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From IR measurements (see Fig. S9) we find that the fibril structure does not change if we exchange the buffers of the fibrils with a different salt concentration after the fibrillization process is finished, in agreement with previous studies7.
If we exchange the horizontal and vertical coordinates of the L-curve, the resulting curve is the mirror image of the S-curve.
Similarly, if we exchange [s1,s2] with [s3,s4] and exchange [s5,s6] with [s7,s8] simultaneously, the structure of R matrix maintains, as well.
For example, if we exchange the occupied levels of a 4 and a 5, as labeled in the outer column in Fig. 5 c, a 4⊕b 5 will be received twice at Rx2.
end{aligned} Therefore, all k-dependent terms in Eq. (9) are the same if we exchange k with j, and hence we arrive at Eq. (16) for thermal states.
In Fig. 2C, we show that if we exchange (f_{{x}}) with (f_{g}) and (tau_{{x}}) with (tau_{g}) such that g is linearly controlled and x is quadratically controlled, then an equilibrium exists for (r_{g }< r_{{{x}} }), but it is not stable.
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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