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The composer's preface contains valuable information about performance: "Should the player find it tedious to play a piece right through he may choose such sections as he pleases, provided only that he ends in the main key.… The opening passages should be played slowly so that what follows may appear more animated.
We may choose, such that.
Since, we may choose such that for and, where satisfies So, (4.18).
In this case, one may choose,, such that (3.6) and (3.7) in Theorem 3.1 hold.
At first, it follows from the condition that we may choose such that (4.2).
By Lemma 3.1, we may choose such that, such that, such that.
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Also by (2.10) one may choose with such that (4.15).
Since are compact and,, we may choose and such that (4.11) .
Since (R_{0}>1), we may
(2.11) Additionally, corresponding to (m k)), we may choose (n k)) such that it is the smallest integer satisfying (2.11) and (n k)>m k geq k).
Previous studies reported that the reasons for DM patients to choose such therapies may be related to the fact that diabetes is a chronic, devastating, and incurable disease.
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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