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Exact(60)
where is a matrix with entries.
where is a diagonal matrix with entries (B2).
and the block diagonal matrix with entries satisfies (3.6).
where N ~ 1 is an N×N matrix with entries i.i.d.i.d
where N1 is an N×N matrix with entries independent and identically distributed (i.i.d).i.d
A weighted mean matrix, written is a lower triangular matrix with entries where is a nonnegative sequence with and as.
where U k is a diagonal matrix with entries U k ( i, i ) = ( u i k ) − 1 / 2. 3.
Here E 2 is a 2 × 2 matrix with entries being 1's, and ∘ means the Hadamard product.
where the matrix models the unused primary subspace (eigenmodes) and is an diagonal matrix with entries (9).
denotes an diagonal matrix with entries along its main diagonal, while is the inverse of a square matrix.
A weighted mean matrix, denoted by, is a lower triangular matrix with entries, where is a nonnegative sequence with, and.
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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