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Symmetry
Special Issues
Geometric Algebra and Its Applications
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Geometric Algebra and Its Applications

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

Deadline for manuscript submissions: closed (15 August 2023) | Viewed by 8349

Special Issue Editors

Prof. Dr. Paul Popescu

E-Mail Website
Guest Editor
Department of Applied Mathematics, University of Craiova, 200585 Craiova, Romania
Interests: geometry of differential equations on manifolds; geometric theory of foliations; Finsler and Lagrange geometry; geometric algebra; geometric methods in mechanics
Special Issues, Collections and Topics in MDPI journals
Dr. Marcela Popescu

E-Mail Website
Guest Editor
Department of Applied Mathematics, University of Craiova, 200585 Craiova, Romania
Interests: geometry of submanifolds; Finsler and Lagrange geometry; geometric methods in mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of our Special Issue, titled Geometric Algebra and Its Applications, is to bring together outstanding theoretical contributions to geometric algebra, adapting special structures and symmetries from various mathematical and physical research areas, but having real-world applications.

Some algebraic structures can be considered coming from mathematical or physical areas, giving outstanding constructions that motivate their abstract form. They can lead to specific symmetries that give sometimes their theoretical force.

Geometric algebra structures are involved in many concrete applications, most associated with singular geometric distributions on manifolds and using vector bundles. Most, for example, in mechanics and control, but also other areas.

The anchored approach used for singular geometric distributions can lead to general structures and symmetries in an analogous way as the classical cases.

Generalizations in a classical way, using different types of algebroids or groupoids, can make the transfer to the study of singular structures and symmetries using classical methods for regular ones.

The submitted manuscripts must fall within the scope of Symmetry.

Prof. Dr. Paul Popescu
Dr. Marcela Popescu
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

  • singular foliation
  • Lie algebroid
  • Lie pseudoalgebra
  • anchored module

Published Papers (6 papers)

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Research

15 pages, 639 KiB  
Article
A Study on a Spacelike Line Trajectory in Lorentzian Locomotions
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2023, 15(10), 1816; https://doi.org/10.3390/sym15101816 - 24 Sep 2023
Viewed by 540
Abstract
In this study, we establish a novel Lorentzian interpretation of the Euler–Savary (E−S) and Disteli (Dis) formulae. Subsequently, we proceed to establish a theoretical structure for a Lorentzian torsion line congruence which is the spatial [...] Read more.
In this study, we establish a novel Lorentzian interpretation of the Euler–Savary (E−S) and Disteli (Dis) formulae. Subsequently, we proceed to establish a theoretical structure for a Lorentzian torsion line congruence which is the spatial symmetry of the Lorentzian circling-point dual curve, in accordance with the principles of the kinematic theory of spherical locomotions. Further, a timelike (Tlike) torsion line congruence is defined and its spatial equivalence is examined. The findings contribute to an enhanced comprehension of the interplay between axodes and Lorentzian spatial movements, which has possible significance in various disciplines, such as the fields of robotics and mechanical engineering. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
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30 pages, 558 KiB  
Article
On the Jacobi Stability of Two SIR Epidemic Patterns with Demography
by Florian Munteanu
Symmetry 2023, 15(5), 1110; https://doi.org/10.3390/sym15051110 - 18 May 2023
Cited by 2 | Viewed by 1491
Abstract
In the present work, two SIR patterns with demography will be considered: the classical pattern and a modified pattern with a linear coefficient of the infection transmission. By reformulating of each first-order differential systems as a system with two second-order differential equations, we [...] Read more.
In the present work, two SIR patterns with demography will be considered: the classical pattern and a modified pattern with a linear coefficient of the infection transmission. By reformulating of each first-order differential systems as a system with two second-order differential equations, we will examine the nonlinear dynamics of the system from the Jacobi stability perspective through the Kosambi–Cartan–Chern (KCC) geometric theory. The intrinsic geometric properties of the systems will be studied by determining the associated geometric objects, i.e., the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the external force εi—the first invariant; the deviation curvature tensor Pji—the second invariant; the torsion tensor Pjki—the third invariant; the Riemann–Christoffel curvature tensor Pjkli—the fourth invariant; the Douglas tensor Djkli—the fifth invariant. In order to obtain necessary and sufficient conditions for the Jacobi stability near each equilibrium point, the deviation curvature tensor will be determined at each equilibrium point. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
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13 pages, 316 KiB  
Article
About the Jacobi Stability of a Generalized Hopf–Langford System through the Kosambi–Cartan–Chern Geometric Theory
by Florian Munteanu, Alexander Grin, Eduard Musafirov, Andrei Pranevich and Cătălin Şterbeţi
Symmetry 2023, 15(3), 598; https://doi.org/10.3390/sym15030598 - 26 Feb 2023
Cited by 3 | Viewed by 1072
Abstract
In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary [...] Read more.
In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi–Cartan–Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
16 pages, 352 KiB  
Article
A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory
by Florian Munteanu
Symmetry 2022, 14(9), 1815; https://doi.org/10.3390/sym14091815 - 01 Sep 2022
Cited by 6 | Viewed by 1258
Abstract
In this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate [...] Read more.
In this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view using the Kosambi–Cartan–Chern (KCC) geometric theory. We then determine the nonlinear connection, the Berwald connection, and the five KCC invariants which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Furthermore, we obtain the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points, and we point these out on a few illustrative examples. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
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22 pages, 352 KiB  
Article
Unified Representation of 3D Multivectors with Pauli Algebra in Rectangular, Cylindrical and Spherical Coordinate Systems
by Ben Minnaert, Giuseppina Monti and Mauro Mongiardo
Symmetry 2022, 14(8), 1684; https://doi.org/10.3390/sym14081684 - 13 Aug 2022
Viewed by 1863
Abstract
In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing [...] Read more.
In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
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10 pages, 270 KiB  
Article
On the Two Categories of Modules
by Marcela Popescu and Paul Popescu
Symmetry 2022, 14(7), 1435; https://doi.org/10.3390/sym14071435 - 13 Jul 2022
Viewed by 1157
Abstract
A new category equivalence is proved in this paper, involving the two distinct categories of modules, the covariant and the contravariant, respectively, released by Higgins and Mackenzie. The equivalence of the two categories is given when restricting to almost finitely generated projective modules [...] Read more.
A new category equivalence is proved in this paper, involving the two distinct categories of modules, the covariant and the contravariant, respectively, released by Higgins and Mackenzie. The equivalence of the two categories is given when restricting to almost finitely generated projective modules and their allowed morphisms, defined in the paper. The equivalence is expressed by using generators. In particular, we obtain the well-known equivalence of the two categories of projective finitely generated modules; thus, our main result extends this classical one. Full article
(This article belongs to the Special Issue Geometric Algebra and Its Applications)
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