Permutation cyclic

Consider a triatomic system with the three nuclei labeled A, Ap, and Ay. Let the arrangement channel -1- A A be called the X arrangement channel, where Xvk is a cyclic permutation of apy. Let Rx,r be the Jacobi vectors associated with this arrangement channel, where r is the vector from A to and the vector from the center of mass of AyA to A . Let R i, rx be the corresponding mass-scaled Jacobi coordinates defined by... 

This equaciOTi, together with the corresponding results obtained from it by cyclic permutation of the suffixes L, 2, and 3, provides the required explicit form of the flux relations. 

E the cyclic group of order and degree 5, generated by cyclic permutations of s objects ... 

Others follow by cyclic permutations all of them can be symbolized conveniently by the vector relation... 

For this purpose, let us use invariance of the matrix product trace under cyclic permutation of factors and represent (7.11) as... 

The other operator is a cyclic permutation g = (13542) on the five ligands, where (did2d, ...,d ) means replace object by dj, dj by d3,...,d by dj. The use of the standard rules for calculating products of permutations yields ... 

Thus, the trace of the commutator [A, B] = AB - BA is equal to zero. Furthermore, the trace of a continued product of matrices is invariant under a cyclic permutation of... 

The corresponding relations for 62 and b- follow by cyclic permutations of the subscripts (see Chapter 8). 

The first term in (13), also called the diagonal term (Berry 1985), originates from periodic orbit pairs (p,p ) related through cyclic permutations of the vertex symbol code. There are typically n orbits of that kind and all these orbits have the same amplitude A and phase L. The corresponding periodic orbit pair contributions is (in general) g n - times degenerate where n is the length of the orbit and g is a symmetry factor (g = 2 for time reversal symmetry). 

We consider first the class structure of Sn. To do this, we note that every permutation may be written as the product of a number of independent cyclic permutations. Thus, the permutation s takes the ligand on site 1 to si, that on si to s(si), that on s(si) to s[s(si)], etc. Following this chain, since n is finite, we must eventually reach a site whose ligand is taken to site 1 by s. The closed chain evidently forms a cyclic permutation, which we will denote by writing the sites concerned in order enclosed in parentheses. Thus, (123.../) denotes a permutation s for which si =2, s2=3,. .. s/ i =/, s/= 1. If this first cycle does not include all the sites, we can do the same thing with the lowest-numbered site not appearing in the cycle, and continue until we have broken down the group element completely into a product of cycles.b>... 

The fifth class listed above contains a 1-cycle and a 2-cycle in its permutation part, with an odd number of reflections (in this case, one reflection) on sites involved in the two-cycle. The seventh class has a permutation part consisting of a cyclic permutation of the three sites, with an odd number of reflections (either one or three). 

P> A transposition is the exchange of two adjacent elements of an ordered set. A cyclic permutation of n symbols contains n—1 transpositions. 

It is easy to verify that the trace of any product of matrices is invariant to cyclic permutations of the matrices, for instance. 

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