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MCA
Special Issues
Geometry of Deterministic and Random Fractals
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Geometry of Deterministic and Random Fractals

A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (30 April 2023) | Viewed by 12515

Special Issue Editors

Prof. Dr. Balázs Bárány

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Guest Editor
MTA-BME Stochastics Research Group, Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary
Interests: geometric measure theory; fractal geometry; ergodic theory; dynamical systems
Dr. István Kolossváry

E-Mail Website
Guest Editor
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, Scotland, UK
Interests: fractals and its applications; random graphs
Dr. Roland Molontay

E-Mail Website
Guest Editor
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, P.O. Box 91, 1521 Budapest, Hungary
Interests: network science; applied probability theory; data science
Prof. Dr. Michał Rams

E-Mail Website
Guest Editor
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
Interests: geometric measure theory; dynamical systems

Special Issue Information

Dear Colleagues,

This Special Issue will collect contributions from the workshop Geometry of Deterministic and Random Fractals (https://simon60.math.bme.hu/). Papers considered to fit the scope of the journal and to be of exceptional quality, after evaluation by our reviewers, will be published free of charge.

Prof. Dr. Balázs Bárány
Dr. István Kolossváry
Dr. Roland Molontay
Prof. Dr. Michał Rams
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematical and Computational Applications is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (6 papers)

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Research

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20 pages, 3841 KiB  
Article
Convexity-Preserving Rational Cubic Zipper Fractal Interpolation Curves and Surfaces
by Vijay and Arya Kumar Bedabrata Chand
Math. Comput. Appl. 2023, 28(3), 74; https://doi.org/10.3390/mca28030074 - 10 Jun 2023
Cited by 4 | Viewed by 1805
Abstract
A class of zipper fractal functions is more versatile than corresponding classes of traditional and fractal interpolants due to a binary vector called a signature. A zipper fractal function constructed through a zipper iterated function system (IFS) allows one to use negative and [...] Read more.
A class of zipper fractal functions is more versatile than corresponding classes of traditional and fractal interpolants due to a binary vector called a signature. A zipper fractal function constructed through a zipper iterated function system (IFS) allows one to use negative and positive horizontal scalings. In contrast, a fractal function constructed with an IFS uses positive horizontal scalings only. This article introduces some novel classes of continuously differentiable convexity-preserving zipper fractal interpolation curves and surfaces. First, we construct zipper fractal interpolation curves for the given univariate Hermite interpolation data. Then, we generate zipper fractal interpolation surfaces over a rectangular grid without using any additional knots. These surface interpolants converge uniformly to a continuously differentiable bivariate data-generating function. For a given Hermite bivariate dataset and a fixed choice of scaling and shape parameters, one can obtain a wide variety of zipper fractal surfaces by varying signature vectors in both the x direction and y direction. Some numerical illustrations are given to verify the theoretical convexity results. Full article
(This article belongs to the Special Issue Geometry of Deterministic and Random Fractals)
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13 pages, 4774 KiB  
Article
On the Fluctuations of Internal DLA on the Sierpinski Gasket Graph
by Nico Heizmann
Math. Comput. Appl. 2023, 28(3), 73; https://doi.org/10.3390/mca28030073 - 7 Jun 2023
Viewed by 1281
Abstract
Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph G, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph, the [...] Read more.
Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph G, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph, the asymptotic shape is known to be a ball in the graph metric. In this paper, we improve the sublinear bounds for the fluctuations known from its known asymptotic shape result by establishing bounds for the odometer function for a divisible sandpile model. Full article
(This article belongs to the Special Issue Geometry of Deterministic and Random Fractals)
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16 pages, 770 KiB  
Article
Sierpiński Fractals and the Dimension of Their Laplacian Spectrum
by Mark Pollicott and Julia Slipantschuk
Math. Comput. Appl. 2023, 28(3), 70; https://doi.org/10.3390/mca28030070 - 17 May 2023
Cited by 3 | Viewed by 1924
Abstract
We establish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated with Sierpiński lattices and infinite Sierpiński gaskets and other post-critically finite self-similar sets. Full article
(This article belongs to the Special Issue Geometry of Deterministic and Random Fractals)
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12 pages, 310 KiB  
Article
A Survey on the Hausdorff Dimension of Intersections
by Pertti Mattila
Math. Comput. Appl. 2023, 28(2), 49; https://doi.org/10.3390/mca28020049 - 22 Mar 2023
Cited by 3 | Viewed by 1934
Abstract
Let A and B be Borel subsets of the Euclidean n-space with dimA+dimB>n. This is a survey on the following question: what can we say about the Hausdorff dimension of the intersections  [...] Read more.
Let A and B be Borel subsets of the Euclidean n-space with dimA+dimB>n. This is a survey on the following question: what can we say about the Hausdorff dimension of the intersections A∩(g(B)+z) for generic orthogonal transformations g and translations by z? Full article
(This article belongs to the Special Issue Geometry of Deterministic and Random Fractals)
22 pages, 606 KiB  
Article
A Survey on Newhouse Thickness, Fractal Intersections and Patterns
by Alexia Yavicoli
Math. Comput. Appl. 2022, 27(6), 111; https://doi.org/10.3390/mca27060111 - 14 Dec 2022
Cited by 2 | Viewed by 2042
Abstract
In this article, we introduce a notion of size for sets, called the thickness, that can be used to guarantee that two Cantor sets intersect (the Gap Lemma) and show a connection among thickness, Schmidt games and patterns. We work mostly in the [...] Read more.
In this article, we introduce a notion of size for sets, called the thickness, that can be used to guarantee that two Cantor sets intersect (the Gap Lemma) and show a connection among thickness, Schmidt games and patterns. We work mostly in the real line, but we also introduce the topic in higher dimensions. Full article
(This article belongs to the Special Issue Geometry of Deterministic and Random Fractals)
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Review

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23 pages, 427 KiB  
Review
Notes on the Transversality Method for Iterated Function Systems—A Survey
by Boris Solomyak
Math. Comput. Appl. 2023, 28(3), 65; https://doi.org/10.3390/mca28030065 - 4 May 2023
Viewed by 2229
Abstract
This is a brief survey of selected results obtained using the “transversality method” developed for studying parametrized families of fractal sets and measures. We mostly focus on the early development of the theory, restricting ourselves to self-similar and self-conformal iterated function systems. Full article
(This article belongs to the Special Issue Geometry of Deterministic and Random Fractals)
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