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Keywords = gyrogroup

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9 pages, 492 KiB  
Article
The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry
by Abraham A. Ungar
Symmetry 2023, 15(8), 1487; https://doi.org/10.3390/sym15081487 - 27 Jul 2023
Cited by 1 | Viewed by 1832
Abstract
Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its [...] Read more.
Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
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21 pages, 650 KiB  
Review
Ptolemy’s Theorem in the Relativistic Model of Analytic Hyperbolic Geometry
by Abraham A. Ungar
Symmetry 2023, 15(3), 649; https://doi.org/10.3390/sym15030649 - 4 Mar 2023
Cited by 2 | Viewed by 2210
Abstract
Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of [...] Read more.
Ptolemy’s Theorem in Euclidean geometry, named after the Greek astronomer and mathematician Ptolemy, is well-known. By means of the relativistic model of hyperbolic geometry, we translate Ptolemy’s Theorem from Euclidean geometry into the hyperbolic geometry of Lobachevsky and Bolyai. The relativistic model of hyperbolic geometry is based on the Einstein addition of relativistically admissible velocities and, as such, it coincides with the well-known Beltrami–Klein ball model of hyperbolic geometry. The translation of Ptolemy’s Theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the relativistic model of analytic hyperbolic geometry gives rise. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics II)
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21 pages, 598 KiB  
Article
On Dihedralized Gyrogroups and Their Cayley Graphs
by Rasimate Maungchang and Teerapong Suksumran
Mathematics 2022, 10(13), 2276; https://doi.org/10.3390/math10132276 - 29 Jun 2022
Cited by 8 | Viewed by 1721
Abstract
The method of constructing the generalized dihedral group as a semidirect product of an abelian group and the group Z2 of integers modulo 2 is extended to the case of gyrogroups. This leads to the study of a new class of gyrogroups, [...] Read more.
The method of constructing the generalized dihedral group as a semidirect product of an abelian group and the group Z2 of integers modulo 2 is extended to the case of gyrogroups. This leads to the study of a new class of gyrogroups, which includes generalized dihedral groups and dihedral groups as a special case. In this article, we show that any dihedralizable gyrogroup can be enlarged to a dihedralized gyrogroup. Then, we establish algebraic properties of dihedralized gyrogroups as well as combinatorial properties of their Cayley graphs. Full article
(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)
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11 pages, 780 KiB  
Article
Hamiltonian Cycles in Cayley Graphs of Gyrogroups
by Rasimate Maungchang, Charawi Detphumi, Prathomjit Khachorncharoenkul and Teerapong Suksumran
Mathematics 2022, 10(8), 1251; https://doi.org/10.3390/math10081251 - 11 Apr 2022
Cited by 2 | Viewed by 2289
Abstract
In this study, we investigate Hamiltonian cycles in the right-Cayley graphs of gyrogroups. More specifically, we give a gyrogroup version of the factor group lemma and show that some right-Cayley graphs of certain gyrogroups are Hamiltonian. Full article
(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)
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12 pages, 302 KiB  
Article
Construction of 2-Gyrogroups in Which Every Proper Subgyrogroup Is Either a Cyclic or a Dihedral Group
by Soheila Mahdavi, Ali Reza Ashrafi, Mohammad Ali Salahshour and Abraham Albert Ungar
Symmetry 2021, 13(2), 316; https://doi.org/10.3390/sym13020316 - 14 Feb 2021
Cited by 12 | Viewed by 2246
Abstract
In this paper, a 2-gyrogroup G(n) of order 2n, n3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup [...] Read more.
In this paper, a 2-gyrogroup G(n) of order 2n, n3, is constructed in which every proper subgyrogroup is either a cyclic or a dihedral group. It is proved that the subgyrogroup lattice and normal subgyrogroup lattice of G(n) are isomorphic to the subgroup lattice and normal subgroup lattice of the dihedral group of order 2n, which causes us to use the name dihedral gyrogroup for this class of gyrogroups of order 2n. Moreover, all proper subgyrogroups of G(n) are subgroups. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
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10 pages, 277 KiB  
Article
On Quasi Gyrolinear Maps between Möbius Gyrovector Spaces Induced from Finite Matrices
by Keiichi Watanabe
Symmetry 2021, 13(1), 76; https://doi.org/10.3390/sym13010076 - 4 Jan 2021
Cited by 3 | Viewed by 1834
Abstract
We present some fundamental results concerning to continuous quasi gyrolinear operators between Möbius gyrovector spaces induced by finite matrices. Such mappings are significant like as operators induced by matrices between finite dimensional Hilbert spaces. This gives a novel approach to the study of [...] Read more.
We present some fundamental results concerning to continuous quasi gyrolinear operators between Möbius gyrovector spaces induced by finite matrices. Such mappings are significant like as operators induced by matrices between finite dimensional Hilbert spaces. This gives a novel approach to the study of mappings between Möbius gyrovector spaces that should correspond to bounded linear operators on real Hilbert spaces. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
23 pages, 373 KiB  
Article
Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups
by Jaturon Wattanapan, Watchareepan Atiponrat and Teerapong Suksumran
Symmetry 2020, 12(11), 1817; https://doi.org/10.3390/sym12111817 - 2 Nov 2020
Cited by 6 | Viewed by 1767
Abstract
A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup G such that G [...] Read more.
