Blokové posloupnosti a (R)-hustota – RNDr. Ferdinánd FILIP

Anotace:

V predloˇzenej dizertaˇcnej pr´aci sa budem zaoberat? (R)-hustotou a blokov´ymi postupnost?ami vytvoren´ymi pomocou prirodzen´ych ˇc´?sel. Ciel?om tejto pr´ace je s´ustredit? v´ysledky z (R)-hustoty a v´ysledky s´uvisiace z blokov´ymi postupnost?ami a tieto doplnit? v´ysledkami z mojho v´yskumu. Doteraz zn´ame poznatky bud´u zhrnut´e najm¨e z (R)-hustoty a vzt?ahu asymptotickej hustoty. ˇDalej bud´u vyˇsetrovan´e distribuˇcn´e funkcie blokov´ych postupnost´? prirodzen´ych ˇc´?sel. T´ato pr´aca hlavn´e nov´e v´ysledky obsahuje v nasleduj´ucich tematick´ych okruchoch: ? Vzt?ah (R)-hust´ych mnoˇz´?n a disperzie: Urˇcili sme najlepˇsie horn´e odhady pre disperziu (R)-hust´ych mnoˇz´?n. Uk´azali sme , ˇze mnoˇzina hodn?ot disperyie je cel´y interval (s´uvisl´a mnoˇzina). Podobn´e tvrdenia sme dok´azali aj pre disperzie ˇspeci´alnych, (R)-hust´ych mnoˇz´?n. ? S´uvis dolnej, hornej asymptotickej hustoty a disperzie: Urˇcili sme najlepˇs´? moˇzn´y horn´y odhad disperzie tak´ych blokov´ych postupnost ´? ktor´e s´u vytvoren´e pomocou mnoˇz´?n danou dolnou a hornou asymptotickou hustotou.Mnoˇzina hodn?ot takejto disperzie je tieˇz s´uvisl´a mnoˇzina. ? Distribuˇcn´e funkcie blokov´ych postupnost´?: V tejto ˇcasti s´u urˇcen ´e nutn´e a postaˇcuj´uce podmienky na to, aby asymptotick´e distribuˇcn´e funkcie blokov´ych postupnost´? boli pr´ave funkcie x . …víceméně

Abstract:

In the offered PhD. Thesis I study (R)-density and block sequences created by means of positive integers. The aim of Thesis is to collect results on (R)-density and results related to block sequences and complete them including the results of my own reaserch. The most of collected results known up to now concern (R)-density and its relation to the asymptotic density. Further are studied distribution functions of block sequences of positive integers. The new results contained in Thesis belong mainly to the following topics: ? Relations of (R)-density to dispresions: We found the best upper estimations of dispersion of (R)-dense sets. It was shown that the set of all possible values of dispersion is a whole interval (i.e. connected set). Similar statements were proved also for dispersion of special (R)-dense sets. ? Relations among the lower and upper asymptotic densities and dispersions: We determined the best possible upper estimation of dispersion of such block sequences which are created by means of sets with given prescribed both lower and upper asymptotic densities. The set of all possible dispersions of such block sequences is again a connected set. ? Distribution sequences of block sequences: In this part are found necessary and sufficient conditions in order that the distribution functions of block sequences were exactly the functions of type x …víceméně