Cauchy data models

We end this section with a comment referring to the Cauchy data for the scalars. In standard Maxwell theory, the Cauchy data are the eight functions A(i,6o<4M, and there is gauge invariance. In this topological model, they are the four complex functions (r, 0), 0 (r. 0), that is, eight real functions, constrained by the two conditions x V< >k) (V0 x V0 ) =0, k = 1,2, to ensure that the level curves of k will be orthogonal to those of 0. It is not necessary to prescribe the time derivatives 9o4>, 000 since they are determined by the duality conditions (138), as explained above. 

An interesting method of fitting was presented with the introduction, some years ago, of the model 310 curve resolver by E. I. du Pont de Nemours and Company. With this equipment, the operator chose between superpositions of Gaussian and Cauchy functions electronically generated and visually superimposed on the data record. The operator had freedom to adjust the component parameters and seek a visual best match to the data. The curve resolver provided an excellent graphic demonstration of the ambiguities that can result when any method is employed to resolve curves, whether the fit is visually based or firmly rooted in rigorous least squares. The operator of the model 310 soon discovered that, when data comprise two closely spaced peaks, acceptable fits can be obtained with more than one choice of parameters. The closer the blended peaks, the wider was the choice of parameters. The part played by noise also became rapidly apparent. The noisy data trace allowed the operator additional freedom of choice, when he considered the error bar that is implicit at each data point. 

Rietveld (g.c.) analysis of the neutron diffraction data on isotactic polypropylene is still in progress. It has afforded the interesting result, already discussed, that the profiles are better approximated by Cauchy than by Gaussian functions. The structural analysis is now restricted to the fourth model (P2 /c, Immirzi), which gives an excellent agreement between observation and calculation, but with the fraction of reversed helices close to 50% instead of 25% and with less chain symmetry. The other models will be tested for a more complete comparison with x-ray results. We cannot exclude, however, the possibility that the two samples used, which have different chemical, thermal and mechanical history, can really have different structures. 

Using a four-phase model consisting of ambient/simple grade/film/ substrate, we fit the data to obtain the dispersion of optical constants for each films in the range of 1.55-6.53 eV. The Cauchy model was used as a model for the substrate and fixed during the fitting. The Cody-Lorentz (CL) model [14] was used as a model for the film. 

This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et al. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a stable region, one could get arbitrarily close to an unstable one. 

Although the extended Cauchy equation fits experimental data well [7], its physical origin is not clear. A better physical meaning can be obtained by the three-band model which takes fliree major electronic transition bands into consideration. 

Figure 6.2 Wavelength-dependent refractive indices of 5CB at T= 25.1 C. Open and closed circles are experimental data for and n , respectively. Solid line represents the three-band model and dashed lines are for the extended Cauchy model. The fitting parameters are listed in Table 6.1. Li and Wu 2004. Reproduced with permission from the American Institute of Physics.

VASE. As in XPS, the data obtained fi-om VASE depend on the model parameters used to fit the measured data Since VASE is very sensitive to small deviations in the layer thiekness and optieal parameters of the substrates used, eaeh sample was measured individually before and after adsorption of the SAM. The differenee between these two measurements was fitted as a Cauchy layer (see Supporting Information) with the only parameter to be fitted being d, the layer thickness on top of a layered structure of Ti02/Si02/Si. For the underlying substrate the parameters have been fitted individually for each sample measured before adsorption and then fixed. Since the refractive index for the phosphate SAM is not known, the obtained layer thicknesses are not absolute. Nevertheless, the value used is unlikely to deviate from the true refractive index by more than... 

A Variable Angle Spectroscopic EUipsometer of the type M200-F (J.A. Woollam Co. Inc., Lincoln, USA) with a spectral range from 245 to 995 nm was used to determine the thickness of the adsorbed polymer layers. Measurements were performed in ambient at three different angles (65, 70, and 75° with respect to the surface normal). For each polymer adlayer, i.e. Sil-PEG (from toluene), Sil-PEG (from acidic aqueous solution), and PLL-g-PEG (from aqueous HEPES buffer), five samples were prepared to obtain statistical data. The measurements were fitted with multilayer models using WVASE32 analysis software. The analysis of optical constants was based on a bulk silicon/ SiOj, layer, fitted in accordance with the Jellison model. After adsorption of the molecules, the adlayer thickness was determined using a Cauchy model A = 1.45, B = 0.01, C = 0). 

Another special case of the Rivlin-Sawyer model that is a generalization of Eq. 10.10 was proposed by Wagner and Demarmels [12] who added a dependency on the Cauchy tensor to Eq. 10.10 to provide for a non-zero second normal stress difference and a better fit to data for planar extension, which is defined in Section 10.9. Their model is shown as Eq. 10.11. 

Schmidt et al. [136] also reported room-temperature spectroscopic ellipsometry results on pulsed laser deposition-grown wurtzite MgxZni xO (0thin films. The refractive index data were fit to a three-term Cauchy approximation type formula (Equation 3.105), and the anisotropic Cauchy model parameters A, B, and C for ZnO were obtained as 1.916,1.76, and 3.9 for E J c and 1.844,1.81, and 3.6 for... 

Among all distribution functions found in the literature, only 25 were chosen to be analyzed in this section. The selected distributions are summarized in Table 12.20. Not included in the list are a few well-known distributions, such as the Tukey-Lambda, Cauchy, and F distributions (Heckert and Filliben, 2003), because they are either seldom used to model empirical data or they lack a convenient analytical form for the CDF. Even though most of the distributions reported in Table 12.20 are not widely applied to engineering applications, they all have the potential to be very useful for describing real-world data sets. Definitions of the probability distribution functions in their two main forms (CDF and PDF) are presented in Table 12.21. The forms of the functions may vary slightly from those reported in the literature. It is also possible that a distribution function could be known with different names. 

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