Reachability matrix

One method of locating these maximal loops is to compute the reachability matrix, R (H1), which is the element by element Boolean union of all of the powers of the adjacency matrix up to the nth, where n is the number of rows of R. An element of the reachability matrix is defined as... 

The maximal loops in the graph can be found from the reachability matrix by finding those sets of vertices that satisfy the following conditions (1) r j = rfi = 1, where i and j take on all possible combinations of the vertex numbers in the set (2) no other vertices, not included in the set, satisfy condition (1). The first condition requires that each vertex in the set is reachable by some path from every other vertex in the set. The second condition requires that there is no path from a vertex in the set to a vertex outside the set... 

In order to obtain in a digital computer all of the powers of the adjacency matrix and then compute the reachability matrix by taking the Boolean sum of all of the powers, each power of the adjacency matrix and the reachability matrix would have to be stored in the computer memory. For large systems of equations an unreasonable amount of storage would be required. A second modification to the methods described in Section II is to drastically reduce... 

Actually, only one matrix need be stored if the adjacency matrix is stored initially and thereafter multiplied by itself. Matrix elements are replaced by the resulting product elements as they are computed. The product matrix obtained in this manner for the fcth power may contain some nonzero elements which correspond to paths longer than k steps instead of strictly k step paths, but this will not affect the final matrix obtained corresponding to the nth power, since these paths would eventually be identified in any case. All of the modifications to the methods of Section II mentioned above simplify the calculations needed to obtain the reachability matrix. The procedure for identifying the maximal loops given in Section II remains the same. 

The required information about the distillation boundary is obtained from the pinch distillation boundary (PDB) feasibility test [8]. The information is stored in the reachability matrix, as introduced by Rooks et al. [9], which represents the topology of the residue curve map of the mixture. A feasible set of linear independent products has to be selected, where products can be pure components, azeotropes or a chosen product composition. This set is feasible if all products are part of the same distillation region. The singular points of a distillation region usually provide a good set of possible product compositions. The azeotropes are treated as pseudo-components. 

Since the truncation of Eq. (4.14) is an approximation, the shifts may be under- or overestimated. When the shifts are added to the original parameter values, that is, X2 = Xj + <5X, the elements of the design matrix A, which depend on X, change. Many iterations may be needed before convergence can be reached, if it is indeed reachable. Strategies for coping with difficult refinements have been discussed by Watkin (1994). 

Application of the purely graphical test to the buck converter example confirms the results obtained by means of the structurally controllability matrix test in the previous section. If switch Sw mi is assumed to be on, its causality can be inverted and the I-element gets integral causality. Then there is a direct causal path from the effort source to the I-element and a causal path from the effort source through the I-element to the C-element. That is, the reachability condition is satisfied. In addition, both storage elements can be assigned derivative causality in the bond graph with preferred derivative causality. Thus, also the sufficient condition is met. Hence, in system mode mi = 1 a m2 = 0, the model is structurally completely controllable with the one effort source. 

Molecular reinforcement is affected by the consequent translation of fiber reinforcement to the molecular level. At the same time, in composite science and technology, it is well known that the mechanical performance of a composite material depends on two main factors, namely, the aspect ratio of the reinforcing component and the adhesion quality between matrix and reinforcement. Analysis of the mechanical data, as well as the superior reachable aspect ratio of single stiff molecules, identifies single molecule fibers as the limit of materials aimed at fiber reinforcement. Lindenmeyer [1] explained the concept of molec-... 

Several procedures for workspace analysis of manipulators have been proposed iterative determination of reachable points by means of matrix formulation of direct problem, [14-16], or through probabilistic techniques, [17, 18], or continuation methods, [19] dynamical evaluation of extreme configuration, [20, 21] determination of boundary surfaces for Jacobian domain, [22-24] algebraic formulation for specific manipulators, [25, 26]. However, in order to facilitate the numerical solution for the design problem of Eqs. (7), (8) and (9) it can be useful to express the involved workspace characteristics by means of a suitable analytical formulation. 

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