Stabilizing functionals

A stabilizing functional (or a stabilizer) is used to select from the space M of all possible models the subset Me, which is a correctness set. 

Definition 10 A nonnegative functional s(m) in some metric space M is called a stabilizing functional if, for any real number c 0 from the domain of functional values, the subset Me of the elements m e M, for which s(m) c, is compact. 

We will give now several examples of stabilizing functionals. 

Example 11 Let us consider a real Hilbert space L2 formed by functions integrable in the interval [a, 6] (see Appendix A). The metric in the space L2 is determined according to the formula 

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Separation. In this function, the geotextile serves to separate two dissimilar materials (Fig. 3), eg, two different soils, landfill material and the native soil, stone material and subgrade sod, old and new pavement, foundation sods and various types of wads, or one of many other similar situations. In some instances, it is difficult to distinguish between the separation and stabilization functions because in both situations the geotextde is serving as a separator. However, in stabilization some additional phenomena occur. 

Organophosphoms compounds, primarily phosphonic acids, are used as sequestrants, scale inhibitors, deflocculants, or ion-control agents in oil wells, cooling-tower waters, and boiler-feed waters. Organophosphates are also used as plasticizers and flame retardants in plastics and elastomers, which accounted for 22% of PCl consumed. Phosphites, in conjunction with Hquid mixed metals, such as calcium—zinc and barium—cadmium heat stabilizers, function as antioxidants and stabilizer adjutants. In 1992, such phosphoms-based chemicals amounted to slightly more than 6% of all such plastic additives and represented 8500 t of phosphoms. Because PVC production is expected to increase, the use of phosphoms additive should increase 3% aimually through 1999. 

These stabilizers function as light-stable antioxidants to protect polymers. Their antioxidant activity is explained by the foUowiag sequence (16) ... 

The tube-current stabilizer is usually put in the grounded return output circuit of the high-voltage transformer. The stabilizer functions by properly adjusting the a-c heating current through the filament (x-ray tube cathode), and in this way regulating the electron emission. 

In general, stabilizers function by reaction with proplnt decompn products. A number of methods have been described and a preliminary evaluation of these methods was conducted by several labs under a cooperative program. Based on results from these screening tests, the PicArsn spec tropho tome trie methods for available stabilizer and primary degradation products were selected for further study. The initial phase of this program was an attempt to standardize the necessary spectrophotometric factors. Significant differences with regard to the factors obtained... 

Hardness stabilization, functioning at substoichiometric levels by crystal growth retardation, (threshold effect). This attribute is useful in low heat-flux HW applications. 

Figure 1. Possible strategies for the synthesis of functional microgel stabilizers functionalization of preformed microgels (left) or copolymerization of suitable functional monomers (right, our approach). (Reprinted from Ref. [13b], 2003, with permission from Wiley-VCH.)...

The fundamental aspects of the structure and stability of carbanions were discussed in Chapter 6 of Part A. In the present chapter we relate the properties and reactivity of carbanions stabilized by carbonyl and other EWG substituents to their application as nucleophiles in synthesis. As discussed in Section 6.3 of Part A, there is a fundamental relationship between the stabilizing functional group and the acidity of the C-H groups, as illustrated by the pK data summarized in Table 6.7 in Part A. These pK data provide a basis for assessing the stability and reactivity of carbanions. The acidity of the reactant determines which bases can be used for generation of the anion. Another crucial factor is the distinction between kinetic or thermodynamic control of enolate formation by deprotonation (Part A, Section 6.3), which determines the enolate composition. Fundamental mechanisms of Sw2 alkylation reactions of carbanions are discussed in Section 6.5 of Part A. A review of this material may prove helpful. 

Solution of the above constrained least squares problem requires the repeated computation of the equilibrium surface at each iteration of the parameter search. This can be avoided by using the equilibrium surface defined by the experimental VLE data points rather than the EoS computed ones in the calculation of the stability function. The above minimization problem can be further simplified by satisfying the constraint only at the given experimental data points (Englezos et al. 1989). In this case, the constraint (Equation 14.25) is replaced by... 

In Equation 14.27, cT, oP and ax are the standard deviations of the measurements of T, P and x respectively. All the derivatives are evaluated at the point where the stability function cp has its lowest value. We call the minimization of Equation 14.24 subject to the above constraint simplified Constrained Least Squares (simplified CLS) estimation. 

