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Opis: The Lelek fan L is usually constructed as a subcontinuum of the Cantor fan in such a way that the set of the end-points of L is dense in L. It easily follows that the Lelek fan is embeddable into the Cantor fan. It is also a well-known fact that the Cantor fan is embeddable into the Lelek fan, but this is less obvious. When proving this, one usually uses the well-known result by Dijkstra and van Mill that the Cantor set is embeddable into the complete Erdős space, and the well-known fact by Kawamura, Oversteegen, and Tymchatyn that the set of end-points of the Lelek fan is homeomorphic to the complete Erdős space. Then, the subcontinuum of the Lelek fan that is induced by the embedded Cantor set into the set of end-points of the Lelek fan, is a Cantor fan. In our paper, we give an alternative straightforward embedding of a Cantor fan into the Lelek fan. We do not use the fact that the Cantor set is embeddable into the complete Erdős space and that it is homeomorphic to the set of end-points of the Lelek fan. Instead, we use our recent techniques of Mahavier products of closed relations to produce an embedding of the Cantor fan into the Lelek fan. Since the Cantor fan is universal for the family of all smooth fans, it follows that also the Lelek fan is universal for smooth fans.
Ključne besede: closed relations, Mahavier products, fans, Cantor fans, Lelek fans
Objavljeno v DKUM: 04.03.2025; Ogledov: 0; Prenosov: 5
Celotno besedilo (659,88 KB)
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Opis: Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the \emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, we introduce different variants of a span of a given connected graph. All the variants model the maximum safety distance kept by two players in a graph traversal, where the players may only move with accordance to a specific set of rules, and their goal: visit either all vertices, or all edges. For each variant, we show that the solution can be obtained by considering only connected subgraphs of a graph product and the projections to the factors. We characterise graphs in which it is impossible to keep a positive safety distance at all moments in time. Finally, we present a polynomial time algorithm that determines the chosen span variant of a given graph.
Ključne besede: strong span of a graph, direct span of a graph, Cartesian span of a graph, safety distance
Objavljeno v DKUM: 07.06.2024; Ogledov: 98; Prenosov: 10
Celotno besedilo (213,47 KB)
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Opis: V magistrskem delu posplošimo tranzitivne dinamične sisteme do tranzitivnih di- namičnih sistemov z zaprtimi relacijami. Predstavimo lastnosti DO-tranzitivnih in tranzitivnih topoloških dinamičnih sistemov z zaprtimi relacijami ter njihove povezave z DO-tranzitivnimi in tranzitivnimi topološkimi dinamičnimi sistemi.
Ključne besede: metrični prostor, kompaktni metrični prostor, zaprte relacije, Ma- havierjev produkt, dinamični sistem, CR-dinamični sistem, tranzitivna točka, DO- tranzitivni CR-dinamični sistem, tranzitivni CR-dinamični sistem.
Objavljeno v DKUM: 05.09.2023; Ogledov: 445; Prenosov: 50
Celotno besedilo (871,08 KB)

Opis: We give new results about the set of all medians, the set of all first quartiles and the set of all third quartiles of a finite dataset. We also give new and interesting results about rela- tionships between these sets. We also use these results to provide an elementary correctness proof of the Langford's doubling method.
Ključne besede: statistic, probability, median, first quartile, third quartile, median set, first quartile set, third quartile set
Objavljeno v DKUM: 19.09.2022; Ogledov: 538; Prenosov: 15
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Opis: V magistrskem delu vpeljemo funkcije, ki ohranjajo konvergenco vrst. To so funkcije \newline $f:\RR \longrightarrow \RR$, za katere velja, da iz konvergence vrste $\sum\limits_{n=1}^{\infty}a_n$ sledi konvergenca vrste $\sum\limits_{n=1}^{\infty}f(a_n)$. Glavni cilj magistrske naloge je podati in dokazati enostavno karakterizacijo takih funkcij. Najprej bomo predstavili osnovne pojme iz Analize in Algebre, ki so potrebni za razumevanje rezultatov, ki jih opisujemo v magistrskem delu. V drugem poglavju podamo in dokažemo glavni izrek magistrskega dela o karakterizaciji funkcij, ki ohranjajo konvergenco vrst. V zadnjem poglavju preučujemo posplošitve takih funkcij. Natančneje, definiramo lastnost, kot so biti $(c, ac)$, $(ac,c)$, $(acp)$ funkcija, in dokažemo zanimive rezultate za opisane razrede funkcij.
Ključne besede: algebra, absolutna vrednost, funkcija, konvergentnost, limita, vrsta, vektorski prostor, zaporedje
Objavljeno v DKUM: 03.05.2022; Ogledov: 1043; Prenosov: 207
Celotno besedilo (434,67 KB)