Prirezani momentni problemi in pozitivno semidefinitne napolnitve matrik

diplomsko delo

Filip Marušič (Author), Aljaž Zalar (Mentor)

Abstract

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.

Keywords

momentni problem;semidefinitna Hanklova matrika;tetivni graf;univerzitetni študij;diplomske naloge;

Data

Language:	Slovenian
Year of publishing:	2022
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FRI - Faculty of Computer and Information Science
Publisher:	[F. Marušič]
UDC:	004:51(043.2)
COBISS:	124326403
Views:	253
Downloads:	36
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Truncated moment problems and positive semidefinite matrix completions
Secondary abstract:	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.
Secondary keywords:	moment problem;semidefinite Hankel matrix;chordal graph;computer science and mathematics;interdisciplinary studies;diploma;
Type (COBISS):	Bachelor thesis/paper
Study programme:	1000407
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za računalništvo in informatiko
Pages:	38 str.
ID:	16479249