scholarly journals Mathieu’s theorem in the Poincar´e disc model of hyperbolic geometry

2011 ◽  
Vol 20 (1) ◽  
pp. 16-19
Author(s):  
CATALIN BARBU ◽  
Keyword(s):  
Hyperbolic Geometry ◽  
Disc Model

In this note, we present a proof of Mathieu’s hyperbolic theorem in the Poincare disc model of hyperbolic geometry.

scholarly journals The Formulas of Möbius-Bretschneider and Möbius-Cagnoli in the Poincaré Disc Model of Hyperbolic Geometry

2021 ◽  
Vol 32 (1) ◽  
pp. 31
Author(s):  
Gülcan Balakan ◽  
Oğuzhan Demirel
Keyword(s):  
Hyperbolic Geometry ◽  
Disc Model

In this paper, we present two gyroarea formulas (Möbius-Bretschneider’s formula and Möbius-Cagnoli’s formula) for Möbius gyroquadrilaterals in the Poincaré disc model of hyperbolic geometry.

scholarly journals The hyperbolic geometry of financial networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Martin Keller-Ressel ◽  
Stephanie Nargang
Keyword(s):  
Hyperbolic Geometry ◽  
Structural Changes ◽  
Banking System ◽  
Financial Networks ◽  
Stress Tests ◽  
Disc Model ◽  
European Banking ◽  
European Banks ◽  
Hyperbolic Embedding ◽  
Embedding Methods

AbstractBased on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we construct networks of European banks and demonstrate that the latent geometry of these financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. Using two different hyperbolic embedding methods, hydra+ and Mercator, this allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincaré disc model of hyperbolic geometry. We show that the latent dimensions of ‘popularity’ and ‘similarity’ in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system, independent of the embedding method that is used. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability. Based on our analysis we argue that embeddings into hyperbolic geometry can be used to monitor structural change in financial networks and are able to distinguish between changes in systemic relevance and other (peripheral) structural changes.

scholarly journals The Beckman–Quarles Theorem in Hyperbolic Geometry

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Oğuzhan Demirel ◽  
Damla Topal ◽  
Leyla Aslan
Keyword(s):  
Hyperbolic Geometry ◽  
Disc Model

In this paper, we present the counterpart of the Beckman–Quarles theorem in the Poincaré disc model of hyperbolic geometry to characterize the gyroisometries (hyperbolic isometries) with a single nonzero distance a ∈ 0,1 satisfying a 2 ∈ ℚ .

Testing the DISC-Model in Service Work: Predicting Emotional Exhaustion

2013 ◽  
Author(s):  
Natasja van Vegchel ◽  
Jan de Jonge ◽  
Christian Dormann ◽  
Wilmar Schaufeli
Keyword(s):  
Emotional Exhaustion ◽  
Service Work ◽  
Disc Model

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

scholarly journals A novel scatterer model for the security of key generation based on DISC model

2021 ◽  
Vol 1873 (1) ◽  
pp. 012075
Author(s):  
Yuanhang Yang ◽  
Xin Zeng ◽  
Xiaoping Xue
Keyword(s):  
Key Generation ◽  
Disc Model
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