Minimum path

The minimum path has been defined as the shortest connection between two grid nodes obtained by walking on the molecular surface . When coupled with the corresponding euclidean distance it provides a novel descriptor derived from each node pair. 

Searching the minimum paths over the entire set of node pairs yields a huge amount of path-distance couples, stored into a path-distance matrix. Its elements count the frequency of the corresponding path-distance pairs therefore, the matrix expresses the probability factor for the existence of t vo points on the molecular surface at a specific distance and path. 

Figure 5.2. GRID positive MIF (a) and path—distance frequency distribution (b) for the methotrexate. Two nodes are used as an example their euclidean distance and minimum path are highlighted with green lines (a) and the point representing that node pair is indicated by P (b).

The minimum paths provide a molecular shape description, coded as a two- vay matrix table. The matrix may then be unfolded into a single array, which describes the molecular shape, as illustrated in Fig. 5.3. 

Only nodes from different chemical groups are connected. Although the minimum paths are calculated by connecting only nodes from negative MIF, the walk still lies on the molecular surface (blue surface on Fig. 5.4) as it does for positive MIF. 

From the detour matrix and the distance matrix, a combined matrix, called detour/ distance matrix A/D (or maximum/minimum path matrix), is defined as [Ivanciuc and Balaban, 1994b] ... 

The maximum/minimum path sum of the i th vertex, denoted by MmPVS, is a local vertex invariant defined as the sum of the lengths of the longest and shortest paths between vertex v, and any other vertex in the molecular graph. It is calculated as the sum of elements over the / th row and / th column in the A/D matrix, or, alternatively, as the sum of the - vertex distance degree o, calculated on the distance matrix D and the maximum path sum MPVS, of the / th vertex calculated on the detour matrix A ... 

Detour-distance combined matrix and distance/detour quotient matrix for the H-depleted molecular graph of ethylbenzene. MmPVS, is the maximum-minimum path sum of the /th vertex VS, and CSj indicate the matrix row and column sums, respectively, w and W are the detour and Wiener index, respectively wAWand D/A are the detour-Wiener combined index and the D/A index, respectively. 

One clue to the appropriate grouping of coordinates are the minimum paths" of the other coordinates along the minimum potentials If the primed... 

The bounds on system reliability using the concepts of minimum paths and cuts can be improved. Further details and derivations for coherent systems can also be found in Barlow and Proschan 1975. 

By inspection of a layout of the shield structure, a danger point is. selected and a line along the path of minimum attenuation determined. This line is generally inadirection from the outer to the inner mouth of the duct. The intensity contribution of the surface element at the source/of the minimum, path is calculated, as well as the contribution of all neighboring surface elements which can be "seen through the effectively thinner j>ortion of the shield. These neighboring contributions usually fall off very rapidly as elements further from the minimum line are considered. 

Assume that a signal-to-noise ratio of 50 has been achieved in measuring the fringe pattern of a Michelson interferometer with one continuously moving mirror. Estimate the minimum path length AL that the mirror has to travel in order to reach an accuracy of 10 " nm in the measurement of a laser wavelength at A, = 600 nm. 

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