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Discover LudwigExact(4)
Let a function (y:mathbb{N}_{a-1}rightarrowmathbb{R}) satisfy (y(a) geq0) and be increasing on (mathbb{N}_{a}).
Let and be increasing, nonnegative continuous functionals on, and let be a nonnegative continuous functional on with such that, for some and, (3.1).
Let and be increasing or decreasing sequences obeying condition then by condition, is a monotonic and order-bounded sequence, so Hence, condition holds.
For example, let X be binary and and be increasing functions of N. Then typical members with tend to come from larger clusters than typical members with, so even though X has no effect on Y within clusters.
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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