Core matrix

An option input of 1 can be used to alter the F maPix called the core matrix. Elements can be entered or altered off the diagonal or, for inclusion of heteroatoms, by designating on-diagonal elements. The prompts and response formats for the E maP ix are similar to those of the G maPix. This opPon is similar to heteroatom inclusion in HMO. We shall use option 1 in Part C below. After all options have been entered, the final option is 0, which causes the calculation to be carried out. In... 

Core matrix elements, H, will be specified with individual methods. Indices k and m refer to closed and open shells, respectively c and y have their usual meaning of expansion coefficients and repulsion integrals, respectively. Numerical values of constants f, a, and b depend on the electronic configuration under study e.g., for a system having an unpaired electron in a nondegenerate... 

The so-called Tucker3 model is defined by the decomposition of a three-way table X into a three-way core matrix Z and three two-way loading matrices A, B, C (one for each mode) ... 

In this way, an nxpxq table X is decomposed into an rxsxt core matrix Z and the nxr, pxs, qxt loading matrices A, B, C for the row-, column- and layer-items of X. The loading matrices are column-wise orthonormal, which means that ... 

The number of factors r, s and t, assigned to each mode, is generally different. They are chosen such as to be less than the dimensions of the original three-way table in order to achieve a considerable amount of data reduction. The elements of Z represent the magnitude of the factors and the extent of their interaction [57]. Computationally, the core matrix Z and the loadings matrices A, B and C are derived such as to minimize the sum of squared residuals. 

The extended matrix notation is represented here by the three dots that surround the core matrix. A graphical example of the TuckerS model is rendered in Fig. 31.19. 

Fig. 31.19. Tucker3 or core matrix decomposition of an nxpxq three-way table X. The matrix Z represents the rxsxt core matrix. A, B and C arethenxr.pxsand xr loading matrices of the row-, column-and layer-items of X, respectively.

The dimensional matrix consists of a square core matrix and a residual matrix. 

Quantities of the square core matrix may eventually appear in all of the dimensionless numbers as fillers, whereas each element of the residual matrix will appear in only one dimensionless number. For this reason the residual matrix should be loaded with essential variables like the target quantity and the most important physical properties and pro-cess-related parameters. 

By the—extremely easy —matrix rearrangement (linear transformations), the core matrix is transformed into a matrix of unity. The main diagonal consists only of ones and the remaining elements are all zero. One should therefore arrange the quantities in the core matrix in a way to facilitate this procedure. 

The dimensional matrix consists of a (square) core matrix and a residual matrix. Based on our relevance list, the dimensional matrix representing a tableting process can be written as shown in Table 1. 

By a simple linear transformation, the core matrix becomes a unity matrix (Table 2). 

The dimensionless numbers are formed as fractions, where each physical quantity indicated in the residual matrix represents the numerator, while a product of all quantities of the core matrix (with the exponents indicated in the residual matrix) constitutes the denominator. This standard procedure yielded the following n set ... 

In deriving these equations Pople separated the core matrix element Hu thus ... 

The above dimensional matrix must be rearranged and divided into two parts, a square core matrix, which contains the dimensions pertaining to the repeating variables, and a residual matrix. Using the rules given in the previous section, the repeating variables are D, p, p and Cp and the dimensional matrix can be written as... 

The next step is to transform the core matrix into a unity matrix. Hence, the order of the physical variables in the core matrix should be such that a minimum amount of linear transformations is required. Adding the M row to the L and T rows eliminates the non-zero term below the diagonal in the core matrix, i.e.,... 

With the above matrix set the dimensionless numbers can be generated, in this case, 4 dimensionless groups, by placing the physical quantities in the residual matrix in the numerator and the quantities in the core matrix in the denominator with the coefficients in the residual matrix as their exponent. Hence,... 

If different repeating variables had been chosen in the core matrix, as for example / , D, u and AT, one would get... 

Once the relevant parameters have been chosen, the dimensional matrix subdivided into core and residual matrix can be obtained. The core matrix is a 3 x 3 matrix, leaving a residual matrix of size 3x1. Since this will result in only one dimensionless number, the target value Pcr is left on the residual side, hence, choosing E, L and I as the repeating quantities... 

Choosing as, p and D as the terms in the core matrix, and arranging them such that the diagonal terms are populated we get... 

The third fundamental hypothesis of the CNDO approximation concerns the core matrix elements H. These correspond to the and parameters of the Pariser-Parr-Pople method, the core including here only the Is electrons and the nuclei. 

A Lysine-base core matrix used for the preparation of a Multiple-An tigen-Peptide... 

With the dimensions of these quantities a dimensional matrix is formed. Their columns are assigned to the individual physical quantities and the rows to the exponents with which the base dimensions appear in the respective dimensions of these quantities (example Ap [M1 L-1 T"2]). This dimensional matrix is subdivided into a quadratic core matrix and a residual matrix, whereby the rank r of the matrix (here r = 3) in most cases corresponds to the number of the base dimensions appearing in the dimensions of the physical quantities. 

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