scholarly journals Directions in orbits of geometrically finite hyperbolic subgroups

2020 ◽  
pp. 1-40
Author(s):  
CHRISTOPHER LUTSKO
Keyword(s):  
Hyperbolic Space ◽  
Nearest Neighbor ◽  
Discrete Subgroup ◽  
Discrete Subgroups ◽  
Statistical Characterization ◽  
Geometrically Finite ◽  
Finite Subgroups ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Apollonian Circle Packings

Abstract We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather they may equidistribute on a fractal subset. Understanding the behavior of these orbits near the boundary is central to Patterson–Sullivan theory and much further work. Our theorem characterises the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish a formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which also applies in the lattice case.

Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension

1996 ◽  
Vol 120 (4) ◽  
pp. 647-662 ◽  
Author(s):  
Sanju L. Velani
Keyword(s):  
Hyperbolic Space ◽  
Finite Groups ◽  
Discrete Subgroup ◽  
Dimensional Euclidean Space ◽  
Simultaneous Diophantine Approximation ◽  
Geometrically Finite ◽  
Geometrically Finite Group ◽  
Fundamental Polyhedron ◽  
Image Position ◽  
Geometrically Finite Groups

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and let be the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

scholarly journals Statistics of Associations Among IR Galaxies

1990 ◽  
Vol 124 ◽  
pp. 353-356
Author(s):  
Jack F. Gallimore ◽  
William C. Keel
Keyword(s):  
Correlation Function ◽  
Sample Size ◽  
Nearest Neighbor ◽  
Control Sample ◽  
Infrared Galaxies ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Luminous Objects ◽  
Tidal Tails

In the course of expanding the search of Kleinmann et.al. (1988) for distant, infrared-luminous objects, we noticed (as is often remarked) that a large number of infrared-selected galaxies have close neighbors or show merger characteristics (e.g. tidal tails, distorted disks). Because the sample size is large (567 infrared galaxies and 2182 field galaxies), this sample is ideal for statistically examining the importance of interactions among infrared galaxies. In particular, we compare the nearest-neighbor distribution and the two-point correlation function of our sample with that of a control sample of field galaxies.

scholarly journals From correlation functions to event shapes in QCD

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
D. Chicherin ◽  
J. M. Henn ◽  
E. Sokatchev ◽  
K. Yan
Keyword(s):  
Correlation Function ◽  
Correlation Functions ◽  
Event Shape ◽  
Proof Of Concept ◽  
Yang Mills ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Event Shapes ◽  
A Charge ◽  
Conserved Currents

Abstract We present a method for calculating event shapes in QCD based on correlation functions of conserved currents. The method has been previously applied to the maximally supersymmetric Yang-Mills theory, but we demonstrate that supersymmetry is not essential. As a proof of concept, we consider the simplest example of a charge-charge correlation at one loop (leading order). We compute the correlation function of four electromagnetic currents and explain in detail the steps needed to extract the event shape from it. The result is compared to the standard amplitude calculation. The explicit four-point correlation function may also be of interest for the CFT community.

scholarly journals Novel metric for hyperbolic phylogenetic tree embeddings

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Hirotaka Matsumoto ◽  
Takahiro Mimori ◽  
Tsukasa Fukunaga
Keyword(s):  
Phylogenetic Tree ◽  
Hyperbolic Space ◽  
Nearest Neighbor ◽  
Cancer Genomics ◽  
Geometric Approach ◽  
Artificial Intelligence Research ◽  
Phylogenetic Methods ◽  
Novel Approach ◽  
Multiple Trees ◽  
Euclidean Embeddings

Abstract Advances in experimental technologies, such as DNA sequencing, have opened up new avenues for the applications of phylogenetic methods to various fields beyond their traditional application in evolutionary investigations, extending to the fields of development, differentiation, cancer genomics, and immunogenomics. Thus, the importance of phylogenetic methods is increasingly being recognized, and the development of a novel phylogenetic approach can contribute to several areas of research. Recently, the use of hyperbolic geometry has attracted attention in artificial intelligence research. Hyperbolic space can better represent a hierarchical structure compared to Euclidean space, and can therefore be useful for describing and analyzing a phylogenetic tree. In this study, we developed a novel metric that considers the characteristics of a phylogenetic tree for representation in hyperbolic space. We compared the performance of the proposed hyperbolic embeddings, general hyperbolic embeddings, and Euclidean embeddings, and confirmed that our method could be used to more precisely reconstruct evolutionary distance. We also demonstrate that our approach is useful for predicting the nearest-neighbor node in a partial phylogenetic tree with missing nodes. Furthermore, we proposed a novel approach based on our metric to integrate multiple trees for analyzing tree nodes or imputing missing distances. This study highlights the utility of adopting a geometric approach for further advancing the applications of phylogenetic methods.

scholarly journals A two-point correlation function for Galactic halo stars

2011 ◽  
Vol 417 (3) ◽  
pp. 2206-2215 ◽  
Author(s):  
A. P. Cooper ◽  
S. Cole ◽  
C. S. Frenk ◽  
A. Helmi
Keyword(s):  
Correlation Function ◽  
Galactic Halo ◽  
Halo Stars ◽  
Point Correlation ◽  
Point Correlation Function

scholarly journals TOWARDS THE PROOF OF AGT-W RELATION: TODA 3-POINT FUNCTION AND ${\mathcal W}_{1+\infty}$ ALGEBRA

2013 ◽  
Vol 21 ◽  
pp. 138-139
Author(s):  
SHOTARO SHIBA
Keyword(s):  
Field Theory ◽  
Correlation Function ◽  
Point Function ◽  
Gauge Groups ◽  
Promising Direction ◽  
Quiver Gauge Theory ◽  
Conformal Theory ◽  
Conformal Field ◽  
Point Correlation ◽  
Point Correlation Function

The AGT-W relation is a conjecture of the nontrivial duality between 4-dim quiver gauge theory and 2-dim conformal field theory. We verify a part of this conjecture for all the cases of quiver gauge groups by studying on the property of 3-point correlation function of conformal theory. We also mention the relation to [Formula: see text] algebra as one of the promising direction towards the proof of the remaining part.

Sign in / Sign up

Export Citation Format

Share Document