Equivalence classes, chemical

Such a procedure corresponding to the permutation-nomenclature system of configuration 7> is in accordance with the fact that a chemical constitution is a representative of a product D, X. .. X Da of the double cosets D, of pairs of subgroups symmetric groups Sn,g. The latter correspond to permutations of atoms of the same element and the symmetry of the adjacent subset of graph points whose degree is compatible with the coordination number of the element equivalence class of the atom. 

Since an R-matrix represents an electron or bond redistribution pattern, it can represent not only one specific reaction but a general class of reactions. Because of the properties of R-matrices, they may be grouped in several classes. Brandt, et al (28) have defined an R-category as an equivalence class of chemical reactions which have in common the same electron relocation... 

For possible cancer-causing chemicals (EPA Class C—some evidence exists that they may cause cancer, but it is not very convincing), a value equivalent to the MCLG is calculated as if they were not carcinogens. This value is then divided by a factor of ten to give the final MCL. Division by ten provides a margin of safety in case the chemical is later determined to be a carcinogen. 

Equivalence Class. In the canonicalization of structures that have some element of symmetry, certain atoms that are topologically equivalent may yield the same canonical number. These atoms are considered to be in the same equivalence class. The concept of equivalence class is used, for example, in the Daylight Chemical Information Systems handling of reactions, to examine equivalent atoms when mapping reactant and product atoms. 

Molecular descriptors calculated as - information content of molecules. Different criteria are used for defining - equivalence classes, i.e. equivalency of atoms in a molecule such as chemical identity, ways of bonding through space, molecular topology and symmetry, - local vertex invariants [Bonchev, 1983]. 

Cluster K3 and equivalence class K3, respectively, contains four out of six samples of Widuchowa (WD). That means, except for June 1997 and April 1999 the pollution pattern by the chemicals considered here is quite similar. The patterns of WD4/99 and WD6/97 are incomparable to these four samples of 1999. 

In contrast to the evaluations above where equivalence classes are a consequence of clustering results (instead of original parameter values samples obtain the values of cluster centres), here equivalence classes are a result of equivalent pattern concerning the three parameter groups (chemicals, ecotox. and biochemical tests). 

Fig. s 10a and 10b shows the Hasse diagrams after single clustering of each attribute (parameter) for AE sediments using a cluster number of FCL=4 and a TMF=0.8. The equivalence classes Kl and K2 in the ecotoxicological evaluation are the only minimal elements, i.e. compared to all samples above their members have the lowest values in all tests. Except K2 all samples are comparable with Kl. The fact that except AE9/94, AE6/94 and AE3/95 all samples above Kl (AE12/92, AE9/93, AE6/95) have been taken at a later date indicates an increasing pollution for AE. Using chemical concentrations for evaluation the temporal trend seems to be contrary. However, both recent samples AE4/01 and AE8/01 are isolated, i.e. not comparable to all other samples. A sensitivity analysis (see e.g. Heininger et al. 2003) shows that the evaluation is most sensitive to the nitrogen content, where omitting this nutrient compensates the isolation of AE4/01 and AE8/01 (see Fig. 11). Moreover, sample AE4/01 is now a minimal element and therefore emphasizes the indication of a decline of the...

Compared to the results of nematodes tests, evaluation by chemical concentrations and ecotoxicological tests present a familiar picture (c.f. Fig. 7 and 13a,b,c), which is dominated by incomparabilities (and no equivalence classes) and therefore a higher diversity of pollution pattern. Again, due to high correlation between the three parameters egg hatch, growth and reproduction, evaluation of samples by nematodes tests results in nearly total order. 

Fig. 5. Linear orders represented in tabular form (ECO > EFD > CIV >...) First column ATF=0 and all other attributes weights 1/26 equivalence classes NRA, SID, PES, UMW, Second and third column Parameters and Chemicals aggregated with different weights (see text) equivalence classes (2nd column) BID, PES, EFD, EXT, HSD, NRA (3rd column) BID, PES ...

Clearly this Hasse diagram comprises a linear order, not a partial order. Three equivalence classes are shown BID PES EFD EXT HSD NRA. The database ECO comes best whereas the databases BID and PES are the least important ones considering the parameters to be of higher importance than the chemicals. UMW is also not recommendable as it is found on the second lowest level. 

CIC, represents the difference between maximum possible complexity of a graph (where each vertex belongs to a separate equivalence class) and the realized topological information of a chemical species as defined by IC,. Figure 6 provides an example of the first order (r = 1) calculations of IC, SIC, and CIC. 

A chemical synthesis of a target molecule could be described in terms of the empirical formulas of the starting materials and some representative intermediates which are characteristic of the conversion of the starting materials into the synthetic target. Such a description would, however, be useless for most synthetic purposes, because it would be representative of a too-large equivalence class of syntheses. Perkins attempt to synthetize quinine from aniline, which led to the discovery of the triphenyl methane dyes, is a well documented example for the need of defining and representing equivalence classes of syntheses in a more detailed manner than by some characteristic empirical formulas. 

Generally, an R-matrix does not only fit one, but many BE-matrices. Accordingly, an R-matrix does not only represent an individual chemical reaction, but a whole category of reactions which have in common the electron relocation pattern represented by the R-matrix ). An R-category is an equivalence class of chemical reactiois which have in common the same electron relocation pattern and certain features of the participating bond systems. The row/column permutation equivalence of BE-matrices implies that R-matrices represent the same reactions when they are interconverted by row/column... 

In order to classify chemical reactions in a consistant manner which is suitable for chemical purposes, we define the following hierarchy of equivalent classes ... 

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