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On a unified reactive mixing plot, the asymptotes for complete mixing (when each of the reactants is limiting) form an upper bound and are linear.
It follows that the formulas derived in Part 1 form an upper bound for E ncis,Comp).
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Since the hexagonal grid model is an ideal case, it has the best 5th percentile SE and forms an upper bound.
We shall prove a generalized form of an upper bound for the dual mixed volume.
Moreover, we obtained a closed-form expression for an upper bound to the normalized information distortion.
The information distortion of the proposed scheme is also investigated and a closed-form expression for an upper bound to the normalized information distortion is derived.
That's sort of an upper bound, a worst case.
Achievable rates are derived in closed-form and compared with an upper bound.
The optimization problem in (3) can be equivalently written in a constrained form: (5) where Cλ is an upper bound for the ℓ1 norm of θ(t )\ u and defines a region Ω in which the parameter lies.
For each plotted extreme value an upper bound, a lower bound and an 'inclusion' probability p are defined such that for each and every point, the probability of it plotting within the upper and lower bounds is equal to p. The loci of the upper and lower bounds then form the control curves.
We then propose a lower bound and an upper bound of the original problem, both in convex forms.
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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