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It is worth saying that if, then (1.2) is false for every even.
For every even integer 2k, let 2 m ( k ) be the least even integer exceeding 2 n ( k ) satisfying (2.10).
By Proposition 5.1, (f^{q}) has no essential fixed point class for every even factor q of n.
Then f has a unique fixed point x = 1 / 2. Note that f n x = x holds for every even natural number n and x in [ 0, 1 ].
Therefore, we have (26) for every even number N. The proof of (26) for odd number is similar and is omitted.
To show the sharpness of several bounds, we construct for every even n an n-vertex pseudo-triangulation of a surface such that every triple is a face exactly once.
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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