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Symmetry
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Symmetry in Differential Geometry and Geometric Analysis
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Symmetry in Differential Geometry and Geometric Analysis

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 May 2025 | Viewed by 5470

Special Issue Editors

Dr. Yongshun Liang

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China
Interests: fractional calculus; fractal geometry; fractal functions; function approximation theory and its applications
Dr. Saurabh Verma

E-Mail Website
Guest Editor
Department of Applied Sciences, IIIT Allahabad, Prayagraj 211015, India
Interests: analysis; fractal geometry; quantization: theory and applications; approximation theory; fixed point theory and applications
Dr. Arulprakash Gowrisankar

E-Mail Website
Guest Editor
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India
Interests: fractals; fractional calculus

Special Issue Information

Dear Colleagues,

Fractal geometry is an important branch of mathematics that allows for the description of sets that are too intricate to fit into classical geometry. The concept of fractals was first introduced by Mandelbrot in the 1970s as a class of highly irregular sets usually presenting with self-similarity, infinite complexity, and a non-integral fractal dimension. Up to now, it has been hugely significant for the development of a variety of sciences. In mathematics, fractals originate from chaos and dynamic systems. Soon after their discovery, they began to appear in almost every field and can be systematically studied using classical and modern methods. In recent years, scholars have mainly focused on the following research objects: fractal dimensions of graphs, fractal interpolation and approximation, fractals and dynamical systems, self-similarity and Lipschitz equivalence, geometric measure theory, fractional calculus of fractal functions, fractal geometry, number theory, etc. In addition, fractals have been widely applied in other academic fields, such as physics, statistics, geology, material science, quantization theory, signal processing, computer image processing, pattern recognition, and more. Therefore, fractal geometry has increasingly shown its tremendous research value for both real life and scientific development.

It is well known that fractals are widely distributed in nature, such as mountains, landforms, cloud clusters, and so on. Mandelbrot once said that fractal geometry is the language of nature. There exist a lot of symmetries in natural objects with fractal characteristics, of which the most prominent one might be the Koch snowflake. Therefore, it is of great interest to explore the phenomena of symmetry or asymmetry in the fractal world.

This Special Issue aims to collect a series of high-quality papers from renowned experts from around the world to present the latest research on fractal geometry and its various applications. While the focus of this issue includes all the aspects mentioned above, we particularly welcome contributions from researchers who use the concepts of symmetry or asymmetry in their methodologies.

Dr. Yongshun Liang
Dr. Saurabh Verma
Dr. Arulprakash Gowrisankar
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. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). 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.

Keywords

  • symmetries in fractal geometry
  • symmetries in geometric analysis
  • fractal dimension
  • graphs of fractal functions
  • fractal curves
  • fractal surfaces
  • fractal interpolation
  • fractal function approximation
  • dynamical systems
  • iterated function systems
  • fractal sets
  • self-similar sets
  • self-affine sets
  • Lipschitz equivalence
  • topological structures
  • multifractals analysis
  • geometric analysis
  • geometric measure theory
  • number theory
  • diophantine approximation
  • fractional calculus
  • fractional differential equations
  • numerical analysis in fractals
  • fractals in nature
  • applications of fractals, e.g., in physics, statistics, geology, material science, quantization theory, signal processing, computer image processing, pattern recognition

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Published Papers (4 papers)

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Research

13 pages, 1247 KiB  
Article
Characterization of Dual Spacelike Curves on Dual Lightlike Cone Q~2 Utilizing the Structure Function
by Pınar Balkı Okullu and Hasan Hüseyin Uğurlu
Symmetry 2024, 16(12), 1574; https://doi.org/10.3390/sym16121574 - 25 Nov 2024
Viewed by 608
Abstract
This study is about the dual spacelike curves lying on the dual lightlike cone, which can be either symmetric or asymmetric. We first establish the dual associated curve, which is related to the reference curve. Using these curves and the derivative of the [...] Read more.
This study is about the dual spacelike curves lying on the dual lightlike cone, which can be either symmetric or asymmetric. We first establish the dual associated curve, which is related to the reference curve. Using these curves and the derivative of the reference curve, we derive the dual asymptotic orthonormal frame. Next, we define the dual structure function, curvature function, and Frenet formulae, and express the curvature function in terms of the dual structure function. This leads to a differential equation that characterizes the dual cone curve in relation to its curvature function. Since curves with constant curvature maintain the same curvature at every point, their geometry is more predictable. Therefore, we assume that the dual cone curvature function is constant and examine how this condition affects the behavior and geometric properties of the dual curves. As a result of this investigation, some new results and definitions are obtained. Full article
(This article belongs to the Special Issue Symmetry in Differential Geometry and Geometric Analysis)
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20 pages, 368 KiB  
Article
On Some Aspects of the Courant-Type Algebroids, the Related Coadjoint Orbits and Integrable Systems
by Anatolij K. Prykarpatski and Victor A. Bovdi
Symmetry 2024, 16(1), 76; https://doi.org/10.3390/sym16010076 - 5 Jan 2024
Viewed by 1384
Abstract
Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect [...] Read more.
Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described. Full article
(This article belongs to the Special Issue Symmetry in Differential Geometry and Geometric Analysis)
15 pages, 584 KiB  
Article
The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral
by Bingqian Wang and Wei Xiao
Symmetry 2023, 15(12), 2158; https://doi.org/10.3390/sym15122158 - 5 Dec 2023
Cited by 2 | Viewed by 1253
Abstract
This article is a study on the (k,s)-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension [...] Read more.
This article is a study on the (k,s)-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the (k,s)-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval I=[0,1], which indicates that the upper Box dimension of the integrand f(x) will not be increased by the σ-order (k,s)-Riemann–Liouville fractional integral ksD−σf(x) where σ>0 on I. Additionally, we prove that the fractal dimension of ksD−σf(x) of one-dimensional continuous functions f(x) is still one. Full article
(This article belongs to the Special Issue Symmetry in Differential Geometry and Geometric Analysis)
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25 pages, 1062 KiB  
Article
On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs
by Amal S. Alali, Shahbaz Ali, Muhammad Adnan and Delfim F. M. Torres
Symmetry 2023, 15(10), 1911; https://doi.org/10.3390/sym15101911 - 12 Oct 2023
Cited by 2 | Viewed by 1302
Abstract
The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications [...] Read more.
The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph G(Zn) is called a zero-divisor graph over the zero divisors of a commutative ring Zn, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded. Full article
(This article belongs to the Special Issue Symmetry in Differential Geometry and Geometric Analysis)
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