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Search Results (8)

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Keywords = Cayley formula

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20 pages, 360 KiB  
Article
Parabolic and Linear Rotational Motions on Cones and Hyperboloids
by Harun Barış Çolakoğlu, Mehmet Duru and Ayşe Yılmaz Ceylan
Axioms 2025, 14(8), 612; https://doi.org/10.3390/axioms14080612 - 6 Aug 2025
Abstract
In this study, we consider the Lorentzian rotation about a lightlike axis. First, we introduce a geometric characterization for the rotation angle between two vectors that can overlap each other under a Lorentzian rotation about a lightlike axis. Then, we give a definition [...] Read more.
In this study, we consider the Lorentzian rotation about a lightlike axis. First, we introduce a geometric characterization for the rotation angle between two vectors that can overlap each other under a Lorentzian rotation about a lightlike axis. Then, we give a definition for the angle measurement between two spacelike vectors whose vector product is lightlike. Later, we generalize the Lorentzian rotation about a lightlike axis, and determine matrices of these transformations using the Cartan frame and the well-known Rodrigues formula, then using the Cayley map, and finally using the generalized split quaternions. We see that such transformations give parabolic rotational motions on general cones or general hyperboloids of one or two sheets, while they also give linear rotational motions on general hyperboloids of one sheet. Full article
(This article belongs to the Section Geometry and Topology)
16 pages, 272 KiB  
Article
Elliptic and Hyperbolic Rotational Motions on General Hyperboloids
by Harun Barış Çolakoğlu and Mehmet Duru
Symmetry 2025, 17(6), 845; https://doi.org/10.3390/sym17060845 - 28 May 2025
Viewed by 337
Abstract
This study proposes a new way to represent elliptic and hyperbolic motions on any general hyperboloids of one or two sheets using the famous Rodrigues, Cayley, and Householder transformations. These transformations are used within the generalized Minkowski 3-space which extends the usual Lorentzian [...] Read more.
This study proposes a new way to represent elliptic and hyperbolic motions on any general hyperboloids of one or two sheets using the famous Rodrigues, Cayley, and Householder transformations. These transformations are used within the generalized Minkowski 3-space which extends the usual Lorentzian geometry by introducing a generalized scalar product. The study is carried out by considering the unit sphere defined in this generalized space along with the use of three-dimensional generalized Lorentzian skew-symmetric matrices that naturally generate continuous rotational motions. The obtained results provide rotational motions on the sphere in Minkowski 3-space as well as elliptic and hyperbolic motions on general hyperboloids in Euclidean 3-space. A numerical example is provided for each of the explored rotation methods. Full article
(This article belongs to the Section Mathematics)
26 pages, 406 KiB  
Article
On Matrices of Generalized Octonions (Cayley Numbers)
by Seda Yamaç Akbıyık
Symmetry 2024, 16(12), 1567; https://doi.org/10.3390/sym16121567 - 22 Nov 2024
Viewed by 855
Abstract
This article focuses on generalized octonions which include real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. We make a classification according to the inner product and vector parts and give the [...] Read more.
This article focuses on generalized octonions which include real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. We make a classification according to the inner product and vector parts and give the polar forms for lightlike generalized octonions. Furthermore, the matrix representations of the generalized octonions are given and some properties of these representations are achieved. Also, powers and roots of the matrix representations are presented. All calculations in the article are achieved by using MATLAB R2023a and these codes are presented with an illustrative example. Full article
(This article belongs to the Special Issue Symmetry in Geometric Mechanics and Mathematical Physics)
13 pages, 336 KiB  
Article
Generalized Galilean Rotations
by Harun Barış Çolakoğlu, İskender Öztürk, Oğuzhan Çelik and Mustafa Özdemir
Symmetry 2024, 16(11), 1553; https://doi.org/10.3390/sym16111553 - 20 Nov 2024
Cited by 1 | Viewed by 905
Abstract
In this article, we give rotational motions on any straight line or any parabola in a scalar product space. To achieve this goal, we first define the generalized Galilean scalar product and determine the generalized Galilean skew symmetric and orthogonal matrices. Then, using [...] Read more.
