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Absolute difference between the rotation number obtained with the 2-dimensional approach and the analytic one, that is | ρ 2 D − ρ |, in the two-parametric space ( ω rel, ϵ ) Fig. 8 Differences among the 1D prediction and the analytic rotation numbers.
Absolute difference between the rotation number obtained with the 1-dimensional approach (phase-reduction hypothesis) and the analytic one, that is | ρ 1 D − ρ |, in the two-parametric space ( ω rel, ϵ ) Fig. 9 Ratio between errors given by the 2D prediction and by the 1D prediction.
In Fig. 7, we plot the absolute difference between the rotation number obtained with the 2-dimensional approach and the analytic one, namely | ρ 2 D − ρ |, whereas in Fig. 8, we plot the error when using the phase-reduction hypothesis, namely | ρ 1 D − ρ |.
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We found a difference between the two shoulder rotation velocity curves, but not during the maximal values or around ball release.
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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