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Finite Bruck loops
"... Let X be a magma; that is X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y ↦ → y ◦ x and L(x) : y ↦ → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are ..."
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Let X be a magma; that is X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y ↦ → y ◦ x and L(x) : y ↦ → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are permutations of X for all x ∈ X. In essence loops are groups without the associative axiom. See [Br, Pf] for further discussion of basic properties of loops. Certain classes of loops have received special attention: A loop X is a (right) Bol loop if it satisfies the (right) Bol identity (Bol): (Bol) or equivalently (Bol2) ((z ◦ x) ◦ y) ◦ x = z ◦ ((x ◦ y) ◦ x). R(x)R(y)R(x) = R((x ◦ y) ◦ x). for all x, y, z ∈ X. In a Bol loop, the subloop 〈x 〉 generated by x ∈ X is a group. Thus we can define x −1 and the order x  of x to be, respectively, the inverse of x and the
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 vecto ..."
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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...
INFINITE SIMPLE BOL LOOPS
, 2004
"... Abstract. If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops. ..."
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Abstract. If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.
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.
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
Comment.Math.Univ.Carolin. 45,2 (2004)275–278 275 Infinite simple Bol loops
"... Abstract. If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops. ..."
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Abstract. If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.
THE HYPERBOLIC SSQUARE AND MÖBIUS TRANSFORMATIONS
"... Abstract. Professor Themistocles M. Rassias ’ special predilection and contribution to the study of Möbius transformations is well known. Möbius transformations of the open unit disc of the complex plane and, more generally, of the open unit ball of any real inner product space, give rise to Mö ..."
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Abstract. Professor Themistocles M. Rassias ’ special predilection and contribution to the study of Möbius transformations is well known. Möbius transformations of the open unit disc of the complex plane and, more generally, of the open unit ball of any real inner product space, give rise to Möbius addition in the ball. The latter, in turn, gives rise to Möbius gyrovector spaces that enable the Poincare ́ ball model of hyperbolic geometry to be approached by gyrovector spaces, in full analogy with the common vector space approach to the standard model of Euclidean geometry. The purpose of this paper, dedicated to Professor Themistocles M. Rassias, is to employ the Möbius gyrovector spaces for the introduction of the hyperbolic square in the Poincare ́ ball model of hyperbolic geometry. We will find that the hyperbolic square is richer in