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For any, we have such that (2.26).
Hence by Theorem 2.3, we have such that (2.29).
Let Then we have, such that (3.4).
Similarly, we have such that and.
By Lemma 3.1, we have such that for all.
From Lemma 3.1, we have such that (3.22).
Therefore, by Theorem 3.4, we have such that.
A singularity will appear in the expansion of whenever we have such that.
It is clear that for all By Lemma 3.1, we have such that for all This implies that (3.7).
For this simple example, we can also apply Theorem 5.2 to conclude that there exists such that since for any and each we have such that (5.9).
So, since this is not uniformly convex with regard to, therefore there exists such that for all we have such that, (2.33).
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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