Conformal transformation/mapping

Conformal Mapping Every function of a complex variable w = f z) = u x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude xnd sense. Every analytic function, except at those points where/ ( ) = 0, is a conformal transformation. See Fig. 3-48. 

Our conformal mapping examples focused on incompressible liquids in isotropic formations for simplicity. Using the formalism of Chapters 1, 2, and 3, extend these mapping results generally to include the effects of anisotropic media and of gases with constants m. Hint recall that simple scale transformations map the anisotropic equation into isotropic form,... 

Thus, x(, ri) and y(, ri) are likewise harmonic, but in the variables % and r. Equations 8-54 and 8-55 are simpler than Equations 8-21 and 8-22, with P = Q = 0. The use of our reciprocity relationships shows that there exists a duality between physical and mapped planes, and vice versa, for conformal transformations that is, Equations 8-50 and 8-51 are mirror images of Equations 8-54 and 8-55. One might have anticipated this type of reversibility, but it is not direetly evident from Equations 8-21 and 8-22. Equations 8-54 and 8-55 are consistent with Thompson s original Equations 8-21 and 8-22. Use of the Cauehy-Riemann eonditions in the transformed plane, that is. Equations 8-52 and 8-53, in Equations 8-14 to 8-16, leads to a = y and 3 = 0. 

Fig. 2. Depiction of conformal mapping of graphene lattice to [4,3] nanotube. B denotes [4,3] lattice vector that transforms to circumference of nanotube, and H transforms into the helical operator yielding the minimum unit cell size under helical symmetry. The numerals indicate the ordering of the helical steps necessary to obtain one-dimensional translation periodicity.

The conformal mapping will transform this lattice operation B,v to a rotation of 1tt/N radians around the nanotube axis, thus generating a C,v subgroup. 

Gordon, L.M., Lee, K.Y.C., Lipp, M.M., Zasadzinski, J.A., Walther, F.J., Sherman, M. A., and Waring, A.J. Conformational mapping of the N-terminal segment of surfactant protein B in lipid using C-13-enhanced Fourier transform infrared spectroscopy. J. Peptide Res. 

Fourier-Transform Infrared (FTIR) spectroscopy as well as Raman spectroscopy are well established as methods for structural analysis of compounds in solution or when adsorbed to surfaces or in any other state. Analysis of the spectra provides information of qualitative as well as of quantitative nature. Very recent developments, FTIR imaging spectroscopy as well as Raman mapping spectroscopy, provide important information leading to the development of novel materials. If applied under optical near-field conditions, these new technologies combine lateral resolution down to the size of nanoparticles with the high chemical selectivity of a FTIR or Raman spectrum. These techniques now help us obtain information on molecular order and molecular orientation and conformation [1],... 

The MWA map has what might be regarded as a drawback. We wish to contain the concentration field that varies during the time Tmax of the simulation, that is, to have distances of about 6 x y/Tmax from all points in the system. This translates, upon conformal mapping, to a certain maximum F value. How this is calculated is described below on page 229. The point is that such a calculation must be made. However, this also applies to the other two transformations, as will be seen below. 

The current integration (12.19) depends on the transformation. For the three conformal mappings described above, the new expressions are as follows. For MWA [394] and VB [557],... 

A major disadvantage is the time required for the computation. The entire algorithm (generation of 3-D structure, conformation analyses, property calculation, property mapping on surface or field and autocorrelation transformation) needs between 100 and 100000 s for one molecule. With today s computer technology, using multi-processor compute servers and highly vectorized software, it is possible to calculate up to several thousand three-dimensional autocorrelation descriptors vtithin a day (Table 7). 

One of the many applications of the theory of complex variables is the application of the residue theorem to evaluate definite real integrals. Another is to use conformal mapping to solve boundary-value problems involving harmonic functions. The residue theorem is also very useful in evaluating integrals resulting from solutions of differential equations by the method of integral transforms. 

Conformal mapping provides a transformation of variables that converts one mathematical problem into another. Its importance lies in the greater ease with which the transformed problem may be solved than the original problem. Applications arise in steady-state conduction or diffusion problems. 

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