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The authors showed that if (2< p<2_{N,m}^:=frac{2 N-m)}{(N-m)-2 N-mthen problem (5) has infinite invariant solutions and one of these solutions is positive.
We executed this procedure for every pair (dataset, cost vector) for which the number of optimal acyclic solutions is positive.
For all datasets whose number of acyclic solutions is positive for unbounded k, we identified k start (minimum k whose optimal cost is equal to the optimal cost obtained for unbounded k) and we searched for the minimum k ′≥ k start whose number of acyclic solutions is non zero.
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Some of such solutions are positive, bounded from above, and monotone in both space and time.
In this section, we prove that the solutions are positive and ultimately bounded.
Faster convergences, better final points which satisfy all constraints imposed on the drilling paths and population diversity maintenance to help the algorithms find better solutions, are positive characteristics of the solutions found using the algorithms proposed.
Possible Thinking Many worriers think the solution is positive thinking.
This solution is positive.
Step 2: We show that the solution is positive.
Without loss of generality, we assume that the solution is positive first.
The following theorem shows that this solution is positive and global.
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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