Cauchy data

Here we summarize a program to find explicitly the Cauchy data of electromagnetic knots [25,27,30-32]. Let < 50, 00 —> S2 be two applications satisfying the following two conditions ... 

Summarizing this subsection, the group-theoretic techniques allow us to obtain three maps S3 —>. S 2 whose velocity vectors are mutually orthogonal, and with the same linking number. Next, we have to build the Cauchy data of the electromagnetic knots based on these maps. 

If we choose the maps [Pg.224]

Consequently, we have obtained the Cauchy data of an electromagnetic knot, a representative of the homotopy class C, for which, according to (63)... 

To find the electromagnetic knot, defined at every time, from the Cauchy data (91), we use the Fourier analysis. The magnetic and electric fields can be written as... 

The important point in the use of Fourier analysis is that the vectors Ri, R2 can be computed from the Cauchy data of the electromagnetic field ... 

For the electromagnetic knot with Cauchy data given by (91), we find... 

Let us now identify the Cauchy data. As the Maxwell equations are of second order in the scalars, the initial data should be the two functions (time derivatives 0o[Pg.232]

Substitution in (128) shows that Sotime derivatives of the scalars can be expressed in terms of the scalars and their space derivatives. In other words, the Cauchy data are just the pair of complex functions cf)(r, 0), 0(r, 0) that verify the condition (126). The system therefore has two degrees of freedom with a differential constraint that is conserved naturally under the time evolution. 

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. 

A Cauchy problem is said to be stable with respect to the initial data and right-hand side if... 

In conformity with the superposition principle ( is a linear operator), the stability of the Cauchy problem with respect to the right-hand side follows from the uniform stability with respect to the initial data... 

As a matter of fact, we will consider the set of solutions ykri )] of Cauchy problem (4) dependent on the input data 2/o/> -... 

Inequality (12) expresses the property of continuous dependence which is uniform in h and t of the Cauchy problem (4) upon the input data. Here and below the meaning of this property is stability. A difference scheme is said to be absolutely stable if it is stable for any r and h (not only for all sufficiently small ones). It is fairly common to distinguish the notion of stability with respect to the initial data and that with respect to the right-hand side. Scheme (4) is said to be stable with respect to the initial data if a solution to the homogeneous equation... 

Only the results for the Cauchy and symmetric contaminated distributions are shown in Figs. 13 and 14, respectively. From these figures, it is clearly shown that the robust approach consistently and successfully performs the data reconciliation, regardless of the distributions of the data. This is a very desirable property in real applications, since in most cases the distribution is unknown or known only approximately. 

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. 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

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. 

In our knowledge robust estimators have not been applied in nonlinear dynamic real plant data yet. The first comparative study among some robust estimators in DR has been presented by Ozyurt and Pike (2004). They conclude that the estimators based on Cauchy and Hampel distributions give promising results, however did not consider dynamic systems. Other earlier studied has been accomplished by Basu and Paliwal (1989) in autoregressive parameter robust estimation issues, showing that for their case the Welsch estimator produced the best results. 

Table 5 shows the experimental specific refractivities, K X) = n(l) l]/ p, and the average polarizability as calculated from equation (1) at a number of frequencies for liquid and vapour phases. The values of the specific refractivity of the vapour have been obtained from the Cauchy dispersion formula of Zeiss and Meath.39 In this paper the authors assess the results of a number of experimental determinations of the refractive index of water vapour and its variation with frequency. Even after some normalization of the data to harmonize the absolute values from different determinations there is a one or two percent spread of results at any one wavelength. Extrapolation of the renormalized data for five independent sets of data leads to zero frequency values of K(7.) within the range (2.985-3.013) x 10-4 m3 kg 1, giving, via equation (1), LL — 9.63 0.10 au. Extrapolation of the earlier refractive index data of Cuthbertson and Cuthbertson40 by Russell and Spackman41 from 8 values of frequency between 0.068 and 0.095 au, leads to a zero frequency value, of y.i, 1,(0) = 9.83 au. While the considerable variation between the raw experimental data reported in different determinations is cause for some uncertainty, it appears that the most convincing analysis to date is that of... 

Table 5 Experimental data for the linear response of water. The gas phase data is from the Cauchy formula of Zeiss and Meith.39 The Cuthbertson and Cuthbertson40 value of a(0) is shown in brackets. The figures in heavy type lie in the range containing the data used in deriving the Zeiss-Meath formula. The liquid phase data is from P. Schiebener, J. Straub, J.M.H. Levelt Sengers and J.S. Gallagher, J. Chem. Phys. Ref. Data, 1990, 19, 677,1617. Polarizability values in all cases have been derived from the refractive index data using equation (1)...

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