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The general problem we address is the following: Given a non-binary gene tree G and a species tree S, find a refinement of G containing the minimum number of NADs with respect to S. Such a refinement of G is called a minimum refinement of G w.r.t. S. Hence, we aim at refining each non-binary node of G.
Then a minimum refinement of G is a minimum refinement of G ′.
Then a minimum refinement of G is a minimum refinement of G ′. Let { G i, for 1 ≤ i ≤ n } be the set of subtrees of G rooted at the n children { x i, for 1 ≤ i ≤ n } of the root of G. Let H m i n (G i, S ) be a minimum refinement of G i w.r.t. S. Let G ′ be the tree obtained from G by replacing each G i by H m i n (G i, S ).
T heorem 1 Let { G i, for 1 ≤ i ≤ n } be the set of subtrees of G rooted at the n children { x i, for 1 ≤ i ≤ n } of the root of G. Let H m i n (G i, S ) be a minimum refinement of G i w.r.t. S. Let G ′ be the tree obtained from G by replacing each G i by H m i n (G i, S ).
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The algorithm was shown to attain finite convergence to the global minimum through successive refinement of a linear relaxation of the feasible region and/or of the objective function and the subsequent solutions of a series of linear programming sub-problems.
The proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of univariate equations.
The proposed algorithm which connects branch-and-bound method with the pruning technique successfully is convergent to the global minimum, according to the successive refinement of the linear relaxation of feasible region of the objective function and the solutions of a series of linear relaxation problems.
Instead, the achievement of both logically correct and high quality switching demands further refinement of the specified minimum requirements and the selection of the key parameters in such way that the Q-factor at the reflection port is optimized.
But this learner may require a large number of trials to map the space and would not prioritize refinement of the global minima.
Arkham Origins, which was developed by Warner Brothers Games Montreal, is at minimum a continuation and in some ways a refinement of the work done by Rocksteady in Arkham City.
However, use of the minimum subset of signs and symptoms represents a refinement of the currently larger set of fifteen signs recommended in Kenyan IMCI guidelines (table 6) 5 and might be expected to be implemented more easily and efficiently.
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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