Related papers: Explicit Formulae to Interchangeably use Hyperplan…
It has been shown beneficial for many types of data which present an underlying hierarchical structure to be embedded in hyperbolic spaces. Consequently, many tools of machine learning were extended to such spaces, but only few…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable…
We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems…
A new method is proposed to improve the numeri- cal simulation of time dependent problems when the initial and boundary data are not compatible. Unlike earlier methods limited to space dimension one, this method can be used for any space…
After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely…
Data uniformity is a concept associated with several semantic data characteristics such as lack of features, correlation and sample bias. This article introduces a novel measure to assess data uniformity and detect uniform pointsets on…
Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and…
We present a versatile formulation of the convolution operation that we term a "mapped convolution." The standard convolution operation implicitly samples the pixel grid and computes a weighted sum. Our mapped convolution decouples these…
We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces…
In the articles [1] and [2] of D. Finch, M. Haltmeier, S. Patch and D. Rakesh inversion formulas were found in any dimension $n\geq2$ for recovering a smooth function with compact support in the unit ball from spherical means centered on…
Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central…
The interaction of neural networks with physical equations offers a wide range of applications. We provide a method which enables a neural network to transform objects subject to given physical constraints. Therefore an U-Net architecture…
Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean…
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…
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and…
The arrangement of network nodes in hyperbolic spaces has become a widely studied problem, motivated by numerous results suggesting the existence of hidden metric spaces behind the structure of complex networks. Although several methods…
We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in [Kublik, Tanushev, and Tsai - J. Comp. Phys. 247: 279-311, 2013]. In this formulation, a boundary integral over…
Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data, upon which various hyperbolic networks have been developed. Existing hyperbolic networks encode geometric priors not…
The object of this study is an integral operator $\mathcal{S}$ which averages functions in the Euclidean upper half-space $\mathbb{R}_{+}^{n}$ over the half-spheres centered on the topological boundary $\partial \mathbb{R}_{+}^{n}$. By…