Weak form

After the application of Green s theorem to the second order term in Equation (2.81) we get the weak form of the residual statement as... 

In addition to heavy hydrocarbons and water vapor, natural gas often contains other contaminants that may have to be removed. Carbon dioxide (CO2), hydrogen sulfide (H2S), and other sulfur compounds such as mercaptans are compounds that may require complete or partial removal for acceptance by a gas purchaser. These compounds are known as acid gases. H2S combined with water forms a weak form of sulfuric acid, while CO2 and water forms carbonic acid, thus the term acid gas. ... 

Figure 3 shows different forms of chemisorption for a C02 molecule. In the weak form of chemisorption the C02 molecule is bound to the surface by two valency bonds, as shown in Fig. 3a. This is an example of adsorption on a Mott exciton which is a pair of free valencies of opposite sign (i.e., an electron-hole pair). This may be either a free exciton wandering about the crystal or a virtual exciton generated in the very act of adsorption. As seen from Fig. 3a, in the case of the C02 molecule the weak form of chemisorption is a valency-saturated and electrically neutral form. As a result of electron capture, this form is transformed into a strong acceptor form shown in Fig. 3b, while as a result of hole capture it becomes a strong donor form shown in Fig. 3c. Both these forms are ion-radical ones. It should, however, be noted that the ion-radicals formed in these two cases are quite different and, having entered into a reaction, may cause it to proceed in different directions.

The evolution of cj) in a flow field is given by the so-called weak-form equation ... 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

It turns out that this analysis applies only to systems with a bounded phase space. It is possible that topological restrictions on the accessible phase space - and not only the form of the particular Hamiltonian -play a crucial role in determining when the weak form of the QCT actually applies. For example, this might explain why the open-system quantum delta-kicked rotor is a counter-example to naive expectations regarding the QCT (S. Habib et.al., 2002). 

For many systems, in particular physisorbed monolayers such as noble gases (He, Ar, Kr, etc.) adsorbed on grafoil " , a lattice gas model description is not microscopically realistic, since the corrugation potential is rather weak, forming only rather shallow minima at the preferred lattice sites. However, rather than studying a more realistic model, it is still possible to infer... 

Referee Laboratories and Spike Recovery Testing. Outside laboratories, with demonstrated performance records, can be used to evaluate the suitability of a candidate method when none of the other accuracy testing options is feasible. However, This technique provides a very weak form of accuracy assessment. Indeed, it provides a comparability check, not an accuracy measure. Similarly, spike recovery tests provide only weak evidence of method accuracy. Quantitative spike recovery only indicates that the added form of the analyte was recovered. If the added form responds differently toward sample preparation or detection the utility of spike recovery testing remains doubtful. 

Since the free electrons and holes in a crystal act. respectively, as free positive and negative valencies (we are dealing with ciystals with more or less sharply defined ionic bonds), it follows that the weak form of chemisorption is that in which the free valencies of the surface play no part, whereas strong chemisorption takes place when the free valence of the surface contributes to the bond, this valence being localized and attached to the valence of the adsorbed particle. We have the acceptor or donor form of strong chemisorption, depending upon which of the free valencies of the surface (positive or negative) is involved. 

The removal of an electron from an acceptor level or a hole from a donor level denotes, as we have seen, not the desorption of the chemisorbed particle but merely its transition from a state of strong to a state of weak bonding with the surface. The neglect of this weak form of chemisorption (i.e., electrically neutral form) which is characteristic of all papers on the boundary-layer theory of adsorption makes it quite impossible to depict the chemisorbed particle in terms of an energy level, i.e., to apply the energy band scheme depicted in Fig. 10 and used in these papers. ... 

In this case, as we see, equilibrium with the gaseous phase is maintained entirely by the weak form of chemisorption. In other words, only weakly bound particles are desorbed, whereas strongly bound ones play practically no part in exchange with the gaseous phase. 

