In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [2]. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddl...

In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation defined in [3]. This relation is deeper than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci of diffe...

We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of principal groupoids such a notion coincide with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids ...

We study the equivalence relation on states of labelled transition systems of satisfying the same formulas in Computation Tree Logic without the next state modality (CTL−X). This relation is obtained by De Nicola & Vaandrager by translating labelled transition systems to Kripke structures, while lifting the totality restriction on the latter. They characterised it as divergence sensitive branch...

We use a relational characterization of binary direct sums to model sequences within the relation-algebraic manipulation and prototyping system RelView in a simple way. As an application we formally derive a RelView program for computing equivalence classes of an equivalence relation, where we combine relation-algebraic calculations with the so-called Dijkstra-Gries program development method. ...

Open Educational Resources are fundamental in Information Technology and Communications (ICT) enhanced learning, due to their capability to be reused in different contexts. However, a current problem is how to decide, given a set of OER which are the best suited to specific requirements. One solution is having criteria to assess values to the properties of the OER. Existing metrics for valuing ...

I. Basic concepts 1. Recursive enumerability and recursiveness. We consider infinite sequences of non-negative integers, free from repetitions. A familiar equivalence relation between such sequences is that based on sets: two sequences are equivalent when they enumerate the same set. The equivalence classes under this relation, with the necessary operations introduced, form a system isomorphic ...

Graph bundles generalize the notion of covering graphs and graph products. In Imrich et al. (Discrete Math. 167=168 (1998) 393) authors constructed an algorithm that 5nds a presentation as a nontrivial cartesian graph bundle for all graphs that are cartesian graph bundles over triangle-free simple base using the relation ∗ having the square property. An equivalence relation R on the edge set of...

In this paper we will introduce a certain collection of fuzzy subsets of a nonempty set X. This certain collection gives raise to a useful new fuzzy partition. This new fuzzy partition gives us a way of generating a fuzzy equivalence relation as in the ordinary case; any partition generates an equivalence relation. Mathematics Subject Classification: 03E72, 46S40, 26E50

We give an elementary proof that there are two topological generators for the full group of every aperiodic hyperfinite probability measure preserving Borel equivalence relation. Our proof explicitly constructs topological generators for the orbit equivalence relation of the irrational rotation of the circle, and then appeals to Dye’s theorem and a Baire category argument to conclude the genera...