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Involutory decomposition of groups into twisted subgroups and subgroups
 J. Group Theory
, 2000
"... Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ..."
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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups
Involutory Decomposition of Groups Into Twisted Subgroups and Subgroups
"... . An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are gro ..."
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. An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are grouplike structures which first arose in the study of Einstein's velocity addition in the special theory of relativity. Particularly, we show that every gyrogroup is a twisted subgroup and that, under general specified conditions, twisted subgroups are gyrocommutative gyrogroups. Moreover, we show that gyrogroups abound in group theory and that they possess rich structure. x1. Introduction Under general conditions, twisted subgroups are near subgroups [1]. Feder and Vardi [4] introduced the concept of a near subgroup of a finite group as a tool to study problems in computational complexity involving the class NP . Aschbacher provided a conceptual base for studying near subgroups demonstrating...
The Relativistic CompositeVelocity Reciprocity Principle
, 2000
"... 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. 881951] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector add ..."
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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. 881951] 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 CompositeVelocity 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 compositevelocity 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...
From the Group SL(2, C) to Gyrogroups and Gyrovector Spaces and Huperbolic Geometry
"... this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and gyrovector spaces. This natural way convincingly demonstrates that the theory of gyrogroups and gyrovector spaces provides a most powerful formalism for dealing with the Lorentz group and hyper ..."
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this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and gyrovector spaces. This natural way convincingly demonstrates that the theory of gyrogroups and gyrovector spaces provides a most powerful formalism for dealing with the Lorentz group and hyperbolic geometry, the geometry that governs the special theory of relativity as well as other areas of physics (see, for instance, [9] and [10]). It is therefore hoped that, following this article, gyrogroup and gyrovector space theoretic techniques will provide standard tools in the study of relativity physics and, as such, will become part of the lore learned by all explorers who are interested in relativity physics. Links between gyrogroups and other mathematical objects are presented in [11] [12] [13] and [14]. Furthermore, our approach to gyrogroups and scalar multiplication in a gyrogroup of gyrovectors can be used as a preparation for the study of a related, but more abstract study of Sabinin's odules in [15]. A related study of quasigroups in differential geometry is presented by Sabinin and Miheev on pp. 357  430 of [16]. 6 2 THE ALGEBRA OF THE SL(2; C) GROUP Let R 3 c be the set of all relativistically admissible velocities, R 3 c = fv 2 R 3 : kvk < cg It is the ball of radius c, c > 0, of the Euclidean 3space R 3 , c being the vacuum speed of light. A boost L(v) is a pure Lorentz transformation, that is, a Lorentz transformation without rotation, parametrized by a velocity parameter v 2 R 3 c . The boost L(v) is a linear transformation of spacetime coordinates which has the matrix representation Lm (v), v = (v 1 ; v 2 ; v 3 ) t , Lm (v) = 0 B B B B B B @ v c 2 v v 1 c 2 v v 2 c 2 v v 3 v v 1 1 + c 2 2 v v +1 v 2 1 c ...
and its Applications In Honor of Stephen Smale’s 80th Birthday 123 Editors
"... the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or ..."
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the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for
HYPERBOLIC BARYCENTRIC COORDINATES
"... ABSTRACT. A powerful and novel way to study Einstein’s special theory of relativity and its underlying geometry, the hyperbolic geometry of Bolyai and Lobachevsky, by analogies with classical mechanics and its underlying Euclidean geometry is demonstrated. The demonstration sets the stage for the ex ..."
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ABSTRACT. A powerful and novel way to study Einstein’s special theory of relativity and its underlying geometry, the hyperbolic geometry of Bolyai and Lobachevsky, by analogies with classical mechanics and its underlying Euclidean geometry is demonstrated. The demonstration sets the stage for the extension of the notion of barycentric coordinates in Euclidean geometry, first conceived by Möbius in 1827, into hyperbolic geometry. As an example for the application of hyperbolic barycentric coordinates, the hyperbolic midpoint of any hyperbolic segment, and the centroid and orthocenter of any hyperbolic triangle are determined.
Computation in Mind 1 COMPUTATION IN MIND
"... “Nous ne sommes que des nains perchés sur les épaules des géants; nous voyons ainsi plus loin et mieux qu’eux, mais nous ne serions rien s’ils ne nous portaient de toute leur hauteur. ” 1 Bernard de Chartres (12th century) Like Philosophy, Mathematics deals with abstract ideas, i.e. immaterial objec ..."
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“Nous ne sommes que des nains perchés sur les épaules des géants; nous voyons ainsi plus loin et mieux qu’eux, mais nous ne serions rien s’ils ne nous portaient de toute leur hauteur. ” 1 Bernard de Chartres (12th century) Like Philosophy, Mathematics deals with abstract ideas, i.e. immaterial objects which inhabit and work in the Mind. The chapter “Computation in Mind ” proposes to use the power of computational mathematics to explore some of the universal ways by which each human mind builds its “imago mundi”, its image of the world. The primary focus is put on epistemology and the use of mathematics is minimal, relegating the necessary technical details to an appendix. The Chapter develops the viewpoint that Science and Mind are mirror images for each other which use specific calculations over three kinds of numbers. It presents some epistemological consequences of the lack of associativity or commutativity for the two basic operations which are × and + when the calculations are performed over vectors or over matrices. The evolutive nature of the scientific logic is illustrated on several examples. In particular induction in computation suggests that any matrix ring can be usefully considered as a structure of macroscalars.