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The following lemma shows that tau_{N^^{j}geqtau_{N^-1}^{j}geq cdotsgeqtau_{1}^{j}> tau_{0}^{j} and hence we have a complete ordering of the bifurcation values (tau_{n}^{j}).
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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