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The adjoint transformations are given by Ad ( exp ( ε X i ) ) X j = X j − ε [ X i, X j ] + 1 2 ε 2 [ X i, [ X i, X j ] ] − ⋯.
The equations for all these transformations are given below.
Indeed, hints for such transformations are given from direct SEM observations of the region that connects the fiber to the droplet.
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An interpretation of univocally defined generalized strain transformations is given in terms of minimization of appropriate norms of the stiffness and compliance matrices of the beam section that univocally define special reference points.
Each matrix was analyzed with and without indel characters; all character state transformations were given equal weight.
Rules for transformation are given in Table 3.
A fixed transformation are given for one-dimensional stationary processes in this paper.
The frequency bin values of each segment using DFT transformation are given by X m + 1 ( k ) = ∑ n = 0 N − 1 x m + 1 ( n ) e − 2 Π jkn / N. (13).
Her transformation was given a jump-start by the editors of Vogue.
The transformation is given by Equation (8).
The Cartesian transformation is given in the following equation.
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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