Minkowski metric

Physicists should take special note of the positive definiteness. Although there are many useful examples of brackets that satisfy all but the positive dehniteness requirement (such as the Minkowski metric on spacetime in special relativity), we are concerned here with positive definite brackets. 

Eq. 5-7 appears as a special case of the so-called MINKOWSKI metrics where m still denotes the dimension of the space spanned by the m features and C is a special parameter ... 

Minkowski metric with k = 1. The shortest path from a to b walking on a co-ordinate axis. Named for the streets of Manhattan, which resemble a co-ordinate axis... 

Minkowski metric with k = 2. This metric takes into account both the shape of a pattern as well as the magnitude of the pattern... 

In summary, the model allows for two types of interactions between the mirror spaces, the weak kinematical perturbation and the adiabatic and sudden limits equivalent to Eq. (17) or Eqs. (29)-(34). The overwhelming rate of particles over antiparticles in the Universe is inferred in this picture once the particular particle state has been selected. The Minkowski metric of the special theory of relativity is represented here by a non-positive definite metric, Eq. (8), bringing about a quantum model with a complex symmetric ansatz. Although the latter permits general symmetry violations, it is nevertheless surprising that fundamental transformations between complex symmetric representations and canonical forms come out unitary. 

The descriptors used for pairwise distance measurements can be continuous, as in a physicochemical property, or binary e.g., the presence or absence of a specific substructure). For continuous chemical spaces, nearly all metrics are based on the generalized Minkowski metric given in (1), where % represents the Mi feature of the ith molecule, k is the total number of features, and r the order of the metric. 

When the descriptors represent counts or sums, the metrics in (3) and (4) have been used, where is (as it is in the generalized Minkowski metric) the value of the th descriptor in the ith compound, and K the total number of descriptors. ... 

Mapping displays 23 Matrix, confusion, 127 determinant, 212 dispersion, 82 identity, 206 inverse, 210 quadratic form, 212 singular, 211 square, 204 symmetric, 204 Matrix multiplication, 207 Mean centring, 17 Mean value, 2 Membership function, 117 Minkowski metrics, 99 Moving average, 36 Multiple correlation, 183 Multiple regression, backward elimination, 182... 

Amorim, R. C Mirkin, B. Minkowski metric, feature weighting and anomalous cluster initializeing in K-Means clustering. Pattern Recognition, 2012, 45 (3) 1061-1075. 

Tanimoto measure . The adoption of bit-encoded information led to the use of association coefficients for comparison of equal-length bit-vectors. In the case of our pattern-based similarity search, it is not the presence or absence of a feature that is being compared, but its frequency of occurrence. Discussions of the different types of similarity measures available for numerical, rather than binary, data can be found in books by Willett and Everitt. From these we chose to test the frequency Tanimoto coefficient and the Minkowski metric ... 

The dissimilarity D(PQ) between fragments P and Q is assessed using the Minkowski metric (equation 2 above), where V(P ) and V(Q ) are the Jth torsion angles for fragments P and Q, and summation is from J = 1 — Nt. Because torsion angles are circular functions, there is a phase restriction of -180< y<180, and the torsional dissimilarity measure must be expressed as ... 

We define a generalized distance in the space of all TD by using the Minkowski metric. The distance between two theoretical descriptions TD, and TDy is defined as... 

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