Number of connections

Thermodynamics provides a set of differential equations that interrelate the properties of PTT systems. Most of these properties are abstract, but the equations provide a limited number of connections with measurable quantities. One of the functions of thermodynamics is to maximize the return in useflil information for any investment in experiment. 

If a vessel is being prepared for entry, the computer could check that the number of slip-plates (blinds) to be fitted (or pipes disconnected) is the same as the number of connections shown on the drawing. 

Detonations can only take place within the detonable limits, which fall inside the flammable limits. Mixture composition may vary widely in some systems, and this can further depend on the number of connections feeding a manifolded header. There are few cases where it can be assumed that a nondetonating flammable mixture will be produced hence, most in-line flame arrester applications involve the use of detonation flame arresters. 

Hydroxy-terminated PDMS, however, has disadvantages. The monofunctional ends limit the number of connections between the polymer (or oligomer) molecule and the glass network to two. This limitation raises the possibility that some PDMS molecules are not tied at both ends to the glass network if the polycondensation does not go to completion i.e. there may be "dangling" or loose PDMS chains in the final sol-gel material. This occurance of free ends would indeed be anticipated since the extent of reaction most likely is not 100%. Hence, the physical properties, specifically the mechanical behavior of the overall material, would be expected to suffer as a result of loose PDMS chains in the system. Disregarding this potential problem, the mechanical behavior of the sol-gel hybrids are, ultimately, influenced by the mechanical behavior of the modifying elastomer ... 

This can be shown by induction on the number of connectives by noting that ... 

As ligands in metal complexes, the anions derived from these acids display a broad variety of coordination patterns (Scheme 2). Usually, the ligands are described as monodentate, bidentate chelating or bridging, and polydentate. We use here a description based upon the number of connections between the... 

The principle of the greatest number of connections (connection principle). To understand the meaning of this principle it may first be necessary to define the concept of connection. To this end we consider a certain crystal structure and imagine connecting each atom with the other atoms present in the structure by straight lines. There will be a shortest segment between any two atoms. We will... 

The most basic element in the molecular structure is the existence of a connection or a chemical bond between a pair of adjacent atoms. The whole set of connections can be represented in a matrix form called the connectivity matrix [249-253]. Once all the information is written in the matrix form, relevant information can be extracted. The number of connected atoms to a skeletal atom in a molecule, called the vertex degree or valence, is equal to the number of a bonds involving that atom, after hydrogen bonds have been suppressed. 

For the various "indices of complexity" which have been proposed and the minimum requirements they must meet in order to be really useful, see Bertz [30a]. The "complexity index" r], proposed by Bertz, is based on the number of connections and is defined as the number of adjacent pairs of bonds existing in one molecule. 

So, the number of connections can be determined either from the "bond graph" [7], or even better, from the number of "propanes" that can be derived from the skeleton of the molecule under consideration. 

The alternative procedure to determine the number of connections is, as we have already mentioned, to calculate the number of "propanes" within the molecular skeleton, bearing in mind that a double bond is equivalent to one "propane" ... 

In the "general index of complexity" (C ) proposed by Bertz, besides the number of connections, symmetry is also taken into account. Such an index, that incorporates concepts from the "graph theory" and the "theory of information" [7] is defined as ... 

In the case of butadiene, the calculation of the number of connections by the "propane method" gives ri = 6 (connections), distributed in two groups one of four equivalent substrucUires and another group of two equivalent substructures. Therefore,... 

Leatherface is a 2-D molecular editor that modifies properties of atoms and bonds in molecular connection tables according to rules specified by the user. Unlike Permute, Leatherface encodes no chemical knowledge and neither processes nor generates 3-D structures. Its real strength is that it allows the user to impose a very detailed and precisely specified chemical view on large numbers of connection tables. 

The U.S. EPA Environmental Research Laboratory-Duluth with the help of their cooperators has developed a data matrix of 90 variables calculated from molecular connectivity Indices (10) for 19,972 of the chemicals In TSCA. Molecular connectivity indices consist of four primary types (paths or the edges between atoms, clusters or branches, path/clusters, and cycles or rings) that are calculated from 0th to 9th order depending on the number of connections between atoms. Path terms can Include as many orders as there are edges between atoms in the molecule, the minimum order for a cluster or a cycle is three, and the minimum for a path/cluster Is four. Therefore, using 0th to 9th order, the number of variables for one set of connectivity Indices is 30 variables. In our data base, we included three sets of... 

The theory of cumulants allows us to partition an RDM into contributions that scale differently with the number N of particles. Because aU of the particles are connected by interactions, the cumulant RDMs scale linearly with the number N of particles. The unconnected terms in the p-RDM reconstruction formulas scale between N and W according to the number of connected RDMs in the wedge product. For example, the term scales as NP since all p particles are statistically independent of each other. By examining the scaling of terms with N in the contraction of higher reconstruction functionals, we may derive an important set of relations for the connected RDMs. 

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