The Experts below are selected from a list of 30 Experts worldwide ranked by ideXlab platform
Abraham A Ungar - One of the best experts on this subject based on the ideXlab platform.
-
Gyrogroups the grouplike loops in the service of hyperbolic geometry and einstein s special theory of relativity
arXiv: Mathematical Physics, 2013Co-Authors: Abraham A UngarAbstract:In this era of an increased interest in loop theory, the Einstein velocity addition law has fresh resonance. One of the most fascinating aspects of recent work in Einstein’s special theory of relativity is the emergence of special grouplike loops. The special grouplike loops, known as Gyrocommutative Gyrogroups, have thrust the Einstein velocity addition law, which previously has operated mostly in the shadows, into the spotlight. We will find that Einstein (Mobius) addition is a Gyrocommutative Gyrogroup operation that forms the setting for the Beltrami-Klein (Poincare) ball model of hyperbolic geometry just as the common vector addition is a commutative group operation that forms the setting for the standard model of Euclidean geometry. The resulting analogies to which the grouplike loops give rise lead us to new results in (i) hyperbolic geometry; (ii) relativistic physics; and (iii) quantum information and computation.
-
Decomposition of Groups into Twisted Subgroups and Subgroups
2007Co-Authors: Tuval Foguel, Abraham A UngarAbstract:.This article is subsequent to our previous one, entitled Involutory decomposition of groups into twisted subgroups and subgroups [7]. The twisted subgroups resulting from the involutory decomposition of groups into twisted subgroups and subgroups in [7] turn out to be Gyrocommutative Gyrogroups. In contrast, the twisted subgroups resulting from the (non-involutory) decomposition of groups into twisted subgroups and subgroups that we present in this article need not be Gyrocommutative. Twisted subgroups arise in the study of problems in computational complexity [1] and in the study of Gyrogroups [7]. Gyrogroups are grouplike structures that first arose in the study of Einstein's velocity addition in the special theory of relativity [23, 24]. We showed in [7] that any Gyrogroup is an extension of a group by a Gyrocommutative Gyrogroup. The Gyrogroups that we construct in this article demonstrate that this extension is not trivial. x1. Introduction Gyrogroup theory is an algebraic theo..
-
the relativistic composite velocity reciprocity principle
Foundations of Physics, 2000Co-Authors: Abraham A UngarAbstract:Gyrogroup theory [A. A. Ungar, Found. Phys. 27, 881–951 (1997)] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of Gyrogroup theory to capture analogies is demonstrated in this article by exposing the relativistic composite-velocity reciprocity principle. The breakdown of commutativity in the Einstein velocity addition ⊕ of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly (i) on one hand, the velocity reciprocal to the composite velocity u⊕v is −(u⊕v) and (ii) on the other hand, it is (−v)⊕(−u). But (iii) −(u⊕v)≠(−v)⊕(−u). We remove the confusion in (i), (ii), and (iii) by employing the Gyrocommutative Gyrogroup structure of Einstein's addition and, subsequently, present the relativistic composite-velocity reciprocity principle with the Thomas rotation that it involves.
-
The Relativistic Composite-Velocity Reciprocity Principle
2000Co-Authors: Abraham A UngarAbstract:Gyrogroup theory [A.A. Ungar, Thomas precession: its underlying Gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), pp. 881-951] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of Gyrogroup theory to capture analogies is demonstrated in this article by exposing the Relativistic Composite-Velocity Reciprocity Principle. The breakdown of commutativity in the Einstein velocity addition # of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly (i) on one hand the velocity reciprocal to the composite velocity u#v is -(u#v) and (ii) on the other hand it is (-v)#(-u). But, (iii) -(u#v) #= (-v)#(-u). We remove the confusion in (i), (ii) and (iii) by employing the Gyrocommutative Gyrogroup structure of Einstein's addition and, subsequ..
Teerapong Suksumran - One of the best experts on this subject based on the ideXlab platform.
-
the algebra of Gyrogroups cayley s theorem lagrange s theorem and isomorphism theorems
2016Co-Authors: Teerapong SuksumranAbstract:Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible Gyrocommutative Gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Mobius addition. In the second part of this paper, we show that the symmetric group of a Gyrogroup admits the Gyrogroup structure, thus obtaining an analog of Cayley’s theorem for Gyrogroups. We examine subGyrogroups, Gyrogroup homomorphisms, normal subGyrogroups, and quotient Gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange’s theorem for Gyrogroups and use this result to prove that Gyrogroups of particular order have the Cauchy property.
