Redundance

This approach to synthesis is one of making a series of best local decisions. Equipment is added only if it can be justified economically on the basis of the information available, albeit an incomplete picture. This keeps the structure irreducible, and features which are technically or economically redundant are not included. 

Figure 1.7 An initial structure for the manufacture of benzene from toluene and hydrogen incorporating some redundant features.

The approach to heat exchanger network design discussed so far is based on the creation of an irreducible structure. No redundant features were included. Of course, when the network is optimized, some of the features might be removed by the optimization. The scope for the optimization to remove features results from the assumptions made during the creation of the initial structure. However, no attempt was made to deliberately include redundant features. 

An alternative approach is to create a reducible structure that deliberately includes redundant features and then subject this to optimization. Redundant features are then removed by the optimization. 

In high permeability reservoirs, wells may produce dry oil for a limited time following a shut-in period, during which gravity forces have segregated oil and water near the wellbore. In fields with more production potential than production capacity, wells can be alternately produced and shut in (intermittentproduction) to reduce the field water cut. This may still be an attractive option at reduced rates very late in field life, if redundant facilities can be decommissioned to reduce operating costs. 

The calculation of characteristic values causes a high amount of values which contain redundant informations. Due to this the forth partial step will be to reduce this amount of values using extraction methods. This can be realized with statistical methods like cross correlation analysis. 

Naturally, such a high probability of detection of theoretically 100% never can be realized in practice. The inspector will not recognize all good visible indications at any time because he cannot always be fiilly concentrated on his task, which is called "human factor". This human factor appears in any visual inspection and may be reduced only by a second redundant inspection or extreme signals as a light flash or an acustic signal. 

The coordinates are redundant since they can be detennined from the condition drat the X, Y, Z)... 

In this way the optimization can be cast m temis of the original coordinate set, including the redundancies. Exactly the same transfomiations between Cartesian and internal coordinate quantities hold as for the non-redundant case (see the next section), but with the generalized inverse replacing the regular inverse. 

The redundant optimization scheme [53] can be applied to natural internal coordinates, which are sometimes redundant for polycyclic and cage compounds. It can also be applied directly to the underlying primitives. 

Pulay P and FogarasI G 1992 Geometry optimization In redundant Internal coordinates J. Chem. Phys. 96 2856... 

Peng C, Ayala P Y, Schlegel H B and Frisch M J 1996 Using redundant Internal coordinates to optimize equilibrium geometries and transition states J. Comput. Chem. 17 49... 

Load balancing can then be achieved in NAMD 2 by moving compute objects and patches between nodes. But what if a compute object and a patch it depends on are on different nodes Compute objects individually communicating with off-node patches would generate a huge amount of redundant communication. Therefore, patches are represented on other nodes by proxy patches, which implement the same interface as home patches for dealing with compute objects and handling dependencies but receive coordinates from and... 

A second notion is that each atom i.s described twice - in a column and in a row. Matrices in which all elements are shown twice are called redundant. A non-redun-dant matrix contains each element only once (e,g., only the top right or bottom left triangle of the matrix, as can be seen later). 

As can be seen in Figure 2-13, the diagonal elements of the matrix are always zero and it is symmetric around the diagonal elements (undirected, unlabeled graph). Thus, it is a redundant matrix and can be reduced to half of its entries (Figure 2-14b. For clarity, all zero entries are omitted in Figures 2-14b-d. 

Figure 2-16. a) The redundant incidence matrix of ethanal can be compressed by b) omitting the zero values and c) omitting the hydrogen atoms, in the non-square matrix, the atoms are listed in columns and the bonds in rows. 

Figure 2-17. a) The redundanl bond malrix of ethanal with ihe zero values omitted, b) It can be compressed by reduction to the top right triangle, c) Omitting the hydrogen atoms provides the simplest non-redundant matrix representation. 

Both tables, the atom and the bond lists, are linked through the atom indices. An alternative coimection table in the form of a redundant CT is shown in Figure 2-21. There, the first two columns give the index of an atom and the corresponding element symbol. The bond list is integrated into a tabular form in which the atoms are defined. Thus, the bond list extends the table behind the first two columns of the atom list. An atom can be bonded to several other atoms the atom with index 1 is connected to the atoms 2, 4, 5, and 6. These can also be written on one line. Then, a given row contains a focused atom in the atom list, followed by the indices of all the atoms to which this atom is bonded. Additionally, the bond orders are inserted directly following the atom in-... 

Figure 2-22. Non-redundant connection table of ethanal. Only non-hydrogen atoms are considered bonds with the lowest indices are counted once (see Figure 2-21).

Almost all chemical information systems work with tlicir own special type of connection table. They often use various formats distinguishing between internal and external connection tables. In most cases, the internal connection tables arc redundant, thus allowing maximum flexibility and increasing the speed of data processing. The external connection tables are usually non-redundant in order to save disk space. Although a connection table can be cprcsented in many different ways, the core remains the same the list of atoms and the list of bonds. Thus, the conversion of one connection table format into another is usually a fairly straightforward task. 

Redundant, isomorphic structures have to be eliminated by the computer before it produces a result. The determination of whether structures are isomorphic or not stems from a mathematical operation called permutation the structures are isomorphic if they can be interconverted by permutation (Eq. (6) see Section 2.8.7). The permutation P3 is identical to P2 if a mathematical operation (P ) is applied. This procedure is described in the example using atom 4 of P3 (compare Figure 2-40, third line). In permutation P3 atom 4 takes the place of atom 5 of the reference structure but place 5 in P2. To replace atom 4 in P2 at position 5, both have to be interchanged, which is expressed by writing the number 4 at the position of 5 in P. Applying this to all the other substituents, the result is a new permutation P which is identical to P]. 

Once the quality of the dataset is defined, the next task is to improve it. Again, one has to remove outliers, find out and remove redundant objects (as they deliver no additional information), and finally, select the optimal subset of descriptors. 

Another misleading feature of a dataset, as mentioned above, is redundancy. This means that the dataset contains too many similar objects contributing no... 

One can find more details on the algorithm in Section 4.3.4. This time the learning yielded essentially improved results. It is sufficient to say that if in the case of the primary dataset, only 21 compoimds from 91 were classified correctly, whereas in the optimized dataset (i.e., that with no redundancy) the correctness of classification was increased to 65 out of 91. 

We have already mentioned that real-world data have drawbacks which must be detected and removed. We have also mentioned outliers and redundancy. So far, only intuitive definitions have been given. Now, aimed with information theory, we are going firom the verbal model to an algebraic one. 

Equation (4) provides a value of AICO = 0.00852. As we see, the populations of classes 3 and 8 bear redundancy in information, as their JCOs are quite low here. In contrast, classes 6 and 7 are clearly outliers. Their JCOs are too high in compar-... 

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