Three-dimensional partition

M. W. Nansteel and R. Greif, An Investigation of Natural Convection in Enclosures With TVre>- and Three-Dimensional Partitions, Int. J. Heat Mass Transfer (27/4) 561-571,1984. 

The sum is from n = 1 to oo so the limits of integration are from 0 to oo in three dimensions. We may consider the molecule to reside in a rectangular box with dimensions, a, b, and c. The three-dimensional partition function then becomes... 

K and G M Crippen 1986. Atomic Physicochemical Parameters for Three-dimensional Struc-directed Quantitative Structure-Activity Relationships. I. Partition Coefficients as a Measure ydrophobicity. Journal of Computational Chemistry 7 565-577. 

Lignin is a high-molecular, mainly three-dimensional aromatic substance, that, in contrast to polysaccharides, can be hydrolyzed only in small partitions [16] ... 

Figure 5. Simplified schematic of the 2-D array of pixels in a focal plane array. The thin wafer of light sensitive material is partitioned into a two-dimensional array of pixels that collect the electric charge produced by the light. Each pixel is a three-dimensional volume that is defined by electric fields within the light sensitive material.

Chen X, Rusinko A, Young SS. Recursive partitioning analysis of a large structure-activity data set using three-dimensional descriptors. J Chem Inf Comput Sci 1998 38 1054-62. 

Ghose, A. K., Crippen, G. M. Atomic physicochemical parameters for three-dimensional strucmre-directed quantitative structure-activity relationships. I. partition coefficients as a measure of hydrophobidty. J. Comp. Chem. 1986, 7, 565-577. 

Waller, C. L. A three dimensional technique for the calculation of octanol-water partition coeffidents. Quant. Struct.-Act. Relat. 1994, 13,172-176. 

If the electron density partitioning results in subsystems without boundaries and with convergence properties which closely resemble the convergence properties of the complete system, then it is possible to avoid one of the conditions of the Holographic Electron Density Fragment Theorem , by generating fuzzy electron density fragments which do not have boundaries themselves, but then the actual subsystems considered cannot be confined to any finite domain D of the ordinary three-dimensional space E3. 

Obviously, the above derivation can be repeated for the other two Cartesian directions. As the energies are additive, the partition function for the three-dimensional translation of the molecule can be written as a product, viz. 

For three-dimensional expansion in the air-filled section with the initial air volume equal to at least 200 times that of the initial water volume and with no partitions, baffles, or other types of compartments present, the air shock pressure is smaller than the final quasi-steady equilibrium pressure buildup. It can therefore be rationalized that the design of containment for a reactor is governed by equilibrium pressure only. 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

Results in Table 31.3 indicate that the combination of TS and TC descriptors resulted in a highly predictive RR model = 0.895) the addition of three-dimensional and QC indices to the set of independent variables did not result in significant improvement in model quality. It may be noted that we have observed such results for various other physicochemical and biological properties including mutagenicity [25,54], boiling point [55], blood air partition coefficient [37], tissue air partition coefficient [46], etc. [24,30,45,56]. Only in limited cases, e.g., halocarbon toxicity [12], the addition of QC indices after TS and TC parameters resulted in significant improvement in QSAR model quality. 

In Equation 4.71 the individual qvib s have been specified qVib(v ) to indicate that these partition functions depend on the normal mode frequencies. It is interesting to note that the partition function for translation, which is usually considered in terms of the problem of the particle in a three dimensional rectangular box, is, itself a product of three partition functions one for motion in the x dimension, one for y, etc. 

In chemoinformatics research, partitioning algorithms are applied in diversity analysis of large compound libraries, subset selection, or the search for molecules with specific activity (1-4). Widely used partitioning methods include cell-based partitioning in low-dimensional chemical spaces (1,3) and decision tree methods, in particular, recursive partitioning (RP) (5-7). Partitioning in low-dimensional chemical spaces is based on various dimension reduction methods (4,8) and often permits simplified three-dimensional representation of... 

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