Ternary schemes

We consider first ternary schemes which are both primal, in the sense that there is a vertex in the new polygon corresponding to each vertex of the old, and dual, in that there is an edge of the new polygon matching each each of the old. 

In this case the mask has an odd number of entries, and the distance reached at the first step is (m— l)/6. Considering subsequent steps multiplies this by a factor of (1 + 1/3 + 1/9 +. ..) which is equal to 3/2. Thus the range of influence is (m — l)/4 on each side. 

Depending on the size of the mask, this can he either at the image of an original control point or half-way in between. 

For example, the ternary scheme whose mask is [1,2,3,2, l]/3 (the ternary linear B-spline) has a support width of exactly (5 — l)/2 = 2 old spans, while the scheme whose mask is [1,3,6,7,6,3, l]/9 (the ternary quadratic B-spline) has a support width of (7 — l)/2 = 3 spans, one and a half on each side of the control point. 

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Case 2 includes many of the example systems studied in this book. For example, reactors with temperature as the only controlled variable fall into this category. Also, the isothermal ternary scheme CS4 shown in Fig. 2.13a has a local composition controller on one of the dominant variables, the composition of component A. However, Case 2 is characterized by the fact that other dominant variables are not controlled at the reactor. Instead, the plantwide control structure plays a significant role in its ability to influence these uncontrolled variables. When the uncontrolled compositions become disturbances and the controlled dominant variables are too weak, we have difficulties. On the other hand, the plantwide control structure can be arranged to provide indirect control of the dominant composition variables, thereby augmenting the unit control loops. The HDA process provides a good illustration. The dominant variables are reactor inlet temperature and toluene composi-... 

This first example has approximately twice as many vertices in the new polygon as in the old. We call it a binary scheme. If there had been three times as many it would have been a ternary scheme, and such generalisations will be discussed in a few pages time. In principle at each refinement we can multiply the number of vertices by whatever we choose, and this number is called the arity and denoted by the letter a. It is also called the dilation factor, which stems from generating function usage. 

One particular ternary scheme, called the ternary quadratic B-spline for... 

It is also possible for the mask of a ternary scheme to have an even number of entries. In this case exactly the same algorithm tells us that the range of an original control point can lie one quarter of the way between point images. The end of the influence of one control point is no longer at the same place as the start of the influence of another. 

This can be seen in the basis function of the ternary scheme [l,3,5,5,3,l]/6 with the control points after one step from cardinal data. 

For example, the ternary scheme whose mask is [1, 3,5, 5,3, l]/6 has a support width of (6 —1)/4 = 5/4 old spans on each side, so that the part of a span near the original control points is influenced by three original vertices, while the part from 1/4 to 3/4 is influenced by only two. 

For ternary schemes which are both primal and dual there is a short cut. There are two mark points, and we can determine the unit row eigenvector for each of them, thus giving the values at the limit points corresponding to both the vertices and the mid-edges. 

In the case of ternary schemes a very similar structure holds. If a scheme is both primal and dual, the structure is exactly the same a linear combination of B-spline masks. But it can be a linear combination of even and odd degree B-splines, because both are both primal and dual. The number of entries in the mask is always odd. 

The expected complication is that the ternary schemes have two matrices each. 

The artifact analysis above is limited, essentially to binary and ternary schemes. Higher arities can bring in artifacts at higher frequencies which might spoil or improve the shapes of the limit curves, and it is not obvious how these can most sensibly be handled. 

The quantitative interpretation of the data may be open to some doubt, since the experiments were carried out at high and variable ionic strengths, but the existence of a measurable contribution from the last term in (42) is probably established. Nevertheless, it has been usually held that the reaction cannot go exclusively or even chiefly by a ternary mechanism, using the following argument due to Pedersen (116). Omitting the terms in [H30+] and [OH-], the remaining terms of (42) will be represented as follows on the ternary scheme ... 

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