Linear inverse

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

JW. Enquire, WE. Deeds, and CV. Dodd. Alternating current distribution between planar conductors. Journal of Applied Physics, 41(10) 3983-3991, September 1970. C. De Mol M. Bertero and E.R. Pike. Linear inverse problems with discrete data. li. stability and regularization. Inverse Problems, 4 pp. 573-594, 1987. 

Lefebvre, J.P., Progress in linear inverse scattering imaging NDE application of Ultrasonic Reflection Tomography, in Inverse Problem in Engineering Mechanies, pp 371-375, (A.A.Balkema/ Rotterdam rookfleld, 1994). 

Evidently a linear inverse relationship exists between molecular weights and initiator concentration. This effect is certainly due to efficient chain transfer to the initiator ... 

In the transition state (i) the coordination planes at M and E are perpendicular to each other. In (ii) E is linearly coordinated and the rotation takes place through a linear inversion at E which allows E to remain 7r-bonded to M throughout the dynamic process. Recent calculations58 have shown that pathway (ii) is of significantly lower energy than (i) for... 

Twomey, S., Comparison of Constrained Linear Inversion and an Iterative Nonlinear Algorithm Applied to the Indirect Estimation of the Particle Size Distribution, J. Comp. Phys. 18 188-200 (1975). 

Fig. 10 relates the composite extraction index (see above) obtained in the low-shear aqueous test system for these Tween surfactants, and adhesion tensions measured against various solids. Adhesion tensions against platinum and bitumen saturated pyrophyllite are irregularly related to tar sand extraction, while the adhesion tension against a fresh pyrophyllite surface is linearly (inversely) related to tar sand extraction. This is the first linear correlation between a measurable property of a surfactant solution and tar sand extraction which we have been able to obtain, and there appears to be no such finding in the literature. Fig. 11 gives the relations between extraction of bitumen with the paddle mill, solvent-aqueous-surfactant extraction and adhesion tensions measured against platinum, bitumen saturated pyrophyllite and hydrated (48 hours in water) pyrophyllite. 

Of course, because of the linear inverse relationship between a and tt charges, the correlation with total charges also implies the existence of individual linear correlations with (T and tt populations, namely, upheld shifts for increasing 7r-electron densities, just as in the case of aromahc hydrocarbons. The latter result also follows from Patiser-Parr-Pople calculahons of tt charge densities reported by Witanowski et al. [161]. AU of these considerahons apply equally well to the series including pyridine, 1,3-diazine, 1,4-diazine, and 1,3,5-triazine, as revealed by their spectra in DMSO soluhon [162] (Fig. 6.8). 

Steele, H. M., and R. P. Turco, Retrieval of Aerosol Size Distributions from Satellite Extinction Spectra Using Constrained Linear Inversion, J. Geophys. Res., 102, 16737-16747 (1997a). 

The linear (inverse) relation between NC moisture content and NC nitrogen content was also noted in Ref 5... 

The constrained deconvolution algorithm produces estimates that cannot be obtained from the data by simple linear inverse filtering. This is most readily seen using the Blass-Halsey weight function as an example. 

The ketoprofen release profiles from these tablets are shown in Fig. 6. The release curves were linear (r=0.99) and the release rate appeared to decrease with increasing coating amounts. On the basis of the linear inverse relation found between the drug release rate and the coat (Fig. 7), the release rate can be varied with predictable effects by varying the coating amount. In particular the KET-R 15 tablet showed a lower rate of drug release than the 2/3 CAP coated tablet (0.96%/h... 

Jamil, H. Madsen, N.B. Phosphorylation state of acetyl-coenzyme A carboxylase. 1. Linear inverse relationship to activity ratios at different citrate concentrations. J. Biol. Chem., 262, 630-637 (1987)... 

Alkaline earth-exchanged samples examined by Ward (211) were more resistant to thermal dehydroxylation hydroxyl bands were present in the spectra after dehydration at 500°C. The concentration of OH groups was, however, much smaller than found in H—Y zeolite, and was dependent on the cation type. An almost linear inverse relationship was found between the alkaline earth cation radius and the concentration of acidic hydroxyl groups (210). 

Figure 3.29 displays once again a nearly linear inverse variation of As with a measure of rDA (here proportional to L + a, where the film thickness (L) was varied from 0-60 A). 

Hansen, P.C., Rank-Deficient and Discrete Ill-Posed Problems Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998. 

Allegre C. J. and Lewin E. (1989) Chemical structure and history of the Earth evidence from global non-linear inversion of isotopic data in a three-box model. Earth Planet. Sci. Lett. 96, 61-88. 

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