-
the algebra of Gyrogroups cayley s theorem lagrange s theorem and isomorphism theorems
2016Co-Authors: Teerapong SuksumranAbstract:Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible Gyrocommutative Gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Mobius addition. In the second part of this paper, we show that the symmetric group of a Gyrogroup admits the Gyrogroup structure, thus obtaining an analog of Cayley’s theorem for Gyrogroups. We examine subGyrogroups, Gyrogroup homomorphisms, normal subGyrogroups, and quotient Gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange’s theorem for Gyrogroups and use this result to prove that Gyrogroups of particular order have the Cauchy property.
Tuval Foguel - One of the best experts on this subject based on the ideXlab platform.
-
Decomposition of Groups into Twisted Subgroups and Subgroups
2007Co-Authors: Tuval Foguel, Abraham A UngarAbstract:.This article is subsequent to our previous one, entitled Involutory decomposition of groups into twisted subgroups and subgroups [7]. The twisted subgroups resulting from the involutory decomposition of groups into twisted subgroups and subgroups in [7] turn out to be Gyrocommutative Gyrogroups. In contrast, the twisted subgroups resulting from the (non-involutory) decomposition of groups into twisted subgroups and subgroups that we present in this article need not be Gyrocommutative. Twisted subgroups arise in the study of problems in computational complexity [1] and in the study of Gyrogroups [7]. Gyrogroups are grouplike structures that first arose in the study of Einstein's velocity addition in the special theory of relativity [23, 24]. We showed in [7] that any Gyrogroup is an extension of a group by a Gyrocommutative Gyrogroup. The Gyrogroups that we construct in this article demonstrate that this extension is not trivial. x1. Introduction Gyrogroup theory is an algebraic theo..
Krzysztof Rózga - One of the best experts on this subject based on the ideXlab platform.
-
ON CENTRAL EXTENSIONS OF Gyrocommutative
2012Co-Authors: Krzysztof RózgaAbstract:Central extensions of Gyrocommutative Gyrogroups (K-loops) are studied in order to clarify the status of a cocycle equation introduced by Smith and Ungar. A sufficient and necessary conditions under which a central invariant extension is a Gyrocommutative Gyrogroup are formulated in terms of a 2-cochain f(x, y). In particular, it is shown that for central invariant extensions of Gyrocommutative Gyrogroups defined by Cartan decompositions of simple Lie algebras, the corresponding f(x, y) satisfies the cocycle equation, provided an extension is a Gyrocommutative Gyrogroup. 1. Introduction. There has been a renewal of an interest in loop theory in recent years, concerning a special non-associative loop structure called a Gyrocommutative Gyrogroup, known also under the name of a K-loop. It began with a paper by A. Ungar [15], who pointed it out that the addition law of relativistic velocitie
Sejong Kim - One of the best experts on this subject based on the ideXlab platform.
-
distributivity on the gyrovector spaces
Kyungpook Mathematical Journal, 2015Co-Authors: Sejong KimAbstract:Abstract. As a vector space provides a fundamental tool for the study of Euclideangeometry, a gyrovector space provides an algebraic tool for the study of hyperbolic ge-ometry. In general, the gyrovector spaces do not satisfy the distributivity with scalarmultiplication. In this article, we see under what condition the distributivity with scalarmultiplication is satis ed. 1. IntroductionIn order to provide an algebraic tool to study Einstein’s relativistic velocitysum, A. A. Ungar [2] has introduced a notion of Gyrogroup and has developed to-gether the study of analytic hyperbolic geometry. The Gyrogroup is a most naturalextension of a group into the nonassociative algebra. The associativity (and thecommutativity) of group operations is salvaged in a suitably modified form, calleda gyroassociativity (and a gyrocommutativity). In Section 2 we introduce a notionof (Gyrocommutative) Gyrogroup with its examples.Later on it is known that Gyrocommutative Gyrogroups are equivalent to Bruckloops (see [1]). To elaborate a precise language, we prefix a