scholarly journals Self-representations of the Möbius group

2019 ◽  
Vol 2 ◽  
pp. 259-280
Author(s):  
Nicolas Monod ◽  
Pierre Py
Keyword(s):  
Möbius Group

scholarly journals MÖBIUS TRANSFORMATIONS OF POLYGONS AND PARTITIONS OF 3-SPACE

2008 ◽  
Vol 17 (11) ◽  
pp. 1401-1413 ◽  
Author(s):  
RICHARD RANDELL ◽  
JONATHAN SIMON ◽  
JOSHUA TOKLE
Keyword(s):  
Simple Closed Curve ◽  
Exact Formula ◽  
Mirror Image ◽  
Line Segments ◽  
Large N ◽  
Möbius Group ◽  
Euclidean Metrics ◽  
Straight Lines ◽  
Figure Eight Knot ◽  
Polygonal Knots

The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon [Formula: see text]. This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group. The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ3. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ3. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n. Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ3 and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ3 and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge.

SUPERSELECTION STRUCTURE AND LOCALIZED CONNES’ COCYCLES

1995 ◽  
Vol 07 (01) ◽  
pp. 133-160 ◽  
Author(s):  
HANS-WERNER WIESBROCK
Keyword(s):  
Field Theories ◽  
Local Theory ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Localization Property ◽  
Algebra Of Observables ◽  
Local Algebras ◽  
Möbius Group ◽  
Modular Groups

Let ρ be a localized endomorphism of the universal algebra of observables of a chiral conformal quantum field theory on a circle, see [16, 17, 23] or Chapter 1. Then ρ transforms covariant under the Möbius group. As was pointed out by D. Guido and R. Longo, [23], the covariance transformations are implemented by [Formula: see text] where Ad ∆it are modular groups to local algebras w.r.t. the vacuum vector, ut is a Connes-Radon-Nikodym-Cocycle. Using the localization property of ρ, one gets, at least for regular nets, localization properties of the cocycles. In this work we will do some steps into the opposite direction. Given a localized Connes’ cocycle of a local algebra. We will construct a localized endomorphism on the whole net. The features of this approach are twofold. Firstly sectors of finite and infinite statistical dimensions are handled on the same footing. Secondly it is a local theory right from the beginning. Moreover, soliton-like sectors can easily be incorporated. We will sketch on the last part. The program is carried through for a special class of conformal quantum field theories, the strongly additive ones.

scholarly journals A Novel Method for Designing Substitution Boxes Based on Mobius Group

2021 ◽  
Author(s):  
Bilal Arshad ◽  
Nasir Siddiqui ◽  
Zamir Hussain
Keyword(s):  
Statistical Analysis ◽  
Significant Role ◽  
Projective Line ◽  
Secret Data ◽  
Private Data ◽  
Substitution Boxes ◽  
Möbius Group ◽  
Novel Method ◽  
Group 1

Abstract It is as of now realized that the use of web in the present period is expanding very quickly, numerous users are sharing open and private data over web/internet. They need to ensure security of information. Encryption is one of the most significant viewpoint which is helpful to verify secret data. In encryption, cryptography assume a significant role. There are many algorithms accessible to shield information from unapproved access. In this work, we will use the strategy for cryptography to build up an algorithm which is increasingly viable for correspondence and information trade. In this study, we use a procedure to assemble new substitution boxes (S-boxes) parallel to coset diagram for the action of Mobius group [1] on projective line over . Afterward, the analyses of constructed S-boxes are performed on the basis of algebraic and statistical analysis.

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