Induction structural representation

Harbome, J.B. (1999B) The comparative biochemistry of ph 4oalexin induction in plants. Biochem. System. Ecol., 27,335-67 (unfortunately this otherwise useful paper is replete of wrong or incomplete structural representations and no reference is given to the original chemical publications). 

As an example, in a cube one has v = 8,e = 12, f = 6, and hence 8 — 12-1-6 = 2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. 

However simple the above considerations may look, the looks are deceiving it turns out that so far mathematics (and logic) has not adequately addressed the issue of structural object representation, for good reasons. I suggest that these reasons are quite profound and have to do with the context within which the development of structural representation should take place. This context, of classes and induction, has never been adequately delineated neither within... 

Consistent with the above two postulates, one should approach the task of developing a formalism for structural representation as that of developing a class-oriented representational formalism , which would automatically ensure that the formalism will be suitable for the purposes of inductive learning—since the object representation would carry much more information about its class representation—and consequently for the purposes of Af in general. 

In what follows, I will proceed under the assumption that the above two postulates are adopted, at least as far as the development of formalism for structural representation is concerned. Again, I would like to emphasize the importance of such adoption without the appropriate guiding considerations, the very notion of object representation becomes far too ambiguous, as can be seen from its development so far. For example, strings and graphs are not appropriate models for structural representation, since, as mentioned above, neither a string nor a graph carry within itself sufEcient information for the inductive recovery of the class from which it is supposed to come. This is, in part, a consequence of the situation when none of the above two postulates have been adopted. 

Figure 7.14. Mesomeric (Mp si,/ Mp u) and Inductive (Ip sh/ Ipuii) effects in carbenes 28 and 29 (Ar = 2,6-bis[trifluoromethyl]phenyl, R = iiso-propyl), and molecular representation of the X-ray structure of 29. (Adapted from reference 87.)...

In DDD, systems are modeled as networks of streams, which are infinite sequences over a type. Streams and other nonfinite structures raise issues in model theory that are only now being ironed out (Barwise Moss 1996). These problems are intrinsic to logics based in well-founded set theory, which must deal with streams indirectly. The typical representation is mapping from natural numbers to values. The problem is that proofs about streams under this representation reduce to inductions over the naturals, instead of "structural coinductions that reflecting the definition style. 

Thus, it should be clear that in contrast to the conventional computational models—where, as was mentioned above, the defining agenda was logical—I propose that it is the (structural) inductive agenda that should now drive the development of the computational framework. This proposal is not really surprising, given that the new framework is aimed at supporting a continuous dynamic interaction between the machine and various natural environments, where, as was hypothesized above, the temperal/structural nature of object representation is ubiquitous. 

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