Interior roots, number

Number of interior roots. System (21) may have both interior (z, 0, i — 1,..., n) and boundary roots (3i z, = 0). As a rule, boundary roots correspond to the zero reaction rate. Some classes of mechanisms can be free of boundary roots at all. For instance, system (21) corresponding to the mechanism of the Basic case type, satisfying the condition (35) does not have boundary roots. All interior roots belong to the algebraic tore. Bemstain theorem (see GeTfand et al., 1994) can be applied to estimate the number of roots in this case. 

Enteric bacteria have been found to differ with respect to the ability to become integrated into the endophytic microflora of alfalfa roots (Dong et al., 2001). A strain isolated from maize, Klebsiella pneumoniae 342, colonizes the interior of several host plants in higher numbers compared to... 

In order to determine the number of roots of the z-transformed characteristic equation that lie outside the unit circle, a procedure analogous to the Routh-Hurwitz approach for continuous systems (Section 7.10.2) can be used. The Routh-Hurwitz criterion cannot be applied directly to the characteristic equation f(z) = 0. However, by mapping the interior of the unit circle in the z-piane on to the left half of a new complex variable -plane, the Routh-Hurwitz criterion can be applied as for continuous systems to the corresponding characteristic equation in terms of the new variable<4,). This mapping can be achieved using the bilinear transformation07 ... 

Now ceand P must be of same sign (otherwise we get the square root of a negative number). Also, since 0 < xx <1 and 0 < x2 < 1, only positive sign is allowed. Thus x2/xj = Joc/[ when dCT jdxj = 0. And only an interior maximum (if oc> 0 and / > 0) or an interior minimum (if ot< 0 and P< 0) can occur, but not both Therefore, van Laar model can not describe the observed behavior. 

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