Shape code vectors

A simple numerical similarity measure is defined by the number of matches between the components of two such shape code vectors. If C(M]) and C(M2> are the shape code vectors of molecules (or conformers) Mj and M2, then the similarity measure sc(Mi, M2) is defined as... 

The corresponding shape code vector c(s(a,b)), a three-dimensional vector, is constmcted in three steps. 

The n diagonal elements ij(a,b) of the matrix s(a,b) are encoded by concatenating them into a single number. For example, a diagonal of elements (2, 2, 1, 2, 0, 2 ) is encoded as the decimal number 221202. (Since the only numerical values that occur along the diagonal are 0, 1, and 2, a ternary number system can also be used). The number so obtained is the second component C2(s(a,b)) of the shape code vector c(s(a,b)). 

The upper off-diagonal triangle of the shape matrix has elements 1 and 0 only, and these n(n-l)/2 numbers concatenated according to columns form a binary number that is the third component C3(s(a,b)) of the shape code vector c(s(a,b)). 

Note that if the element s(a,b)i,i of the shape matrix is different from zero, then the information on the dimension n of the matrix can be deduced from the second element C2(s(a,b)) of the shape code vector c(s(a,b)). The special case of s(a,b)i 1 = 0 seldom occurs, since this implies that the largest domain is a locally concave Dq domain relative to the curvature parameter b. Nevertheless, in order to avoid ambiguity in such cases, the dimension n is specified as the first component ci(s(a,b)) of the shape code vector c(s(a,b)). 

This shape code vector can be decoded easily by simply reversing the above process. In view of the relation n = C (s(0.0l,0)), the reconstruction of the diagonal elements of matrix s(0.0l,0) from C2(s(0.0l,0)) is a trivial task, whereas the... 

The very same procedure that has been used to construct the global C(Mi) codes for the global shape matrices s(a,b,M]) along the parameter map (a,b) can also be applied to the set of local shape matrices Ib(a,b,Mi) along the parameter map (a,b), resulting in a local shape code vector... 

For either code, c(a,b) or c (a,b), the corresponding number can be assigned to the (a,b) location of the parameter map (a,b). Since most small changes in the values of a and b do not change the shape groups of the actual truncated molecular surfaces, the entire (a,b) map will contain only a finite number of different values for the c(a,b) or c (a,b) code. A list of these code values can be regarded as a vector, providing a numerical shape code for the entire (a,b) map (i.e., for all relevant electron density values a and test curvature values b). 

The code axialdisp4DaPerunbackw. m consists of a comments block, followed by the general default and initialization block, as well as the BVP initialization and solver block. The actual solution is found inside the central try. . . catch. . . end lines of code. This is followed by two blocks of plotting code and the coded DE in dydx with its boundary conditions in Rand, both expressed as vector valued functions. Finally four initial guess functions for the shape of the solution are given that are used inside the BVP solver for different parameter data. 

The molecular surface concept is not only useful for a representation of the bulkiness and the shape of molecules. These surfaces can also be used as screens for the visualization of many properties by means of color coding techniques. Color coding is a popular means of displaying scalar information on a surface. " " Every three-dimensional scalar or vector field that may be generated on the basis of the position of atomic or molecular fragments can be visualized by color coding on a given surface. 

In practice, only a finite number of parameter pairs (a,b) are considered, for example, those at the grid points of the 41 x 21 grid of the parameter map (a,b) described above. The entire map of shape matrix codes can then be represented by three 861-dimensional vectors, C( ), C(2), and C(3), containing all first, all second, and all third components, respectively, of the individual c(s(a,b)) vectors. Alternatively, a single (3 x 861) - dimensional vector C can be assigned to the (a,b) parameter map, where C is obtained by concatenating the components of C(l), C(2), and C(3) into a single vector. 

Figure 4.8 Three-dimensional velocity field of of-planecomponentofthevelocity,z-component stereo- X-PIV measurements in the mixing zone is displayed colour coded. Only the lower half of of a T-shaped micromixer at Re = 120. The flow is the 3D scan from z = 22 to lOOpm in the center of laminar and station a 7. The in-plane velocity the channel ofthe 800 x 200 Xm cross-section is distribution is presented as vectors and the out- shown [12] (by courtesy of Springer-Verlag).

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