Defining path maximal

Definition 6.2.9 A maximal defining path of a relevant anchor r R vi) of a vertex Vi is the defining path / (r,v,) whose length is maximal among all defining paths from r-to-Vi. 

Therefore, the maximal defining path of a relevant anchor r G R vi) is the longest path by which r is defined to be relevant Obviously, there can be more than one defining path for a particular relevant anchor since there can be more than one path from r to v,-. Similarly, the maximal defining paths may not be unique. 

By Theorem 6.2.5, the irredundant anchors are also relevant anchors. Therefore, there exists a maximal defining path p r,vi) of r IR(vi), where the w,) is equal to length r,Vi). By Theorem 6.2.3, length r,Vi) is equal to the minimum offset Let T "(vi) and be the start times... 

Note thato is now added to A(vj), where the maximal defining path p a, vf) trivially reduces to a path containing only the newly created forward edge. Since the original edges remain unchanged, and since the created forward edges satisfy... 

In many processes multiple reaction pathways are possible and it is the role of the catalyst and reaction conditions to maximize the rate along the desired path. Selectivity is defined as the amount of desired product divided by reactant converted. A practical example of the catalyst directing reactants to a selective product is shown by the oxidation of ammonia to nitric oxide, which is used in the production of fertilizers. 

Is it possible to bypass these models, and determine the nature of the bonds directly from the charge density Bader has developed a useful approach which defines both which pairs of atoms are joined by chemical bonds, and the nature of the bonds thus formedA characteristic of all chemical bonds is that they possess a bond critical point at which the electron density is a minimum along the path between the nuclei, and is a maximum in all direction perpendicular to that path. The path between the nuclei which contains the bond critical point, and maximizes the charge density throughout its length is known as the bond path. 

QTAIM provides a definitive answer to the question of whether two atoms are bonded or not even in ambiguous cases [87, 88] and, as a consequence, the molecular graph, i.e., the chemical structure, is readily defined by this theory. Two atoms are bonded if their nuclei are linked in space by a line of maximal electron density termed the bond path [89, 90] (see Fig. 1). A single bond path links the nuclei of chemically bonded atoms irrespective of the mode of the bonding covalent (single or multiple), hydrogen, van der Waals, ionic, metallic, etc. The properties of the eleetron density determined at the point of lowest density along the bond path, where... 

The problem is to define the path of x versus time which will minimize this integral. For this reaction, minimization of the integral corresponds to maximization of the rate of conversion to B at each instant. Mathematically we can state the requirement... 

The reduction of permeability arises from the longer diffusive path that the penetrants must travel in the presence of the filler (layered silicate in the present case). A sheet-like morphology is particularly efficient as it maximizes the path length. The tortuosity factor (f or T depending on the symbology) is defined as the ratio of the actual distance (d ) that a penetrant must travel to the shortest distance (d) that it would have traveled in the absence of the layered silicate and is expressed in terms of the length (L), width (W)/ and volume fraction of the sheets ((j)). 

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