Browsing by Author "Zarichnyi, M."
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Item Frechet distance on the set of compact trees(2015) Lozinska, O.; Savchenko, А.; Zarichnyi, M.; Савченко, О. Г.We introduce a counterpart of the Fr echet distance for the rooted trees in a metric space. Some properties and possible generalizations of this distance are discussed.Item FUZZY METRIZATION OF THE SPACES OF IDEMPOTENT MEASURES(2020) Brydun, V.; Savchenko, A.; Zarichnyi, M.; Савченко, О. Г.In idempotent mathematics, the idempotent measures (Maslov measures) are counterparts of the probability measures. We provide a fuzzy metrization of the set of idempotent measures on fuzzy metric spaces. We prove that this fuzzy metrization determines a monad in the category of fuzzy metric spaces and non-expanding maps.Item Fuzzy ultrametrics on the setofprobability measures(2009) Savchenko, A.; Zarichnyi, M.; Савченко, О. Г.Weintroduceafuzzyultrametriconthesetofprobabilitymeasureswithcompactsupport defined on a fuzzy metric space. The construction is a counterpart, in the realm of fuzzy ultrametricspaces,oftheconstructionduetoVinkandRuttenofanultrametricontheset ofprobabilitymeasureswithcompactsupportsonanultrametricspace. It is proved that the set of probability measures with finite supports is dense in the naturaltopologygeneratedbythedefinedfuzzyultrametric.Item Hyperspaces and spaces of probability measures on R-trees(2014) Lozinska, O.; Savchenko, A.; Zarichnyi, M.; Савченко, О. Г.We prove that the "sliced" hyperspaces and spaces of probability measures of the rooted R-trees are also rooted R-trees.Item Metrization of free groups on ultrametric spaces(2010) Savchenko, A.; Zarichnyi, M.; Савченко, О. Г.We consider ultrametrizations of free topological groups of ultrametric spaces. A construction is defined that determines a functor in the category UMET1 of ultrametric spaces of diameter ≤ 1 and nonexpanding maps. This functor is the functorial part of a monad in UMET1 and we provide a characterization of the category of its algebrasItem Probability measure monad on the category of fuzzy ultrametric spaces(2011) Savchenko, А.; Zarichnyi, M.; Савченко, О. Г.It is proved that the probability measure functor comprises a monad on the category of fuzzy ultrametric spaces and nonexpanding maps. It is also proved that the G-symmetric power functor admits an extension on the Kleisli category of this monad (i.e. the category of fuzzy ultrametric spaces and nonexpanding measure-valued maps).Item STRONG TOPOLOGY ON THE SET OF PERSISTENCE DIAGRAMS(2019) Zarichnyi, M.; Savchenko, A.; Kiosak, V.; Савченко, О. Г.We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described. Also, we prove that the space of persistence diagrams with the bottleneck metric has infinite asymptotic dimension in the sense of Gromov.Item Triples of infinite iterations of hyperspaces of maxнplus compact convex sets(2016) Savchenko, А.; Zarichnyi, M.; Савченко, О. Г.Geometrу of the innite iterated hуperspace of compact max-plus convex sets, their completions and compactications is investigated.