New Representations from Old

I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation. It is too long ago. I was in the middle before I knew that I had begun. 

In this chapter we discuss several natural ways to construct representations from other representations. 

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In this section we show how to construct a new representation from an old one hy restricting the domain of the linear transformations. One cannot restrict the domain to any old subspace, only to invariant subspaces. 

Figure 10-5 Diagrammatic Representation of Amylopectin Structure. Lines represent a-D-glucan chains linked by 1,4-bonds. The branch points are 1,6-a glucosidic bonds. Source From J.J. Marshall, Starch Degrading Enzymes, Old and New, Starke, Vol. 27, pp. 377-383, 1975.

Figure 4.1 Schematic representation of the detachment of a crystal from a substrate showing the transformation of the contact interface Oeft) into two new interfaces (right) (i) ciystal/ambient phase and (ii) substrate/ambient phase. The detachment energy is the sum of the surface energies for creation of the new surfaces and annihilation of the old one.

This time, as a result of the cyclic nature of this molecule, it is possible to draw three new double bonds, and not just two. This arises, because what were the two unpaired electrons that were at the ends of the chain, are now adjacent to each other and so may interact to form the third bond. Superficially, the new molecule looks the same as the old one, except that the double bonds have moved one position around the ring. However, even though this looks like a small difference, in fact it belies the fact that the cyclic version of the triene has completely different properties from its linear cousin. This cyclic triene molecule is benzene. Draw a representation for the time-averaged redistribution of the electrons present in these three double bonds within this cyclic molecule. 

The Commission, in conjunction with the Notice to Applicants Working Party, completed its work on the incorporation of the ICH Common Technical Document (CTD) into a revision of Volume 2B of the Notice to Applicants. A schematic representation of the correspondence between the five modules of the CTD and the four parts of the European registration dossier has also been prepared in order to facilitate the transition from the old application format to the new one. The non-clinical and clinical overviews and summaries are equivalent to the present expert report described above. The clinical summary may replace the Integrated Summary of Efficacy and Integrated Summary of Safety Required by the FDA, and the Clinical Overview and Clinical Summary have been designed to replace the Japanese GAIYO. 

This representation of the dynamical path probability emphasizes specific realizations of the random force It enables an interesting class of trial moves in trajectory space A new path is obtained from an old path... 

The Legendre transform coimects the point-representation of the data to the tangent-representation. To construct the transform, note first that we want to replace x with s as the independent variable. Since /is convex, each value of the slope s is unique. Further note that if we introduce a new independent variable, then we must also introduce a new function, call it g(s). The Legendre transformation performs the mapping from the old function to the new one. 

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