Numerical Methods for the 3-dimensional 2-body Problem in the Action-at-a-Distance Electrodynamics

We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses St\"urmer's extrapolation formula and assumes that a step of integration can be taken as a step of light ladder, which limits its use to shallow energies. The second method is an improvement of pre-existing iterative schemes, designed for stronger convergence and can be used at high-energies.Comment: 17 pages, 11 figure

Mitochondrial DNA analysis of eneolithic trypillians from Ukraine reveals neolithic farming genetic roots

The agricultural revolution in Eastern Europe began in the Eneolithic with the Cucuteni-Trypillia culture complex. In Ukraine, the Trypillian culture (TC) existed for over two millennia (ca. 5,400–2,700 BCE) and left a wealth of artifacts. Yet, their burial rituals remain a mystery and to date almost nothing is known about the genetic composition of the TC population. One of the very few TC sites where human remains can be found is a cave called Verteba in western Ukraine. This report presents four partial and four complete mitochondrial genomes from nine TC individuals uncovered in the cave. The results of this analysis, combined with the data from previous reports, indicate that the Trypillian population at Verteba carried, for the most part, a typical Neolithic farmer package of mitochondrial DNA (mtDNA) lineages traced to Anatolian farmers and Neolithic farming groups of central Europe. At the same time, the find of two specimens belonging to haplogroup U8b1 at Verteba can be viewed as a connection of TC with the Upper Paleolithic European populations. At the level of mtDNA haplogroup frequencies, the TC population from Verteba demonstrates a close genetic relationship with population groups of the Funnel Beaker/ Trichterbecker cultural complex from central and northern Europe (ca. 3,950–2,500 BCE)