Related papers: Maximal distance minimizers for a rectangle
A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in…
If $D$ is a partially filled-in $(0,1)$-matrix with a unique completion to a $(0,1)$-matrix $M$ (with prescribed row and column sums), we say that $D$ is a {\em defining set} for $M$. A {\em critical set} is a minimal defining set (the…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its…
Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a…
The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum…
Minimax distance measure extracts the underlying patterns and manifolds in an unsupervised manner. The existing methods require a quadratic memory with respect to the number of objects. In this paper, we investigate efficient sampling…
The minimum distance of a code is an important concept in information theory. Hence, computing the minimum distance of a code with a minimum computational cost is a crucial process to many problems in this area. In this paper, we present…
Finite mixture models have long been used across a variety of fields in engineering and sciences. Recently there has been a great deal of interest in quantifying the convergence behavior of the \emph{mixing measure}, a fundamental object…
In this paper we prove that the shape optimization problem $$\min\left\{\lambda_k(\Omega):\ \Omega\subset\R^d,\ \Omega\ \hbox{open},\ P(\Omega)=1,\ |\Omega|<+\infty\right\},$$ has a solution for any $k\in\N$ and dimension $d$. Moreover,…
Let $X$ be a compact metric space which is locally absolutely retract and let $\phi: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence of $n$…
In this article we study point configurations minimizing the discrete energy on a compact Riemannian manifold, where the energy kernel is taken to be the Green's function for the Laplacian. We show that every point in a minimizing…
Let $M$ be a simple 3-manifold such that one component of $\partial M$, say $F$, has genus at least two. For a slope $\alpha$ on $F$, we denote by $M(\alpha)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood…
An expander code is a binary linear code whose parity-check matrix is the bi-adjacency matrix of a bipartite expander graph. We provide a new formula for the minimum distance of such codes. We also provide a new proof of the result that…
In this article we introduce the notion of badly approximable matrices of higher order using higher sucessive minima in $\mathbb R^d$. We prove that for order less than $d$, they have Lebesgue measure zero and the gaps between them still…
A maximal commutative subalgebra is a substructure in algebra with the greatest commutative property. By studying the lengths of maximal commutative subalgebras, one can more clearly characterize the structure of commutative subalgebras in…
We prove that the minimal size $M(\pi_n)$ of a maximal matching in the permutahedron $\pi_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(\pi_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron.…
A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many…
This work investigates minimal parametric networks in hyperspaces of closed subsets of metric spaces endowed with the Hausdorff distance. It is shown that the problems of finding such networks are nontrivial only within finiteness classes,…
The purpose of this paper is to extend the symmetry of maximals of the ring of a germ of reducible plane curve proved by Delgado to a relation between the relative maximals of a fractional ideal and the absolute maximals of its dual for any…
That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the \v{C}ech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that…