Related papers: Finding Fibonacci in the Hyperbolic Plane
We introduce and study a new random surface which we call the hyperbolic Brownian plane and which is the near-critical scaling limit of the hyperbolic triangulations constructed in arXiv:1401.3297. The law of the hyperbolic Brownian plane…
A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a…
We study the Dirichlet problem for harmonic maps between hyperbolic planes, under the assumption that the Euclidean harmonic extension of the boundary map is quasiconformal.
Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to…
Hyperbolic spaces, which have the capacity to embed tree structures without distortion owing to their exponential volume growth, have recently been applied to machine learning to better capture the hierarchical nature of data. In this…
We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In…
In this note, we show that the half-plane capacity of a subset of the upper half-plane is comparable to a simple geometric quantity, namely the euclidean area of the hyperbolic neighborhood of radius one of this set. This is achieved by…
In this paper we study volumes of moduli spaces of hyperbolic surfaces with geodesic, cusp and cone boundary components. We compute the volumes in some new cases, in particular when there exists a large cone angle. This allows us to give…
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the…
We survey several methods of extending quasisymmetric homeomorphisms of the real line to bi-Lipschitz diffeomorphisms of the upper half-plane with respect to the hyperbolic metric.
Using the method of C. V\"or\"os, we establish results on hyperbolic plane geometry, related to triangles. In this note we investigate the orthocenter, the concept of isogonal conjugate and some further center as of the symmedian of a…
In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is…
We discuss the known results and methods for determining root multiplicities for hyperbolic Kac--Moody algebras.
Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of…
We present a large scale hyperbolic recommender system. We discuss why hyperbolic geometry is a more suitable underlying geometry for many recommendation systems and cover the fundamental milestones and insights that we have gained from its…
Hyperbolic geometry plays an important role within function theory of the disk. For example, via the Schwarz-Pick Lemma, the isometries of the unit disk $\mathbb D$ with respect to this geometry are the conformal self-maps of $\mathbb D$.…
The notions of mass and center of mass are extended to laminae of the hyperbolic plane. The resulting formulae contain many surprises.
Hyperbolic elliptic parabolic disks can be described by the inequality $\frac{x^2}{C^2}+2y^2-2y\leq0$ ($0<C<1$) in the unit disk based Beltrami--Cayley--Klein model of the hyperbolic geometry, up to hyperbolic congruences. The hyperbolic…
In this work we investigate the following isoperimetric problem in the hyperbolic plane: to find the regions of prescribed area with minimal perimeter between two parallel horocycles. We give an explicit and detailed description of all such…
We examine a simple hard disc fluid with no long range interactions on the two dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is…