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A square matrix of order $n$ with $n\geq 2$ is called a \textit{permutative matrix} or permutative when all its rows (up to the first one) are permutations of precisely its first row. In this paper, the spectra of a class of permutative…

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of…

A $\delta$-vector $\delta(\Pc)= (\delta_0, \delta_1, ..., \delta_d)$ is called shifted symmetric if $\delta_{d-i} = \delta_{i+1}$ for each $0 \leq i \leq [(d-1)/2]$. A natural family of $(0,1)$-polytopes with shifted symmetric…

Based on work presented in [4], we define $S^2$-Upper Triangular Matrices and $S^2$-Lower Triangular Matrices, two special types of $d\times d(2d-1)$ matrices generalizing Upper and Lower Triangular Matrices, respectively. Then, we show…

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the…

Spatial networks are networks where nodes are located in a space equipped with a metric. Typically, the space is two-dimensional and until recently and traditionally, the metric that was usually considered was the Euclidean distance. In…

This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ be a positive semidefinite $d\times d$ complex matrix and let $\mathbf a=(a_i)_{i\in\mathbb{I}_k}\in \mathbb{R}_{>0}^k$, indexed by $\mathbb{I}_k=\{1,\ldots,k\}$, be a $k$-tuple of positive numbers. Let…

We study the notion of a distance between different AdS vacua of string theory. The distance is measured by a metric that is derived by taking an off-shell quadratic variation of the effective action, and evaluating it over families of…

We present the simulation of the time evolution of the distance matrix. The result is the node-node distance distribution for various kinds of networks. For the exponential trees, analytical formulas are derived for the moments of the…

Let ${\cal T}=(T,w)$ be a positive-weighted tree with at least $n$ vertices. For any $i,j \in \{1,...,n\}$, let $D_{i,j} ({\cal T})$ be the weight of the unique path in $T$ connecting $i$ and $j$. The $D_{i,j} ({\cal T})$ are called…

Matrix Factorization plays an important role in machine learning such as Non-negative Matrix Factorization, Principal Component Analysis, Dictionary Learning, etc. However, most of the studies aim to minimize the loss by measuring the…

The Earth movers distance (EMD) is a measure of distance between probability distributions which is at the heart of mass transportation theory. Recent research has shown that the EMD plays a crucial role in studying the potential impact of…

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…

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…

We introduce a metric on the set of permutations of given order, which is a weighted generalization of Kendall's $\tau$ rank distance and study its properties. Using the edge graph of a permutohedron, we give a criterion which guarantees…

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each…

In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices…

A 1989 result of Duarte asserts that for a given tree T on n vertices, a fixed vertex i, and two sets of distinct real numbers L, M of sizes n and n-1, respectively, such that M strictly interlaces L, there is a real symmetric matrix A such…

For an $n\times n$ diagonally dominant matrix $T=(t_{i,j})_{n\times n}$ with positive elements satisfying certain bounding conditions, we propose to use a diagonal matrix $S=(s_{i,j})_{n\times n}$ to approximate the inverse of $T$, where…