We prove a type of converse of the Banach contraction mapping theorem for metric spaces: if X is a T 1 topological space and f : X ! X is a function with unique xed point a such that f n (x) converges to a for each x 2 X , then there exists a distance function d on X such that f is a contraction on the complete ultrametric space (X; d) with contractivity factor 1 2. We explore properties of the...

Given a matroid M on the ground set E, the Bergman fan B̃(M), or space of M -ultrametrics, is a polyhedral complex in RE which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point ω in RE , we wish to find an M -ultrametric which is closest to it in the ...

The accuracy of the Fitch method for reconstructing ancestral states on ultrametric phylogenetic trees is studied. Two recurrence relations for computing the accuracy are given here. Using these relations, we analyze the convergence of the accuracy of the Fitch method for reconstructing the root state on a complete binary tree of 2( n ) leaves as n goes to infinity, present a closed-form formul...

Abstract We show that the group of isometries an ultrametric normed space can be seen as a kind fractal. Then, we apply this description to study counterparts some classical problems in Archimedean analysis, such so called Problème des rotations de Mazur or Tingley’s problem. In particular, it turns out that, contrast with case real spaces, between spaces very far from being linear.

The aim of this study is to prove the existence and uniqueness fixed point common theorems for self-mappings in modular ultrametric spaces. These are proved under varying contractive circumstances without property spherical completeness. As a consequence, examples problems correctly formulated. an application, well-posedness problem proved. This expands on prior research space provide more comp...

Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to space $(X, d)$. The main results paper completely describe for equality $$ \rho(x, y) = f(d(\Phi(x), \Phi(y))) holds every $(Y, \rho) \in WS(X, d)$, weak similarity $\Phi \colon Y \to X$, and all $x$, $y Y$ with some (pseudoultrametric) preserving function $f$ depending on $\Phi$.

Given a distance matrix M that specifies the pairwise evolutionary distances between n species, the phylogenetic tree reconstruction problem asks for an edge-weighted phylogenetic tree that satisfies M, if one exists. We study some extensions of this problem to rooted phylogenetic networks. Our main result is an O(n(2) log n)-time algorithm for determining whether there is an ultrametric galled...