Parameter mapping

By adding or subtracting parameter maps (see Figure 6.3 in Section 6.1.2) additional information can be obtained. They show trends in the parameters and are used to optimise reserves development and management. 

In Fig. 2 conditions (11) and (12) are plotted on a parameter map together with the parameter values reported for NaZn13 by Zintl Hauke, those reported for KCd13 by Ketelaar, and those determined for NaZn13 in the present investigation. The uncertainties are indicated by the radii of circles drawn around the points determined by the parameter values since Zintl Hauke reported no uncertainty value, the uncertainty reported by Ketelaar (0-003) was assumed. It is seen that the parameter values obtained in the present work lie between those of Zintl Hauke and those of Ketelaar, and that they differ considerably from the values required either by a regular icosahedron or by a regular snub cube. 

Phase encoding of higher order terms (locally time dependent accelerations) is feasible in principle with pulse trains such as the example shown in Figure 2.9.4(d). However, the relatively long acquisition times of NMR parameter maps conflict with snapshot records that would be needed for such maps. In any case, in Ref. [27] fast imaging techniques are described and discussed that could be employed for moderate versions of snapshot experiments. 

Steady state equations fXr the adiabatic case corresponding to (1) through (4) were solved by the parameter mapping technique combined with the Newton-Fox shooting algorithm. The steady state nonadiabatic problems were solved by the finite-difference approach. 

Griine M, Pillekamp F, Schwindt W, Hoehn M (1999) Gradient echo time dependence and quantitative parameter maps for somatosensory activation in rats at 7 T. Magn Reson Med 42 118-126... 

For both types of FIDCO surfaces, the usual Shape Group method [2] of electron density shape analysis is applicable. The additional formal domain boundaries AD i(Ga b(3)) and AD i(GA(B)(a)) introduce one additional index -1, which can be treated the same way as relative curvature indices. The one-dimensional homology groups obtained by truncations using all possible index combinations are the shape groups of FIDCO surfaces. The (a,b)-parameter maps and shape codes are generated the same way as for complete molecules [2],... 

Such a decreasing sequence is associated with each point of a grid on the (a,b) parameter map. The grid itself can be regarded as a matrix with the sequences of Betti numbers as elements. This matrix is a shape code for the molecule that can be used for storing shape information in molecular data banks and can be retrieved for shape similarity assessment by numerical methods. 

This numerical code, representing shape information for the given (a,b) parameter pair, is assigned to the associated (a,b) point of the (a,b) parameter map. 

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). 

As an example, the shape matrix s(0.0l,0) of the b=0 shape domain partitionings of the allyl alcohol MIDCO G(O.OI) shown in Figure 5.6 belongs to the (0.01,0) point of the parameter map (a,b). If the index ordering of the various D i domains follows the order of decreasing size of their surface area, then the shape matrix s(0.01,0) has the form given by Equation (5.12) ... 

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. 

More detailed shape comparison is possible if the decoded elements of the two vectors C(Mi) and C(M2) are compared directly. For example, by taking the number of matches along the diagonals and within the off-diagonal upper triangles of the two shape matrices s(a,b,Mi) and s(a,b,M2), divided by n(n-i-l)/2, where n is the dimension of the larger of the two matrices, an elementary similarity measure s(a,b) is obtained, characteristic to the point (a,b) of the parameter map. Clearly,... 

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... 

Whereas the curvature types for truncation are complementary, the above two (a,b) maps cannot yet be compared directly, since in a direct comparison of these maps, identical, and not complementary, a and b values occur for the two molecules. However, the complementarity of density thresholds and curvatures can be taken into account by a simple transformation by inverting the (a,b) parameter map of molecule M2 centrally with respect to the point (ao,0), and by comparing the centrally inverted (a,b) map of M2 to the original (a,b) map of M ]. This transformation ensures that domain types, density thresholds, and curvature parameters are matched properly, as required by the pairing scheme (6.77) - (6.79). For example, the locally convex domains of MIDCO G(ao-a, M ) relative to the reference curvature b are tested for shape complementarity against the locally concave domains of MIDCO G(ao+a, M2) relative to a reference curvature - b. 

