Generalized Metrics

TL;DR

This paper explores various generalizations of metric spaces, including partial and strong partial metrics, and applies these concepts to DNA sequence scoring and fixed point theorems.

Contribution

It introduces new generalized metric frameworks that accommodate negative distances and self-distances, and applies them to biological data and fixed point theory.

Findings

Generalized metrics can model DNA sequence scoring.

Topological properties like convergence are studied in these new spaces.

Fixed point theorems are extended to these generalized metric spaces.

Abstract

The distance on a set is a comparative function. The smaller the distance between two elements of that set, the closer, or more similar, those elements are. Fr\'echet axiomatized the distance into what is today known as a metric. In this thesis we study the generalization of Fr\'echet's axioms in various ways including a partial metric, strong partial metric, partial $n-\mathfrak{M}$etric and strong partial $n-\mathfrak{M}$etric. Those generalizations allow for negative distances, non-zero distances between a point and itself and even the comparison of $n-$tuples. We then present the scoring of a DNA sequence, a comparative function that is not a metric but can be modeled as a strong partial metric. Using the generalized metrics mentioned above we create topological spaces and investigate convergence, limits and continuity in them. As an application, we discuss contractiveness in the language of our generalized metrics and present Banach-like fixed, common fixed and coincidence point theorems.