Triangular inequality

Finally, by appeal to the triangular inequality from (38) we obtain (37). Thus, Theorem 5 is completely proved. 

These vanish unless // + /, + 1 equals an even integer and the triangular inequalities are satisfied,... 

The operator Tn + is called the transition operator from the layer j to the layer n + 1, while the operator o refers to the resolving operator. The triangular inequality yields... 

Distance-based metrics quantify the diversity of a set of compounds as a function of their pairwise (dis)similarities in a descriptor space. It is important to mention that distance coefficients are analogous to distances in multidimensional geometric space, although they are usually not equivalent to such distances. For a distance coefficient to be described as a metric, it must possess the following four properties (1) Distance values must be nonzero and the distance from an object to itself must be zero. (2) Distance values must be symmetric. (3) Distance values must obey the triangular inequality. (4) Distances between nonidentical objects must be greater than zero. A coefficient containing only the first three properties is dubbed a pseudometric, and one without the third property is a nonmetric. 

With respect to metric properties, the Tanimoto coefficient obeys all four properties if dichotomous variables are used. Complement does not obey the triangular inequality in the Dice coefficient. 

Substituting equation (A.6) into (A.5), we obtain (A.4). In a general case of n dimensional Euclidean space, the triangular inequality comes from the Cauchy inequality, which we will discuss below. 

We can use several other indices beside the Garbo index. One of the most interesting indices is the Euclidean distance, which constitutes a real metric, in the sense that it obeys the triangular inequality.So, a further requirement that must be fulfilled is that the MQSM matrix should not contain any inconsistencies. Inconsistency checking should always be carried out to increase the reliability of the MQSM in the similarity matrix. ... 

Distance values must obey the triangular inequality... 

The Tanimoto coefficient or Jaccard similarity coefficient is a statistic rrsed for comparing the similarity and dissimilarity of stractmes [74]. It is one of the most commonly used similarity coefficient used in chemoinformatics, because it allows rapid calculation due to its simple nature and absence of complex mathematical operators. In general, the complement of the Tanimoto coefficient does not follow the triangular inequality. The Tanimoto coefficient is calculated as follows ... 

Distance tables illustrate the mathematical properties of the distance measure distance values are positive and zero only if the same impedance spectra are compared, the matrix is symmetric and the triangular inequality is vahd for all distances. In this particular case it can be observed that not only the triangular inequality holds, but the distance from 10°C... 

It has been proven that the triangular inequality (4) follows from the three axioms. In the case of Hamming distance, the second axiom on distance allows that distance be zero even for cases that x y. Hence, as in the case of pseudo-distance, the second axiom is relaxed and replaced by the requirement ... 

The two components of J need not point in the same direction and so its magnitude J satisfies the triangular inequality, L— j specific reactive collision depends on the system. For the F -b H2 system the reduced mass fi is atypically low and the reaction cross-section in... 

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