Singularity analysis

Using the singularity analysis again, the condition for the appearance or disappearance of a hysteresis loop is given in terms of the two unfolding parameters K and k2 as... 

Table 1. Singularity analysis of the energy function of the BH molecule from quadratic approximants...

As in the case of the esterification process presented before, we study first a one-stage column by singularity analysis to gain some physical insight and then continue with a numerical bifurcation study of an industrial size RD column. 

A model of a one-stage distillation process for ethylene glycol synthesis is reported by Gehrke and Marquardt [23]. Singularity analysis reveals a codimension 4 singularity for a Damkohler number Da = 0.0247, a heat duty Q/(AHvF) = 0.2871 and feed concentrations = 0.4597, = 0.072, = 0.3961, Xj q = 0.072. How-... 

The findings from singularity analysis of the one-stage column will be checked for their relevance for the nonlinear behavior of an industrial size RD column by means of numerical bifurcation analysis using DIVA [53, 63]. The model used is of moderate complexity. The major assumptions are constant liquid holdup on every tray negligible vapor holdup, constant heat capacities and heats of vaporization of all the species, ideal gas phase, almost ideal liquid phase, perfect mixing on... 

This singularity analysis is applicable to any Schrodinger equation for a central potential, and the resulting singularity structure depends only on the behavior of the potential in the limit r — 0. Any system whose potentied at small r is proportional r will have a second-order pole according to Eq. (30). However, it is important to note that this second-order pole is only the most singular behavior at i = 3 — 2n in general, one can also expect a coincident first-order pole. 

We have been able to fit the parameter 6 even more precisely using a modified version of the Fade singularity analysis [10]. Fade, in his original treatise [35], suggested a generalization of his approximants that can explicitly model branch-point singularities. Let P S) and Q(S) be the numerator and denominator polynomials, respectively. Then Eq. (15) can be written in the form of a linear equation in E,... 

Fade approximants were used in Section 3.B. as a tool for singularity analysis, but their primary function is as a summation method. Unfortunately, their convergence with dimensional expansions tends to be rather slow and uneven. They can be useful if the expansion is known to high order, but they are not appropriate as a low-order summation technique. However, their rate of convergence can be improved considerably if they are used in conjunction with the hybrid expansion, of Eq.(38). The Fade approximants of the E converge quite well even at low order. We will refer to this technique as hybrid Fade summation. 

For n = 0 the shifted expansion is identical to the rescaled expansion, Eq. (41), with a second-order pole at = 1, but they differ at higher order since the shifted expansion adds additional higher-order poles at <5 = 1. We know from dimensional singularity analysis [18] that there are no poles at = 1 of order higher than two, so we can expect that the shifted expansion will be less and less accurate than the rescaled expansion as n increases. This is indeed what happens in practice [11]. The shifted expansion procedure has been rather popular [31], but now that we have information about the dimensional singularity structure of the problem this method should probably be considered obsolete. 

Figure 3. Singularity analysis of the [7/8] Fade approximants of the four energy expansions, x represents a pole of the approximant while o represents a zero. The panels are labeled Inan n ) according to the quantum numbers of the large-dimension limit.

Singularities can be thought of as either an abrupt change or impulse in a signal, or the sudden shift of the signal s mean value to a different level. The good time-frequency localisation property provides wavelet in singularity analysis (Mallat and... 

In this chapter, based on the signal theory, wavelet theory and sparse decompositions are introduced with their applications to TCM. Because of the non-stationary property of the machining process, wavelet analysis and sparse decomposition are more effective thanks to Fourier methods in TCM signal analysis. Real case studies with wavelet singularity analysis and sparse decomposition for de-noising are also presented and discussed with their suitability and... 

Elastoplastic analyses of bimaterial comers have also been conducted, albeit more recently. A singularity analysis based on 72-deformation theory was devel-... 

---