In mathematical morphology, connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from persistent homology and Morse theory that represents stability function. Since these two concepts seem be closely related, in this paper we examine their relationship, prove they equal n-D functions, $$n\ge 1$$ . More exactly, pairing minimu...

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, k-abelian equivale...

This paper is the third in a series of papers studying equivalence classes of fuzzy subgroups of a given group under a suitable equivalence relation. We introduce the notion of a pinned flag in order to study the operations sum, intersection and union, and their behavior with respect to the equivalence. Further, we investigate the extent to which a homomorphism preserves the equivalence. Whenev...

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every Σ1 equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence...

We present some applications of Popa’s Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, we show that the universal countable Borel equivalence relation E∞ is not essentially free.