Mathematical concepts matrices vectors

Now we come to a totally different method for producing matrix representations of a point group a method which involves the concept of a function space. The word space is used in this context in a mathematical sense and should not be confused with the more familiar three-dimensional physical space. A function space is a collection or family of mathematical functions which obeys certain rules. These rules are a generalization of those which apply to the family of position vectors in physical space and in order to help in understanding them, the corresponding vector rule will be put in square brackets after each function rule. 

Whilst a single residual may be sufficient to detect faults, a vector of residuals is usually required for fault isolation. For isolation purposes, structured residuals [8, 17] can be generated, i.e., each residual is affected only by a specific subset of faults, and each fault only affects a specific subset of residuals. This concept can be expressed in a mathematical form by introducing a boolean fault code vector and a boolean structure matrix [8],... 

In solid-state NMR, a very important concept is that the resonance frequency of a given nucleus within a particular crystallite depends on the orientation of the crystallite [3—5]. Considering the example of the CSA of a nucleus in a carboxyl group, Fig. 9.1 illustrates how the resonance frequency varies for three particular orientations of the molecule with respect to the static magnetic field, Bq. At this point, we note that the orientation dependence of the CSA, dipolar, and first-order quadrupo-lar interactions can all be represented by what are referred to as second-rank tensors. This simply means that the interaction can be described mathematically in Cartesian space by a 3 X 3 matrix (this is to be compared with scalar and vector quantities, which are actually zero- and first- rank tensors, and are specified by a single element and a 3 X 1 matrix, respectively). For such a second-rank tensor, there exists a principal axes system (PAS) in which only the diagonal elements of the matrix are non-zero. Indeed, the orientations illustrated in Fig. 9.1 correspond to the orientation of the three principal axes of the chemical shift tensor with respect to the axis defined by Bq. 

Another important concept in molecular quantum similarity is associated with convex conditions. The idea underlying convex conditions, associated with a numerical set, a vector, a matrix, or a function, has been described previously in the initial work on VSS and related issues.Convex conditions correspond to several properties of some mathematical objects. The symbol X(x) means that the conditions (x) = 1 A x V(R ) hold simultaneously for a given mathematical object x, which is present as an argument in the convex conditions symbol. Convex conditions become the same as considering the object as a vector belonging to the unit shell of some VSS. For such kind of elements,... 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

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