Spectrum of the Laplacian on the Vicsek Set “With No Loose Ends”

Strichartz, Robert S.; Zhu, Jiangyue

doi:10.1142/s0218348x17500621

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…Finding out properties of the spectrum of the Laplacian in the fractal setting is a problem with long history in the literature, see e.g. [13,24,9,1,8,29] and the second part of this paper is devoted to the investigation of the spectrum of the Laplacian on the Hanoi attractor and on the hybrid H previously mentioned and studied in detail in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Analysis on hybrid fractals

Ruiz¹,

Chen²,

Gu³

et al. 2018

Preprint

Self Cite

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We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the 3-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the 3-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space are investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.

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Section: Introductionmentioning

confidence: 99%

Analysis on hybrid fractals

Ruiz¹,

Chen²,

Gu³

et al. 2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finding out properties of the spectrum of the Laplacian in the fractal setting is a problem with long history in the literature, see e.g. [13,24,9,1,8,30] and the second part of this paper is devoted to the investigation of the spectrum of the Laplacian on the Hanoi attractor and on the hybrid H previously mentioned and studied in detail in Section 2.…”

mentioning

confidence: 99%

Analysis on hybrid fractals

Ruiz¹,

Chen²,

Gu³

et al. 2020

Communications on Pure &Amp; Applied Analysis

View full text Add to dashboard Cite

We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the 3-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the 3-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space is investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.

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On the Correlation Between Fractal Dimension and Robustness of Complex Networks

Chen

Yao

et al. 2019

Fractals

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In recent years, because complex networks can be used to model real-world complex systems, such as the Internet, urban infrastructure networks, and gene interaction networks, such research has been widely applied in engineering, social sciences, and life sciences and has caused widespread concern. Fractal dimension, as a concept concerning the filling ability and complexity of an object space, has great significance for the study of the robustness of complex networks. This paper studies the relationship between fractal dimension and the robustness of different types of complex networks from the perspective of network structure and network scale. We find that fractal dimension is strongly correlated with robustness under certain conditions and can be used as an important index to evaluate the robustness of complex networks.

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Spectrum of the Laplacian on the Vicsek Set “With No Loose Ends”

Cited by 3 publications

References 11 publications

Analysis on hybrid fractals

Analysis on hybrid fractals

Analysis on hybrid fractals

On the Correlation Between Fractal Dimension and Robustness of Complex Networks

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