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   Antil, Harbir; Dondl, Patrick; Striet, Ludwig (January 2021, SIAM Journal on Scientific Computing)
2. Fractional coalescent

   https://doi.org/10.1073/pnas.1810239116  

   Mashayekhi, Somayeh; Beerli, Peter (March 2019, Proceedings of the National Academy of Sciences)

   An approach to the coalescent, the fractional coalescent (f-coalescent), is introduced. The derivation is based on the discrete-time Cannings population model in which the variance of the number of offspring depends on the parameter α. This additional parameter α affects the variability of the patterns of the waiting times; values of $\alpha <1$lead to an increase of short time intervals, but occasionally allow for very long time intervals. When $\alpha =1$, the f-coalescent and the Kingman’s n-coalescent are equivalent. The distribution of the time to the most recent common ancestor and the probability that n genes descend from m ancestral genes in a time interval of length T for the f-coalescent are derived. The f-coalescent has been implemented in the population genetic model inference software Migrate. Simulation studies suggest that it is possible to accurately estimate α values from data that were generated with known α values and that the f-coalescent can detect potential environmental heterogeneity within a population. Bayes factor comparisons of simulated data with $\alpha <1$and real data (H1N1 influenza and malaria parasites) showed an improved model fit of the f-coalescent over the n-coalescent. The development of the f-coalescent and its inclusion into the inference program Migratefacilitates testing for deviations from the n-coalescent. 

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5. Approximation of fractional harmonic maps

   https://doi.org/10.1093/imanum/drac029  

   Antil, Harbir; Bartels, Sören; Schikorra, Armin (July 2022, IMA Journal of Numerical Analysis)

   Abstract This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields. We devise and analyze numerical methods for the approximation of various partial differential equations related to fractional harmonic maps. The compactness results imply the convergence of numerical approximations. Numerical examples on spin chain dynamics and point defects are presented to demonstrate the effectiveness of the proposed methods. 

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