Method upper relaxation

The upper relaxation method. In order to accelerate the iteration process in view, we are forced to revise Seidel method by inserting in (5) the iteration parameter u> so that... 

This method falls within the category of relaxation methods and gives rise to Seidel method in one particular case where w = 1. In the modern literature the iteration process (9) with w > 1 is known as the upper relaxation method. 

As shown above, Seidel method is quite applicable for any operator A = A >0. However, the extra restriction 0 < w < 2 is necessary for the convergence of the upper relaxation method. This is certainly true under condition (8) with a known operator Bq. Along these lines, it is straightforward to verify that B = u> A + ), r, = w and... 

By appeal once again to the model problem of interest it is plain to show that the upper relaxation method is a perfect tool in such matters, since the work and storage require... 

As a matter of fact, the upper relaxation method and Seidel method are nothing more than the implicit scheme (6) with B E incorporated. Still using the usual framework of implicit iterative methods, the value yk+i is determined from the equation... 

The upper relaxation method. In order to accelerate the iteration... 

Here, pi 2 is the surface relaxivity for longitudinal and transverse relaxations. In addition, it has been assumed that the ratio is directly proportional to the pore radius Rp (as for cylindrical pores). Therefore, the distribution of the relaxation times observed from the porous sample reflects the PSD. The prerequisite for using this equation is that the pore is so small that the fluid molecules collide many times with the pore wall during their contribution to the NMR signal, and this imposes the upper size limit of the method. Surface relaxivity is material dependent, and it must be determined separately for each material before pore sizes can be determined. The chemical shift of xenon depends on the surface-to-volume ratio of the pocket. On the basis of the geometry of the pocket, it can be shown that the surface-to-volume ratio is proportional to pore size, and therefore the following relationship exists between the chemical shift, 8, and the pore radius, Rf. 

Fig. 7. A C-13 relaxation time measurement of solid state wetted cellulose acetate (6% by weight water) using the inversion recovery (IR) method at 50.1 MHz and spinning at 3.2 kHz at the magic angle (54.7 deg) with strong proton decoupling during the aquisition time (136.3 ms), (upper part of the Figure). Tau represents the intervals between the 180 deg (12.2 us) inverting and 90 deg (6.1 us) measuring pulse. 2200 scans were collected and the pulse delay time was 10 s, Cf. Table 3 and Ref.281...

The modern branch-and-bound algorithms for MILPs use branch-and-bound with integer relaxation, i.e., the branch-and-bound algorithm performs a search on the integer components while lower bounds are computed from the integer relaxation of the MILP by linear programming methods. The upper bound is taken from the best integer solution found prior to the actual node. 

Fig. 28. FFC Inversion Recovery sequence. In the upper case the sample is first prepolarized in a filed Bp, then switched to the acquisition field Ba where the first RF pulse of 180° is applied and the sample magnetization is inverted. The field is then switched to B,. and the sample is allowed to relax for the variable time t. Finally, the field is switched again to the acquisition value and the magnetization is sampled by any of the sample-detection methods (here, a simple FID following a 90° RF pulse). Notice that, as shown in the lower diagram, in the special case when Bp = Ba it is possible to neatly avoid the extra switching interval prior to the inversion pulse.

For the present work, we chose the constrained method described by Jansson (1968) and Jansson et al (1968, 1970). See also Section V.A of Chapter 4 and supporting material in Chapter III. This method has also been applied to ESCA spectra by McLachlan et al (1974). In our adaptation (Jansson and Davies, 1974) the procedure was identical to that used in the original application to infrared spectra except that the data were presmoothed three times instead of once, and the variable relaxation factor was modified to accommodate the lack of an upper bound. Referring to Eqs. (15) and (16) of Section V.A.2 of Chapter 4, we set k = 2o(k)K0 for 6(k) < j and k = Kq exp[3 — for o(k) > This function is seen to apply the positivity constraint in a manner similar to that previously employed but eliminates the upper bound in favor of an exponential falloff. We also experimented with k = k0 for o(k) > j, and found it to be equally effective. As in the infrared application, only 10 iterations were needed. 

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