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By sequentially integrating (4) by parts, we obtain the following identities: (5) u (t ) = α 1 〈 x (t ) 〉 α 2 = α 1 α 2 x (t ) − α 1 α 2 〈 d x d t (t ) 〉 α 2. where the brackets < > denote the convolution introduced in relationship (4).
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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