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Mathematics
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Fractal and Computational Geometry
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Fractal and Computational Geometry

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (25 November 2023) | Viewed by 11359

Special Issue Editor

Dr. Vasileios Drakopoulos

E-Mail Website
Guest Editor
Department of Computer Science and Biomedical Informatics, University of Thessaly, 35131 Lamia, Greece
Interests: chaos theory; computer graphics; fractal and computational geometry; mathematical modelling; computational complex analysis; nonlinear dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

From one point of view, fractal geometry is a field in science that unifies mathematics, theoretical physics, art, and computer science. Therefore, it is not difficult to find applications of fractals in almost every scientific field wherein the information available in a finite number of grid points has to be modelled with a continuous function. Applications of this theory include geometric design, data visualization, reverse engineering, physics, geology, image encoding and compression, metallurgy, signal processing, and wavelet theory. On the other hand, computational geometry is a branch of computer science aiming to design efficient algorithms for solving geometric problems. Therefore, a number of methods to describe, generate, and encode a wide variety of images, possibly with grey or color tones, which might have fractal (self-similar, self-affine, etc.) characteristics could be investigated. Moreover, it could be focused on how the developed techniques yield novel and quite general methods to analyze the time complexity of divide-and-conquer algorithms, by geometrically capturing the dynamic structure of such algorithms as fractals. This book will contain state-of-the-art contributions to these rapidly growing research areas. It will be of essential value to mathematicians, physicists, and engineers working in the fields of fractal and computational geometry as well as of related phenomena, and to researchers working in medicine and the life sciences.

Prof. Dr. Vasileios Drakopoulos
Guest Editor

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Keywords

  • fractal
  • analysis
  • geometry
  • dynamic system
  • interpolation
  • scientific computation
  • computer graphics
  • computational geometry
  • geometric methods
  • visualization

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Published Papers (5 papers)

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16 pages, 1411 KiB  
Article
Parameter Identification of Bivariate Fractal Interpolation Surfaces by Using Convex Hulls
by Vasileios Drakopoulos, Dimitrios Matthes, Dimitrios Sgourdos and Nallapu Vijender
Mathematics 2023, 11(13), 2850; https://doi.org/10.3390/math11132850 - 25 Jun 2023
Cited by 2 | Viewed by 983
Abstract
The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some [...] Read more.
The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented. Full article
(This article belongs to the Special Issue Fractal and Computational Geometry)
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14 pages, 356 KiB  
Article
Iteration of Operators with Contractive Mutual Relations of Kannan Type
by Ram N. Mohapatra, María A. Navascués, María V. Sebastián and Saurabh Verma
Mathematics 2022, 10(15), 2632; https://doi.org/10.3390/math10152632 - 27 Jul 2022
Cited by 9 | Viewed by 1628
Abstract
Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the [...] Read more.
Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the properties of the operators satisfying this type of condition; in particular, we study the case where the space is normed, and the maps are linear. Then we generalize some theorems proposed by this author on the existence of a fixed point of one operator or a common fixed point for two operators. Our results first prove the existence of a common fixed point of a set of self-maps of any cardinal number (countable or uncountable) satisfying the conditions of Kannan type in metric spaces. The same is proved for a set of maps satisfying the mutual relations of classical contractivity. We prove in both cases the convergence of iterative schemes involving operators with mutual relations of contractivity, proposing sufficient conditions for the iteration of the operators on any element of the space to converge to the common fixed point when a different operator is taken in each step. The results obtained are applied to operators acting on real functions, coming from the fractal convolution with the null function. Full article
(This article belongs to the Special Issue Fractal and Computational Geometry)
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17 pages, 3647 KiB  
Article
Introduction to the Class of Prefractal Graphs
by Rasul Kochkarov and Azret Kochkarov
Mathematics 2022, 10(14), 2500; https://doi.org/10.3390/math10142500 - 18 Jul 2022
Cited by 1 | Viewed by 2013
Abstract
Fractals are already firmly rooted in modern science. Research continues on the fractal properties of objects in physics, chemistry, biology and many other scientific fields. Fractal graphs as a discrete representation are used to model and describe the structure of various objects and [...] Read more.
Fractals are already firmly rooted in modern science. Research continues on the fractal properties of objects in physics, chemistry, biology and many other scientific fields. Fractal graphs as a discrete representation are used to model and describe the structure of various objects and processes, both natural and artificial. The paper proposes an introduction to prefractal graphs. The main definitions and notation are proposed—the concept of a seed, the operations of processing a seed, the procedure for generating a prefractal graph. Canonical (typical) and non-canonical (special) types of prefractal graphs are considered separately. Important characteristics are proposed and described—the preservation of adjacency of edges for different ranks in the trajectory. The definition of subgraph-seeds of different ranks is given separately. Rules for weighting a prefractal graph by natural numbers and intervals are proposed. Separately, the definition of a fractal graph as infinite is given, and the differences between the concepts of fractal and prefractal graphs are described. At the end of the work, already published works of the authors are proposed, indicating the main backlogs, as well as a list of directions for new research. This work is the beginning of a cycle of works on the study of the properties and characteristics of fractal and prefractal graphs. Full article
(This article belongs to the Special Issue Fractal and Computational Geometry)
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17 pages, 873 KiB  
Article
Binary Operations in Metric Spaces Satisfying Side Inequalities
by María A. Navascués, Pasupathi Rajan and Arya Kumar Bedabrata Chand
Mathematics 2022, 10(1), 11; https://doi.org/10.3390/math10010011 - 21 Dec 2021
Cited by 3 | Viewed by 2582
Abstract
The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was [...] Read more.
The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space. Full article
(This article belongs to the Special Issue Fractal and Computational Geometry)
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35 pages, 12612 KiB  
Article
Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm
by Michel L. Lapidus, Machiel van Frankenhuijsen and Edward K. Voskanian
Mathematics 2021, 9(6), 591; https://doi.org/10.3390/math9060591 - 10 Mar 2021
Cited by 1 | Viewed by 2168
Abstract
The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of [...] Read more.
The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lovász. Full article
(This article belongs to the Special Issue Fractal and Computational Geometry)
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