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Yuching You - One of the best experts on this subject based on the ideXlab platform.
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Equivalence of Two Gyrogroup Structures on Unit Balls
Results in Mathematics, 1995Co-Authors: Yuching YouAbstract:In special relativity all possible velocities form an open ball, R_c ^3, of radius c in R^3 with a binary operation, ⊕, given by Einstein’s velocity Composition Law. The concept of gyrogroups arose in the study of the algebraic structure underlying the resulting groupoid, (R_c ^3, ⊕), giving rise to a generalization of the usual abelian group theory. Two different gyrogroup structures were defined on the unit ball of the Euclidean space resulting in the so-called standard and nonstandard Lorentzian groups [ 14 ]. The aim of this paper is (i) to define gyrogroup isomorphism and hence (ii) to show that the two structures are isomorphic in the gyrogroup sense; and (iii) to show that the resulting two Lorentzian groups are isomorphic, in the ordinary group sense, as pointed out by Urbantke [ 23 ]. Our results also include determining the group of all continuous automorphisms of the unit ball, when the latter is considered as a gyrogroup.
Abraham A. Ungar - One of the best experts on this subject based on the ideXlab platform.
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parametric realization of the lorentz transformation group in pseudo euclidean spaces
arXiv: Mathematical Physics, 2015Co-Authors: Abraham A. UngarAbstract:The Lorentz transformation group $SO(m,n)$ is a group of Lorentz transformations of order $(m,n)$, that is, a group of special linear transformations in a pseudo-Euclidean space of signature $(m,n)$ that leave the pseudo-Euclidean inner product invariant. A parametrization of $SO(m,n)$ is presented, giving rise to the Composition Law of Lorentz transformations of order $(m,n)$ in terms of parameter Composition. The parameter Composition, in turn, gives rise to a novel group-like structure called a bi-gyrogroup. Bi-gyrogroups form a natural generalization of gyrogroups where the latter form a natural generalization of groups. Like the abstract gyrogroup, the abstract bi-gyrogroup can play a universal computational role which extends far beyond the domain of pseudo-Euclidean spaces.
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Thomas precession: a kinematic effect of the algebra of Einstein's velocity addition Law. Comments on 'Deriving relativistic momentum and energy: II. Three-dimensional case'
European Journal of Physics, 2006Co-Authors: Abraham A. UngarAbstract:The authors of a recently published paper (Sonego S and Pin M 2005 Eur. J. Phys. 26 851–6) have erroneously asserted that Einstein's velocity addition Law is associative. Moreover, they have attributed the alleged associativity of Einstein's velocity addition Law to 'The relativity principle[, which] requires that [Einstein's velocity addition] gives the Composition Law of a group'. Accordingly, we note that Einstein's velocity addition is non-associative and demonstrate that the breakdown of associativity and commutativity in Einstein's velocity addition Law results from the presence of Thomas precession.
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gyrogroups modeled on einstein s addition
2001Co-Authors: Abraham A. UngarAbstract:As we have seen in Chapter 1 and in the exercises of Section 13 of that chapter, the interplay between Einstein’s addition and Thomas precession is extraordinarily rich. We therefore extend it by abstraction thereby arriving at the notion of the gyrogroup, a concept which generalizes the notion of the group. The gyrogroup definition is modeled on the Einstein groupoid of relativistically admissible velocities and their Thomas precessions, where the Composition Law is given by Einstein’s velocity addition. Following the definition of a group, we will select key features of Einstein’s addition to extend the group notion to that of the gyrogroup.
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Extension of the Unit Disk Gyrogroup into the Unit Ball of Any Real Inner Product Space
Journal of Mathematical Analysis and Applications, 1996Co-Authors: Abraham A. UngarAbstract:Abstract The group of all holomorphic automorphisms of the complex unit disk consists of Mobius transformations involving translation-like holomorphic automorphisms and rotations. The former are called gyrotranslations . As opposed to translations of the complex plane, which are associative-commutative operations (i.e., their Composition Law is associative and commutative) forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations (i.e., their Composition Law is gyroassociative and gyrocommutative) forming a gyrogroup . The complex unit disk gyrogroup has previously been studied by the author ( Aequationes Math. 47 , 1994, 240–254). Employing analogies shared by complex numbers and linear transformations of vector spaces, we extend in this article the complex disk gyrogroup and its Mobius transformations into the ball of any real inner product space and its generalized Mobius transformations. A gyrogroup is a mathematical object which first arose in the study of relativistic velocities which, under velocity addition, form a nongroup gyrogroup, as opposed to prerelativistic velocities, which form a group under velocity addition. It has been discovered that the mathematical regularity, seemingly lost in the transition from prerelativistic to relativistic velocities, is concealed in a relativistic effect known as Thomas precession. In its abstract context, Thomas precession is called Thomas gyration, giving rise to our “gyroterminology.” Our gyroterminology, developed by the author ( Amer. J. Phys. 59 , 1991, 824–834), involves terms like gyrogroups, gyroassociative-gyrocommutative Laws, and gyroautomorphisms, in which we extensively use the prefix “gyro.”
