Second-order complex roots

The determination of the number of the SHG active complex cations from the corresponding SHG intensity and thus the surface charge density, a°, is not possible because the values of the molecular second-order nonlinear electrical polarizability, a , and molecular orientation, T), of the SHG active complex cation and its distribution at the membrane surface are not known [see Eq. (3)]. Although the formation of an SHG active monolayer seems not to be the only possible explanation, we used the following method to estimate the surface charge density from the SHG results since the square root of the SHG intensity, is proportional to the number of SHG active cation com-... 

In establishing the relationship between time-domain and Laplace-domain, we use only first and second order differential equations. That s because we are working strictly with linearized systems. As we have seen in partial fraction expansion, any function can be "broken up" into first order terms. Terms of complex roots can be combined together to form a second order term. 

This is the idea behind the plotting of the closed-loop poles—in other words, construction of root locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple first and second order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 

As discussed, the first-order neuron is the afferent neuron that transmits impulses from a peripheral receptor toward the CNS. Its cell body is located in the dorsal root ganglion. This neuron synapses with the second-order neuron whose cell body is located in the dorsal horn of the spinal cord or in the medulla of the brainstem. The second-order neuron travels upward and synapses with the third-order neuron, whose cell body is located in the thalamus. Limited processing of sensory information takes place in the thalamus. Finally, the third-order neuron travels upward and terminates in the somatosensory cortex where more complex, cortical processing begins. 

The roots of the characteristic equation can be real or complex. But if they are complex they must appear in complex conjugate pairs. The reason for this is illustrated for a second-order system with the characteristic equation... 

There is a quantitative relationship between the location of roots in the s plane and the damping coefficient. Assume we have a second-order system or, if it is of higher order, assume it is dominated by the second-order roots closest to the imaginary axis. As shown in Fig. 10,5 the two roots are Si and and they are, of course, complex conjugates. From Eq. 6.68) the two roots are... 

Several metal insertion mechanisms have been proposed, but none of them is conclusive.18 The rate of metallation varies from square root to second order in metal salt from one system to another, and apparently there exists more than one pathway. Where the rate law is second order in metal salt, a so called sitting-atop metal ion-porphyrin complex intermediate or metal ion-deformed porphyrin intermediate, which then incorporates another metal ion into the porphyrin centre, has been postulated (Figure 3).19 For the reactions with the square root dependence on the metal salt concentration, the aggregation of metal salts is suggested.18 Of course, there are many examples which follow simple kinetics, i.e. d[M(Por)]/df = k[M salt][H2Porj. 

The overall rate of polymerization Rp of barium acrylate (BA) polymerization sensitized by MB in the presence of varying amounts of j>-toluene sulfinate sodium sslt (PTSS) was shown to depend on the square root of the absorbed light intensity and to exhibit a complex second order dependence on the concentration of sulfinate and monomer (43). The latter dependence was shown to result from a desensitization process described by eq. 16 in which PTSS undergoes Michael addition to the acrylate monomer. 

Manometers and pressure springs may be described dynamically to a first approximation by second-order differential equations for which the roots of the characteristic equation are conjugate complex. As shown in Section III, 8, lc, since the roots are complex, these systems have an oscillatory mode, and the response of the system to step forcing, for example, is a damped sinusoid. 

The polynomial P(s) is of second order and has two distinct roots which are not real (as in the previous case) but complex conjugates ... 

This filter has two poles and two zeroes. Depending on the values of the a and b coefficients, the poles and zeroes can be placed in fairly arbitrary positions around the z-plane, but not completely arbitrary if the a and b coefficients are real numbers (not complex). Remember from quadratic equations in algebra that the roots of a second order polynomial can be foimd by the formula (-a +/- / 2 for the zeroes, and similarly for the... 

We will now show how polynomial analysis can be applied to the transfer function. A polynomial defined in terms of the complex variable z takes on just the same form as when defined in terms ofx. The z form is actually less misleading, because in general the roots will be complex (e.g. /(z) = z + z - 0.5 has roots 0.5 + 0.5J and 0.5 - 0.5J.). The transfer function is defined in terms of negative powers of z - we can convert a normal polynomial into one in negative powers by multiplying ly z. So a second-order polynomial is... 

Equation 3.19, Equation 3.29, and Equation 3.33 give the general solutions of the second order, constant coefficient, homogeneous, and linear differential equation for the respective cases of real unequal, repeated, and complex characteristic roots. However, the actual steps that are used in deriving a solution to the homogeneous problem are as follows ... 

When the complexity of the mechanism is increased to a two step reaction, then the solutions of the Laplace forms of the equations involve the two roots of a quadratic as indicated above for second order differential equations. Recently Zhang, Strand White (1989) have suggested how a general matrix solution of rate equations in the Laplace form can be used to model kinetic mechanisms. Zhang et al. (1989) suggest this method as an alternative to numerical integration, but its use is, of course, restricted to linear equations like that of the more elegant matrix method described in section 4.2. 

It is clear from this example that the Laplace transform solution for complex or repeated roots can be quite cumbersome for transforms of ODEs higher than second order. In this case, using numerical simulation techniques may be more efficient to obtain a solution, as discussed in Chapters 5 and 6. 

The root locus diagram can be used to provide a quick estimate of the transient response of the closed-loop system. The roots closest to the imaginary axis correspond to the slowest response modes. If the two closest roots are a complex conjugate pair (as in Fig. 11.28), then the closed-loop system can be approximated by an underdamped second-order system as follows. 

A very important point is that, contrary to methods based on a Hartree-Fock zero-order wave function, those rooted in the Kohn-Sham approach appear equally reliable for closed- and open-shell systems across the periodic table. Coupling the reliability of the results with the speed of computations and the availability of analytical first and second derivatives paves the route for the characterization of the most significant parts of complex potential energy surfaces retaining the cleaness and ease of interpretation of a single determinant formalism. This is at the heart of more dynamically based models of physico-chemical properties and reactivity. 

From Eq. (7.174), we see that the ideal terms obtained with infinite GB have been corrupted by additional terms caused by finite GB. To see the effects graphically, we select several values of Q and then factor the polynomial in Eq. (7.175) for many values of GB in order to draw loci. The polynomial in Eq. (7.175) is fifth order, but three of the roots are in the far left-half normalized s plane. The dominant poles are a pair of complex poles that correspond to the ideal poles in Eq. (7.172) but are shifted because of finite GB. Figure 7.117 shows the family of loci generated, one locus for each value of Q selected. Only the lod of the dominant second quadrant pole are plotted. The other dominant pole is the conjugate. 

The second observation is that unlike the acids previous cases the actual bases ordering looks quite similar for all employed finite difference, Ghosh-Biswas and density functional electronegativity recipes. Therefore, the present suggested soft-to-hard Lewis bases classification is that recommended by DFEP hierarchy - rooting on the most complex conceptual-computational containing algorithm. 

---