Related papers: Euclidean traveller in hyperbolic worlds

Complete hyperbolicity of small Euclidean balls with respect to a C^1-smooth almost complex structure standard at origin is improved to give a complete hyperbolicity of strictly pseudoconvex domains. More precise (and lower) regularity…

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove…

Using black-hole arguments with widely accepted premises, we show that it is extremely improbable that space is Euclidean, and that it is unspeakably improbable that space is hyperbolic. Independently, using an argument which makes no…

Graph-structured data are widespread in real-world applications, such as social networks, recommender systems, knowledge graphs, chemical molecules etc. Despite the success of Euclidean space for graph-related learning tasks, its ability to…

A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are…

This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the…

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean…

I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. G\"odel in…

The study of relativistic Coulomb systems in velocity space is prompted by the fact that the study of Newtonian Kepler/Coulomb systems in velocity space, although less familiar than the analytic solutions in ordinary space, provides a much…

We show that $3$-graphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing…

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $\pi$. In the case of general angles, we prove that there…

This second part on polygons in the hyperbolic plane is based on the first part which deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The topic here is the maximum question for the area of these…

We survey the distributional properties of progressively dilating sets under projection by covering maps, focusing on manifolds of constant sectional curvature. In the Euclidean case, we review previously known results and formulate some…

This paper is devoted to the problem of choosing the most suitable model of a geometrical system for describing the real crystallographic space. It has been shown that all 230 crystallographic groups used to describe the crystalline…

We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the…

How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…

Recent studies have demonstrated the potential of hyperbolic geometry for capturing complex patterns from interaction data in recommender systems. In this work, we introduce a novel hyperbolic recommendation model that uses geometrical…

Given a geodesic line $\gamma$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to…

In this study, we define a brief description of the hyperbolic and elliptic rotational surfaces using a curve and matrices in 4-dimensional semi Euclidean space. That is, we provide different types of rotational matrices, which are the…

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…