Določanje mere v nesingularnem kvartičnem momentnem problemu s sledjo v dveh spremenljivkah

Abstract
Končen momentni problem s sledjo je vprašanje, kdaj lahko dano zaporedje realnih števil predstavimo kot integracijo po neki meri, ki jo računamo kot sled ovrednotenj nekomutativnih polinomov na neki množici simetričnih matrik. Za reševanje problema lahko uporabimo orodja linearne algebre, tako da zaporedju priredimo momentno matriko in prek obravnave njenih lastnosti sklepamo o obstoju mere. V diplomskem delu statistično preverjamo domnevo, da v kvartičnem primeru v dveh spremenljivkah s pozitivno definitno 7 × 7 momentno matriko, obstaja mera iz enega atoma velikosti 2 in največ šestih atomov velikosti 1. Glavno tehnika za to je odštevanje večkratnika momentne matrike ranga 1, tako da pridemo do momentne matrike ranga 6, za katero pa uporabimo znane rezultate, ki prevedejo problem na problem dopustnosti nekaj semidefinitnih programov.

Language: Slovenian momentni problem, mera, semidefinitno programiranje Bachelor thesis/paper 2.11 - Undergraduate Thesis FRI - Faculty of Computer and Information Science FMF - Faculty of Mathematics and Physics 2020 20.500.12556/RUL-120060 31862787 15.09.2020 796 144 Copy citation

## Secondary language

Language: English Determining a type of a measure in a nonsingular bivariate quartic tracial moment problem The truncated tracial moment problem asks to characterize when a finite sequence of real numbers can be represented with tracial moments of matrices, i.e., the measure is the trace of the evaluations of monomials on the set of symmetric matrices. To tackle the problem we can use the tools from linear algebra. We associate to the sequence the truncated moment matrix and study its properties, which ether prove or disprove existence of the measure. In the diploma thesis we statistically analyze the conjecture, that in the quartic bivariate case with a positive definite 7 × 7 moment matrix, there exists a measure consisting of one atom of size 2 and at most six atoms of size 1. The main technique is to subtract the multiple of a moment matrix of rank 1, such that we get a moment matrix of rank 6, for which we can use known results about the existence of a measure, that translate the problem to the feasibility of certain semidefinite programs. moment problem, measure, semidefinite programming

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