We then calculated the proportion of simulated data sets, as a function of the admixture rate, for which the deduced origin was the ancestral population of the two actual source populations (i.e. the native area in the case of the invasion of Europe by HA), considering either FST or assignment likelihood values (see Fig. S2).
In this study, the macro-, micro-, and nano- scale fragmentation of concrete was studied as a function of different admixtures and inclusions.
We derive a recursive expression for the expectation of the X-chromosomal admixture fraction as a function of sex-specific admixture parameters, demonstrating that the X-chromosomal admixture is obtained from a more complex formula than in the simple 2/3 and 1/3 weighting (Equation 1).
In an admixed population descended from two source populations, we derive the moments of the distribution of the fraction of autosomal admixture from a specific source population, as a function of sex-specific admixture parameters and time.
Because we consider an admixture process that is constant from one generation to the next and we assume h ≠ 0 and h ≠ 1, we can apply Theorem 3.1.2 of Cull et al. (2005) to Equation 29 to obtain the unique solution for E[ H1, g]: Figures 3 and 4 illustrate the expected admixture fraction as a function of g under constant admixture, as determined in Equation 30.
Therefore, the female X-chromosomal admixture fraction can equivalently be written as a function of the female X-chromosomal admixture fractions in the previous generation, H 1, g − 1, f X, and from two generations ago, H 1, g − 2, f X.
Analysis of the moments of the distribution of admixture as a function of time g can provide a way of understanding features of the distribution and its determinants in the historical admixture process itself.
(8 Using Equations 7 and 8, we can analyze the distribution of the fraction of admixture as a function of the time g and the parameters s 1, g f, s 1, g m, s 2, g f, s 2, g m.
In Figure 8, we consider the variance of the fraction of admixture as a function of s 1 f, the female contribution from S1, on the x-axis, and s 1 m, the male contribution from S1, on the y-axis, computed using Equation 53.
Using Equation 47, we can plot the long-term limit of the variance of admixture proportions as a function of s1 and s2. Figure 8 illustrates that the long-term limit of V[ H1, g] is greater when s1 = s2 and s1 + s2 ≈ 1 (and thus h ≈ 0).
