Orthogonal processes

If a 33 = 0, we have a i3 = 0, and the function 03 is then a linear combination of the functions 0X and 0 2 and should be omitted in the orthogonalization process, which is here simply accomplished by means of the Gaussian elimination technique developed for solving equation systems. The connection between the matrices a and a may be written in the form ... 

Rates of addition to carbonyls (or expulsion to regenerate a carbonyl) can be estimated by appropriate forms of Marcus Theory. " These reactions are often subject to general acid/base catalysis, so that it is commonly necessary to use Multidimensional Marcus Theory (MMT) - to allow for the variable importance of different proton transfer modes. This approach treats a concerted reaction as the result of several orthogonal processes, each of which has its own reaction coordinate and its own intrinsic barrier independent of the other coordinates. If an intrinsic barrier for the simple addition process is available then this is a satisfactory procedure. Intrinsic barriers are generally insensitive to the reactivity of the species, although for very reactive carbonyl compounds one finds that the intrinsic barrier becomes variable. ... 

Figure 6.9 A schematic representation of orthogonal process for nanoparticles self-assembly (a) a patterned sihcon wafer with Thy-PS and PVMP polymers fabricated through photolithography and (b) orthogonal surface functionahzation through Thy-PS/DP-PS recognition and PVMP/acid-nanoparticle electrostatic interaction. Reprinted with permission from Xu et al. (2006). Copyright 2006 American Chemical Society.

If we choose our basis functions for a particular function space to be orthonormal (orthogonal and normalized) i.e. (/ /,) = J/, /, dr then, since the transformation operators are unitary ( 5-7), the representation created will consist of unitary matrices. This is proved in Appendix A.6-1. It should be stated that it is always possible to find an orthonormal basis and one way, the Schmidt orthogonalization process, is given in Appendix A.6-2. 

Gram - Schmidt orthogonalization process Thus, equation (B.6) takes the form... 

The orthogonalization process plays an important role in the solution of the inverse problem. 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

It can be seen that, in doing the orthogonalization process, the coefficient of y in (Equatrions 2.304) becomes identically zero, and we obtain... 

We have found [1] that orthogonalization of the correlated vectors not only takes care of the above problem very effectively but also gives vital insight into the phenomenon of memory storage in the brain. The orthogonalization process in fact acts like a decision making process wherein when a new object is compared with the stored ones, the nature of its correlation with them decides, in an efficient (or economical) manner, in what form it should be stored. The noise is also eliminated using this approach. 

An alternative approach has been proposed by Philipp and Friesner [33] and Murphy et al. [34]. It differs from the previous one by the introduction of modified Roothaan equations to compute the electronic density and energy of the QM part, which avoids the orthogonalization process, and by the treatment of the interaction of... 

Thus the orthogonalizing Process of (4.99) (or rather one possible orthogonalization process, Lowdin orthogonalization) is the use of an orthogonalizing matrix to transform Hby pre- and postmultiplication (Eq. 102) into H. H satisfies the standard eigenvalue equation (Eq. (4.103)), so... 

Orthogonality theorem for irreducible representations, 341/. Orthogonalization process, Schmidt, 212 Out-of-plane bending, 58 Overtone frequencies, 9, 36, 246 selection ndes, infrared, 160 Uaman, 161 Overtone levels, 36... 

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