University of KashanMathematics Interdisciplinary Research2538-36391120160101Abraham A. Ungar's Autobiography131263210.22052/mir.2016.12632ENAbraham A.UngarNorth Dakota State UniversityJournal Article20160115This autobiography presents the scientific living of Abraham Ungar and his role in Gyrogroups and Gyrovector spaces.University of KashanMathematics Interdisciplinary Research2538-36391120160101The Intrinsic Beauty, Harmony and Interdisciplinarity in Einstein Velocity Addition Law: Gyrogroups and Gyrovector Spaces5511363610.22052/mir.2016.13636ENAbraham A.UngarNorth Dakota State UniversityJournal Article20160122The 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.University of KashanMathematics Interdisciplinary Research2538-36391120160101Special Subgroups of Gyrogroups: Commutators, Nuclei and Radical53681390710.22052/mir.2016.13907ENTeerapongSuksumranDepartment of Mathematics,
North Dakota State University0000-0002-1239-5586Journal Article20160129A 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 $[Gcolon H]$ is the smallest prime dividing |G|, is normal in G.University of KashanMathematics Interdisciplinary Research2538-36391120160101Gyroharmonic Analysis on Relativistic Gyrogroups691091390810.22052/mir.2016.13908ENMiltonFerreiraPolytechnic Institute of Leiria, PortugalJournal Article20151207Einstein, 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.University of KashanMathematics Interdisciplinary Research2538-36391120160101Bi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups1111421391110.22052/mir.2016.13911ENTeerapongSuksumranDepartment of Mathematics,
North Dakota State University,
Fargo, ND 58105, USA0000-0002-1239-5586Abraham A.UngarDepartment of Mathematics,
North Dakota State University,
Fargo, ND 58105, USAJournal Article20160120The 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 $so{1,n}$, $ninN$, in pseudo-Euclidean spaces of signature $(1, n)$. The study in this article is motivated by generalized Lorentz groups $so{m, n}$, $m, ninN$, 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.University of KashanMathematics Interdisciplinary Research2538-36391120160101Normed Gyrolinear Spaces: A Generalization of Normed Spaces Based on Gyrocommutative Gyrogroups1431721391210.22052/mir.2016.13912ENToshikazuAbeNiigata University, JapanJournal Article20151126In this paper, we consider a generalization of the real normed spaces and give some examples.<br /> University of KashanMathematics Interdisciplinary Research2538-36391120160101Gyrovector Spaces on the Open Convex Cone of Positive Definite Matrices1731851392210.22052/mir.2016.13922ENSejongKimChungbuk National UniversityJournal Article20160202In 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. University of KashanMathematics Interdisciplinary Research2538-36391120160101An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach1871981392310.22052/mir.2016.13923ENMahfouzRostamzadehUniversity of Kurdistan,
416 Sanandaj, IranSayed-GhahremanTaherianDepartment of Mathematical Sciences,
Isfahan University of Technology,
84156 Isfahan, I R IranJournal Article20151129The 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.University of KashanMathematics Interdisciplinary Research2538-36391120160101The Principle of Relativity: From Ungar’s Gyrolanguage for Physics to Weaving Computation in Mathematics1992281392410.22052/mir.2016.13924ENFrançoiseChatelinUniversite Toulouse 1 Capitole,Journal Article20151119This 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.University of KashanMathematics Interdisciplinary Research2538-36391120160101From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups2292721392510.22052/mir.2016.13925ENAbraham A.UngarNorth Dakota State UniversityJournal Article20160126The Lorentz transformation of order $(m=1,n)$, $ninNb$, 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 $vb$, $vbinRcn$, whose domain is the $c$-ball $Rcn$ of all relativistically admissible velocities, $Rcn={vbinRn:|vb|<c}$, 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 $nge1$ to all $m,ninNb$, 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.