20.500.12556/RUL-140542 Prirezani momentni problemi in pozitivno semidefinitne napolnitve matrik Truncated moment problems and positive semidefinite matrix completions Prirezani momentni problem sprašuje po karakterizaciji linearnih funkcionalov na prostoru polinomov dane stopnje, ki jih lahko predstavimo kot integracijo po pozitivni Borelovi meri $\mu$ z nosilcem na dani zaprti podmnožici v $\real^n$. To se lahko rešuje z opazovanjem lastnosti pripadajoče momentne matrike $\mathcal{M}$. V delu se ukvarjamo s primerom dveh spremenljivk. Stolpce matrike $\mathcal{M}$ indeksiramo z monomi $x^i y^j$. Vsak element jedra matrike $\mathcal M$ lahko v tej identifikaciji izrazimo kot simbolno ničelno množico nekega polinoma. V našem pristopu bomo privzeli singularnost matrike $\mathcal{M}$ in se rešili ene od spremenljivk, nato pa reševali pripadajoč enodimenzionalni problem. Zaradi nekaterih manjkajočih momentov v zaporedju bomo ključno uporabili rezultate, ki sledijo z uporabo teorije grafov. The truncated moment problem asks to characterize linear functionals over the space of polynomials of a given degree, which can be represented as integration over the positive Borel measure $\mu$ with support on a given closed subset of $\real^n$. We can solve this by observing the properties of the corresponding moment matrix $\mathcal{M}$. In this work we are going to study the cases that have two variables. We then label the columns of $\mathcal{M}$ with monomials $x^i y^j$. In this way, every element of the kernel of $\mathcal{M}$ can be expressed as a symbolic zero set of some polynomial. In our approach we will assume that $\mathcal{M}$ is singular and in this way we will get rid of one of the variables. Afterwards we will solve the corresponding one dimensional problem. We are going to crucially rely on some results from graph theory, since there are certain moments in the sequence that are missing. momentni problem semidefinitna hanklova matrika tetivni graf moment problem semidefinite Hankel matrix chordal graph true false false Slovenski jezik Angleški jezik Diplomsko delo/naloga 2022-09-15 14:25:00 2022-09-15 14:25:02 2023-11-27 09:48:43 0000-00-00 00:00:00 2022 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 35180 124326403 Marusic_Filip_-_Koncni_momentni_problemi_in_pozitivno_semidefinitne_napolnitve_matrik.pdf Marusic_Filip_-_Koncni_momentni_problemi_in_pozitivno_semidefinitne_napolnitve_matrik.pdf 1 48A1B311398D29D489FC7198FA3454FD 17ae7949e4fc302cb34e14d71f4e2929851905318ba4ec532f71785b75f1af4a 5b05ee53-34f1-11ed-92af-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=161295 Fakulteta za računalništvo in informatiko Fakulteta za matematiko in fiziko 0 0 0