Operator substitution

Consider an arbitrary set of Boolean input values, Xi,..., Xn- Using only the Boolean operators AND and NOT, we can write down an expre.ssion which takes the value 1 if and only if a particular configuration of O s and I s appears in the input. Suppose that n = 6 and we want to construct an expression that equals one if and only if xi = X3 = X4 = X5 = 1 and. X2 = xe = 0. A little thought will show that the required expression is given by joining the string of six Boolean variables with the AND operator, substituting NOT(xi) for all values x)i that are required to equal zero ... 

Formally, we recall that this equation can be derived by inserting the operator substitutions... 

Suppose that the inverse operators and A exist. Moreover, we assume that A and A are self-adjoint positive operators. Substitutions of u = A f and u = A f into (15) yield... 

The first term of (197) stands for contraction (196) and the second term represents diagrams where operators X and Xh are directly contracted with W operators. Substituting (197) into (195) gives us... 

The expansion given above is exact, except for r12 = 0. If the terms resulting from the regions II-IV are neglected, one recovers the standard multipole expansion of the interaction operator. Substituting the bipolar expansion of 1 /r12 and analogous expansions for other terms of the operator V into the matrix elements (0 V J), the bipolar expansion of a given polarization correction is obtained. 

Here the superscript 0 represents the trace with respect to the non-interacting density matrix. The zeroth order Green functions are given in Eq. (55). The terms coming from the lead-molecule coupling (V. ) vanish because they are odd in creation and annihilation operators. Substituting Eq. (C34) in Eq. (C25) gives for the phonon contribution... 

Let us now explain the detailed expression of Vf,-corJ in Eq. (12) and the approximations of operators leading to it. The individual components of the spin-free CGWB-AIMP are obtained by the following operator substitutions ... 

In 1980, Permea, with its hydrogen-separating Prism membrane, launched the first large industrial application of GS membranes. Since then, membrane-based operations, substituting or to be integrated with the traditional ones, have had a rapid growth, with many companies, such as Cynara-NATCO, UOP Separex, GMS, Generon, Praxair, Air Products, and Ube, involved in this field. ... 

The systematic surveillance detects the loss of systems which are required for implementing the operating strategy. An alternate system is then proposed to the operator (substitution). 

The above data do not take into account the increased knowledge of the potential (and actual) apphcations of PI since the strategies were formulated. In some instances the opportunities will have increased - in others they may have been superseded by other process improvements. An important factor is that major process changes/plant replacement would be needed to realise the savings, and a second critical observation is that integration is necessary in aU except minor unit operation substitution. Effective integration can maximise emission reductions. 

This is simply equation 10.13, with the wavefunction and the position operator substituted. 

This result for the density matrix elements is well known from the SCF theory. The same expression can also be derived directly from Eq. (7.5), by establishing the specific commutation rules between the MO and AO operators. Substitution of the MO expansion of Eq. (7.8) into the commutators yields ... 

To determine this expectation value for the case of a one-determinantal wave function one may proceed exactly in the same manner as in Sect. 7.1. That is, the AO operators are expressed in terms of the MO operators. Substituting Eq. (7.12) into Eq. (7.19) one gets ... 

Mathcad symbol mathematics works with units as with variables neglecting that 1 m amounts to 100 cm or 60 min are in an hour, etc. Symbol mathematics is foreign element in Mathcad it was taken from Maple where unit appeared in the eighth version. Although, symbol mathematics is auxiliary tool in Mathcad integrated seldom into a calculation but operates auxiliary functions. If it is necessary to have units worked in symbol mathematics, we should make efforts enter to Mathcad via operator substitute that 1 m = 100 cm, 1 h = 60 min, etc. 

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