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#### Open Access in DiVA

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#### Authority records

Tsougkas, Konstantinos
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Tsougkas, Konstantinos
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Mathematics
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Combinatorial and analytical problems for fractals and their graph approximationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2019 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics , 2019. , p. 37
##### Series

Uppsala dissertations in mathematics, ISSN 1401-2049 ; 112
##### Keywords [en]

Fractal graphs, energy Laplacian, Kusuoka measure
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:his:diva-23101Libris ID: 6gw0px6t4lt2dtbjISBN: 978-91-506-2739-8 (print)OAI: oai:DiVA.org:his-23101DiVA, id: diva2:1788082
##### Public defence

2019-02-15, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, 13:15 (English)
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt681",{id:"formSmash:j_idt681",widgetVar:"widget_formSmash_j_idt681",multiple:true}); Available from: 2023-08-15 Created: 2023-08-15 Last updated: 2023-08-15Bibliographically approved
##### List of papers

The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs.

The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself.

In paper I, we study the number of spanning trees on the sequence of graphs approximating a self-similar set admitting spectral decimation.

In paper II, we study harmonic functions on p.c.f. self-similar sets. Unlike the standard Dirichlet problem and harmonic functions in Euclidean space, harmonic functions on these sets may be locally constant without being constant in their entire domain. In that case we say that the fractal has a degenerate harmonic structure. We prove that for a family of variants of the Sierpinski gasket the harmonic structure is non-degenerate.

In paper III, we investigate properties of the Kusuoka measure and the corresponding energy Laplacian on the Sierpinski gaskets of level k.

In papers IV and V, we establish a connection between the discrete combinatorial graph Laplacian determinant and the regularized determinant of the fractal itself. We establish that for a certain class of p.c.f. fractals the logarithm of the regularized determinant appears as a constant in the logarithm of the discrete combinatorial Laplacian.

Ett av fem delarbeten (övriga se rubriken Delarbeten/List of papers): Tsougkas, K. Connections between discrete and regularized determinants on fractals. Manuscript.

1. Counting spanning trees on fractal graphs and their asymptotic complexity$(function(){PrimeFaces.cw("OverlayPanel","overlay1788074",{id:"formSmash:j_idt781:0:j_idt791",widgetVar:"overlay1788074",target:"formSmash:j_idt781:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Non-degeneracy of the harmonic structure on Sierpiński gaskets$(function(){PrimeFaces.cw("OverlayPanel","overlay1788000",{id:"formSmash:j_idt781:1:j_idt791",widgetVar:"overlay1788000",target:"formSmash:j_idt781:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. The Kusuoka measure and the energy Laplacian on level-k Sierpiński gaskets$(function(){PrimeFaces.cw("OverlayPanel","overlay1788087",{id:"formSmash:j_idt781:2:j_idt791",widgetVar:"overlay1788087",target:"formSmash:j_idt781:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Regularized Laplacian determinants of self-similar fractals$(function(){PrimeFaces.cw("OverlayPanel","overlay1788066",{id:"formSmash:j_idt781:3:j_idt791",widgetVar:"overlay1788066",target:"formSmash:j_idt781:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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