Compressed adjacency matrix

Lukovits (2000, 2002, 2004) and Lukovits and Gutman (2002) offered an approach by which the vertex-adjacency matrix of an acyclic structure can be replaced by a single number, called the compressed (vertex-) adjacency matrix code, denoted by CAM. Here we present, besides the CAM code, the N-tuple code of trees that induces the unique labeling of trees (Aringhieri et al 1999). A graph is acyclic if it does not contain cycles. A tree is a connected acyclic graph. 

Figure 6.5 The characteristic deviation in packing fraction (Acj)), = < )c - ())j at which the adjacency matrix deviates from that at (pj for each of the 20 MS packings at jamming onset for N = 6 bidisperse systems using small successive compressions 10 (circles) and 10 (crosses). The dashed line indicates the average ((A( ))c) = 0.028 over the 20 MS packings, (b) The average ((Acf)) ) over 100 2D bidisperse MS packings as a function of N. The solid line has slope -1.9.

In the in situ consolidation model of Liu [26], the Lee-Springer intimate contact model was modified to account for the effects of shear rate-dependent viscosity of the non-Newtonian matrix resin and included a contact model to estimate the size of the contact area between the roller and the composite. The authors also considered lateral expansion of the composite tow, which can lead to gaps and/or laps between adjacent tows. For constant temperature and loading conditions, their analysis can be integrated exactly to give the expression developed by Wang and Gutowski [27]. In fact, the expression for lateral expansion was used to fit tow compression data to determine the temperature dependent non-Newtonian viscosity and the power law exponent of the fiber-matrix mixture. 

In the general approach, the loads are applied incrementally until first-ply failure occurs. The type of failure, matrix or fiber, determines which properties of the failed plies must change to reflect the damage created. This is subjective and can cover a range of possibilities. The most conservative approach would completely discard affected properties for the failed plies. So for fiber failure, E would be set to zero. For matrix failure, E22 and G12 would be set to zero. Then, the loads would be incremented until another ply fails, and the procedure would be repeated to complete failure of the laminate. Less conservative approaches attempt to only partially discount stiffness values of the failed ply and even differentiate between tension and compression moduli. These methods can be reasonably accurate if they are accompanied by selected tests that help better define adjustment factors for the stiffness properties of failed plies. However, they are limited in applicability and accuracy because they are affected by the first-ply failure criterion used to trigger the failure sequence and because they do not correctly capture damage modes such as delamination and the interaction between them such as matrix cracks causing delaminations in adjacent ply interfaces. 

For Cl on Ta and W(IOO), the rows of adatoms along the [011] and [011] directions compress [80K1, 83S]. Simultaneously, adjacent adatom rows shift with respect to one another as shown in Figure 4.d. In this way, a c(2x2) stracture can be converted into the (1x1) stractures observed at saturation. Again, the compression is continuous. Using the matrix notation, the surface periodicity can be described by the matrix ... 

The results of mechanical tests indicate that the genipin stent has a significantly higher ultimate compression load (1,123 77 mN) and collapse pressure (2.5 0.1 bar) than the epoxy stent (856 148 mN, 1.9 0.1 bar). These results are attributable to the various crosslinking structures that are formed in stent matrices (Fig. 17) [176]. The two epoxide functional groups in the epoxy compound that was used in the study crosslinked the amine groups of chitosan in the stent matrix [177]. In this way, a linearly crosslinked structure between the adjacent chitosan molecules may be formed intermolecularly. 

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