Network rank

For reactions that include irreversible steps of higher reaction orders, the same procedure is used. However, a distinction between the Delplot rank and network rank of a participant must now be made. The Delplot rank R is related to the intercepts as for networks with first-order steps only and is obtained by inspection of the plots. The network rank N indicates the provenance N = 1 for primary participants, N = 2 for secondary participants, etc. The Delplot rank of the participant L of a step nKK — L (n th order in K) is related to the Delplot rank of its parent K by... 

For example, if the step K — P in the network 7.1 were replaced by a second-order step 2K — P, the Delplot rank of P would become 1 -I- 2 1 =3, although the network rank of P as a secondary product is 2. 

This procedure introduces uncertainties because the deduction of the network rank from the Delplot rank requires some prior knowledge about the network. For example, suppose the Delplot ranks of K and L are 1, that of M is 2, and that of N is 3, that M is likely to be formed from K, and N from K and L. If so, K and L could react directly to N in a second-order step, and K to M in a separate first-order step (Case I). Instead, K could form M, which then reacts with L in a two-step reaction that is first order in M and zero order in L (Case II, with X + L — N much faster than M — X) ... 

Some of the features of GO (EPRI NP-3123) are given in Table 3.4.6-2. A GO model is networks GO operators to represent a system. It can be constructed from engineering drawings by replacing system elements (valves, switches, etc.) with one or more GO symbols. The GO computer code quantifies the GO model for system reliability, availability, identification of system fault sequences, and relative importance in rank of the constituent elements. 

The structure of a perfect network may be defined by two variables, the cycle rank and the average junction functionality (f>. Cycle rank is defined as the number of chains that must be cut to reduce the network to a tree. The three other parameters used often in defining a network are (i) the number of network chains (chains between junctions) v, (ii) the number of junctions p, and (iii) the molecular weight Mc of chains between two junctions. They may be obtained from and using the relations... 

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

The stoichiometric matrix N consists of m rows, corresponding to m metabolic reactants, and r columns, corresponding to r biochemical reactions or transport processes (see Fig. 5 for an example). Within a metabolic network, the number of reactions (columns) is usually of the same order of magnitude as the number of metabolites (rows), typically with slightly more reactions than metabolites [138]. Due to conservation relationships, giving rise to linearly dependent rows in N, the stoichiometric matrix is usually not of full rank, but... 

Covalent polymer networks or (Class II) crosslinked macromolecular architecture polymers rank among the largest molecules known. Their molecular weight is given by the macroscopic size of the object for instance, a car tire made of vulcanized rubber or a crosslinked layer of protective coating can be considered one crosslinked molecule. Such networks are usually called macronetworks. On the other hand, micronetworks have dimensions of several nanometers to several micrometers (e.g. siloxane cages or microgels). 

Telechelic polymers rank among the oldest designed precursors. The position of reactive groups at the ends of a sequence of repeating units makes it possible to incorporate various chemical structures into the network (polyether, polyester, polyamide, aliphatic, cycloaliphatic or aromatic hydrocarbon, etc.). The cross-linking density can be controlled by the length of precursor chain and functionality of the crosslinker, by molar ratio of functional groups, or by addition of a monofunctional component. Formation of elastically inactive loops is usually weak. Typical polyurethane systems composed of a macromolecular triol and a diisocyanate are statistically simple and when different theories listed above are... 

The national network of air pollution measurements is keyed to the 247 air quality control regions (aqcr s), which were classified according to the relative severity of their pollution problems. The classification is a ranking of measured ambient air concentrations or the estimated air quality in the area of maximal severity. The priorities for air quality problem severity are as follows ... 

The fact that I am the only African American at a senior rank is due to the way the company recmits. We go to, for example, Harvard and Cal Tech. We are in contact with faculty members there. .. I don t think we make an overt effort to develop networks which would identify African American Ph.D. chemists and recmit them. 

In the context of the present discussion the term "mobile phase will be used to describe those components which can be thermally extracted ("distilled", "desorbed ) under vacuum at temperatures below the thermal degradation range of the coal. The residue, designated as the nonmobile ("network ) phase, is thermally degraded in the pyrolysis temperature range. Of course, the onset of pyrolysis may vary considerably, depending on heating rate, rank and coal type (10). 

The above observations point to the importance of rank and depositional environment in defining the molecular characteristics of the mobile phase as well as the network phase of coals. 

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