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After defining the auxiliary reference frequency, the proposed approach directly uses the classical symmetrical component transformation matrix at any harmonic or interharmonic.
But it can also be useful for conformal maps, in which case the matrix at any point can be transformed to the identity matrix multiplied by a scalar.
Under these conditions, the Jacobian matrix at any fixed point has a negative trace and positive determinant, therefore, the fixed point must be stable.
Finally, its results are very same to the real condition, because of its flexibility which makes us consider effects and parameters of fracture and matrix at any direction (X, Y, and Z direction).
Interestingly, while morphology of the sphenoid was not found to reflect the molecular distance matrix at any ontogenetic stage evaluated (Table 9), it did perform quite well in the DFAs (Table 8).
When individuals age, their contacts gradually change such that the population statistics of the age-dependent contact distribution remain in agreement with the POLYMOD matrix at any time (see Appendix 2 for details).
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The final global coefficient matrices at any frequency can be simply formed by a summation of the frequency-independent global matrices and the need to compute the numerical integration at each frequency is eliminated.
Frantz, C., Stewart, K. M. & Weaver, V. M. The extracellular matrix at a glance.
Let S be the state matrix at a given time.
The two dimensional (2D) matrix at a specific wavenumber was extracted from the map file.
Moreover we were able to demonstrate, that the alary muscles are not directly connected to the heart tube but by extracellular matrix fibers at any stage of development.
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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