Related papers: Distances on a one-dimensional lattice from noncom…
For any sequence of matrix algebras that converge to a coadjoint orbit we give explicit formulas that show that the distances between the matrix algebras (viewed as quantum metric spaces) converges to 0. In the process we develop a general…
The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds…
A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The goal of this paper is to initiate a systematic study of the geometry…
An introduction is given to some selected aspects of noncommutative geometry. Simple examples in this context are provided by finite sets and lattices. As an application, it is explained how the nonlinear Toda lattice and a discrete time…
We establish the equivalence between the regularity (rectifiability) of sets and suitable estimates on the oscillation of the gradient for smooth non-local distance functions. A prototypical example of such a distance was introduced, as…
In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of…
Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The…
This paper is a companion paper to [Lipman and Daubechies 2011]. We provide numerical procedures and algorithms for computing the alignment of and distance between two disk type surfaces. We provide a convergence analysis of the discrete…
Let $d_1\leq d_2\leq\ldots\leq d_{n\choose 2}$ denote the distances determined by $n$ points in the plane. It is shown that $\min\sum_i (d_{i+1}-d_i)^2=O(n^{-6/7})$, where the minimum is taken over all point sets with minimal distance $d_1…
Let P_1 and P_2 be two sets of points in the plane, so that P_1 is contained in a line L_1, P_2 is contained in a line L_2, and L_1 and L_2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs…
Effective estimates for the lattice point discrepancy of certain planar and three-dimensional domains. This paper provides estimates, with explicit constants, for the lattice point discrepancy of o-symmetric ellipse discs and ellipsoids in…
The general expression of the angular distance between two point sources as measured by an arbitrary observer is given. The modelling presented here is rigorous, covariant and valid in any space-time. The sources of light may be located at…
It is shown that $N$ points on a real algebraic curve of degree $n$ in $\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the…
We report on some findings concerning Connes' noncommutative distance $d$ on a weighted undirected graph $G$. Our main result is the lower bound $\ell/\Delta(G)\le d$ where $\ell$ is the geodesic distance and $\Delta(G)$ the degree of $G$.…
Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we…
We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a…
We study the distance on the Bruhat-Tits building of the group $\mathrm{SL}_d(\mathbb{Q}_p)$ (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with…
We study the hexagonal lattice $\mathbb{Z}[\omega]$, where $\omega^6=1$. More specifically, we study the angular distribution of hexagonal lattice points on circles with a fixed radius. We prove that the angles are equidistributed on…
The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$ belongs to a special class of algebraic cross-polytopes and simplices. It is shown that…
For any graph $G = (V,E)$ and positive integer $d$, the exact distance-$d$ graph $G_{=d}$ is the graph with vertex set $V$, where two vertices are adjacent if and only if the distance between them in $G$ is $d$. We study the exact…