Support vector machines (SVMs) are regularly used for classification of unbalanced data by weighting more heavily the error contribution from the rare class. This heuristic technique is often used to learn classifiers with high F-measure, although this particular application of SVMs has not been rigorously examined. We provide significant and new theoretical results that support this popular he...

We introduce the notion of ∧and ∨-pairs of functions on lattices as an abstraction of the notions of metric and its related entropy for probability distributions. This approach allows us to highlight the relationships that exist between various properties of metrics and entropies and opens the possibility of extending these concepts to other algebraic structures.

We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.

scott and szewczyk in technometrics, 2001, have introduced a similarity measure for twodensities f1 and f2 , by1, 21 21 1 2 2( , ), ,f fsim f ff f f f< >=< >< >wheref1, f2 f1(x, θ1)f2(x, θ2)dx.+∞−∞< >=∫sim(f1, f2) has some appropriate properties that can be suitable measures for the similarity of f1 and f2 .however, due to some restrictions on the value of parameters and the kind of densities, ...

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$int_{S}f(sigma(y)xt)dmu(t)-int_{S}f(xyt)dmu(t) = 2f(x)f(y), ;x,yin S,$$ where $S$ is a semigroup, $sigma$ is an involutive morphism of $S$, and $mu$ is a complex measure that is linear combinations of Dirac measures $(delta_{z_{i}})_{iin I}$, such that for all $iin I$, $z_{...