This Centrally Inverted Map Method (CIMM) of molecular shape complementarity analysis allows one to use the techniques of similarity measures. In fact, the problem of shape complementarity is converted into a problem of similarity between the original (a,b) parameter map of shape groups HP (a,b) of molecule M] and the centrally inverted (a,b) parameter map of the complementary HP2-ii(a,b) shape groups of molecule M2. 

For most practical applications, CIMM is used within the framework of local measures. These measures are based on local shape matrices or on the shape groups of local moieties, defined either by the density domain approach mentioned earlier, or by alternative conditions, such as the simple truncation condition replacing the "remainder" of the molecule by a generic domain [192], For proper complementarity, identity or close similarity of the patterns of the matched domains is an advantage, hence the parts Cl HM)) and Cl KM2) of the corresponding local shape codes are compared directly. For shape complementarity only the specified density range [bq - Aa, Bq + Aa] and a specified curvature range of the (a,b) parameter maps is considered. A local shape complementarity measure, denoted by... 

Isopotential contours of the composite nuclear potentials (NUPCO s, see Chapter 4), provide an inexpensive, approximate shape representation that can be computed easily even for very large molecules. Although NUPCO s only approximate the MIDCO s of molecules, the family of NUPCO s of a molecule describes an important molecular property that has a major effect on the actual molecular shape. Consequently, NUPCO s can be used for direct comparisons between molecules, and similar NUPCO s are likely to be associated with similar molecular shapes. All the shape analysis techniques originally developed for MIDCO s are equally applicable to NUPCO s. The shape groups, the (a,b) parameter maps [where a is the nuclear potential threshold of a NUPCO G(a)], the shape matrices, shape codes, and the shape globe invariance maps of NUPCO s of molecules can serve as inexpensive methods for the detection and evaluation of a particular aspect of molecular similarity. 

Shape codes [43,109,196,351,408]. The simplest topological shape codes derived from the shape group approach are the (a,b) parameter maps, where a is the isodensity contour value and b is a reference curvature against which the molecular contour surface is compared. Alternative shape codes and local shape codes are derived from shape matrices and the Density Domain Approach to functional groups [262], as well as from Shape Globe Invariance Maps (SGIM). 

Rowe, R.C. The prediction of compatibihty/incompatibility in blends of ethyl cellulose with hydroxypropyl methylcellulose or hydroxypropyl cellulose using 2-dimen-sional solubility parameter maps. J. Pharm. Pharmacol. 1986, 38, 214-215. 

Khinast and Luss presented a rigorous method for constructing parameter maps of the RFR which contain different bifurcation diagrams. Using the... 

Shape complementarity, as expressed by the pairings (b, b) and (a0-a, a0 + a ), can be represented using a single condition based on the (a,b) parameter maps. With respect to the most informative one-dimensional shape groups for both molecules, we shall consider the shape group with reference to the /t = 2 truncation for... 

Evidently, in these two (a,b) maps, the curvature types for the truncation in the two fragments are complementary. Unfortunately, the above two (a,b) maps cannot be compared directly since if a direct comparison is made by simply overlaying these maps, identical and not complementary, a and b values occur for the two molecular fragments. Nevertheless, a simple transformation of either one of these two maps ensures complementarity of the density threshold and reference curvature values all one needs to do is to carry out a central inversion of one of these maps. Such a central inversion of the (a,b) parameter map of molecular fragment F2 with respect to the point (u0,0) of the map ensures a proper match between complementary parameter values. Consequently, direct comparison of the original (a,b) map of fragment F, and the centrally inverted (a,b) map of fragment F2 is suitable to evaluate shape complementarity. 

---