Introduction

This document mathematically defines the general meaning of lineal admixture time. A simple special case of lineal admixture time is defined non-mathematically for a broad inter-disciplinary audience in Lineal admixture time: an interdisciplinary definition [1]. Some of the benefits of the more general mathematical meaning are:

1. the option to condition on specific genetic regions (e.g. X chromosome), and

2. compatibility with convenient mathematical models such as stationary processes.

Definition

We formally define lineal admixture time in terms of two givens:

1. a fertilization function $\mathrm{Fert}$ [2], and

2. a diploid categorization function $\mathrm{Cat}$.

A haploid retrograde sequence is a strictly decreasing sequence through $\mathrm{Hap}$ relative to $\mathrm{Fert}_H$, as defined in [2].

A diploid categorization function $\mathrm{Cat}: \mathrm{Dip}\mapsto \mathbb{Z}_{\ge 0}$ assigns each diploid to either zero (indexing an admixed population) or a positive integer (indexing non-admixed populations).

Given diploid categorization $\mathrm{Cat}$, the Most Recent Lineal Transition is the function $$\mathrm{Mrlt}: D \mapsto \Re \cup \{ -\infty \}$$ where $D$ is the domain of all haploid retrograde sequences and $$\mathrm{Mrlt}(S) := \sup \big\{ \mathrm{Fert}_H(S_i) : \exists i, \, \mathrm{Cat}(S_{i+1}) \not= \mathrm{Cat}(d), \, (d, s) = S_i \big\} \text{ .}$$

Given an observation time $\tau$, lineal admixture time is the function $$\mathrm{Lat}_{\tau}(S) := 0$$ when all haploids in $S$ fertilize diploids of the same non-admixed category, otherwise $$\mathrm{Lat}_{\tau}(S) := \tau - \mathrm{Mrlt}(S) \text{ .}$$

References