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Therefore, after simplifying: Note that if we let n grow without bound, the expression (12) becomes the well-known range formula.
We describe the method only for two-SNP haplotypes or four-allele markers to simplify notation, noting that extension to markers with any number of alleles or haplotypes composed of more SNPs is straightforward.
Thus, θ n, r can be interpreted as the proportion of the transcripts in the nth tumor arising from the 'tissue' represented by profile b r. (Note that to simplify notation, we occasionally indicate θ n,1, θ n,2,..., θ n, R and α n using the vector θ n of length R+1 whose rth element is θ n, r for r < R+1 and whose R+1st element is set to α n).
To simplify the notation, we note: θ R = θ R xz, θ G = θ G xz et θ B = θ B yz.
To simplify notations, we note τ M, k = C k g M, k.
This yields: K k a km, b km = 2 π α k 1 σ x 2 + β k a km σ x 2 a km 2 + β k + 1 × exp − b km 2 2 1 − σ x 2 1 + σ x 2 β k + a km 2. Again, a Taylor series expansion can be used to simplify this expression.Note that in the biasless case we have: K k a km, 0 = 2 π α 1 σ x 2 + β k a km σ x 2 a km 2 + β k + 1 .
We can further simplify (44) by noting that,, and : (45).
We can simplify this equation, noting that every node of the tree except the root has exactly one incoming edge.
In order to simplify the equations, we note and.
We can further simplify the system by noting that the change in intragranular porosity during extraction is small relative to the final (or initial) value of intragranular porosity.
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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