In this article, we give rotational motions on any straight line or any parabola in a scalar product space. To achieve this goal, we first define the generalized Galilean scalar product and determine the generalized Galilean skew symmetric and orthogonal matrices. Then, using the well-known Rodrigues, Cayley, and Householder maps, we produce the generalized Galilean rotation matrices. Finally, we show that these rotation matrices can also be used to determine parabolic rotational motion. Full article
(This article belongs to the Section Mathematics)
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11 pages, 250 KiB  
Article
Explicit Parameterizations of Ortho-Symplectic Matrices in R4
by Clementina D. Mladenova and Ivaïlo M. Mladenov
Mathematics 2024, 12(16), 2439; https://doi.org/10.3390/math12162439 - 6 Aug 2024
Cited by 2 | Viewed by 904
Abstract
Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional [...] Read more.
Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension. Full article
(This article belongs to the Section B: Geometry and Topology)
29 pages, 3044 KiB  
Article
Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory
by Agustín Moreno Cañadas, Ismael Gutierrez and Odette M. Mendez
Symmetry 2024, 16(5), 570; https://doi.org/10.3390/sym16050570 - 6 May 2024
Cited by 4 | Viewed by 1772
Abstract
Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration [...] Read more.
Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. Full article
(This article belongs to the Special Issue Symmetry in Graph Algorithms and Graph Theory III)
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23 pages, 331 KiB  
Article
Tricomi Continuants
by Emanuele Munarini
Mathematics 2024, 12(3), 401; https://doi.org/10.3390/math12030401 - 26 Jan 2024
Cited by 1 | Viewed by 1060
Abstract
In this paper, we introduce and study the Tricomi continuants, a family of tridiagonal determinants forming a Sheffer sequence closely related to the Tricomi polynomials and the Laguerre polynomials. Specifically, we obtain the main umbral operators associated with these continuants and establish some [...] Read more.
In this paper, we introduce and study the Tricomi continuants, a family of tridiagonal determinants forming a Sheffer sequence closely related to the Tricomi polynomials and the Laguerre polynomials. Specifically, we obtain the main umbral operators associated with these continuants and establish some of their basic relations. Then, we obtain a Turan-like inequality, some congruences, some binomial identities (including a Carlitz-like identity), and some relations with the Cayley continuants. Furthermore, we show that the infinite Hankel matrix generated by the Tricomi continuants has an LDU-Sheffer factorization, while the infinite Hankel matrix generated by the shifted Tricomi continuants has an LTU-Sheffer factorization. Finally, by the first factorization, we obtain the linearization formula for the Tricomi continuants and its inverse. Full article
13 pages, 343 KiB  
Article
Kirchhoff Index and Degree Kirchhoff Index of Tetrahedrane-Derived Compounds
by Duoduo Zhao, Yuanyuan Zhao, Zhen Wang, Xiaoxin Li and Kai Zhou
Symmetry 2023, 15(5), 1122; https://doi.org/10.3390/sym15051122 - 20 May 2023
Cited by 2 | Viewed by 1892
Abstract
Tetrahedrane-derived compounds consist of n crossed quadrilaterals and possess complex three-dimensional structures with high symmetry and dense spatial arrangements. As a result, these compounds hold great potential for applications in materials science, catalytic chemistry, and other related fields. The Kirchhoff index of a [...] Read more.
Tetrahedrane-derived compounds consist of n crossed quadrilaterals and possess complex three-dimensional structures with high symmetry and dense spatial arrangements. As a result, these compounds hold great potential for applications in materials science, catalytic chemistry, and other related fields. The Kirchhoff index of a graph G is defined as the sum of resistive distances between any two vertices in G. This article focuses on studying a type of tetrafunctional compound with a linear crossed square chain shape. The Kirchhoff index and degree Kirchhoff index of this compound are calculated, and a detailed analysis and discussion is conducted. The calculation formula for the Kirchhoff index is obtained based on the relationship between the Kirchhoff index and Laplace eigenvalue, and the number of spanning trees is derived for linear crossed quadrangular chains. The obtained formula is validated using Ohm’s law and Cayley’s theorem. Asymptotically, the ratio of Kirchhoff index to Wiener index approaches one-fourth. Additionally, the expression for the degree Kirchhoff index of the linear crossed quadrangular chain is obtained through the relationship between the degree Kirchhoff index and the regular Laplace eigenvalue and matrix decomposition theorem. Full article
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