Typical rules generated

We need to point out that, if the wavelengths of laser radiation are less than the size of typical structures on the optical element, the Fresnel model gives a satisfactory approximation for the diffraction of the wave on a flat optical element If we have to work with super-high resolution e-beam generators when the size of a typical structure on the element is less than the wavelengths, in principle, we need to use the Maxwell equations. Now, the calculation of direct problems of diffraction, using the Maxwell equations, are used only in cases when the element has special symmetry (for example circular symmetry). As a rule, the purpose of this calculation in this case is to define the boundary of the Fresnel model approximation. In common cases, the calculation of the diffraction using the Maxwell equation is an extremely complicated problem, even if we use a super computer. 

Rules may be combined by composition i.e., two rules, (f>i and 2, may be combined to form the rule (f) = 4>i(t>2- The set of rules obtained in this way is closed under composition, although the number of sites in the neighborhood will typically have to be increased. If a rule is composed with itself, then = (fxp) generates patterns consisting of alternate time steps of the patterns of . In general, composition is noncommutative (f>i(p2 ... 

Figures 3.38 and 3.39 show typical space-time patterns generated by a few r = 1 reversible rules starting from both simple and disordered initial states. Although analogs of the four generic classes of behavior may be discerned, there are important dynamical differences. The most important difference being the absence of attractors, since there can never be a merging of trajectories in a reversible system for finite lattices this means that the state transition graph must consist exclusively of cyclic states. We make a few general observations.

The Rule of 10 is better still. If we use 10 times the number of samples as there are components, we will usually be able to create a solid calibration for typical applications. Employing the Rule of 10 will quickly sensitize us to the need we discussed earlier of Educating the Managers. Many managers will balk at the time and money required to assemble 40 calibration samples (considering the example, above, where temperature variations act like a 4th component) in order to generate a calibration for a "simple" 3 constituent system. They would consider 40 samples to be overkill. But, if we want to reap the benefits that these techniques can offer us, 40 samples is not overkill in any sense of the word. 

Conformation analysis methods. In many cases in the process of building a 3D structure from scratch, decisions have to be made between multiple alternatives with similar energy. A typical example is an sp -sp torsion angle with similar energies for the alternatives of -i-60°, -60° and 180°. In many cases, rules are used to decide (e.g. stretch an open chain portion as much as possible to avoid clashes). Sometimes, the best result cannot be determined without a conformation analysis (e.g. complex ring systems with exocycHc substituents). Despite conformation analysis being a topic of its own covered in the next chapter, many automatic 3D structure generators have to fall back in certain situations to a limited conformation search in order so solve a specific problem and to come up with a reasonable solution. 

In addition to value tables and fragment data, a certain part of the knowledge about 3D structures can be expressed in more general rules for solving specific problems in the 3D generation process. Here, some typical examples will be illustrated. 

The process controller is the master of the process control system. It accepts a set point and other inputs and generates an output or outputs that it computes from a rule or set ofrules that is part of its internal configuration. The controller output serves as an input to another controller or, more often, as an input to a final control element. The final control element typically is a device that affects the flow in the piping system of the process. The final control element serves as an interface between the process controller and the process. Control valves and adjustable speed pumps are the principal types discussed. 

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