Abraham A. Ungar's Autobiography
Abraham A.
Ungar
North Dakota State University
author
text
article
2016
eng
This autobiography presents the scientific living of Abraham Ungar and his role in Gyrogroups and Gyrovector spaces.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
1
3
https://mir.kashanu.ac.ir/article_12632_468d1999dd2516581663c3bdfdf27361.pdf
dx.doi.org/10.22052/mir.2016.12632
The Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces
Abraham A.
Ungar
North Dakota State University
author
text
article
2016
eng
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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
5
51
https://mir.kashanu.ac.ir/article_13636_01f4395f33b5311416ea86da2a1e38d4.pdf
dx.doi.org/10.22052/mir.2016.13636
Special Subgroups of Gyrogroups: Commutators, Nuclei and Radical
Teerapong
Suksumran
Department of Mathematics,
North Dakota State University
author
text
article
2016
eng
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 $[G\colon H]$ is the smallest prime dividing |G|, is normal in G.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
53
68
https://mir.kashanu.ac.ir/article_13907_7d64c578f99c83315fe22b9317d61813.pdf
dx.doi.org/10.22052/mir.2016.13907
Gyroharmonic Analysis on Relativistic Gyrogroups
Milton
Ferreira
Polytechnic Institute of Leiria, Portugal
author
text
article
2016
eng
Einstein, M\"{o}bius, and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time $\bkR^{n,1}.$ 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 $t,$ $t \rightarrow +\infty,$ the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
69
109
https://mir.kashanu.ac.ir/article_13908_cf1541b1fb78615f996b6bd5130e01e4.pdf
dx.doi.org/10.22052/mir.2016.13908
Bi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups
Teerapong
Suksumran
Department of Mathematics,
North Dakota State University,
Fargo, ND 58105, USA
author
Abraham A.
Ungar
Department of Mathematics,
North Dakota State University,
Fargo, ND 58105, USA
author
text
article
2016
eng
The decomposition $\Gamma=BH$ of a group $\Gamma$ into a subset B and a subgroup $H$ of $\Gamma$ 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 \mbox{operation} given by the Einstein velocity addition law of special relativity theory. The latter leads to the Lorentz transformation group $\so{1,n}$, $n\in\N$, in pseudo-Euclidean spaces of signature $(1, n)$. The study in this article is motivated by generalized Lorentz groups $\so{m, n}$, $m, n\in\N$, in pseudo-Euclidean spaces of signature $(m, n)$. Accordingly, this article explores the bi-decomposition $\Gamma = H_LBH_R$ of a group $\Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of $\Gamma$, along with the novel bi-gyrogroup structure of $B$ induced by the bi-decomposition of $\Gamma$. As an example, we show by methods of Clifford \mbox{algebras} that the quotient group of the spin group $\spin{m, n}$ possesses the bi-decomposition structure.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
111
142
https://mir.kashanu.ac.ir/article_13911_6682eb4d26d8342b14fa567f9cd92575.pdf
dx.doi.org/10.22052/mir.2016.13911
Normed Gyrolinear Spaces: A Generalization of Normed Spaces Based on Gyrocommutative Gyrogroups
Toshikazu
Abe
Niigata University, Japan
author
text
article
2016
eng
In this paper, we consider a generalization of the real normed spaces and give some examples.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
143
172
https://mir.kashanu.ac.ir/article_13912_e5b0cf6391ede1b9194da5c1e6957797.pdf
dx.doi.org/10.22052/mir.2016.13912
Gyrovector Spaces on the Open Convex Cone of Positive Definite Matrices
Sejong
Kim
Chungbuk National University
author
text
article
2016
eng
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 M\"{o}bius 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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
173
185
https://mir.kashanu.ac.ir/article_13922_cb42b70fd53473b27d5c95a2db3e19ba.pdf
dx.doi.org/10.22052/mir.2016.13922
An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach
Mahfouz
Rostamzadeh
University of Kurdistan,
416 Sanandaj, Iran
author
Sayed-Ghahreman
Taherian
Department of Mathematical Sciences,
Isfahan University of Technology,
84156 Isfahan, I R Iran
author
text
article
2016
eng
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 $SL(2,\mathbb C)$ 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 Poincar\'e model of hyperbolic geometry. For our purpose we consider the Poincar\'e plane model of hyperbolic geometry inside the complex open unit disc $\mathbb{D}$. Also we prove that this model is isomorphic to the Poincar\'e 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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
187
198
https://mir.kashanu.ac.ir/article_13923_45adafb5f2e2797a3bb789d969d94705.pdf
dx.doi.org/10.22052/mir.2016.13923
The Principle of Relativity: From Ungar’s Gyrolanguage for Physics to Weaving Computation in Mathematics
Françoise
Chatelin
Universite Toulouse 1 Capitole,
author
text
article
2016
eng
This paper extends the scope of algebraic computation based on a non standard $\times$ 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 $\pluscirc$ 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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
199
228
https://mir.kashanu.ac.ir/article_13924_66d8c3b9adb9b68702310db250cb14db.pdf
dx.doi.org/10.22052/mir.2016.13924
From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups
Abraham A.
Ungar
North Dakota State University
author
text
article
2016
eng
The Lorentz transformation of order $(m=1,n)$, $n\in\Nb$, is the well-known Lorentz transformation of special relativity theory. It is a transformation of time-space coordinates of the pseudo-Euclidean space $\Rb^{m=1,n}$ of one time dimension and $n$ space dimensions ($n=3$ in physical applications). A Lorentz transformation without rotations is called a {\it boost}. Commonly, the special relativistic boost is parametrized by a relativistically admissible velocity parameter $\vb$, $\vb\in\Rcn$, whose domain is the $c$-ball $\Rcn$ of all relativistically admissible velocities, $\Rcn=\{\vb\in\Rn:\|\vb\|<c\}$, 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 $n\ge1$ to all $m,n\in\Nb$, obtaining algebraic structures called a {\it bi-gyrogroup} and a {\it 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.
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
229
272
https://mir.kashanu.ac.ir/article_13925_125300b1d08e6cd1712f89d4ebf0423b.pdf
dx.doi.org/10.22052/mir.2016.13925
Completed Issue 2016-1
text
article
2016
eng
Mathematics Interdisciplinary Research
University of Kashan
2538-3639
1
v.
1
no.
2016
1
272
https://mir.kashanu.ac.ir/article_54858_6f1efd929c388572d8520914cebe37d5.pdf
dx.doi.org/10.22052/mir.2016.54858