2022-05-17T06:06:02Z
https://mir.kashanu.ac.ir/?_action=export&rf=summon&issue=2154
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Abraham A. Ungar's Autobiography
Abraham A.
Ungar
This autobiography presents the scientific living of Abraham Ungar and his role in Gyrogroups and Gyrovector spaces.
Gyrogroup
Gyrovector space
2016
01
01
1
3
https://mir.kashanu.ac.ir/article_12632_468d1999dd2516581663c3bdfdf27361.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
The Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces
Abraham A.
Ungar
The only justification for the Einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in Einstein addition remained for a long time a mystery to be conquered. Accordingly, the aim of this expository article is to present (i) the Einstein relativistic vector addition, (ii) the resulting Einstein scalar multiplication, (iii) the Einstein relativistic mass, and (iv) the Einstein relativistic kinetic energy, along with remarkable analogies with classical results in groups and vector spaces that these Einstein concepts capture in gyrogroups and gyrovector spaces. Making the unfamiliar familiar, these analogies uncover the intrinsic beauty and harmony in the underlying Einstein velocity addition law of relativistically admissible velocities, as well as its interdisciplinarity.
Einstein addition
Gyrogroup
gyrovector space
hyperbolic geometry
special relativity
2016
01
01
5
51
https://mir.kashanu.ac.ir/article_13636_01f4395f33b5311416ea86da2a1e38d4.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Special Subgroups of Gyrogroups: Commutators, Nuclei and Radical
Teerapong
Suksumran
A gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by Einstein's velocity addition law. In this article, we present a few of groups sitting inside a gyrogroup G, including the commutator subgyrogroup, the left nucleus, and the radical of G. The normal closure of the commutator subgyrogroup, the left nucleus, and the radical of G are in particular normal subgroups of G. We then give a criterion to determine when a subgyrogroup H of a finite gyrogroup G, where the index [G: H] is the smallest prime dividing |G|, is normal in G.
Gyrogroup
Commutator subgyrogroup
nucleus of gyrogroup
subgyrogroup of prime index
radical of gyrogroup
2016
01
01
53
68
https://mir.kashanu.ac.ir/article_13907_7d64c578f99c83315fe22b9317d61813.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Gyroharmonic Analysis on Relativistic Gyrogroups
Milton
Ferreira
Einstein, Möbius, and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time Rn,1. Using the gyrolanguage we study their gyroharmonic analysis. Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them. Our study focus on the translation and convolution operators, eigenfunctions of the Laplace-Beltrami operator, Poisson transform, Fourier-Helgason transform, its inverse, and Plancherel's Theorem. We show that in the limit of large t, t⇒ +∞, the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on Rn, thus unifying hyperbolic and Euclidean harmonic analysis.
Gyrogroups
gyroharmonic analysis
Laplace Beltrami operator
eigenfunctions
generalized Helgason-Fourier transform
Plancherel's theorem
2016
01
01
69
109
https://mir.kashanu.ac.ir/article_13908_cf1541b1fb78615f996b6bd5130e01e4.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Bi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups
Teerapong
Suksumran
Abraham A.
Ungar
The decomposition Γ=BH of a group Γ into a subset B and a subgroup H of Γ induces, under general conditions, a group-like structure for B, known as a gyrogroup. The famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by the Einstein velocity addition law of special relativity theory. The latter leads to the Lorentz transformation group So(1,n), n∈N, in pseudo-Euclidean spaces of signature (1, n). The study in this article is motivated by generalized Lorentz groups So(m, n), m, n∈N, in pseudo-Euclidean spaces of signature (m, n). Accordingly, this article explores the bi-decomposition Γ= HLBHR of a group Γ into a subset B and subgroups HL and HR of Γ, along with the novel bi-gyrogroup structure of B induced by the bi-decomposition of Γ. As an example, we show by methods of Clifford algebras that the quotient group of the spin group Spin(m, n) possesses the bi-decomposition structure.
Bi-decomposition of group
Bi-gyrogroup
gyrogroup
Spin group
pseudo-orthogonal group
2016
01
01
111
142
https://mir.kashanu.ac.ir/article_13911_6682eb4d26d8342b14fa567f9cd92575.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Normed Gyrolinear Spaces: A Generalization of Normed Spaces Based on Gyrocommutative Gyrogroups
Toshikazu
Abe
In this paper, we consider a generalization of the real normed spaces and give some examples.
Gyrogroups
gyrovector spaces
2016
01
01
143
172
https://mir.kashanu.ac.ir/article_13912_e5b0cf6391ede1b9194da5c1e6957797.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Gyrovector Spaces on the Open Convex Cone of Positive Definite Matrices
Sejong
Kim
In this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional Euclidean spaces, which are the Einstein and Möbius gyrovector spaces. We introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and explore its interesting applications on the set of invertible density matrices. Finally we give an example of the gyrovector space on the unit ball of Hermitian matrices.
