Congruence

Cycloaddition reactions are close to the heart of many chemists - these reactions have fascinated the chemical community for generations. In a series of communications in the sixties. Woodward and Hoffmann [2] laid down the fundamental basis for the theoretical treatment of all concerted reactions. The basic principle enunciated was that reactions occur readily when there is congruence between the orbital symmetry characteristics of reactants and products, and only with difficulty when that congruence is absent - or to put it more succinctly, orbital symmetry is conserved in concerted reactions [3]. 

Re-examining the four conditions, one notices that the congruence of graphs as defined here establishes a reflexive, symmetric, and transitive relation between graphs. Observe that (IV) does not involve an arbitrary set but a group of permutations. 

If the principal node K is an endpoint, the planted tree consists of a , the root and the stem, which connects these two points. There are no vertices of degree 4, there are no principal branches. There are no two noncongruent planted trees of this type, whether we deal with topological, spatial, or planar congruence. Hence... 

According to the nature of the congruence, topological, spatial, or planar, the generating functions of the planted C-H trees are given by... 

For the spatial congruence, that is, with group and its cycle index (cf. Sec. 21), we find... 

Spatial congruence of C-H graphs is applied essentially only in chemical formulas which represent a compound of carbon atoms and atoms of valence 1 (or radicals of valence 1). In this case condition (IV), besides (I), (II), (III), adds another restriction not only the relationships are important but also the spatial arrangement of the bonds. The spatial interpretation of the congruence of C-H graphs coincides with the interpretation of the chemical formula as stereoformula. I use stereoisomers in this sense. For example, the number of different stereoisomers is equal to the number of spati-... 

Not all chemistry textbooks provide a clear description of what stereo- and structural formulas imply. Maybe the concepts of spatial and topological congruence of graphs could contribute to the clarification of the chemical terminology. 

Figure 69, Spectral and concentration data plotted side-by-side to show the congruence of the points in the two different data spaces.

When we calculate the eigenvectors for the two different data spaces (concentration and spectral spaces) we find the corresponding spectral and concentration vectors are shifted by different amounts in different directions. This is a consequence of the independence of the noises in the concentration and spectral spaces. So, just as the noise destroyed the perfect congruence between the noise-free spectral and concentration data points, it also destroyed... 

The whole idea behind PLS is to try to restore, to the extent possible, the optimum congruence between the each spectral factor and its corresponding concentration factor. For the purposes of this concept, optimum congruence is defined as a perfectly linear relationship between the projections, or scores, of the spectral and concentration data onto the spectral and concentration factors as exemplified in Figure 73. Since the spectral noise is independent from the concentration noise, a perfectly linear relationship is no longer possible. So, the best we can do is restore optimum congruence in the least-squares sense. 

PLS attempts to restore optimal congruence between each spectral factor and its corresponding concentration factor by rotating them towards each other until the angle between them is zero. In other words, PLS will search for a single... 

Manos, P. S. 1997. Systematics of Nothofagus (Nothofagaceae) based on rDNA spacer sequences (ITS) taxonomic congruence with morphology and plastic sequences. Amer. J. Bot. 84 1137-1155. 

Commonly the absence of systematic errors is tested by recovery studies. The estimation of recovery by analyzing only one standard, as frequently be done, can give misleading results as can be seen from Fig. 4.2. In case of combined and nonlinear systematic deviations there can occur points of accidental congruence that can be hit. Therefore, if at all possible, there should be estimated the entire recovery function. However, at least two different standards should be analyzed. 

This pH-congruence can be explained (Lyklema, 1987) by inferring that the process of charge formation is independent of the nature of the oxide hence it must be determined by the solution side of the double layer. On the other hand, the specific part, determined by the surface side, is determined by the chemical nature and is reflected only by the pHpzc. 

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