Occurrence matrix reduced

Ledet s algorithm involves two phases. In the first phase an ordering of the rows and columns of the occurrence matrix takes place according to certain optimality criteria. The second phase involves reordering the occurrence matrix to reduce the number of torn variables. The first phase carries out an initial tearing, and the second phase improves the results of the first phase. However, instead of tearing individual variables as in Steward s algorithm, Ledet s method systematically reorders the occurrence matrix as described below. 

Once all of the rows have been removed from the reduced occurrence matrix, the tears corresponding to the minimum number of variables that must be assumed in the solution of the process material and energy balances have been found. 

M Reduced occurrence matrix used in y System variable... 

The observed reversal in the thermal stability of the copolymer at a critical composition, which appears to be between 30 and 40 mol% of ethylene, may be explained on the basis of the emergence of phase-separation between the nonpolar ethylene and polar vinyl chloride blocks. Although crystallization of the ethylene blocks in the copolymer is only observed when more than 70 mol% ethylene units are present, the possibility of phase-separation occurring at lower contents of ethylene units cannot be excluded. Also, round about the critical copolymer composition, the Tg of the copolymer may be reduced to a level that would facilitate separation between the unlike phases by increased molecular mobility within the polymer matrix. As has been discussed earlier, occurrence of phase-separation in the copolymer would not only make the mechanism of stabilization due... 

E. Brandas, C.A. Chatzidimitriou-Dreismann, On the Connection Between Certain Properties of the Second-Order Reduced Density Matrix and the Occurrence of Coherent-Dissipative Structures in Disordered Condensed Matter, Int. J. Quant. Chem. 40 (1991) 649. 

It is a common occurrence that we wish to evaluate the reduced matrix element of an operator which acts on only one part of a coupled scheme. For example, the general formula for the reduced matrix element of an operator Tk(A ) which acts only on part 1 of a coupled scheme j + j2 = j is ... 

In the spin-free formulations of the UGA-MRCC theories, the use of CSFs entails that both the MRCC equations are in matrix form and the associated effective Hamiltonians will involve various w-body spin-free reduced density matrices (n-RDMs). n-RDMs are product separable and hence not size-extensive. From now on, we will refer to the spin-free RDMs as simply the ROMs. When spinorbital-based RDMs are discussed, we will explicitly indicate this. So, no confusions should arise. It is non-trivial to establish the extensivity of both the cluster operators and the effective Hamiltonian in spite of the occurrence of these n-RDMs. This paper will briefly review the formulation of the UGA-MRCC theories mentioned above and will present a comprehensive account of the aspects of connectivity which leads to extensivity. Although in some of our earlier papers [47] we sketched how size extensivity emerges after the cumulant decomposition of the n-RDMs, we will present here a detailed and thorough analysis of the underlying issues. 

Clays compatibilized and evenly dispersed in the rubber matrix tend to build networks at low concentration. Rheological measurements allow us to observe the occurrence of the filler networking phenomenon at low filler content (even 4 phr) in rubber matrices such as IR, ENR, SBR, EPR. At zero shear, the viscosity of RCN is thus higher than the one of the neat elastomer. However, OC reduces the steady shear viscosity of RCN, with pronounced shear-thinning behaviour, increasing with the clay content, a higher extent of extrudate, a... 

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