>''q''<sub>''j''</sub></sub> (''q''<sub>''i''</sub>) ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''n'' does not exist.
+<nowiki>#1</nowiki> holds immediately following the definition of the round function, and the point is ''n'' &#61; (''v''<sub>''i''</sub> + 1/2)/log<sub>2</sub> (''q''<sub>''i''</sub>).
+To prove <nowiki>#2</nowiki>, let us assume there exists such an ''n''. By the definition of the round function, an increment of ''y'' &#61; round (''x'') occurs only if 2''x'' ∈ '''Z'''. Thus, for any distinct primes ''q''<sub>''i''</sub>, ''q''<sub>''j''</sub> in Q, 2''n'' log<sub>2</sub> (''q''<sub>''i''</sub>) ∈ '''Z''', and 2''n'' log<sub>2</sub> (''q''<sub>''j''</sub>) ∈ '''Z'''. If that is the case, then their quotient (2''n'' log<sub>2</sub> (''q''<sub>''i''</sub>))/(2''n'' log<sub>2</sub> (''q''<sub>''j''</sub>)) &#61; log<sub>''q''<sub>''j''</sub></sub> (''q''<sub>''i''</sub>) ∈ '''Q''', which contradicts [[Wikipedia: Gelfond–Schneider theorem|Gelfond–Schneider theorem]]. Therefore, the hypothesis is false, and such an ''n'' does not exist.
+}}
 == Application ==