Perturbation analysis

In Sect. 8 we have used the small-load approximation to derive frequency shifts for various geometries. Evidently, the linearization in A/, which was applied in order to derive the small-load approximation, has its limits of 

For an illustration of this shortcoming, assume that the film has the exact same acoustic properties as the quartz crystal. In this case the fractional frequency shift must be strictly the same on all overtone orders  

Equation 94 holds regardless of the overtone order and is a simple result. However, Eq. 94 is not reproduced when applying the small-load approximation (Eq. 51) and using the load impedance of a viscoelastic film as expressed in Eq. 72  

Unless the compliance of the film, /f, is zero (which is physically impossible) there is a nontrivial dependence of A/ on overtone order, contradicting Eq. 94. This error occurs because the term - 2iAZq cot(fcqfiq) in Eq. 113 was linearized inf -fi (Appendix A), whereas the impedance of the load was expanded to third order in thickness (Sect. 8.2.3). In order to do the derivation consistently, we have to omit the linearization in Af = f - f. Requiring that the mechanical impedance on resonance, Zm, be zero amounts to an imphcit equation in A/. The entire equation has to be Taylor-expanded in Af/f and solved iteratively. In the following, we sketch the argument. The full derivation is given in [104]. 

Superscripts in brackets ( ) denote the respective coefficient of the Taylor expansion. Importantly, the zeroth-order term on the right-hand side van- 

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Forward Analysis In this type of analysis, we are interested in the propagation of initial perturbations Sxq along the flow of (1), i.e., in the growth of the perturbations 5x t xo) = (xo -h Sxq) — xq. The condition number K,(t) may be defined as the worst case error propagation factor (cf. textbook [4]), so that, in first order perturbation analysis and with a suitable norm j ... 

Exact computability in this sense, however, is achieved only at the cost of being able to obtain approximate solutions. Perturbation analysis, for example, is rendered virt ially meaningless in this context. It is not s irprising that traditional investigatory methodologies are not very well suited to studies of complex systems. Since the behavior of such models can generally be obtained only through explicit simulation, the computer becomes the one absolutely indispensable research tool. 

It is now well-established that for atomic fluids, far from the critical point, the atomic organisation is dictated by the repulsive forces while the longer range attractive forces serve to maintain the high density [34]. The investigation of systems of hard spheres can therefore be used as simple models for atomic systems they also serve as a basis for a thermodynamic perturbation analysis to introduce the attractive forces in a van der Waals-like approach [35]. In consequence it is to be expected that the anisotropic repulsive forces would be responsible for the structure of liquid crystal phases and numerous simulation studies of hard objects have been undertaken to explore this possibility [36]. 

Barcilon, V, Singular Perturbation Analysis of the Fokker-Planck Equation Kramer s Underdamped Problem, SIAM Journal of Applied Mathematics 56, 446, 1996. 

For the case of a thread breaking during flow, the analysis is complicated because the wavelength of each disturbance is stretched along with the thread. This causes the dominant disturbance to change over time, which results in a delay of actual breakup. Tomotika (1936) and Mikami et al. (1975) analyzed breakup of threads during flow for 3D extensional flow, and Khakhar and Ottino (1987) extended the analysis to general linear flows. Each of these works uses a perturbation analysis to describe an equation for the evolution of a disturbance. 

Using a perturbative analysis of the time-dependent signal, and focusing on the interference term between the one- and two-photon processes in Fig. 14, we consider first the limit of ultrashort pulses (in practice, short with respect to all time scales of the system). Approximating the laser pulse as a delta function of time, we have... 

The main results of our first-order regular perturbation analysis are the expressions for the constant thin film thickness, hQ, and for the total hydrodynamic pressure drop across the entire... 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

Thus, these orbitals can be used to represent exactly any property of the system in localized terms. The NAOs divide naturally into a leading high-occupancy set (the natural minimal basis ) and a residual low-occupancy set (the natural Rydberg basis ), where the occupancies of the latter orbitals are usually quite negligible for chemical purposes. Thus, even if the underlying variational basis set is of high dimensionality (6-311++G for the applications of this book), a perturbative analysis couched in NAO terms has the simplicity of an elementary minimal-basis treatment without appreciable loss of chemical accuracy. 

The corresponding estimate from NBO second-order perturbative analysis is... 

Figure 4.105 A schematic perturbative-analysis diagram for occupied valence MOs (center) of PtH42 (B3LYP/LANL2DZ level), showing tie-lines for analysis in terms of AO basis functions (left) versus NAOs (right). (A tie-line is shown when the AO or NAO contributes at least 5% to the connected MO.) Note the much smaller number of contributing orbitals, the sparser tie-line patterns, and the more realistic physical range of atomic orbital energies for NAOs than for standard Gaussian-basis AOs.

The effect of external field on reactivity descriptors has been of recent interest. Since the basic reactivity descriptors are derivatives of energy and electron density with respect to the number of electrons, the effect of external field on these descriptors can be understood by the perturbative analysis of energy and electron density with respect to number of electrons and external field. Such an analysis has been done by Senet [22] and Fuentealba [23]. Senet discussed perturbation of these quantities with respect to general local external potential. It can be shown that since p(r) = 8E/8vexl, Fukui function can be seen either as a derivative of chemical potential... 

Fox, R. O. (1989). Steady-state IEM model Singular perturbation analysis near perfectmicromixing limit. Chemical Engineering Science 44, 2831-2842. 

Multiple-scale perturbation analysis and numerical simulation of the unsteady-state IEM model. Chemical Engineering Science 45, 2857-2876. 

W. Kutzelnigg and D. Mukherjee, Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. IV. Perturbative analysis. J. Chem. Phys. 120, 7350 (2004). 

There are three possibilities that are more easily understood in a perturbative analysis, which we discuss in the next section. 

D. Perturbative Analysis and Relation to Coupled-Cluster Theory... 

The perturbative analysis in Section II.D showed that single-reference L-CTSD is exact through third order in the fluctuation potential, much like L-CCSD theory, and our results are consistent with this analysis. This suggests that one of the things we... 

In effect, near the minimum of E, the 6Tk obtained through a perturbative analysis of (32) when substituted in (31) gives 8 k< 0. 

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