Kronecker’ symbol

To simplify the formulae below we assume that Cijki = SlSf, 5 is the Kronecker symbol. Nevertheless, all the results obtained in the section are valid in the general case (5.74). 

The operator symmetrizes with respect to permutation of the operators A and B together with the accompanied Cauchy orders m, n 6 is the Kronecker symbol. Similar as for the first-order Cauchy vectors the second-order Cauchy vectors are obtained as solutions of a recursive set of equations ... 

Eq. (10) is block diagonalized as a function of an electronic quantum number, Sjg (Kronecker symbol) factor. Equations (9) and (10) are related. These equations for each electronic subspace differ only by a local linear transformation, then, one may write ... 

Here the <5 s are Kronecker symbols and the primed summations extend only over nonvanishing k-vectors. In deriving this result we used the orthonormality of the plane waves Q-l/2e-ifcr... 

Table 1 Summary of the notation. In these definitions we use the transverse Kronecker symbol = ij — riitij. Due to the thermodynamic stability of the systems the following combinations of constants must be positive Bq,B, K, Lq, L s L, and Mq —KL. For the last relation we used the equivalence of K and ...

The transverse Kronecker symbols 5A in (17) and (21) guarantee the normalization of A. This implies that only two of the (17) are independent. For the following calculations it turned out to be useful to guarantee the normalization of the director by introducing two angular variables 9 and 0 to describe the director ... 

For the reader that would not know about covariant derivative, suppose that one is in the Euclidean case (K = 0) and that the spatial part of the unperturbed manifold has Cartesian coordinates, so that 7ij is simply a Kronecker symbol, and the Vi are simply ordinary derivative with respect to the coordinate xl. 

Next, observe that, because of the Kronecker symbols involved in Eq. (B.30), the summation over j appearing in Eq. (B.33) is dealing only with two terms not far from unity. Therefore, according to Eq. (B.31), it is possible to perform the following approximation ... 

Kronecker symbol, unity when i = j and zero otherwise... 

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