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An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Since there is no effective formula to compute the distance of a point and a set, it is hard to…
An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points only.…
Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have…
Discrete Euclidian Spaces (DESs) are the starting point in the study of the major fields of the Isodimensional Discrete Mathematics (IDM). The isodimensional analysis is not an exception, being particularly interesting and fruitful the…
We prove a central limit theorem concerning the number of critical points in large cubes of an isotropic Gaussian random function on a Euclidean space.
A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show…
We study the problem of finding, in a real algebraic matrix group, the matrix closest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results; and among the…
For general data, the number of complex solutions to the likelihood equations is constant and this number is called the (maximum likelihood) ML-degree of the model. In this article, we describe the special locus of data for which the…
The classification loss functions used in deep neural network classifiers can be grouped into two categories based on maximizing the margin in either Euclidean or angular spaces. Euclidean distances between sample vectors are used during…
Categorical attributes with qualitative values are ubiquitous in cluster analysis of real datasets. Unlike the Euclidean distance of numerical attributes, the categorical attributes lack well-defined relationships of their possible values…
Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except 4-dimensional cases and 4-dimensional standard spheres. The class of such maps also…
We present two algorithms for large-scale low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive…
When we represent a network of sensors in Euclidean space by a graph, there are two distances between any two nodes that we may consider. One of them is the Euclidean distance. The other is the distance between the two nodes in the graph,…
This paper addresses the numerical computation of critical angles between two convex cones in finite-dimensional Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary…
We consider two disjoint sets of points. If at least one of the sets can be embedded into an Euclidean space, then we provide sufficient conditions for the two sets to be jointly embedded in one Euclidean space. In this joint Euclidean…
We establish upper bounds for the size of two-distance sets in Euclidean space and spherical two-distance sets. The main recipe for obtaining upper bounds is the spectral method. We construct Seidel matrices to encode the distance relations…
Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees…
We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify…
By using the metric projection onto a closed self-dual cone of the Euclidean space, M. S. Gowda, R. Sznajder and J. Tao have defined generalized lattice operations, which in the particular case of the nonnegative orthant of a Cartesian…
Random optical fields with two widely different correlation lengths generate far field speckle spots that are themselves highly speckled. We call such patterns speckled speckle, and study their critical points (singularities and stationary…