Three dimension automatic

Assuming that a polynomial has been found which adequately represents the response behavior, it is now possible to reduce the polynomial to its canonical form. This simply involves a transformation of coordinates so as to express the response in a form more readily interpreted. If a unique optimum (analogous to a mountain peak in three dimensions) is present, it will automatically be located. If (as is usual in multidimensional problems) a more complex form results, the canonical equation will permit proper interpretation of it. 

The computational labor associated with two-dimensional Fourier syntheses is not too formidable, and two-dimensional Fourier maps can be constructed without machine help. The labor associated with two-dimensional Patterson sysntheses is even less, and a two-dimensional vector map can often be obtained from measured intensities in a few hours. For Fourier and Patterson syntheses in three-dimensions, however, machine help is virtually indispensable. Before application of automatic computers to x-ray diffraction, the main obstacle standing in the way of a structure determination was generally the computational effort involved. In the 1950 8, the use of computers became commonplace, and the main obstacle became the conversion of measured intensities to amplitudes (the so-called phase problem ). There is still no general way of attacking this problem that is applicable in all situations, but enough methods have been developed so that by use of one, or a combination of them, all but very complicated structures may, with time and ingenuity, be determined. 

Pure rotary diffusion of rigid dipoles in two or three dimensions, then, gives exponential decay of polarization with a single relaxation time, provided the sites are uniformly distributed and D is constant. The description of the motion in terms of D alone breaks down, as we shall see, for very short times. A three-dimensional rigid body in any case executes a more complex motion. Even an internally uniform model of rectilinear charge-carrier difiurion automatically shows multiple relaxation. More realistic models must take account of the dynamic s of molecular motion. 

Figure 34.33. Charting a Course. This projection of the track of an E. coli bacterium was obtained with a microscope that automatically follows bacterial motion in three dimensions. The points show the locations of the bacterium at 80-ms intervals. [After H. C. Berg. Nature 254(1975) 390.]...

The processes of physical and chemical activation of carbons, Chapters 5 and 6, extract carbon atoms from the interior of the network of porosity in carbons. It would be an advantage to have some idea of the structure of this network, in three dimensions, in order to understand the extraction (gasification) process. This three-dimensional network or porosity can be described as a labyrinth and this automatically takes us into children s maze games, some of which are complex (computer generated) as illustrated in Figure 3.28(b, c), while others are quite simple, as shown in Figure 3.28(d), designed for children under five. 

For matrices of modest dimensions 1024 matrix diagonalizations may not be a serious CPU problem for a PC, but if we include (as we will in the next chapter) distributions in the spin Hamiltonian parameters the required CPU time goes up by, say, two orders of magnitude, and if we want to implement automatic minimization, we must pay with another two or three orders of magnitude in CPU-time. 

In Gunn s procedure the matrix of split-fraction coefficients is represented by three vectors a vector D containing the non-zero coefficients, in column order within consecutive rows an integer vector Z, of the same dimensions as D, containing the column address of each non-zero element and an integer vector L giving the position in the other vectors of the first element in each row. The program MM3 contains a sub-routine that automatically reads the values from the data file into these vectors for the calculation procedure. 

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