The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (σ-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of triangle ABR vanishes. The triangle ar...

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...

We introduce a general method for automatic diagnostic evaluation of the pronunciation of individual non-native speakers based on a model of the human auditory system trained with native data stimuli. For each phoneme class, the Euclidean geometry similarity between the native perceptual domain and the non-native speech power spectrum domain is measured. The problematic phonemes for a given sec...

Deep learning presents notorious computational challenges. These challenges include, but are not limited to, the non-convexity of learning objectives and estimating the quantities needed for optimization algorithms, such as gradients. While we do not address the non-convexity, we present an optimization solution that exploits the so far unused “geometry” in the objective function in order to be...

There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We pres...

in this paper, we study fuzzy substructures in connection withhv-structures. the original idea comes from geometry, especially from thetwo dimensional euclidean vector space. using parameters, we obtain a largenumber of hyperstructures of the group-like or ring-like types. we connect,also, the mentioned hyperstructures with the theta-operations to obtain morestrict hyperstructures, as hv-groups...

By “elementary” plane geometry I mean the geometry of lines and circles—straightedge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s a...