Coset factorization

It may also be observed that the coset factorization of Eq. (229) is valid for any orbital pair regardless of the CSF space. However, the completely arbitrary transfer of parameters from the orbital variation space to the CSF space may not result in a CSF partitioning as in the above examples. For example, consider the n -I- l)-term expansion case considered previously in which Cl2 is allowed to be non-zero. The transfer of the orbital rotation parameter from the orbital space into this CSF space requires the introduction of two new CSF expansion terms, C13 and C23. However, these two terms may not be varied independently one of the terms may be written as a function of the other terms in the expansion. Although the analysis of this transfer is straightforward in the two-electron case through Eq. (230), it is more difficult in the general case and this type of constrained CSF coefficient optimization will not be considered/further. 

Folic Acid. Evidence for recognizing folic acid (pteroylglutamic acid, vitamin Be, L. coset factor or vitamin M) as an essential human nutrient is presented in the text. The quantitative requirement cannot be closely estimated from evidence now available. 

We briefly mention the case where the number of points m is less than n, i.e., less than the number of elements in the symmetry group G with respect to which we measure symmetry. In this case, m should be a factor of n such that there exists a subgroup H of G with n/m elements. In this case, we duplicate each point trim times and fold/unfold the points with elements of a left coset of G with respect to H. Following the folding/unfolding method, the relocated points will align on symmetry elements of G (on a reflection plane or on a rotation axis for example). Further details of this case and proof can be found in Ref. 2. 

Equation (2.59) is thus a representation of an arbitrary group element expressed as a left coset of this subgroup. Expressed in other words, the unitary transformation that is described by exp(i2) may alternatively be described by the unitary transformation exp(i/l )exp(iX ). It should be pointed out that there exist no simple relations between the parameters and the A ., and parameters. With the above factorization of the redundant part (A"), the unitary transformation of the reference state may be written as... 

The factor group of D2 consists of the cosets of an invariant subgroup. ... 

It is easy to see that the cosets form a group, since by multiplying any two elements of the factor group, say,... 

If a group is homomorphic onto the matrix representation then the representation is a faithful representation of the factor group. A factor group of Cjv consists of the cosets of the invariant subgroup iJ, = [Pg.234]

---