Boolean matrix

The adjacency matrix of a molecule con.si.sting of n atom.s i.s a square (n / n) matrix. with the entric.s giving all the connectivities of the atoms. The intersection of a row and a column obtains a value of 1 if the corresponding atoms are connected. If there is no bond between the atoms being considered, the position in the matrix obtains the value 0. Thus, this matrix representation is a Boolean matrix with bits (0 or I) (Figure 2-13). 

Boolean matrix multiplication is carried out according to the usual matrix rules, expressed in terms of the logical product and logical sum as defined above. 

Since the systems of equations to be considered are quite large, it is necessary to use some compact method to represent the information flow among them. A very convenient technique is to relate the system equations to a digraph and its associated Boolean matrix, which represent the structure of the information flow in the system. The Boolean matrix to be used is called the occurrence matrix (HI, S3), and is defind as follows (1) each row of the occurrence matrix corresponds to a system equation, and each column corresponds to a system variable (2) an element of the matrix, s -, is either a Boolean 1 or 0 according to the rule... 

One method of partitioning the system equations is to compute the maximal loops using powers of the adjacency matrix as discussed in Section II. Certain modifications to the methods of Section II are needed in order to reduce the computation time. The first modification is to obtain the product of the matrices using Boolean unions of rows instead of the multiplication technique previously demonstrated to obtain a power of an adjacency matrix. To show how the Boolean union of rows can replace the standard matrix multiplication, consider the definition of Boolean matrix multiplication, Eq. (2), which can be expanded to... 

This modification for Boolean matrix multiplication permits use of the Boolean union operation (logical OR operation or logical sum) instead of regular multiplication and union operations. The Boolean union operation can be executed much faster on a digital computer. Experience has shown that performing the Boolean union of rows instead of the standard Boolean multiplication of matrices can reduce the computation time by as much as a factor of four. 

Form a new j by n Boolean matrix, M(0) as follows For each zero entry in column k, reproduce the corresponding row as a row in M(0). For example, the second row of the occurrence matrix in Fig. 12a contains a zero in column k = 1 and therefore the element = 0, element — 1, to 3 = 0, and to 4 = 1 comprise the first row of M(0). The second row of M(0) would be exactly like the third row of that occurrence matrix. This process is continued until all of the rows with zero entries in column k have been included as rows of M(0). A final row is added to M(0), whch is the Boolean union of all of the rows in the occurrence matrix which contain nonzero entries in column k. For example, rows 1 amd 4 of the occurrence matrix contain nonzero elements in column 1 so that the elements of the last row of M(0) are m3I = 1, m32 = 0, m33 = 1, m34 = 0. Figure 13a illustrates M(0). 

A Boolean matrix in general P Adjacency matrix of information... 

Figure 2.11 The Boolean matrix describes the Jacobian structure and the function dependency from the variables of the nonlinear system.

Computer-Generation of Reaction Schemes, The structure of hydrocarbons can be represented in a Boolean matrix. 

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