855 KB0 KB0 KB2013Kališnik Verovšek, SaraRepovš, DušanSmrekar, Jaka90 str., 30 cmCOBISSID_HOST:16756057PID:https://repozitorij.uni-lj.si/IzpisGradiva.php?id=95849URN:URN:NBN:SI:doc-J892ERK1enS. Kališnikvisokošolska delaAleksandrova dualnostAlexander dualitycikcak vztrajnostcikcak vztrajnost za nivojnicelevelset zigzag persistenceparametrized homologyparametrizirana homologijapersistent homologyvztrajna homologijazigzag persistencePersistent homology and duality| doctoral thesis|An important problem with sensor networks is that they do not provide information about the regions that are not covered by their sensors. If the sensors in a network are static, then the Alexander Duality Theorem from classic algebraic topology is sufficient to determine the coverage of a network. However, in many networks the nodes change position with time. In the case of dynamic sensor networks, we consider the covered and uncovered regions as parametrized spaces with respect to time. Parametrized homology is a variant of zigzag persistent homology that measures how the homology of the levelsets of the space changes as we vary the parameter. We present a few theorems that extend different versions of classical Alexander Duality theorem to the setting of parametrized homology theories. This approach sheds light on the practical problem of 'wandering' loss of coverage within dynamic sensor networksEden izmed večjih problemov pri preučevanju senzorskih omrežij je, da nudijo le informacijo o področju, ki ga senzorji pokrivajo. V statičnih senzorskih omrežjih klasična Aleksandrova dualnost zadošča kot kriterij za pokritost, ampak v mnogo omrežjih se položaj senzorjev spreminja s časom in ta izrek ni dovolj. V primeru dinamičnih senzorskih omrežij sta območji pokritosti in nepokritosti parametrizirana prostora glede na čas. Parametrizirana homologijaje različica cikcak vztrajne homologije, ki meri, kako se homologijanivojnic prostora spreminja, če spreminjamo parameter. V disertaciji predstavimo parametrizirane ekvivalente nekaj različic klasične Aleksandrove dualnosti. Parametrizirana Aleksandrova dualnost nam tudi pomaga pri razumevanju 'problema vsiljivca'TEXTvisokošolska delatheses and dissertationsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza v Ljubljani, Fakulteta za matematiko in fiziko