Analysis of the Process Topology

A measured process variable, belonging to subset x, is called redundant (overdetermined) if it can also be computed from the balance equations and the rest of the measured variables. 

Based on the preceding formulation, the following problems can be defined  

Define the subset of redundant equations to be used for the adjustment of measurements 

In the following sections, the basic tools for the structural evaluation of the process equations are briefly discussed. They allow us to systematically analyze the topological structure of the balance equations and to solve the three problems defined earlier. 

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This section presents results of the space-time analysis of the above-mentioned motional processes as obtained by the neutron spin echo technique. First, the entropically determined relaxation processes, as described by the Rouse model, will be discussed. We will then examine how topological restrictions are noticed if the chain length is increased. Subsequently, we address the dynamics of highly entangled systems and, finally, we consider the origin of the entanglements. 

This chapter has shown that the analysis of the topological structure of the balance equations allows classification of the measured and unmeasured process variables, finally leading to system decomposition. 

The analysis of the dynamics consists of (i) the calculation of the dressed eigenenergy surfaces of the effective quasienergy operator as a function of the two Rabi frequencies flj and if, (ii) the analysis of their topology, and (iii) the application of adiabatic principles to determine the dynamics of processes in view of the topology of the surfaces. 

The steps, in general, can be summarized as follows (i) simplification process, (ii) identification and separation of the individual motifs, (iii)topological analysis of these motifs, and (iv) topological analysis of the whole entanglement. 

Before analysis of the interactions of the nucleic acid bases with the clay minerals in the presence of water and cation one needs to understand the individual interactions of NAs with isolated water and with a cation. Such theoretical study was performed for 1 -methylcytosine (MeC) [139]. The study revealed influence of water and cation in the proton transfer for this compound. This leads to the formation of imino-oxo (MeC ) tautomer. Topology of the proton transfer potential surface and thermodynamics of stepwise hydration of MeCNa+ and MeC Na+ complexes is further discussed. The one dimensional potential energy profile for this process followed by the proton transfer with the formation of hydrated MeC Na+ is presented in Fig. 21.2. One-dimensional potential energy profile for amino-imino proton transfer in monohydrated N1-methylcytosine (this represents the situation when tautomerization is promoted by a single water molecule without the influence of Na+ cation) and for the case of pure intramolecular proton transfer (tautomerization is not assisted by any internal interactions) is also included. The most important features of this profile do not depend on the presence or absence of Na+ cation. All the potential energy curves have local minima corresponding to MeC and MeC. However, the significant difference is observed in the relative position of local minima and transition state, which results in a different thermodynamic and kinetic behavior for all presented cases (see Fig. 21.2). 

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