The set of points over which the minimum of

gradient method for the location of the minimum of the stability function, we advocate the use of direct search. The rationale behind this choice is that first we avoid any local minima and second the computational requirements for a direct search over the interpolated and the given experimental data are rather negligible. Hence, the minimization of Equation 14.24 should be performed subject to the following constraint... 

Using the values (-0.2094, -0.2665) for the parameters ka and kd in the EoS, the stability function, stability function at each isotherm are shown in Figure 14.5. The stability function at 311.5 K is shown in Figure 14.6. As seen,

liquid phases. This is also evident in Fig. 14.7 where the EoS-based VLE calculations at 311.5 K are shown. 

Hgure 14.5 The minima of the stability function at the experimental temperatures for the diethylamine-water system [reprinted from Computers Chemical Engineering with permission from Elsevier Science]. 

Figure 14.6 The stability function calculated with interaction parameters from unconstrained least squares estimation.

Therefore, although the stability function was found to be positive at all the experimental conditions it becomes negative at mole fractions between 0 and the first measured data point. Obviously, if there were additional data available in this region, the simplified constrained LS method that was followed above would have yielded interaction parameters that do not result in prediction of false liquid phase splitting. 

New interaction parameter values were obtained and they are also shown in Table 14.5. These values are used in the EoS to calculate the stability function and the calculated results at 311.5 K are shown in Figure 14.10. 

As seen from the figure, the stability function does not become negative at any pressure when the hydrogen sulfide mole fraction lies anywhere between 0 and 1. The phase diagram calculations at 311.5 K are shown in Figure 14.11. As seen, the correct phase behavior is now predicted by the EoS. 

The improved method guarantees that the EoS will calculate the correct VLE not only at the experimental data but also at any other point that belongs to the same isotherm. The question that arises is what happens at temperatures different than the experimental. As seen in Figure 14.10 the minima of the stability function increase monotonically with respect to temperature. Hence, it is safe to assume that at any temperature between the lowest and the highest one, the EoS predicts the correct behavior of the system. However, below the minimum experimental temperature, it is likely that the EoS will predict erroneous liquid phase separation. 

The above observation provides useful information to the experimenter when investigating systems that exhibit vapor-liquid-liquid equilibrium. In particular, it is desirable to obtain VLE measurements at a temperature near the one where the third phase appears. Then by performing CLS estimation, it is guaranteed that the EoS predicts complete miscibility everywhere in the actual two phase region. It should be noted, however, that in general the minima of the stability function at each temperature might not change monotonically. This is the case with the C02-n-Hexane system where it is risky to interpolate for intermediate temperatures. Hence, VLE data should also be collected at intermediate temperatures too. 

Boldyrev, A. A., Koldo, A., Kurella, E., Maltseva, V., and Stvolinski, S. (1993). Natural histidine-containing dipeptide carnosine as a potent hydrophilic antioxidant with membrane stabilizing function. A biomedical report. Mol. Chem. Neuropathol. 19,185-192. 

Kasai H, Matsuzaki M, Noguchi J, et al (2003) Structure-stability-function relationships of dendritic spines. Trends Neurosci 26 360-368... 

Capping at the 5 -end of the pre-mRNA occurs immediately after incorporation of about 30 nucleotides in the primary transcript. The 5 cap structure is required for the binding of the mRNA to the 40S subunit of the ribosome during the initiation of translation. Capping is also ascribed a stabilizing function for mRNA. 

Shan Yao is sweet and neutral, and enters the Lung, Spleen and Kidney meridians. It can either tonify the Qi or nourish the Yin. Its astringent property enhances its tonifying and stabilizing function. This herb is particularly useful in chronic conditions of Qi and Yin deficiency, such as in Thirst and West syndrome (diabetes) to moisten the internal organs and stabilize the body essence. 

Cell-cell interactions and cell-scaffold interactions have been shown to unprove viability and stabilize function. Complete understanding of the mechanisms required to stabilize hepatocyte function will have a broad impact on this technology. 

Lin, C. F. 1971. The casein stabilizing function of sulfated polysaccharide. Ph.D. thesis, Ohio State University, Columbus. 

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