Redundant points

Plots for 11 isoelectronic sequences having more than three data, or having three centered on IZ1-Z2I = zero, were prepared, one of which is shown in Fig. 8.3. Six of the plots have six data points, and three more have five points (one in the center and two duplicated on each side) the remainder have three non-redundant points. The average standard deviation of fitting for these is 2.595°. Figure 8.4 has the largest scatter around its trend line (a = 3.951°), so it is used as an example of what the ( — 1) standard deviations look like. There is no theoretical basis for using quadratic trend lines they are used for sake of simplicity. 

It should be pointed out that it is probably not necessary to make as many redundant taps as called for in a code test. However, as long as taps... 

When the point values are average probabilities, the overall result from combining system.s as combinations of sequences and redundancies is found by simply combining the mean probabiliiies according to the arithmetic operations. 

Figures 6.4 shows some of the variety of possible shapes of P f) for elementary rules shown in the figures are the power spectra for rules Rll, R56, R150 and R200. The plots were generated for lattice size N = 2048, ignoring the first 15 transient steps and averaging a total of 20 runs. Also, since there are only N data points but 2N real Fourier components, half of the components are redundant. Thus, only the first half of the components are shown (see [H89b] or [H87] for a complete set of power spectra).

Fig. 1 Results from a redundancy analysis of stream nutrient concentrations vs. catchment characteristics in 31 headwater catchments in Catalonia (Spain). Note how phosphorus (SRP) and ammonium (NH4) appear associated with urban point and diffuse sources (Urban percent urban land use, Inhabs number of inhabitants, eq. inhab inhabitant equivalents), whereas nitrate (NO3) concentration is positively associated with irrigated agriculture (IrrAg) and bovine cattle units (ECU), and negatively with mean catchment slope (slope) and percent forest land (Forest). The other variables are Area log area, P precipitation, NonIrrAg non irrigated agriculture. Data are from the Catalan water agency (http //ww.gencat.cat/ac)...

It is to be noticed that three criteria depend on this criterion which seems somehow redundant. It is particularly true in the case of C4 and C, which from a thermodynamic point of view are exactly the same. Knowing the dangerous nature of these compounds, it would seem that C4 is more representative of instability risk than C. ... 

The set point tracking controller not only becomes redundant as soon as we add feedback control, but it also unnecessarily ties the feedforward controller into the closed-loop characteristic equation. 

Constraints (2.13) and (2.14) imply that state. v can only be used in a particular unit, at any time point, after all the previous states have been processed. Constraint (2.13) is only relevant in situations where more than one task can be conducted in one unit, otherwise it is redundant in the presence of constraints (2.14) and (2.15). Constraint (2.15) stipulates that a state can only be processed at a particular time point p in a particular unit j after it has been produced from another unit /. In case of a recycle, j is the same as /. It is worthy of note that constraints (2.14) and... 

The assignment constraint is aimed at ensuring that only one task is conducted in a unit at any time point. It is, therefore, apparent that the assignment constraint is only necessary if more than one task can be performed in a given unit. Otherwise, it is also redundant. 

In this section, the above mathematical model is applied to a literature example shown in Fig. 2.2 (Ierapetritou and Floudas, 1998). The SSN representation is given in Fig. 2.3b. Table 2.1 gives data for this example. 5 time points and a 12-h time horizon were used. Using less time points leads to a suboptimal solution with an objective value of 50, and using more time points than 5 did not improve the solution. It is worthy of note that, in this particular example, constraint (2.13) is redundant as mentioned earlier, since each unit is only performing one task. 

To ensure that a unit is only used for either processing or storage at a particular time point, constraint (3.32) is required. Constraint (3.33), ensures that a batch can not be split. Constraint (3.11) is redundant in the presence of constraint (3.32). 

In this chapter we will present a discussion of those points, leading us directly to the decomposition of the general problem into estimable, nonestimable, redundant, and nonredundant subsystems. This allows us to reduce the size of the commonly used least squares estimation technique and allows easy classification of the process variables the topic of the next chapter. 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

The use of hybrid atomic orbitals in qualitative valence theory has, in the past, rested on two points (i) an empirical justification of their use involving the concept of the valence state of an atom and (ii) a simple linear transformation technique for the construction of the explicit forms of the orbitals. In this section we show that both of these points can be replaced. The justification can be replaced by a derivation and this derivation can be used to suggest variational forms which render the linear transformation technique redundant. 

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