Conformal maps inversions

SCHEME 1. Conformational map of the RcRn and RcSn diastereomers of /V-ethyl-iV-methyl-2-aminobutane (EMAB). Interconversions among conformers within dashed boxes are fast on the NMR time scale at 104 K. Those between dashed boxes occur via rotations about the methine carbon-nitrogen bond with barriers which are DNMR-visible. The interconversion between the solid boxes occurs via nitrogen inversion (disstereomeric interconversion). The values in parentheses are MM2-80 results. Reprinted with permission from Reference 71. Copyright (1988) American Chemical Society... 

The key observation is that Levi-Civita s conformal map (7), u i—> x = u2, not only regularizes collisions at x = 0 but also analogous singularities at x = oo. This is seen by closing the complex planes to become Riemann spheres (by adding the point at infinity) and using inversions x = 1/x, u = 1/u. 

In the spatial case we choose v, u, x, y e U to be quaternions, x = x, y = y being quaternions with vanishing fourth components, associated with 3-vectors x, y. Then the mappings (44), (45), being shifted inversions in 4 or 3 dimensions, are both conformal maps, in fact the only conformal maps existing in those dimensions, except for the translations, magnifications, and rotations. Composing these with the KS or Hopf map (42), uhx = uu, yields... 

As the skewed section is a loxodrome, i.e. a path over Planet Earth under constant compass direction, it is well known that the length of this section equals / sin(/). As to the length of the parallel section, consider the mercator projection of Planet Earth. It is well known that on this projection the projected distance of the -parallel to the equator equals gmercator projection is a conformal mapping of Planet Earth s surface, i.e. it preserves angles, and that it also preserves distances... 

Figure 10. Relaxed (adiabatic) conformational energy map for p-maltose as computed by Brady and coworkers.i3 Contours are drawn at 2,4,6, 8, and 10 kcal/mol above the minimum near < ), y = -60°, -40°. The p-maltose structure may be derived from that of p-cellobiose in Fig. 1 by inversion of the stereochemical configuration at Cl.

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