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Let Z be the K×K matrix given in (7).
The D w matrix given below is the watermarked D matrix given in Equation (2).
It can be calculated by the method of next generation matrix given in [32].
The MIMO LE matrix given in (12) has implicit forms which are difficult for evaluation.
For the sufficiency, let A be the (ntimes n) symmetric matrix given in (2.1).
The starting point is the Fisher information matrix given in (25).
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The method is based on the concept of the so-called impedance matrix giving in the frequency domain, the linear relation between the generalized displacements of the boundaries and the generalized forces exerted on these boundaries.
Now we focus our attention to the case of (1.2) for being weighted mean matrices given in (1.3).
The proof is based on the uniformity property of transfer matrices given in Lemma 1 together with counting.
The following discussion makes use of the important properties of circulant matrices given in the appendices (Facts 1 and 2).
We also thank the anonymous reviewers for their comments and one in particular for suggesting the useful formal definition of disambiguation matrices given in the Methods section.
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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