1,3-Transposition

The transposed matrix A1 is defined as the interchange of rows and columns of A. This can also be seen as the reflection of all elements of A at its main diagonal (along ay, i=j), according to 

the row dimension of A becomes the column dimension of A4, the column dimension of A will be the row dimension of A4. 

Note that Matlab uses the quote ( ) as the transposition operator. 

In an Excel spreadsheet, the [TRANSPOSE function can be applied to an array of data. For this, we need to become familiar with the two most important rules to perform matrix operations in Excel  

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It is again appropriate to use inversely oriented coordinates as shown in Figs. 15 and 17. The addition theorem valid for the spherical potentials (5.15) is found by repeated differentiation of the addition theorem for spherical Bessel functions of zero order, Eqs. (10.1.45) and (10.1.46) in 

with respect to separation and angles. This is described in full detail in Section 5.6. If f (Q is a spherical Bessel function y (Q of the first or the second kind, we obtain (see Fig. 24) 

Applying this scalar addition theorem to the magnetic and electric potentials Zj(r — r ) with s= 1,2 we find  

Addition theorem (5.36) obviously couples magnetic and electric modes. 

In the following we distinguish between the coefficients F (0) mniO which arise on transposing spherical Bessel functions j Q, y (0 of the first or the second kind by adding a second argument VH, C,J), W L(Cj ) where =1,2. 

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In the example, we proceed from the right (rotated molecule) to the left (reference molecule). The first transposition is done with ligands 4 and 2, in order to obtain the fir.st part of the reference. scqttencc 1 2". Then, only a pcrmtttation of ligands 3 and 4 has to be done to obtain the reference matrix on the left-hand side. Thus, in total, we have executed two transpositions (4 2) and (3 4),... 

The two structures in our example are identical and are rotated by only 1 20 h Clearly, rotation of a molecule docs not change its stereochemistry, Thus, the permutation descriptor of both representations should be (+ I). On this basis, we can define an equation where the number of transpositions is correlated with the permutation descriptors in an exponential term (Eq. (9)). 

Figure 2-81. The permutation matrices oftwo structures that differ through rotation by 120 T The permutation matrix of the rotated Isomer can be brought Into correspondence with the permutation matrix of the reference isomer by two Interchanges of two ligands (transpositions),...

Figure 2-82. The permutation matrices of two structures that differ by reflection through the plane of the drawing. One transposition is necessary to bring these matrices into correspondence.

The. same rcfcTcncc molecule s now reflected at a plane. spanned by the Cl, Br, and C atoms, to give the other enantiomer, (n the same manner as previously, we write down the permutation matrices of the two structures, and then determine the transpositions (Figure 2-82). 

In this case, two transpositions have to be performed, resulting in a permutation descriptor of (+ 1) for the right-hand subunit. 

We sec from Figure 3-22 that we need three transpositions to transform the isomer of the product of this reaction into the reference isomer. Thus, for Eq. (7 we obtain = (-1) (+ 1 = (-1) and everything seems fine as the descriptor of the... 

Experience has shown that is better to obtain basis sets in electronic form than paper form since even slight errors in transposition will affect the calculation results. Some basis sets are included with most computer programs that require them. There is also a form page on the Web that allows a user to choose a basis and specify a format consistent with the input of several popular computational chemistry programs at http //www.emsl.pnl.gov 2080/forms/basisform.htm. The basis set is then sent to the user in the form of an e-mail message. 

Irradiation with ultraviolet light of arylthiazoles in different solvents gave the transpositions described in Table III-39. (215, 216). 

Much more information can be obtained by examining the mechanical properties of a viscoelastic material over an extensive temperature range. A convenient nondestmctive method is the measurement of torsional modulus. A number of instmments are available (13—18). More details on use and interpretation of these measurements may be found in references 8 and 19—25. An increase in modulus value means an increase in polymer hardness or stiffness. The various regions of elastic behavior are shown in Figure 1. Curve A of Figure 1 is that of a soft polymer, curve B of a hard polymer. To a close approximation both are transpositions of each other on the temperature scale. A copolymer curve would fall between those of the homopolymers, with the displacement depending on the amount of hard monomer in the copolymer (26—28). 

Fig. 5. Initial piping transposition (a) process flow diagram and (b) plotplan (1).

In addition to reduction, ring transposition of the type (177) (184) is observed in the... 

Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A or A. ... 

The influence of skin effects in a multi-core cable is almost the same as that of a multiphase busbar system, discussed in Sections 28.7 and 28.8. However, unlike a busbar system, the resistance and inductive reactance for various sizes of cables can be easily measured and are provided by leading manufacturers as standard practice in their technical data sheets. To this extent, making an assessment of skin effects in cables is easy compared to a busbar system. Since all the phases in a cable, of a 3-core or 3 72-core are in a regularly twisted formation throughout the length of the cable, they represent the case of an ideal phase transposition (Section 28.8.4(3)) and almost nullify the effect of proximity. 

Transposing the overhead communication lines, i.c. reversing the respective positions of the two sides of the lines every I km or so, to avoid continuous parallelism (due to electrostatic and electromagnetic inductions), as illustrated in Figure 23,8.. See also Section 28.8.4(3) on phase transposition. 

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