Remarks on Ultrametrics and Metric-Preserving Functions

Several Remarks on Dissimilarities and Ultrametrics

We investigate the relationships between tolerance relations, equivalence relations, and ultrametrics. The set of spheres associated to an ultrametric space has a tree structure that reflects a hierarchy on the set of equivalences associated to that space. We show that every ultrametric defined on a finite space is a linear combination of binary ultrametric and we introduce the notion of ultram...

Introduction to Metric-preserving Functions

Linear Functions Preserving Sut-Majorization on RN

Suppose $textbf{M}_{n}$ is the vector space of all $n$-by-$n$ real matrices, and let $mathbb{R}^{n}$ be the set of all $n$-by-$1$ real vectors. A matrix $Rin textbf{M}_{n}$ is said to be $textit{row substochastic}$ if it has nonnegative entries and each row sum is at most $1$. For $x$, $y in mathbb{R}^{n}$, it is said that $x$ is $textit{sut-majorized}$ by $y$ (denoted by $ xprec_{sut} y$) if t...

Linear Functions Preserving Multivariate and Directional Majorization

Let V and W be two real vector spaces and let &sim be a relation on both V and W. A linear function T : V → W is said to be a linear preserver (respectively strong linear preserver) of &sim if Tx &sim Ty whenever x &sim y (respectively Tx &sim Ty if and only if x &sim y). In this paper we characterize all linear functions T : M_{n,m} → M_{n,k} which preserve or strongly preserve multivariate an...

Remarks on bootstrap percolation in metric networks

We examine bootstrap percolation in d-dimensional, directed metric graphs in the context of recent measurements of firing dynamics in 2D neuronal cultures. There are two regimes, depending on the graph size N . Large metric graphs are ignited by the occurrence of critical nuclei, which initially occupy an infinitesimal fraction, f∗ → 0, of the graph and then explode throughout a finite fraction...