Transformational independence

The complete vibrational wave function (9.157) is the product of functions (9.158) of different frequencies since functions of different frequencies are transformed independently of one another, the function v/ transforms according to the ordinary direct product of the representations of the functions (9.158). Hence vib belongs to the representation... 

Sequences that allow combinatorial variations to be introduced in immediate succession may also be verified by starting from compounds with functional groups that can be transformed independently. As an example, 4-amino-benzylamine was first selectively ac-ylated at the aliphatic amino group. Subsequently the aromatic amino group was converted to a thioureido or a secondary amino group (Scheme 3.8, [9]). 

The derivation of Eqs. (1) and (2) will be given below. First let us make some observations. Most importantly note that the ifi/s, from an informational point of view, at a minimum need be only point-by-point solutions inside the range of the potential. Their behavior outside is irrelevant, as is their norm. Equations similar to (1) are known to have been used to transform independent solutions of close-coupled differential equations into proper outgoing (incoming) solutions of the Schrodinger equation. In effect (as will be seen) the matrices A give the transformation from if/, to properly normalized if / as... 

Is as / —V oo). This is because in Eq. (141) a spin-orbitals transform independently of ones. The He - He case is particularly simple, there being no aa intra-atomic pairs. The calculation of 13 and 34 is then just the same as obtaining the correlation energy of H in the state.At larger r, neglecting the difference between aa and pairs. 

The squares of the deviations and therefore function J/i 0 itself transform independently of i. 

The reaction of the ferrio-silanediols 16a-c with 13b,c, respectively, results in the formation of the heterosiloxanols 17a-e (Eq. 4). Only one OH function of 16a-c is transformed, independently of the amount of trimethylgallane or -indane used. 

Depending on the distribution of values it may make sense to perform nonlinear transformations on the data, such as n-th root or logarithm. Nonlineeirly transformed independent variables may be used as additional predictors. Further, new variables may be obtained by applying arithmetic operators to pedrs or leirger subsets of predictors Xj, j n. This is called a base extension. Quadratic base extensions are often used, where the squares X, j e n, eind products X X , k + l,k,l e neire used as predictors along with X. ... 

Step 3 is repeated using transformed independent variables. Note that transformation of the variables requires adding a free parameter to the model even if it was omitted in previous steps. The most appropriate model is selected by comparing the models obtained in steps I - 4 on the basis of the variance, R, the residual plots and additional practical considerations (complexity of the model, derivatives etc.)... 

An alternative coding strategy would be to treat one or more of the various forms of pleurocarpy found in grade R as separate states of a single character in which acrocarpy and conventional pleurocarpy were also states. This would inappropriately link characters that appear to vary independently, however, such as pleurocarpy sensu stricto and production of subperichaetial innovations. The result would be a highly artificial character with a large number of states, in which transformational independence would be compromised (Lee and Bryant, 1999). 

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