Homology groups with coefficients

Definition 3.14. Let A be an arbitrary trisp, and let n > 0. The nth sim-piiciai homology group with integer coefficients of A is defined by... 

In general, passing between the homology groups with different coefficients requires some work. Just to give an example, let A be the real projective plane... 

In order to deal with nonorientable geometric Poincare complexes we define generalized homology groups with twisted coefficients in the following manner. 

This lemma is equivalent to saying that X/G is a rational homology manifold. It holds because we have the Poincare duality isomorphism for the cohomology groups with rational coefficients on X/G. 

The coefficients A and B along with their standard deviations sA and sB are shown in Table 8.3.1. Equation 8.3.1 is invalid for nitrilo- and acetylalkanes. Homologous series with multiple functional groups, such as dialkyl sebacate esters and triglyceride esters, have been studied by Kishore et al. [8]. 

Homology Groups of Trisps with Integer Coefficients... 

Given a 7i-space X define the reduced homology groups of X/tt with w-twisted coefficients... 

The HLB numbers decrease with increasing chain length, e.g., from 13.25 for sodium decane 1-sulfonate to 9.45 for the C18 homolog [72]. Typical HLB numbers for positional isomers range from 12.3 for sodium dodecane 1-sulfonate to 13.2 for the more hydrophilic 6 isomer [73]. The HLB numbers of alkanesulfonates are less influenced by the isomeric position of the functional group and by substituents than the cM values [68]. HLB numbers can be correlated with partition coefficients for the distribution of a surfactant between the aqueous and oily phases, which emphasizes that the partition coefficient is dependent on the carbon number [68]. 

The correlation coefficients changed from -.90 to 1.00 with a mean of. 216. Even if specific groups of substances like the homologous series of the fatty-acids or a group of aromatics were singled out, the mean of the correlation coefficients rose only to. 219 and. 280 respectively. 

Skubla [9] has designed a group contribution scheme which applies for various homologous series. His method relies on eq. 6.2.1 where both coefficients a0 and a have to be derived from group contributions and with respect to Nc. The model applies for n-alkanes, 1-alkenes, n-alkylcyclopentanes and n-alkylcyclohexanes, alkylbenzenes, 1-bromoalkanes, 1-alkanols, di-n-alkyl ethers, carboxylic acids and esters, 1-alkanethiols, 1-aminoalkanes, dialkylamines, alkaneamides, and some... 

Reference Eq. (6-20) for an infinite chain of covalently bonded methylene groups can be considered to be an asymptotic limit for the homologous series of n-alkanes. By substitution of w into the exponent of Eq. (6-20) by the corresponding term, wI (., which represents a matrix composed of a paraffin with i carbon atoms, an equation for the diffusion coefficient Dski for trace amounts of a paraffin with k carbon atoms results ... 

An often helpful principle is that of "shortsightedness" of reaction steps The reactivity of a group on a molecule usually does not depend on the size and shape of the molecule more than two atoms away. Pathways involving like chemistry, such as conversion of aldehyde groups on secondary carbon atoms of molecules with different substituents, can thus be assumed to involve closely similar rate coefficients. In large networks this can significantly reduce the number of coefficient values to be determined. The principle can also provide predictions of still unknown rate behavior, particularly in homologous series. It is not valid without exceptions, and must therefore be tested before use. 

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