Structural incidence matrix

In the structural incidence matrix given above, there are seven variables columns terms on abscissa, i.e., on horizontal axis viz., JCi-JCy) and four relationships (numbered rows terms on ordinate, i.e., on horizontal axis viz., 1-4). Therefore, three variables must be assigned as inputs to specify the system. They are X2, x, and x. Rewriting the structural matrbc by moving the input variables to the right-hand side and delineating them with a vertical line, we have... 

Unique output set assignment occurs when the structural incidence matrix is triangular, such as the above. As a consequence, the variable-influence path leading to the top-level output variable (e.g., x,) is also unique and can be affected only by inputs that occur along the path. 

Fig. 18. The structural incidence matrix for the reactor-quench example.

THE WEICHTED-HEXACON-KEKULE-STRUCTURE INCIDENCE MATRIX... 

The incidence matrix that connects the numeric and geometric Kekule structures can be constructed using expression (3.1). It is called the weighted-hexagon-Kekule-structure incidence matrix and is denoted by HK. 

The construction of the weighted-hexagon-Kekuld-structure incidence matrix is exemplified for chrysene. In Figure 3.4, the geometric Kekule structures of... 

The simplest form to represent the chemical information contained in a molecular graph is by a -> matrix representation of a molecular structure. Examples are -+ adjacency matrix A, - edge adjacency matrix E, vertex - distance matrix D, -> edge distance matrix D, - incidence matrix I, - Wiener matrix W, -> Hosoya Z-matrix Z, - Cluj matrices CJ, - detour matrix A, - Szeged matrix SZ, -> distance/distance matrix DD, and - detour/distance matrix A/D. 

The concepts of molecular and fragment structure were defined in the previous chapter, Section 1.1, in terms of incidence matrices. In structure correlation we compare molecules with the same incidence matrix coming from different crystal structures, or fragments with the same incidence matrix coming from different molecules. More often than not, the molecules or fragments in which we are interested show little or no symmetry. Why then should a book on structure correlation contain a chapter dealing with symmetry aspects What symmetry aspects ... 

Using a DAE solver that accepts the Jacobian existence matrix, that is, an incidence matrix only. This provides the solver with the possibility to completely exploit the system s sparsity, but not its overall structure. Nevertheless, it is often not possible to provide the Jacobian incidence matrix, especially when the system is very large or when the incidence matrix changes as part of an iterative process. 

A physical process can be modelled by a set of physical quantities or variables and a set of physical equations or constraints that link these variables. In a structural approach, an incidence matrix is used to represent the set of constraints. Each row corresponds to a constraint and each column to a variable. A 1 in position (i, j) indicates that variable j appears in constraint i. 

For instance, in the previous example of three connected pipes, flowmeters may be implemented at 3 instrumentation points. 3 constraints and 3 known variables are thus added to the structural model (Table 2). A rapid analysis of the graph (or incidence matrix)... 

The non-invertibiUly of a constraint with regard to a variable is indicated by putting 1 instead of 1 in the structural model. When this value appears in the incidence matrix, the corresponding variable cannot be determined by the other variables. 

Thanks to the incidence matrix, the exhaustive set of ways to reach a given physical quantity can be determined (Ploix 2005). It consists in determining all the paths in the structural graph (incidence matrix) that go from the known variables to the unknown variable that has to be estimated. The paths must respect the invertibihty constraints i.e. must correspond to possible calculability sequences. 

This paper presents a relatively easy-to-use design method. Indeed, it requires a reduced amount of data, that is to say, an incidence matrix built by a structural modeUmg, the cost of the potential usable sensors and a... 

Let R- = [j J define a literal causality matrix of size nxn whose entries follow Eq (2). In graph theory (Deo 1974), R corresponds to a node-to-node incidence matrix. Moreover, the transposed form of the R matrix (i.e. R ) has an equivalent structure to that of a DSM(Aleisa Lin 2009). This allows us to exploit the well-established methods of DSM to structure literal spaces, while still remaining consistent with the previous literature on state-space literature, the theory of ordered relations (Dartmouth College Writing Group. Cogan 1958) and Markov Chains by considering the transposed form. 

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] ... 

Taking the previous remark into account we can use a oup technology algorithm (see for example [3],[4] ) in order to transform the initial incidence matrix [m ] of P into a more structured (possibly block-diagonal) form. Thus when a block-diagonal form is found for [niij] then we can decompose the original problem P into independent sub-problems P, P2.-(see figure 2). 

The matrix of a structure with n atoms consists of an array of n / u entries. A molecule with its different atoms and bond types can be represented in matrix form in different ways depending on wbat kind of entries are chosen for the atoms and bonds. Thus, a variety of matrices has been proposed adjacency, distance, incidence, bond, and bond-electron matrices. 

---