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20 pages, 3841 KiB

Open AccessArticle

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

Viewed by 776

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

Open AccessArticle

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 - 07 Jun 2023

Viewed by 584

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

Open AccessArticle

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 1 | Viewed by 965

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

Open AccessFeature PaperArticle

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

Viewed by 886

Abstract

Let A and B be Borel subsets of the Euclidean n-space with

dim A + dim B > 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

dim A + dim B > n

. This is a survey on the following question: what can we say about the Hausdorff dimension of the intersections

A \cap (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

Open AccessFeature PaperArticle

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 1 | Viewed by 1189

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

Open AccessFeature PaperReview

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 - 04 May 2023

Viewed by 1010

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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