The Thin Residue

In this section, S is assumed to have finite valency. The letter T will always stand for a closed subset of S. 

We define O 1 (T) to be the intersection of all strongly normal closed subsets of T and call it the thin residue of T. 

Since p and q have been chosen arbitrarily in T, we have shown t Rt C (R) for each element t in T. Thus, by Lemma 3.1.4, (R) is strongly normal in T. Thus, by definition, 0 (T) C (R). 

In order to show that R) C 0 T) it suffices to show that R C 0 T). (This is because 0 T) is closed.) 

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In the second section, we introduce the thin residue of a closed subset. The thin residue is a specific commutator subset. We look at the thin residue of a complex product of two closed subsets, at the complex product of the thin residue and the thin radical, and at multiple thin residues of closed subsets. 

In the second section, we shall relate specific closed subsets of S containing a finite closed subset T to the corresponding closed subsets of the quotient scheme of S over T. Among other issues we focus on the relationship between commutators and quotient schemes. This leads naturally to the connection between the thin residue of S and the thin residue of quotient schemes of S. This relationship will be described in Theorem 4.2.8, a result which depends on Lemma 3.2.7. Theorem 4.2.8 turns out to be useful in Section 5.5 where we discuss residually thin schemes. 

Let us now look how the thin residue works together with quotient schemes. 

From Theorem 3.2.1 (i) we know that O U J ) is strongly normal in T. Thus, by Lemma 4.2.5(h), T//0 T) is thin. Thus, we obtain from Lemma 4.2.7(i) that the thin residue 0 S(T) of T is the uniquely defined smallest closed subset of S having a thin quotient scheme. 

The physico-chemical characterization of processed flat photoresist layers showed that the properties of the photoresist change significantly if it is processed with the threshold exposure dose that makes the photoresist soluble in the developer. Caused by a deprotection reaction, the photoresist becomes acidic and more hydrophilic. It was postulated that the thin residual photoresist layers processed with the threshold dose are appropriate model surfaces for the sidewalls of the photoresist structures in the photolithographic process. 

Global cleaning with wet N2 sweeping to treat the thin residues of metallic sodium (module s position 30°) ... 

Dry Ice. Refrigeration of foodstuffs, especially ice cream, meat products, and frozen foods, is the principal use for soHd carbon dioxide. Dry ice is especially useful for chilling ice cream products because it can be easily sawed into thin slabs and leaves no Hquid residue upon evaporation. Cmshed dry ice may be mixed directly with other products without contaminating them and is widely used in the processing of substances that must be kept cold. Dry ice is mixed with molded substances that must be kept cold. For example, dry ice is mixed with molded mbber articles in a tumbling dmm to chill them sufficiently so that the thin flash or rind becomes brittle and breaks off. It is also used to chill golf-ball centers before winding. 

From the third reactor the polymer is then run into a devolatilising ( stripping ) vessel in the form of thin strands. At a temperature of 225°C the solvent, residual monomer and some very low molecular weight polymers are removed, condensed and recycled. The polymer is then fed to extruder units, extruded as filaments, granulated, lubricated and stored to await dispatch. 

The thin film of oxide which remains in the casserole is dissolved by treating with a little constant-boiling hydrobromic acid (Note 7) the solution obtained may then be evaporated and the residue added to the next fusion mixture. 

Si element ATR-FTIR spectroscopy was used to analyze this residue, and its spectrum, along with the closest library matches, are shown in Figure 41. The absorbance of this residue is low as a consequence of the thin layer present on the plate. This makes matching the sample spectrum with a reference spectrum somewhat difficult. The closest matches extracted from the library interrogated are to ester-based plasticizer materials, which is consistent with a phthalate-plasticized PVC. A more specific identification could have been made with further testing such as subjecting the residue to GC-MS analysis, but the information suggested by the ATR-FTIR analysis was, in this case sufficient. 

The thin-layer configuration and its associated diffusion problems means that it is possible to oxidise (or reduce) all of the electroactive species in the thin layer before they can be replenished to any marked degree. Consider, for example, the 0"+/0 couple, with a standard redox potential well within the "electrochemical window of the solvent, so that the current in the absence of the couple is small and can easily be accounted for. With the electrode pushed against the window the potential is stepped cathodic enough to ensure the rapid reduction of the 0" + and the current measured as a function of time, the concentration such that the time for the current to reach zero, or a steady residual value, is small. If the area under the I ft curve is A ampere seconds, then the charge passed Q = A coulombs. Thus, the number of moles of 0"+ reduced, N0, is given by ... 

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