Related papers: On Lobachevsky's trigonometric formulae
Object detection, for the most part, has been formulated in the euclidean space, where euclidean or spherical geodesic distances measure the similarity of an image region to an object class prototype. In this work, we study whether a…
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a…
We show the existence of group-theoretic sections of the "etale-by-geometrically abelian" quotient of the arithmetic fundamental group of hyperbolic curves over $p$-adic local fields relative to a proper and flat model which are…
The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical…
The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is…
This is a tale describing the large scale geometry of Euclidean plane domains with their hyperbolic or quasihyperbolic distances. We prove that in any hyperbolic plane domain, hyperbolic and quasihyperbolic quasi-geodesics are the same…
A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as…
We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain N=2 supersymmetric gauge dual…
Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks…
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
In this paper, we intend to bring together the hyperbolic spinors, which are useful frameworks from mathematics to physics, and both spacelike and timelike framed curves in Minkowski 3-space $\mathbb{R}_1^3$, which are new type attractive…
Geometric representation learning has recently shown great promise in several machine learning settings, ranging from relational learning to language processing and generative models. In this work, we consider the problem of performing…
It is a classical fact in Euclidean geometry that the regular polygon maximizes area amongst polygons of the same perimeter and number of sides, and the analogue of this in non-Euclidean geometries has long been a folklore result. In this…
We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron…
We introduce decorated piecewise hyperbolic and spherical surfaces and discuss their discrete conformal equivalence. A decoration is a choice of circle about each vertex of the surface. Our decorated surfaces are closely related to…
We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by…
Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of…
Trajectory similarity is a cornerstone of trajectory data management and analysis. Traditional similarity functions often suffer from high computational complexity and a reliance on specific distance metrics, prompting a shift towards deep…
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which…
In a conservative and partially hyperbolic three-dimensional setting, we study three representative classes of diffeomorphisms: those homotopic to Anosov (or Derived from Anosov diffeomorphisms), diffeomorphisms in neighborhoods of the…