Exact(12)
(d) Analyzing the statement of Theorem 6 (the case (29) of a double real root), we obtain the following subcases: .
(c) Analyzing the statement of Theorem 4 and Theorem 5 (the case (28) of a double real root), we obtain the following subcases: .
Moreover, when these isotropy subgroups contain the m-fold rotation group (mathbf {C}_{m}) with (mgeq3), these equilibria are foci (attractive or repulsive) or nodes (the case when the Jacobian matrix has a double real eigenvalue).
For D < 0, there are three different real roots; for D = 0, there are triple or double real roots; and for D > 0, there is only one real root and two imaginary roots.
Controller synthesis is designed on the basis of the pole placement method (a double real pole and a pair of complex conjugated poles with the same real component has been chosen in the characteristic polynomial).
if (P<0) there are two double real zeros of (P_{4}); if (P>0) and (Q=0) there are two double complex conjugate zeros of (P_{4}); if (Delta _{0}=0), then all four zeros of (P_{4}) are real and equal to (-b/4).
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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.

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