Weighted distance

The squared generalized distance is a weighted distance of the general form ... 

It has been shown in Chapter 29 that the set of vectors of the same dimension defines a multidimensional space S in which the vectors can be represented as points (or as directed line segments). If this space is equipped with a weighted metric defined by W, it will be denoted by the symbol S. The squared weighted distance between two points representing the vectors x and y in is defined by the weighted scalar product ... 

Wiener-type index from Z weighted distance matrix (Barysz matrix) Wiener-type index from mass weighted distance matrix Wiener-type index from van der Waals weighted distance matrix Wiener-type index from electronegativity weighted distance matrix Wiener-type index from polarizability weighted distance matrix Balaban J index... 

Balaban-type index from polarizability weighted distance matrix connectivity index chi-0... 

A continuous factor is a factor that can take on any value within a given domain. Similarly, a continuous response is a response that can take on any value within a given range. Examples of continuous factors are pressure, volume, weight, distance, time, current, flow rate, and reagent concentration. Examples of continuous responses are yield, profit, efficiency, effectiveness, impurity concentration, sensitivity, selectivity, and rate. 

W AXS-pattern of a polymer melt if these Boltzmann-weighted distance fluctuations are not accounted for. They characterize a relevant feature of polymer melt structures. We treat this in terms of the Rotational Isomeric State Approximation (RISA) discussed in the next section. 

In order for Equations (2.153) and (2.154) to yield meaningful results, the fragments Aj and A2 have to be aligned. This can be done by a quaternion rotation chosen so that the sum of the squares of the mass-weighted distances between the nuclei one wants to superpose is minimal [12,60],... 

Explosive (30mm in diameter, lOOg in weight) Distance between j packages in center (cm) Testing results... 

Schuitz weighted distance matrices -> weighted matrices... 

The X matrix [Hall, 1990] is the most popular weighted distance matrix, based on the path contributions arising in constructing -> connectivity indices. Using the vertex degree 8 of the atoms, each path pij between the vertices v, and vj is weighted by the path connectivity defined as ... 

TTie Barysz distance matrix is a weighted distance matrix accounting simultaneously for the presence of heteroatoms and multiple bonds in the molecule it is defined as [Barysz et al, 1983a] ... 

Moreover, the covalent radius-weighted distance matrix was also defined [Ivanciuc et al, 1998a] as ... 

The vertex weights are collected into a diagonal matrix W which is used as a premultiplier of the distance matrix D in order to give a vertex weighted distance matrix D as ... 

The third weight is related to vertex valence and takes into account the number of bonds incident to a vertex and its pairs of unshared electrons. Each present electron pair counts one, each missing electron pair contributes -1 to the total valence of heteroatoms, and in the case of free radicals the unique electron present in the outer valence shell contributes half a bond to the total valence. The valence weights are collected into a diagonal matrix W used to give a valence weighted distance matrix D ... 

Analogously, edge-valence and vertex-valence weighted distance matrices can be calculated. 

The most common topological indices can be derived from the above defined weighted distance matrices, e.g. the Schultz indices were calculated as well as the - MTV index, -> determinant, permanent, - product of row sums and the -> hafnian. 

One weighted distance measure which does occur frequently in the scientific literature is the Mahalanobis distance ... 

Calculation of DAI indices for4-methyl-3-pentanol. VS,- indicates the matrix row sums D(r ) is the bond length-weighted distance matrix. 

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