Measures of distance or how data points are positioned relative to each other fundamental in pattern recognition. The concept depth measures deep an arbitrary point is a dataset, and interesting this regard. However, while has received lot attention the statistical literature, its application within recognition still limited. To increase applicability recognition, we address well-known computat...

Extending recent investigations on the structure of Tukey types of ultrafilters on P(ω) to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras. These include the classes of free Boolean algebras, superatomic Boolean algebras generated by almost disjoint families, interval algebras, tree algebras, and pseudo-tree algebras. W...

In [1], it was shown that the generic ultrafilter G2 forced by P(ω × ω)/(Fin ⊗ Fin) is neither maximum nor mimumum in the Tukey order of ultrafilters, but left open where exactly in the Tukey order it lies. We prove that G2 is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each k ≥ 2, the collection of all nonprincipal ultrafilters Tukey reducible to...

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This extends the Pudlak-Rödl Theorem canonizing equivalence relatio...

The generic ultrafilter G2 forced by P(ω × ω)/(Fin ⊗ Fin) was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove that G2 is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each k ≥ 2, the collection of all nonprincipal ultrafilters Tukey re...

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical is a natural Ellentuck space [5]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This is analogous to the Pudlak-Rödl Theorem canonizin...