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1.
INVERZNE LIMITE Z ENOLIČNIMI IN VEČLIČNIMI VEZNIMI PRESLIKAVAMI
Matej Merhar, 2009, undergraduate thesis

Abstract: V diplomskem delu bomo najprej predstavili osnovne primere kontinuumov. Nato bomo predstavili inverzne limite inverznih zaporedij kompaktnih metričnih prostorov in enoličnih zveznih veznih funkcij ter dokazali njihove osnovne lastnosti. Definirali bomo tudi inverzne limite inverznih zaporedij kompaktnih metričnih prostorov in navzgor polzveznih večličnih veznih funkcij in si ogledali nekatere njihove lastnosti.
Keywords: Inverzno zaporedje, Inverzna limita, Navzgor polzvezna funkcija, Kontinuum
Published: 11.05.2009; Views: 2405; Downloads: 251
.pdf Full text (885,34 KB)

2.
Limite inverznih limit
Matej Merhar, 2013, doctoral dissertation

Abstract: V doktorski disertaciji se obravnava vprašanje ali iz konvergence grafov navzgor polzveznih veznih funkcij sledi konvergenca ustreznih pripadajočih inverznih limit za konstantna inverzna zaporedja kompaktnih metričnih prostorov. V uvodnem delu se vpeljejo osnovni pojmi kot so navzgor polzvezne funkcije, inverzna zaporedja in inverzne limite. V osrednjem delu se na konkretnih primerih pokaže, da je odgovor na zgoraj zastavljeno vprašanje v splošnem negativen in v obliki izrekov poda dodatne pogoje za vezne funkcije, ki zagotavljajo, da iz konvergence njihovih grafov sledi konvergenca pripadajočih inverznih limit. Med drugim se dokaže, da če so vezne funkcije surjektivne in funkcija h kateri njihovi grafi konvergirajo enolična, tedaj tudi zaporedje pripadajočih inverznih limit konvergira. Te pogoje se v nadaljevanju nekoliko omili in posploši na splošna inverzna zaporedja. Predstavi se tudi uporaba navedenih rezultatov za konstrukcijo poti v hiperprostorih. V zaključnem poglavju se navede še nekatera odprta vprašanja, ki odpirajo možnost nadaljnjega raziskovanja.
Keywords: kontinuum, hiperprostor, limita, inverzna limita, zvezna preslikava, navzgor polzvezna preslikava, pot
Published: 08.10.2013; Views: 1333; Downloads: 77
.pdf Full text (305,50 KB)

3.
Paths through inverse limits
Iztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2009

Abstract: In [I.Banič, M. Črepnjak, M. Merhar, U. Milutinović, Limits of inverse limits, Topology Appl. 157 (2010) 439-450] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions ▫$f_n: X rightarrow 2^X$▫ converges to the graph of a continuous single-valued function ▫$f: X rightarrow X$▫, then the sequence of corresponding inverse limits obtained from ▫$f_n$▫ converges to the inverse limit obtained from ▫$f$▫. In this paper a more general result is presented in which surjectivity of ▫$f_n$▫ is not required. Also, the result is generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.
Keywords: matematika, topologija, kontinuumi, limite, inverzne limite, navzgor polzvezne večlične funkcije, poti, loki, mathematics, topology, continua, limits, inverse limits, upper semi-continuous set-valued functions, paths, arcs
Published: 10.07.2015; Views: 527; Downloads: 36
URL Link to full text

4.
Towards the complete classification of tent maps inverse limits
Iztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2010

Abstract: We study tent map inverse limits, i.e. inverse limits of inverse sequences of unit segments ▫$I$▫ with a tent map being the only bonding function. As the main result we identify an infinite family of curves in ▫$I^2$▫ such that if top points of graphs of tent maps belong to the same curve, the corresponding inverse limits are homeomorphic, and if they belong to different curves, the inverse limits are non-homeomorphic. The inverse limits corresponding to certain families of top points are explicitly determined, and certain properties of the inverse limit are proved in the case of ▫$(0,1)$▫ as the top point.
Keywords: matematika, topologija, kontinuumi, inverzne limite, mathematics, topology, continua, inverse limits, tent maps, Knaster continua
Published: 10.07.2015; Views: 383; Downloads: 16
URL Link to full text

5.
Tent maps inverse limits and open problems
Matevž Črepnjak, Iztok Banič, Matej Merhar, Uroš Milutinović, 2011, published scientific conference contribution abstract

Keywords: mathematics, topology, continua, inverse limits, tent maps, Knaster continua
Published: 07.06.2012; Views: 669; Downloads: 24
URL Link to full text

6.
Limits of inverse limits and applications
Matej Merhar, Iztok Banič, Matevž Črepnjak, Uroš Milutinović, 2011, published scientific conference contribution abstract

Keywords: mathematics, topology, continua, inverse limits, upper semi-continuous set-valued functions
Published: 07.06.2012; Views: 694; Downloads: 14
URL Link to full text

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Paths through inverse limits
Iztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović, 2011, original scientific article

Abstract: In Banič, Črepnjak, Merhar and Milutinović (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions ▫$f_n : X to 2^X$▫ converges to the graph of a continuous single-valued function ▫$f : X to X$▫, then the sequence of corresponding inverse limits obtained from ▫$f_n$▫ converges to the inverse limit obtained from ▫$f$▫. In this paper a more general result is presented in which surjectivity of ▫$f_n$▫ is not required. The result is also generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications, it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.
Keywords: mathematics, topology, continua, limits, inverse limits, upper semi-continuous set-valued functions, paths, arcs
Published: 07.06.2012; Views: 877; Downloads: 56
URL Link to full text

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