Permutation index

The final result contains a sum over four different indicies with 81 terms, several of which vanish due to the nature of the permutation index. Equations (8-69) and (8-70) yield ... 

Evaluation of the Divergence of the Gradient (i.e., Laplacian) of the Curl of the Velocity Vector. Begin by calculating the curl of the velocity vector with assistance from the permutation index ... 

Although they are only indirectly concerned with information retrieval from the literature, it is worth noting that the computer has also made significant contributions to the production of printed indexes. Much of the work has involved the production of traditional indexes by computer methods, ranging from simple Uniterm and keyword indexes to fully articulated subject indexes and dual dictionaries. Products more particularly of the computer age are permuted indexes such as the Permuterm Index of the Institute for Scientific Information and the KWIC, KWAC, KWOC family. 

Figure 2-80. The permutation matrix oFthe reference isomer the second line gives the indices of the sites of the skeleton and the first line the indices of the ligands (e.g.. the ligand with index 3 is on skeleton site 3).

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

An additional complication in the PIMC simulations arises when Bose or Fermi statistics is included in the formalism. The trace in the partition function allows for paths which may end at a particle index which is different from the starting index. In this way larger, closed paths may build up which eventually spread over the entire system. All such possible paths corresponding to the exchange of indistinguishable particles have to be taken into account in the partition function. For bosons these contributions are summed up for fermions the number of permutations of particle indices involved decides whether the contribution is added (even) or subtracted (odd) in the partition function. 

Similarly, the increase in the number of isomers in other homologous series (e.g., in the series starting with naphthalene and anthazene) is asymptotically proportional to the number of isomers of the alcohol series. The proportionality factor can easily be derived from the cycle index of the permutation group of the replaceable bonds of the basic compound. 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

The polynomial (1.5) which I called cycle index is, if H is the symmetric group, equal to the principal character of H in representation theory. Professor Schur informed me that the cycle index of an arbitrary permutation group being really a subgroup of a symmetric group is of importance for the representation of this symmetric group. We will, however, not expand on the relationship between representation theory and our subject. 

The problem of Sec. 12 can be stated in this special situation as follows Let It be an arbitrary permutation group of degree 5 and /cj, /cj,. .., denote n non-negative integers whose sum is s. How many nonequivalent ways modulo H are there to place /Cj balls of the first, balls of the second,. .., k balls of the n-th color in 5 slots According to Sec. 16 the solution is established by introducing m cycle index of H and expanding the... 

Relations Between Cycle Index and Permutation Group... 

The main theorem, stated in Sec. 16 and proved in Sec. 19, combined with the proposition of Sec. 25 yields the following proposition Tvfo permutation groups are combinatorially equivalent if and only if they have the same cycle index. 

Referring to the definition (1.5) of the cycle index we find further two permutation groups are combinatorially equivalent if and only if there exists a unique correspondence between the permutations of the two groups such that corresponding permutations have the same type of cycle decomposition. 

A third permutation group of the graph of cyclopropane obtains if the regular prism discussed as a model for (a) is subjected to rotations as well as to reflections which leave it invariant. The six vertices are thus subject to a permutation group of order 12. We call it the extended group of the stereoformula. Its cycle index is... 

The example demonstrates that the concepts in chemistry rely heavily on notions from group theory, specifically the concept, introduced in Sec. 11, of the equivalence of configurations with respect to a permutation group. The cycle index and the main theorem of Sec. 16 play a role. 

To make use of the group G we need some way of summarizing those properties of the group that are relevant to the problem. This was provided by Polya in the form of the "cycle index". It is well known that a permutation can be expressed as a product of disjoint... 

Now the cycle index of the group can be found for any given p by considering each of the various types of permutations, and finding the type of permutation induced on the edges. In each case the coefficient will be the same as in S. For details see [HarF55] and many other references. 

Only the so called first quantisation in particular Schrodinger s formulation, starts off heavily on the wrong foot, by assuming an index, label or name to each particle. But this individualisation is immediately wiped out by systematically permuting the labels of all like particles Any one labelled particle occurs simultaneously in all positions occupied by like particles (Post [1963]). 

The index is augmented by successively permuted versions of all empirical formulas. As an example, C3H3AIO9 will appear as such and, at the appropriate positions in the alphanumeric sequence, as H3A109 C3, AIO/C3H3 and 09 C3H3A1. The asterisk identifies a permuted formula and allows the original formula to be reconstructed by shifting to the front the elements that follow the asterisk. 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

This is a true permutation as each index appears once and once only in the bottom row. To show that the n elements gai are all distinct suppose that ga2 = ga and multiply on the left by g x to give Oj = a. However, this is impossible since the elements a are all distinct. Moreover, the correspondence is invertible, i.e. a given permutation 11(5) cannot arise from any other group element g, since gai = g ai => g = g on multiplication from the right by a l. These relationships imply a 1 1 correspondence g n(g), which respects the group structures of G and Sn. Thus if gi -> n i) and g2 O 11(52) then gig2 n(5i)II(52), the permutation resulting from successive left multiplications by g2 and g. There are n distinct permutations... 

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