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Lemma 3. Given a source string x, a target string y, If the cost of duplication is 1 per duplicate operation, and the cost of deletion is 1 per delete operation, then (x, y ) = d x, y ).
This change should cut the cost of duplication and bureaucracy.
Suddenly, the cost of duplication was lower than the cost of appropriation.
(3) Copies of transcripts, electronic recordings, or minutes of Commission meetings closed to public observation containing material nonexempt pursuant to § 375.206(f) of this chapter are also available at the actual cost of duplication or transcription.
In both cases, the originators property right is measured by the savings in cost of duplication that the copier could obtain by taking out a license from the originator.
Copies of such transcript, or minutes, or a transcription of such recording disclosing the identity of each speaker, shall be furnished to any person at the actual cost of duplication or transcription.
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The total cost of duplications on edge V is calculated as C d = c d · (| R V1)|+| R V2)|–| R V)|–| r V1, V2)|), where | | denotes the cardinality of a set, and c d is the cost of one duplication (the algorithm parameter).
On the other hand, our Constraint 1 does not require that the cost of mutations be higher than the cost of duplications and contractions.
The classic and still commonly used distance was introduced in [ 33] as the total cost of duplications and losses (and transfers, if allowed).
As shown in Figure 7, the average fitness cost of duplications remains low over the entire evolution and does not depend on pathway size.
According to Lemma 5, the cost c(V) includes the total cost of duplications in all paralogous subtrees over all G j over all species from V. Therefore, the costs of singletons can be any constants, as the optimal tree S V) does not depend on them.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com