Vztrajnost lastnih podprostorov endomorfizma

magistrsko delo

Juš Kosmač (Author), Jaka Smrekar (Mentor)

Abstract

Namen dela je predstaviti uporabo vztrajne homologije pri preučevanju diskretnih dinamičnih sistemov. Pojem vztrajnosti obravnavamo v bolj splošnem kategoričnem okviru kot pri klasični definiciji vztrajne homologije. Posvetimo se tudi algebraični korespondenci med vztrajnimi moduli nad poljem in končno generiranimi stopničenimi moduli nad kolobarjem polinomov v eni spremenljivki. Odraža se v enostavnem opisu vztrajnosti prek vztrajnega diagrama. Pojasnimo, kako lahko iz danega končnega vzorca točk in vzorčne preslikave, s pomočjo vztrajnosti lastnih podprostorov induciranega endomorfizma na homologiji, sklepamo o globalnem značaju neznanega prostora in preslikave. Podamo algoritem za izračun vztrajnega diagrama stolpa lastnih podprostorov in utemeljimo njegovo stabilnost. Preizkusimo ga na nekaj enostavnih primerih in pojasnimo vpliv različnih parametrov (npr. velikosti vzorca, prisotnosti šuma) na rezultate.

Keywords

matematika;vztrajna homologija;diskretni dinamični sistemi;lastni podprostori endorfizma;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[J. Kosmač]
UDC:	515.14
COBISS:	19119107
Views:	1618
Downloads:	334
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Persistence of eigenspaces of an endomorphism
Secondary abstract:	The aim of this work is to show a use of persistent homology in the study of discrete dynamical systems. The notion of persistence is presented in a more general categorical setting compared to the standard definition of persistent homology. An algebraic correspondence between persistent modules over a field and finitely generated graded modules over the ring of polynomials in one variable is established. As a consequence, a simple description of persistence in terms of the persistent diagram is obtained. With the use of persistence of eigenspaces of an induced endomorphism on homology, a procedure is developed which allows us to describe the global behaviour of an unknown self-map just from a given finite sample of points and a sampled map. An algorithm to compute the persistent diagram of a tower of eigenspaces is given and its stability is proven. Lastly, it is tested on a few basic examples and the effect of different parameters (e.g. sample size, noise) on the results is explained.
Secondary keywords:	persistent homology;discrete dynamical systems;eigenspaces of an endorphism;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages:	VII, 66 str.
ID:	11454193