Algebraic models coupling

In the previous sections the correspondence between the Schrodinger picture and the algebraic picture was briefly reviewed for some special cases (dynamical symmetries). In general the situation is much more complex, and one needs more elaborate methods to construct the potential functions. These methods are particularly important in the case of coupled problems. This leads to the general question of what is the geometric interpretation of algebraic models. 

Kellman, M. E. (1983), Dynamical Symmetries in a Unitary Algebraic Model of Coupled Local Modes of Benzene, Chem. Phys. Lett. 103,40. 

The short time scale model is similar to the case with reaction extents. Table 5.4 shows the short time scale result in the left hand column. Deriving the correct slow time-scale model is less obvious and sometimes leads to confusion. The equilibrium assumption is made by adding the algebraic constraint that rz - 0 or KiCb - cc 0. This algebraic equation coupled with Equation 5.44 and the mole balance for component A, in which the second reaction does not appear, then constitute a complete set of equations for the slow time scale, shown in the lower, right portion of Table 5.4. If we prefer differential equations, we can add the... 

Figure 34. Local-mode coupling (according to the three-dimensional algebraic model) in a bent triatomic molecule for the first two vibrational polyads.

To begin with, we recall that in certain cases, the algebraic model has been already put in a one-to-one correspondence with a specific potential function for the usual space coordinates. We have already studied dynamic symmetries providing exact solutions for the one-, two-, and three-dimensional truncated harmonic oscillators, the Morse and Poschl-Teller potential functions. When we consider more complicated algebraic expansions in terms of Casimir operators, or when we deal with coupled... 

To characterize the effective turbulent viscosity of liquid in the bath, two models, namely, differential models and algebraic models, have frequently been used. The differential models also can also be categorized into two. In the first group, the effective viscosity /tteff is determined by solving a differential equation, which expresses the conservation of turbulent energy coupled with a prescribed length scale. Szekely and co-workers [22,42] used this approach in their earlier models. However, it had been realized that this model is not valid for recirculatory turbulent flow [42,48]. 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

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