Matrices colon operator

It is also possible to extract single rows or columns from a matrix, by using a colon operator. The second row of matrix X is denoted by X (2, ). This is exemplified in Figure A.35. It is possible to define any rectangular region of a matrix, using the colon operator. For example, if S is a maffix having dimensions 12 x 8 we may want a sub-matrix between rows 7 to 9 and columns 5 to 12, and it is simply necessary to define S (7 9, 5 12 ). 

The mode specific vibrational root mean squared displacement vector is thus replaced by the mean square displacement tensor of that atom in that specific mode, it has units of A. The tensor is represented in Eq. (A2.85) by a matrix, in more compact notation it is written, for a specific vibrational mode and atom Bi. The colon-operator ( ) or tensor contraction operation is found by taking the trace (Tr) of the product of the two tensors, see Eq. (2.50). Further we write the total mean square displacement tensor of an atom as the sum over all the individual vibrational contributions. 

The colon operator fsee Sec. 1.7.3) is extremely powerful, and provides a very efficient way of handling matrices, e.g., if Z is the matrix defined by ... 

During start-up, the microbial population distribution in the biofilm varies with time. Initial colonization of the particle may be by one or more species that alter the surface favorably for colonization by other species. For instance, in the operation of a butyrate-degrading fluidized bed bioreactor, methanogens attached to the sand particles early in the start-up process and produced a primary matrix of heteropolysaccharides that allowed attachment of other bacterial species (Sreekrishnan et al., 1991 Zellner et al., 1991 Yongming et al., 1993). This is contrary to findings in an acetate-propionate-butyrate degrading reactor, in which facultative anaerobes were found to be the initial colonizers (Lauwers et al., 1990). 

The system being described in Figure 2-3 is a Kalman filter, a realtime controller consisting of several loops implementing matrix multiplication and other operations. The input to the system is the y array, and the output is the v array. First to be described are the carriers that will take part in the computation. Second, the procedures that specify the actions on the variables are listed. In this case, there is only one procedure (filt) consisting of several nested, labeled blocks and repeat loops. The "Imain)" qualifier indicates that filt is the starting point for execution. A quote mark (") indicates a hexadecimal constant, labeled blocks are named with a colon-equal ( =), and the is the binary concatenation operator. 

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