Related papers: A generalization of the 3d distance theorem

The normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}(G)$ of a graph $G$ is a natural generalization of the normalized Laplacian matrix, arising from the matrix of pairwise distances between vertices rather than the adjacency…

We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications…

In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational…

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic…

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\leq \dim A$. We show that if the map $A^k\to CH_0(A)$ has a $d$-dimensional fiber then $k\geq d+(\dim A+1)/2$. This extends results of the…

Let D be the distance matrix of a connected graph G and let nn(G), np(G) be the number of strictly negative and positive eigenvalues of D respectively. It was remarked in [1] that it is not known whether there is a graph for which np(G) >…

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the…

A finite set X in the d-dimensional Euclidean space is called an s-distance set if the set of Euclidean distances between any two distinct points of X has size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance set…

We investigate the problem of reconstructing a set $P\subseteq \mathbb{R}$ of distinct points, where the only information available about $P$ consists of the distances between some of the pairs of points. More precisely, we examine which…

This paper presents the generalization of weighted distances to modules and their computation through the chamfer algorithm on general point lattices. The first part is dedicated to formalization of definitions and properties (distance,…

A beautiful theorem of Zeckendorf states that every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. Such decompositions exist more generally, and much is known about them. First, for any positive…

A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes…

We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional…

Let G(x,y) and G_D(x,y) be the Green functions of rotationally invariant symmetric \alpha-stable process in R^d and in an open set D respectively, where 0<\alpha < 2. The inequality G_D(x,y)G_D(y,z)/G_D(x,z) \le c(G(x,y)+G(y,z)) is a very…

The paper introduces a class of distances for linear behaviors over finite time horizons. These distances allow for comparisons between finite-horizon linear behaviors represented by matrices of possibly different dimensions. They remain…

We prove a quantitative local limit theorem for the number of descents in a random permutation. Our proof uses a conditioning argument and is based on bounding the characteristic function $\phi(t)$ of the number of descents. We also…

Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erd\H{o}s-Falconer distance problem: If $|A|\gg…

Continuing our work on group-theoretic generalizations of the prime Ax-Katz Theorem, we give a lower bound on the $p$-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\ldots,f_r)$ of $r$ maps…

An $L(2,1)$-labelling of a finite graph $\Gamma$ is a function that assigns integer values to the vertices $V(\Gamma)$ of $\Gamma$ (colouring of $V(\Gamma)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least…

We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$…