Permutation of Indices

Although a formal solution of the A-representability problem for the 2-RDM and 2-HRDM (and higher-order matrices) was reported [1], this solution is not feasible, at least in a practical sense [90], Hence, in the case of the 2-RDM and 2-HRDM, only a set of necessary A-representability conditions is known. Thus these latter matrices must be Hermitian, Positive semidefinite (D- and Q-conditions [16, 17, 91]), and antisymmetric under permutation of indices within a given row/column. These second-order matrices must contract into the first-order ones according to the following relations ... 

It must be noted that, due to the arbitrary symmetry under permutation of indices of this second-order matrix, a larger set of contractions into the 0- and 1 -body space must be taken into account. 

The derivation of this Permutation of Indices procedure, PI, has been published elsewhere7. 

From analyzing this expression, the symmetry of coefficients M p in respect to the permutation of indices a and P occurs necessarily only in the case of independent rushes fij of internal parameters (concentrations of intermediates involved in the reaction groups). It was shown previ ously that these are namely the systems allowing the construction of the Lyapunov function to describe their evolution through the entire region of applicability of the thermodynamic approach. It is usually impossible to find the Lyapunov functional for the intermediate nonlinear schemes. 

When the power is small enough, there is also a set of fiilly antisymmetric combinations (determinantally antisymmetric), defined by the property that they are reversed in sign by every pairwise permutation of indices. For a set of given dimension d, combinations of this type exist for all those powers that lie between 2 and d, i.e. (d+l) > p > 1. The number of totally antisymmetric combinations is... 

Displacement parameters are included in Eq. 2.97 with all possible permutations of indices. Thus, for a conventional anisotropic approximation after considering the diagonal symmetry of the corresponding tensor (Eq. 2.96)... 

TREOR90 is a semi-exhaustive trial-and-error indexing program, which is based on the permutation of indices in a selected basis set of lowest Bragg angles peaks. TREOR90 includes an analysis of the dominant axial zones (i.e. / 00, 0 and 00/)- In the case of a monoclinic crystal system, the so-... 

The algorithm is called semi-exhaustive because certain limitations on the possible permutations of indices are incorporated into the program in order to increase its speed. 

Equation (121) was obtained from Eq. (120) by use of the antisymmetry of the test function h. Now let us consider the permutation of indices between topologically distinct regions ... 

There is no convenient 2n + 1 rule to compute fourth-order quadruples. It is possible to create second-order Tf by taking all 18 distinct permutations of indices in the (T 0)2 term. This, however, although easy to program, would require an 8 basis set size dependence asymptotically. It turns out that the more efficient way to program the (T2)2 contribution to the energy is to use so-called vertical factorization of the diagram, which makes it possible to do the computations with still an n6 basis set size dependence. This was described in detail by Bartlett and Purvis.6 The more exhaustive discussion of the different types of factorization of the CC and MBPT diagrams will be presented in subsequent subsections and systematized in Appendix C. 

The first equation always applies, while the second only applies for samples without electric currents inside and through their surface. Thus, the second equation is valid for nonconducting materials, but not necessarily for conductors. It establishes a relation between the Cartesian gradient components which are obtained by permutation of indices. 

From equations (4.56) and (4.61), it follows that the terms of S and C are equal for the following permutation of indices ... 

Here, the matrices H and V are symmetric and positive definite matrices, which are each, after suitable permutation of indices, tridiagonal matrices. The matrix S is a non-negative diagonal matrix. Recalling that tridiagonal matrix equations are efficiently solved by the Gauss elimination method, we consider now the Peaceman-Rachford iterative method [27], a particular variant of the lAD methods, which is defined by... 

Show that the cyclic permutation of indices p, v, p or that of X, a, x in Problem 5.2 does not affect the value of the matrix element. 

A closer look at relationship [11.24] shows an invariance of apparent conductivity by the permutation of indices. Both the materials must therefore play equivalent roles. This is the case of a biphasic material, made of grains of similar size but of different kinds. The same does not apply if one of the phases is made of inclusions in a matrix. This is, however, a microstracture frequently observed when as a result of liquid phase sintering. It is also possible, however, to define and calculate an apparent conductivity. Let us consider a spherical inclusion, consisting of a nucleus of the dispersed phase and a shell of the continuous phase o in adequate proportions. Similar developments to those of the previous section lead to the following expression, proved by Wagner ... 

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