Statistical weight factors matrices

Here L0 and Lv are the effective absorption path lengths at line frequencies vQ and vv, respectively g is a statistical weight factor m//I is the dipole matrix element of the rotational transition r+/, and Ap is the halfwidth of the line at half intensity. 

The matrix ai comprises the elements of (see Eq. (15)) joined with the A matrix for the rotational state of bond i as prescribed by the column index. It will be apparent that serial multiplication of the ai according to Eq. (28) generates the statistical weight factor U2U3... for every configuration of the chain in the same... 

The states potentially available to each residue are denoted c, h, and h. States denoted by h differ from those denoted by h in that they merit an additional weighting factor that arises from helix-helix Interaction In the cross-linked dimer. For a single chain, a 3 x 3 matrix U S,n9 e Is used in which rows index the states of residue i-1 and i, while columns index the states of residues / and/ + 1. The order of indexing is etc or h), he, hh. The required statistical weight matrices for the A Bj pair are of dimensions 10 x 10 and are of different formulation depending upon whether ix (see below). For indexing, see original paper. 

Therefore, we have to analyse the variation of the rate of permeation according to the temperature (zj), the trans-membrane pressure difference (Z2) and the gas molecular weight (Z3). Then, we have 3 factors each of which has two levels. Thus the number of experiments needed for the process investigation is N = 2 = 8. Table 5.13 gives the concrete plan of the experiments. The last column contains the output y values of the process (flow rates of permeation). Figure 5.8 shows a geometric interpretation for a 2 experimental plan where each cube corner defines an experiment with the specified dimensionless values of the factors. So as to process these statistical data with the procedures that use matrix calculations, we have to introduce here a fictive variable Xq, which has a permanent +1 value (see also Section 5.4.4). 

Molecular architectures can be structurally classified as being more comb-like or Cayley tree-like. Structure has impact on the radius of gyration, which is larger for linear molecules than for branched molecules of the same weight (number of monomer units), since the latter are more compact. The ratio between branched and linear radius is usually described by a contraction factor . Furthermore, Cayley tree-like structures are more compact than comb-like structures [33, 56]. We will show here how to obtain the contraction factor from the architectural information. The squared radius of gyration is expressed in monomer sizes. According to a statistical-mechanical model [55] it follows from the architecture as represented in graph theoretical terms, the KirchhofF matrix, K, which is derived from the incidence matrix, C [33] ... 

---