20.500.12556/RUL-95849 Persistent homology and duality doctoral thesis Vztrajna homologija in dualnost 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 networks. Eden 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'. Alexander duality persistent homology zigzag persistence levelset zigzag persistence parametrized homology Aleksandrova dualnost vztrajna homologija cikcak vztrajnost cikcak vztrajnost za nivojnice parametrizirana homologija true false false [S. Kališnik] Angleški jezik Slovenski jezik Doktorsko delo/naloga 2017-09-22 02:53:37 2017-10-24 11:32:10 2022-08-10 06:57:49 0000-00-00 00:00:00 2013 0 Ljubljana 0 90 str. 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.14(043.3) 16756057 RAZ_Kalisnik_Verovsek_Sara_i2013.pdf RAZ_Kalisnik_Verovsek_Sara_i2013.pdf 1 794BD25A6F70BA3E0073275BD8646095 548e8dcec129a4a7dc48a79eb2516eee983d02d221d199aa3640eda5ecac4a1c 25a12c31-a1b4-11eb-a523-00155dcfd717 20.500.12556/rul/eb6643dd-e796-4088-a8c9-58bb60770ff1 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=106326 Fakulteta za matematiko in fiziko 0 0 0