Gyrogroup
gyrovector space
gyroline
gyromidpoint
positive definite matrix
density matrix
2016
01
01
173
185
https://mir.kashanu.ac.ir/article_13922_cb42b70fd53473b27d5c95a2db3e19ba.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach
Mahfouz
Rostamzadeh
Sayed-Ghahreman
Taherian
The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. In [1], Ungar and Chen showed that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the Lorentz group and its underlying hyperbolic geometry. They defined the Chen addition and then Chen model of hyperbolic geometry. In this paper, we directly use the isomorphism properties of gyrovector spaces to recover the Chen's addition and then Chen model of hyperbolic geometry. We show that this model is an extension of the Poincaré model of hyperbolic geometry. For our purpose we consider the Poincaré plane model of hyperbolic geometry inside the complex open unit disc D. Also we prove that this model is isomorphic to the Poincaré model and then to other models of hyperbolic geometry. Finally, by gyrovector space approach we verify some properties of this model in details in full analogue with Euclidean geometry.
Hyperbolic geometry
gyrogroup
gyrovector space
Poincar'e model
analytic hyperbolic geometry
2016
01
01
187
198
https://mir.kashanu.ac.ir/article_13923_45adafb5f2e2797a3bb789d969d94705.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
The Principle of Relativity: From Ungar’s Gyrolanguage for Physics to Weaving Computation in Mathematics
Françoise
Chatelin
This paper extends the scope of algebraic computation based on a non standard × to the more basic case of a non standard +, where standard means associative and commutative. Two physically meaningful examples of a non standard + are provided by the observation of motion in Special Relativity, from either outside (3D) or inside (2D or more), We revisit the "gyro"-theory of Ungar to present the multifaceted information processing which is created by a metric cloth W, a relating computational construct framed in a normed vector space V, and based on a non standard addition denoted ⊕ whose commutativity and associativity are ruled (woven) by a relator, that is a map which assigns to each pair of admissible vectors in V an automorphism in Aut(W). Special attention is given to the case where the relator is directional.
Relator
noncommutativity
nonassociativity
Induced addition
organ
metric cloth
weaving information processing
cloth geometry
hyperbolic geometry
special relativity
liaison
geodesic
organic line
action at a distance
2016
01
01
199
228
https://mir.kashanu.ac.ir/article_13924_66d8c3b9adb9b68702310db250cb14db.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups
Abraham A.
Ungar
The Lorentz transformation of order (m=1,n), n∈N, is the well-known Lorentz transformation of special relativity theory. It is a transformation of time-space coordinates of the pseudo-Euclidean space Rm=1,n of one time dimension and n space dimensions (n=3 in physical applications). A Lorentz transformation without rotations is called a boost. Commonly, the special relativistic boost is parametrized by a relativistically admissible velocity parameter v, v ∈ Rcn, whose domain is the c-ball Rcn of all relativistically admissible velocities, Rcn={v ∈ Rn: ||v||<c}, where the ambient space Rn is the Euclidean n-space, and c>0 is an arbitrarily fixed positive constant that represents the vacuum speed of light. The study of the Lorentz transformation composition law in terms of parameter composition reveals that the group structure of the Lorentz transformation of order (m=1,n) induces a gyrogroup and a gyrovector space structure that regulate the parameter space Rcn. The gyrogroup and gyrovector space structure of the ball Rcn, in turn, form the algebraic setting for the Beltrami-Klein ball model of hyperbolic geometry, which underlies the ball Rcn. The aim of this article is to extend the study of the Lorentz transformation of order (m,n) from m=1 and n≥1 to all m,n∈N, obtaining algebraic structures called a bi-gyrogroup and a bi-gyrovector space. A bi-gyrogroup is a gyrogroup each gyration of which is a pair of a left gyration and a right gyration. A bi-gyrovector space is constructed from a bi-gyrocommutative bi-gyrogroup that admits a scalar multiplication.
Bi-gyrogroup
Bi-gyrovector space
eigenball
gyrogroup
Inner product of signature (m,n)
Lorentz transformation of order (m,n)
Pseudo-Euclidean space
special relativity
2016
01
01
229
272
https://mir.kashanu.ac.ir/article_13925_125300b1d08e6cd1712f89d4ebf0423b.pdf
Mathematics Interdisciplinary Research
Math. Interdisc. Res.
2538-3639
2538-3639
2016
1
1
Completed Issue 2016-1
2016
01
01
1
272
https://mir.kashanu.ac.ir/article_54858_6f1efd929c388572d8520914cebe37d5.pdf