Let A; B be two diagonal endomorphisms of the d-dimensional torus with corresponding eigenvalues relatively prime. We show that for any A-invariant ergodic measure , there exists a projection onto a torus T r of dimension r dim , that maps-almost every B-orbit to a uniformly distributed sequence in T r. As a corollary we obtain that the Hausdorr dimension of any bi-invariant measure, as well as...

The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...

This paper presents a new similarity measure to be used for general tasks including supervised learning, which is represented by the K-nearest neighbor classifier (KNN). The proposed similarity measure is invariant to large differences in some dimensions in the feature space. The proposed metric is proved mathematically to be a metric. To test its viability for different applications, the KNN u...

In this paper, we give an extension of the Wendel's theorem on KPC-hypergroups. We also show that every translation invariant mapping is corresponding with a unique positive measure on the KPC-hypergroup.