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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 ...
Hyperbolic Geometry, Nehari’s Theorem, Electric Circuits, and Analog Signal Processing
"... Abstract. Underlying many of the current mathematical opportunities in digital signal processing are unsolved analog signal processing problems. For instance, digital signals for communication or sensing must map into an analog format for transmission through a physical layer. In this layer we meet ..."
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Abstract. Underlying many of the current mathematical opportunities in digital signal processing are unsolved analog signal processing problems. For instance, digital signals for communication or sensing must map into an analog format for transmission through a physical layer. In this layer we meet a canonical example of analog signal processing: the electrical engineer’s impedance matching problem. Impedance matching is the design of analog signal processing circuits to minimize loss and distortion as the signal moves from its source into the propagation medium. This paper works the matching problem from theory to sampled data, exploiting links between H ∞ theory, hyperbolic geometry, and matching circuits. We apply J. W. Helton’s significant extensions of operator theory, convex analysis, and optimization theory to demonstrate new approaches and research
NONEUCLIDEAN PYTHAGOREAN TRIPLES, A PROBLEM OF EULER, AND RATIONAL POINTS ON K3 SURFACES
, 2006
"... Abstract. We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a nonEuclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an alg ..."
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Abstract. We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a nonEuclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler’s work on the second problem with a modern point of view. 1. Problem I: Pythagorean triples An ordinary Pythagorean triple is a triple (a, b, c) of positive integers satisfying a 2 + b 2 = c 2. Finding these is equivalent, by the Pythagorean theorem, to finding right triangles with integral sides. Since the equation is homogeneous, the problem for rational numbers is the same, up to a scale factor. Some Pythagorean triples, such as (3, 4, 5), have been known since antiquity. Euclid [6, X.28, Lemma 1] gives a method for finding such triples, which leads to a complete solution of the problem. The primitive Pythagorean triples are exactly the triples of integers (m 2 − n 2, 2mn, m 2 + n 2) for various choices of m, n (up to