Permutation diagram

Each of the N permutations may be represented by a permutation diagram, made by drawing an arrow from each orbit in Pab e to the orbit whose place in ab- -e it occupies. Thus if Pab- -e = bd- -e, an arrow is drawn from b to a, from d to b, etc. A closed polygon of arrows is called a cycle of the permutation.3 A cycle may... 

To find the coefficient of a given exchange integral in a matrix element, (I/F/PII), draw the vector-bond diagram for structure II, change it as indicated by the permutation diagram for P, and form the superposition pattern of I and PI I. The coefficient is then given, except for the factor (—l)p, by the above rules for the Coulomb coefficient that is, it is (— 1)F(— V)r2 in t>. 

FlG. 3. The permutation diagram for P abcdef=bcfdae, and the superposition pattern for I PII. 

Fig. 1. "Permutation diagram" Two sequences of objects according to two different characteristics.

Obviously, the concept of partially ordered sets appears rather useful in environmental sciences. The usual order, namely the order in which each object can be compared with each other, can be considered as a special case of partial order, i.e., the term "linear" or "total" order is used. Permutation diagrams become confusing if many objects are included and especially if more than two attributes characterize the objects. In such cases a corresponding number of sequences may arise and for each pair of... 

Fig. 2. The Hasse diagram as an alternative to the permutation diagram shown in Fig. 1...

ABSTRACT The objective of the approach is to calculate the probability of a failure coincidence or maintenance conflict, respectively, in an -system-single-maintenance-unit scenario. Beside operation with conventional systems. Reliability-adaptive Systems can be considered as well in this scenario. The approach applies multiple integrals over products of probability density functions and discusses the permutation of coincidence patterns explicitly. So-called staple graph coincidence permutation diagrams and coincidence permutation trees are introduced as graphical representations. 

The staple graph coincidence permutation diagram (see Fig. 8) shows the permutations of coincidence patterns as given in Section 2. This diagram gives an impression of a 6-system scenario failure coincidence permutation if single-clusters are considered only. 

As the staple graph coincidence permutation diagram opens the first level of abstraction, then the second level is represented by the coincidence permutation tree, see Fig. 9. Here, the number of involved systems and the maximum number of overlapping maintenance interval are graphically described. Note that the permutations with non-involved systems are not considered here. The purpose of both diagrams is to simplify the collecting of pattern probabilities as shown in Section 5. 

Figure 8. Staple graph coincidence permutation diagram of a 6-system scenario considering single-clusters only.

One way of proceeding is shown in the flow diagram of figure 2 for the ease of = 8, P = 3. The operation labeled PERMUTE rearranges the sequence of data. The /th member is placed into theyth position where] is calculated from i as follows... 

For the case of chiral ligands (with which this section is exclusively concerned), it is easy to convince oneself that the representation (n)u of <5n, whose diagram is shown in Fig. 16, is chiral for every skeleton. For, the representation is totally symmetric under all pure permutations, in particular under those belonging to , and antisymmetric under all group elements involving to, in particular under those in the coset of D in . The representations of [Pg.63]

Pe), Note the permutation of the chains in the first diagram of the last line... 

Fig. 4,12. Diagrams contributing to G 23 (0,0,0,0) to second order. Permutation of the polymer lines yields the weight factor of 2 for the last three contributions...

Jones symbols for the set IR are obtained by changing the sign of the symbols for R). The principal axis has been chosen along v for the choices z orx, use cyclic permutations of xyz, or derive afresh, using the appropriate projection diagram. 

Figure 8.15. Correlation diagram between levels of a rigid rotor K = 0 (water dimer with Cs symmetry in the nontunneling limit), a rotor with internal rotation of the acceptor molecule around the C2 axis (permutation-inversion group G ), and group G16. The arrangement of levels is given in accordance with the hypothesis by Coudert et al. [1987], The arrows show the allowed dipole transitions observed in the (H20)2 spectrum. The pure rotational transitions E + (7 = 0) - E (J = 1) and E (7 = 1) <- E + (/ = 2) have frequencies 12 321 and 24 641 MHz, respectively. The frequencies of rotationtunneling transitions in the lower triplets AI (7 = 1) <- A,+ (7 = 2) and A," (7 = 3) <- A,+ (7 = 4) are equal to 4863 and 29 416 MHz. The transitions B2(7 = 0)<- B2(7 = l) and BJ(7 = 2) <- B2 (7 = 3) with frequencies 7355 and 17123 MHz occur in the higher multiplets.

It must be added that for each of the great (but finite) number of possibilities of distribution on the levels in circular diagrams or diagrams of the lemniscate type a number of permutations between the different reactants and resultants appearing in a sequence of given overall reaction are possible. 

Central to freeon dynamics is the indistinguishability of electrons this property is a symmetry which is expressed in terms of the symmetric group, Sn> the group of permutations on the indices of the N identical electrons. The irreducible-representation-spaces (IRS) of Sn are uniquely labeled by Young diagrams denoted YD[X] where [X] is a partition of N and where YD[X] is an array of N boxes in columns of nondecreasing lengths. The Hamiltonian for a system of N identical particles commutes with the elements of Sn- By the... 

The basic elements of the diagrams are shown in Figure 1. Figure 1 (a) shows the diagrammatic representation of a one-electron operator matrix element. Figure 1 (b) shows the representation of a two-electron matrix which in the Brandow scheme includes permutation of the two electrons involved. Upward (downward) directed lines represent particles (holes) created above (below) the Fermi level when an electron is excited. 

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