Related papers: Spherical Localisation in Convex and Metric Geomet…

We survey different perspectives on the stochastic localization process of Eldan, a powerful construction that has had many exciting recent applications in high-dimensional probability and algorithm design. Unlike prior surveys on this…

A new formalism is presented for analytically obtaining the probability density function, \( P_{n}(s) \), for the distance between two random points in an \( n \)-dimensional sphere of radius \( R \). Our formalism allows \( P_{n}(s) \) to…

A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. We solve this problem using algorithms that employ…

This paper introduces a method of calculating and rendering shapes in a non-Euclidean 2D space. In order to achieve this, we developed a physics and graphics engine that uses hyperbolic trigonometry to calculate and subsequently render the…

Circular and radial geodesics are studied in the spacetime described by the $\gamma$ metric. Their behaviour is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical…

We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…

This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…

A new gridding technique for the solution of partial differential equations in cubical geometry is presented. The method is based on volume penalization, allowing for the imposition of a cubical geometry inside of its circumscribing sphere.…

The area method is a decision procedure for geometry developed by Chou et al. in the 1990's. The method aims to reduce the specified hypothesis to an algebraically verifiable form by applying elimination lemmas. The order in which the…

T-SNE is a well-known approach to embedding high-dimensional data and has been widely used in data visualization. The basic assumption of t-SNE is that the data are non-constrained in the Euclidean space and the local proximity can be…

In this article, we describe and numerically implement a method for relativistic location in slightly curved but otherwise generic spacetimes. For terrestrial positioning in the context of Global Navigation Satellite Systems, our algorithm…

Here we analyze three dimensional analogues of the classical Crofton's formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of…

In this article, we present a space-frequency theory for spherical harmonics based on the spectral decomposition of a particular space-frequency operator. The presented theory is closely linked to the theory of ultraspherical polynomials on…

In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of…

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…

Convex clustering is a recent stable alternative to hierarchical clustering. It formulates the recovery of progressively coalescing clusters as a regularized convex problem. While convex clustering was originally designed for handling…

We consider the Neumann version of the spherical mean value operator and its variants in the space of smooth functions, distributions and compactly supported ones. Surjectivity and range characterization issues are addressed from the…

Spectral clustering is a fundamental technique in the field of data mining and information processing. Most existing spectral clustering algorithms integrate dimensionality reduction into the clustering process assisted by manifold learning…

A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation…