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Tatishvili pulled even and for the fifth time in the six matches they have played, the women tumbled into a third and final set.
Also for if is even and for if is odd.
If is odd, then is even and for some odd integer.
Secondly, let where Recall (6.15), we see that for all By our assumption, it is easy to check that is even and for some odd integer.
Let be a -periodic traveling wave solution with velocity Since is odd, by Lemma 4.4 ii), we have that is even and for some odd integer From Lemma 4.3 iii), we also have (5.24).
For (y(0)=1), the solutions are even and for (1leq mleq n), the number of nodes is {mathcal{N}}(n, m) = left { textstylebegin{array}{l@{quad}l} n - m + 2, & mbox{if } n-m mbox{ is even}, n - m + 1, & mbox{if } n-m mbox{ is odd}, end{array}displaystyle right.
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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