A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup G such that G contains a gyro-isomorphic copy of G. We then prove that every strongly topological gyrogroup G can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup G. We also study several properties shared by G and G, including gyrocommutativity, first countability and metrizability. As an application of these results, we prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
30 pages, 389 KiB  
Article
Isomorphism of Binary Operations in Differential Geometry
by Nikita E. Barabanov
Symmetry 2020, 12(10), 1634; https://doi.org/10.3390/sym12101634 - 3 Oct 2020
Cited by 2 | Viewed by 2237
Abstract
We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from [...] Read more.
We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
36 pages, 360 KiB  
Article
Differential Geometry and Binary Operations
by Nikita E. Barabanov and Abraham A. Ungar
Symmetry 2020, 12(9), 1525; https://doi.org/10.3390/sym12091525 - 16 Sep 2020
Cited by 5 | Viewed by 2374
Abstract
We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic [...] Read more.
We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
26 pages, 350 KiB  
Article
A Gyrogeometric Mean in the Einstein Gyrogroup
by Takuro Honma and Osamu Hatori
Symmetry 2020, 12(8), 1333; https://doi.org/10.3390/sym12081333 - 10 Aug 2020
Cited by 5 | Viewed by 1935
Abstract
In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We [...] Read more.
In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
40 pages, 500 KiB  
Review
A Spacetime Symmetry Approach to Relativistic Quantum Multi-Particle Entanglement
by Abraham A. Ungar
Symmetry 2020, 12(8), 1259; https://doi.org/10.3390/sym12081259 - 30 Jul 2020
Cited by 6 | Viewed by 3606
Abstract
A Lorentz transformation group SO(m, n) of signature (m, n), m, n  N, in m time and n space dimensions, is the group of pseudo-rotations of a [...] Read more.
A Lorentz transformation group SO(m, n) of signature (m, n), m, n  N, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature (m, n). Accordingly, the Lorentz group SO(1, 3) is the common Lorentz transformation group from which special relativity theory stems. It is widely acknowledged that special relativity and quantum theories are at odds. In particular, it is known that entangled particles involve Lorentz symmetry violation. We, therefore, review studies that led to the discovery that the Lorentz group SO(m, n) forms the symmetry group by which a multi-particle system of m entangled n-dimensional particles can be understood in an extended sense of relativistic settings. Consequently, we enrich special relativity by incorporating the Lorentz transformation groups of signature (m, 3) for all m  2. The resulting enriched special relativity provides the common symmetry group SO(1, 3) of the (1 + 3)-dimensional spacetime of individual particles, along with the symmetry group SO(m, 3) of the (m + 3)-dimensional spacetime of multi-particle systems of m entangled 3-dimensional particles, for all m  2. A unified parametrization of the Lorentz groups SO(m, n) for all m, n  N, shakes down the underlying matrix algebra into elegant and transparent results. The special case when (m, n) = (1, 3) is supported experimentally by special relativity. It is hoped that this review article will stimulate the search for experimental support when (m, n) = (m, 3) for all m  2. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
44 pages, 455 KiB  
Article
Binary Operations in the Unit Ball: A Differential Geometry Approach
by Nikita E. Barabanov and Abraham A. Ungar
Symmetry 2020, 12(7), 1178; https://doi.org/10.3390/sym12071178 - 16 Jul 2020
Cited by 8 | Viewed by 2983
Abstract
Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n N , and discover the properties that qualify these operations to the title addition despite the fact that, [...] Read more.
Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
13 pages, 244 KiB  
Article
Ordered Gyrovector Spaces
by Sejong Kim
Symmetry 2020, 12(6), 1041; https://doi.org/10.3390/sym12061041 - 22 Jun 2020
Cited by 9 | Viewed by 2558
Abstract
The well-known construction scheme to define a partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space we construct the partial order, called a gyro-order. We also give several inequalities of gyrolines and [...] Read more.
The well-known construction scheme to define a partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space we construct the partial order, called a gyro-order. We also give several inequalities of gyrolines and cogyrolines in terms of the gyro-order. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
37 pages, 1366 KiB  
Article
Orthogonal Gyrodecompositions of Real Inner Product Gyrogroups
by Milton Ferreira and Teerapong Suksumran
Symmetry 2020, 12(6), 941; https://doi.org/10.3390/sym12060941 - 3 Jun 2020
Cited by 13 | Viewed by 2704
Abstract
In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner [...] Read more.
In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups. Full article
(This article belongs to the Special Issue Symmetry and Geometry in Physics)
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9 pages, 256 KiB  
Article
Left Regular Representation of Gyrogroups
by Teerapong Suksumran
Mathematics 2020, 8(1), 12; https://doi.org/10.3390/math8010012 - 19 Dec 2019
Cited by 2 | Viewed by 2380
Abstract
In this article, we examine a subspace L gyr ( G ) of the complex vector space, L ( G ) = { f : f   is   a   function   from   G to C } , where G is [...] Read more.
In this article, we examine a subspace L gyr ( G ) of the complex vector space, L ( G ) = { f : f   is   a   function   from   G to C } , where G is a nonassociative group-like structure called a gyrogroup. The space L gyr ( G ) arises as a representation space for G associated with the left regular representation, consisting of complex-valued functions invariant under certain permutations of G. In the case when G is finite, we prove that dim ( L gyr ( G ) ) = 1 | γ ( G ) | ρ γ ( G ) | Fix ( ρ ) | , where γ ( G ) is the subgroup of Sym ( G ) generated by a class of permutations of G and Fix ( ρ ) = { a G : ρ ( a ) = a } . Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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