Noah J. Cowan - One of the best experts on this subject based on the ideXlab platform.
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Navigation Functions on Cross Product Spaces
2010Co-Authors: Noah J. CowanAbstract:Abstract—Given two compact, connected manifolds with corners, and a navigation function (NF, a refined artificial potential function) on each manifold, this paper presents a simple Composition Law that yields a new NF on the cross product space. The method provides tunable “hooks ” for shaping the new potential function while still guaranteeing obstacle avoidance and essentially global convergence. The Composition Law is associative, and successive Compositions fold into a single, computational simple expression, enabling the practical construction of NFs on the Cartesian product of several manifolds. Index Terms—Morse theory, navigation functions, potential shaping. I
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Composing Navigation Functions on Cartesian Products of Manifolds with Boundary
2004Co-Authors: Noah J. CowanAbstract:Given two compact, simply connected manifolds with boundary, and a navigation function (NF) on each manifold, this paper presents a simple Composition Law that yields a new NF on the cross product space. The method provides tunable "hooks" for shaping the new potential function while still guaranteeing obstacle avoidance and essentially global convergence. The Composition Law is associative, and successive Compositions fold into a single, computational simple expression, enabling the practical construction of NFs on the Cartesian product of several manifolds
Manjul Bhargava - One of the best experts on this subject based on the ideXlab platform.
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higher Composition Laws i a new view on gauss Composition and quadratic generalizations
Annals of Mathematics, 2004Co-Authors: Manjul BhargavaAbstract:Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful Law of Composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow. Even today, two centuries later, this Law of Composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders. It is hence only natural to ask whether higher analogues of this Composition Law exist that could shed light on the structure of other algebraic number rings and fields. This article forms the first of a series of four articles in which our aim is precisely to develop such “higher Composition Laws”. In fact, we show that Gauss’s Law of Composition is only one of at least fourteen Composition Laws of its kind which yield information on number rings and their class groups. In this paper, we begin by deriving a general Law of Composition on 2×2×2 cubes of integers, from which we are able to obtain Gauss’s Composition Law on binary quadratic forms as a simple special case in a manner reminiscent of the group Law on plane elliptic curves. We also obtain from this Composition Law on 2× 2 × 2 cubes four further new Laws of Composition. These Laws of Composition are defined on 1) binary cubic forms, 2) pairs of binary quadratic forms, 3) pairs of quaternary alternating 2-forms, and 4) senary (six-variable) alternating 3-forms. More precisely, Gauss’s theorem states that the set of SL2(Z)-equivalence classes of primitive binary quadratic forms of a given discriminant D has an inherent group structure. The five other spaces of forms mentioned above (including the space of 2 × 2 × 2 cubes) also possess natural actions by special linear groups over Z and certain products thereof. We prove that, just like Gauss’s space of binary quadratic forms, each of these group actions has the following remarkable properties. First, each of these six spaces possesses only a single polynomial invariant for the corresponding group action, which we call the discriminant. This discriminant invariant is found to take only values that
Tolga Yarman - One of the best experts on this subject based on the ideXlab platform.
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thomas wigner rotation and thomas precession actualized approach
Canadian Journal of Physics, 2014Co-Authors: A L Kholmetskii, Tolga YarmanAbstract:We show that the explanation of Thomas–Wigner rotation and Thomas precession (TP) in the framework of special theory of relativity (STR) contains a number of points of inconsistency, in particular, with respect to physical interpretation of the Einstein velocity Composition Law in successive space–time transformations. In addition, we show that the common interpretation of TP falls into conflict with the causality principle. To eliminate such a conflict, we suggest considering the velocity parameter, entering into the expression for the frequency of TP, as being always related to a rotation-free Lorentz transformation. Such an assumption (which actually resolves any causal paradoxes with respect to TP), comes however to be in contradiction with the spirit of STR. The results obtained are discussed.