By itself, hydralazine is used for hypertension however, it is not even the drug of choice for weak forms of this condition. Synonyms of this drug are apressin, hypatol, depressan, and others. 

The first urge of envy is not "I want what he has," but T want him not to have what he has, because it makes me feel that I am less." There are different kinds of envy. A weak form is "If I can t have it, nobody shall." A more malignant form involves the willingness to cut off one s nose to spite one s face to accept less for oneself if others arc thereby brought down to one s own level. Inhabitants of small towns everywhere will recognize the "Law of Jante," written down (in 1953) by one who got away ... 

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Finally, we receive a system of equations with 5 unknown quantities which are the motion of the solid phase us, the realistic pressure of the Liquid pLR and the Gas phase pGR and the temperature 9. Thus, in order to determinate these quantities we use the weak forms of the balance of momentum and of the balance of energy for the mixture ... 

Unlike ethylene glycol, a-hydroxycarboxylic acids, and p-mercaptoethanol, ligands such as ethanolamine, ethylenediamine, amino acids, and 2-aminoethanethiol do not form V2L2 complexes with vanadate [46], However, weakly formed products of VL stoichiometry (—546 ppm, —556 ppm) are observed with a number of a-amino acids [47], Because the amine functionality in amino acids is not suitable for formation of the cyclic [VN]2 core expected for a dimer, then one or the other of these VL compounds may correspond to the monomeric precursor to such a dimeric product. Certainly, with the a-hydroxycarboxylic acids, the monomeric complex can be a major component in solution. Similar VL complexes have not been reported for the amino alcohols. Perusal of the available reports suggests that the amine derivatives were studied under conditions where the nitrogen functionality was protonated, so the results may be somewhat misleading. 

When all soil acidity has converted to Ca-soil, the pH would be about 8.3. Commonly, only exchangeable acidity is easily neutralized. A significant fraction of the titratable acidity may remain intact owing to its extremely weak form or high apparent pKa values. Soil acidity is for the most part produced in soils under high rainfall regimes (tropical or temperate regions). 

The overall goal can be derived in one of two ways it can either be based on the estimates from the due diligence - a top-down goal set by the companies management - or it can be built from the bottom up by the individual project teams. A pure example of either case is very rare in practice usually, a weak form of the top-down variety prevails, in which the compan/s management formulates objectives in cooperation with the relevant project staff. 

Field et al. [157] introduced the concept of critical flux in membrane filtration. They proposed that upon start-up, there exists a flux below which a decline of flux with time does not occur. Although a concentration polarization layer is present, solid deposition on the membrane that gives rise to cake layer formation does not take place, so that a nonfoufing or cake-free operation is achieved. This flux is the critical flux and it may either be in strong form, in which flux is identical to the clean water flux at the same TMP, or in weak form, in which flux varies linearly with TMP but the slope of the fine differs from that of clean water [6,157,161]. 

The critical flux value depends largely on the hydrodynamic conditions in the process, the membrane pore size, and the feed physicochemical condition [161,168]. Appropriate manipulation of these parameters, specifically the hydrodynamic condition, may lead to the reduction or even the elimination of both reversible and irreversible fouling of the membrane. Youravong et al. [152] estimated the critical flux in the UF of skimmed milk to lie between 55 and 60 kg h m and that the average critical TMP was 0.22 bar (22 kPa) at 50°C using PVDF membranes with MWCO of 200 kDa. In the UF of WPC and sodium caseinate suspension, Youravong et al. [168] reported that both gave the weak form of critical flux, which increases with increase in crossflow velocity. 

For singlet reactions, the direction of ring closure is the one that involves the most stable 1,3-biradical, as shown in Scheme 43. This is equivalent to the assumption that in a concerted mode of reaction the bond between centers C-3 and C-5 is only weakly formed during the initial stages of the reaction. 

Next, the weak forms of the mass and momentum conservation equations ( 1) and (2) are written using these same basis functions as weighting functions ... 

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