Composition series

Note that the Homomorphism Theorem and the First Isomorphism Theorem deal with factorizations over arbitrary closed subsets, whereas the Second Isomorphism Theorem deals with factorizations over normal closed subsets. 

We shall generalize the Second Isomorphism Theorem in the next section. 

The Homomorphism Theorem and the two Isomorphism Theorems were first proved in [35]. The thin case was already proved in 1929 by Emmy Noether cf. [32 I. 2], 

Note that, by Lemma 2.5.1(h), the hypothesis T C Ns(U) in Corollary 5.3.5 says exactly that U is a normal closed subset of S. 

In this short section, S is assumed to have finite valency. We start with a generalization of a theorem of Hans Zassenhaus cf. [41 II. 5]. 

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Stoichiometric saturation defines equilibrium between an aqueous solution and homogeneous multi-component solid of fixed composition (10). At stoichiometric saturation the composition of the solid remains fixed even though the mineral is part of a continuous compositional series. Since, in this case, the composition of the solid is invariant, the solid may be treated as a one-component phase and Equation 6 is the only equilibrium criteria applicable. Equations 1 and 2 no longer apply at stoichiometric saturation because, owing to kinetic restrictions, the solid and saturated solution compositions are not free to change in establishing an equivalence of individual component chemical potentials between solid and aqueous solution. The equilibrium constant, K(x), is defined identically for both equilibrium and stoichiometric saturation. 

In a series of experiments carried out on a composition series of glassy Sb cSei- c alloys, it was found that the time-dependent dark-decay rate of the potential to which... 

Burnham, 1962 Keller and Hanson, 1968 Rose, 1970 Lowell and Guilbert, 1970 Keller, 1963). These studies indicate that kaolinite can be formed by hydrothermal alteration at the surface as well as to depths of several kilometers. Although information is lacking for low temperatures, intermediate conditions of pressure and temperature are known to permit the stability of the potassic mica-beidellite mixed layered composition series which excludes the stable coexistence of K-feldspar and kaolinite. If one accepts the argument that both beidellite-sodic and potassic are... 

Experimental work in the systems K-Mg-Si-Al-Fe- O concerning celadonites has also produced expandable minerals (Velde, 1972 Velde, unpublished). In both the muscovite-MgAl celadonite and MgFe-MgAl celadonite compositional series, fully expandable phases were produced below 300°C at 2Kb pressure. These expandable phases can coexist with a potassic feldspar (Figure 23). Their (060) reflection near 1.50 X indicates a dioctahedral structure which can apparently be intimately... 

Mg which is calculated as octahedral ions. It is nevertheless quite possible that the analyses of the fully expandable montmorillonites do show a valid chemical variation and not just analytical error of one sort or another. A remarkable point, in comparing the mixed layered and fully expandable bulk compositions is that the former defines two compositional series while the latter is found just between these two series. If indeed, this is the result of not only analytical errors, the relations would suggest that the fully expandable series are mixtures of the two extreme compositional types beidellite and montmorillonite. Since neither these nor the two forms are found alone, one would suspect the above deduction to be true. The possibility of the coexistence of two fully expanding phases has important implications in the phase relations as we will see. 

The major aluminous clay minerals, alkali zeolites and feldspars which are most commonly associated in nature can be considered as the phases present in a simplified chemical system. Zeolites can be chemiographically aligned between natrolite (Na) and phillipsite (K) at the silica-poor, and mordenite-clinoptilolite at the silica-rich end of the compositional series. Potassium mica (illite), montmorillonite, kaolinite, gibbsite and opal or amorphous silica are the other phases which can be expected in... 

Figure 41. Phase diagram for the extensive variables R -R -Si combining the data for synthetic magnesian chlorites and the compositional series of natural sepiolites and palygorskites. Numbers represent the major three-phase assemblages related to sepiolite-palygorskite occurrence in sediments. Chi = chlorite M03 = trioctahedral montmorillonites M02 = dioctahedral montmorillonite Sep = sepiolite Pa = palygorskite Kaol = kaolinite T = talc.

In Section 5.4, we define composition series of schemes having finite valency. The main result will be a generalization of a group theoretic theorem due to Camille Jordan and Otto Holder. This generalization says that any two composition series of a closed subset of a scheme of finite valency are isomorphic. 

Composition series lead naturally to the notion of a composition factor, and, according to Lemma 4.2.4(i), composition factors must be simple. Thus, the above theorem on composition series gives reason to consider simple schemes as crucial in scheme theory. 

Theorem 5.4.2 Let T be a closed subset of S. Then any two composition series ofT are isomorphic. 

On the other hand, as above, the fact that V is a composition series of T yields Uv = UuV n UVUVV. Therefore, Uu C Uvv would lead to Uu n UvV C Uu, and that contradicts the above given isomorphism. Thus, Uu % U"v. 

Fig. 9. Dependence of mechanical properties of highly-loaded composites (series 7, Table 2) on their free volume. Pp, — breaking stress, MPa x 10 2.

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