Branching index, calculation

The concept of the molecular connectivity index (originally called branching index) was introduced by Randic [266]. The information used in the calculation of molecular connectivity indices is the number and type of atoms and bonds as well as the numbers of total and valence electrons [176,178,181,267,268]. These data are readily available for all compounds, synthetic or hypothetical, from their structural formulas. All molecular connectivity indices are calculated only for the non-hydrogen part of the molecule [269-271]. Each non-hydrogen atom is described by its atomic 6 value, which is equal to the number of adjacent nonhydrogen atoms. For example, the first-order Oy) molecular connectivity index is calculated from the atomic S values using Eq. (38) ... 

The Mark-Houwlnk parameters K and a, where appropriate for the sample of interest, are calculated from a plot of log [h] vs. log M(ld) for linear polymers. For branched polymers, the branching index, g, is calculated from... 

The branching index, g, calculated from the Intrinsic viscosity measurement is related to the g value calculated from the light scattering experiment by the following equations... 

A molecular branching index was proposed for acyclic graphs, based on the double invariant approach and called - AA/ branching index. A calculation example [Randic, 1997d Randic, 1998a] for 2-methylpentane is given in Box D-7, where the symbol " p represents a path of length m. 

Ngs) [Gordon and Scantlebury, 1964] or Bertz branching index Bl), is the simplest graph-invariant obtained from the edge adjacency matrix which considers both vertices and edges and is calculated as ... 

Connectivity indices are among the most popular —> topological indices and are calculated from the —> vertex degree 5 of the atoms in the —> H-depleted molecular graph. The Randic connectivity index was the first connectivity index proposed [Randic, 1975b, 2008 Li and Gutman, 2006] it is also called connectivity index or branching index, and is defined as... 

ETA indices are an extension of the TAU indices (or Topochemically Arrived Unique indices) [Pal, Purkayastha et al, 1992 Pal, Sengupta et al, 1988,1989,1990], which were defined some years before in the framework of a previous version of the Valence Electron Mobile environment (VEM environment). TAU indices are calculated from previous definitions of core count and VEM count and include four indices the composite topochemical index, denoted by T (similar to the composite ETA index), the functionality index, denoted by F, the skeletal index, denoted by Tr, and the simple branching index, denoted by B. In QSAR studies, these indices were used in combination with —> STIMS indices, connectivity indices, and some information indices [Roy, Pal et al, 1999, 2001]. 

Figure 4.6-7 Weight average, M, of the MWD and long-chain branching index, /lcb. calculated for polyethylenes produced at 200 °C and 2000 bar in a CSTR at different conversions. The rate coefficients used for this simulations are A ,p = 256 LmoP s and kp = 1976 s . Values for kp and kt are taken from Schweer [24] and ka,M from Reference [38].

R. Emenwein, M. Benard, and I. Shavitt, Vectorizing a Sequence of Conditional Branches the Calculation of the Class Index of Two-Electron Repulsion Integrals on Cray Computers, Comput. Phys. Com-mun. 48, 175-180 (1988). 

For a harmonic crystal the phonon lifetime is infinite and there is no scattering of thermal phonons.To understand the mechanism on how the guest-host interactions lead to the anomalous temperature dependence of the thermal conductivity, the lifetimes were calculated for phonon-phonon scatterings as a result of the anharmonic terms in the xenon-water potential of xenon hydrate in the small and large cage. The inverse relaxation time (lifetime), of a lattice vibration with frequency C0j q) (/ is the branch index and q is the direction of the momentum transfer) is related to the transition rate, W, of the lattice wave scattered from state qj q f by a defect according to, ... 

The water density reactivity coefficient corresponds to the void reactivity coefficient of BWRs or PWRs and it is an important index parameter when judging the inherent safety characteristics of the Super LWR. The density reactivity coefficient for a typical fuel is shown with respect to the water density (average of the coolant and moderator densities) in Fig. 2.60 [9]. The coefficients are derived from the change in the infinite multiplication factor of the fuel when the average density is instantaneously changed at a particular bumup using the branching bumup calculations (Sect. 2.3.1). 

The point of interest is the path of highest pressure drop or index leg. Other parallel branches can be designed of appropriate size to pass the required amount of air, those of lower resistance than the index leg being throttled by dampers. The pressure drop is the sum of the drops caused by the following and is calculated to determine the pressure against which the fan must operate ... 

In Section II, we presented the computational model involved in branching from a node, cr, to a node aa,. In this model, it was necessary to interpret the alphabet symbol a, and ascribe it to a set of properties. In the same way, we have to interpret o- as a state of the flowshop, and for convenience, we assigned a set of state variables to tr that facilitated the calculation of the lower-bound value and any existing dominance or equivalence conditions. Thus, we must be able to manipulate the variable values associated with state and alphabet symbols. To do this, we can use the distinguishing feature of first-order predicates, i.e., the ability to parameterize over their arguments. We can use two place predicates, or binary predicates, where the first place introduces a variable to hold the value of the property and the second holds the element of the language, or the string of which we require the value. Thus, if we want to extract the lower bound of a state o-, we can use the predicate Lower-bound Ig [cr]) to bind Ig to the value of the lower bound of cr. This idea extends easily to properties, which are indexed by more than just the state itself, for example, unit-completion-times, v, which are functions of both the state and a unit... 

Since the measurements most commonly used as an index of long branching are those of intrinsic viscosities, whereas the most readily calculable theoretical measures of branching are the purely geometrical quantities or g0, and since the theoretical problem of relating these is not completely solved (Section 4), one of the applications of model branched polymers is to test proposed relationships. 

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