Related papers: Lattice Embedding of Heronian Simplices

Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural…

We introduce the n-th Heron variety as the realization space of the (squared) volumes of faces of an n-simplex. Our primary goal is to understand the extent to which Heron's formula, which expresses the area of a triangle as a function of…

Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the…

We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to…

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of…

Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We…

Lattice QCD was invented thirty years ago but only in the last few years has it finally fulfilled its promise as a precision tool for calculations in hadron physics. This review will cover the fundamentals of discretising QCD onto a…

In this paper, we explain a simple and uniform construction of a smooth integral model associated to a quadratic, (anti)-hermitian, and (anti)-quaternionic hermitian lattice defined over an arbitrary local field. As one major application,…

A metric polygon is a metric space comprised of a finite number of closed intervals joined cyclically. The second-named author and Ntalampekos recently found a method to bi-Lipschitz embed an arbitrary metric triangle in the Euclidean plane…

This paper is concerned with a covering problem of Euclidean space by a particular arrangement of cones that are not necessarily full and are allowed to overlap. The problem provides an equivalent geometric reformulation of the solvability…

We say a lattice tetrahedron whose centroid is its only non-vertex lattice point is lattice barycentric. The notation T(a,b,c) describes the lattice tetrahedron with vertices {0, e_1, e_2, a e_1 + b e_2 + c e_3}. Our result is that all such…

For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish…

Polyhedral semantics is a recently introduced branch of spatial modal logic, in which modal formulas are interpreted as piecewise linear subsets of an Euclidean space. Polyhedral semantics for the basic modal language has already been well…

We provide an algorithm for computing an effective basis of homology of elliptic surfaces over the complex projective line on which integration of periods can be carried out. This allows the heuristic recovery of several algebraic…

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice ${\bf Z}_n$. We also consider some relatives of those 28 partitions,…

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…

Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain an ${\mathsf M}_3$-sublattice). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are complementary $a, b \in I$ such that $a$ is to the…

We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of…

We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our…

In their paper about a dual of MacMahon's classical theorem on plane partitions, Ciucu and Krattenthaler proved a closed form product formula for the tiling number of a hexagon with a "shamrock", a union of four adjacent triangles, removed…