Solving integral equations in η→3π\eta\to 3\pi

A dispersive analysis of $\eta\to 3\pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for $\omega\to 3\pi$.Comment: 11 pages, 10 Figures. Version accepted for publication in EPJC. The ancillary files contain an updated set of fundamental solutions. The numerical differences to the former set are tiny, see the READMEv2 file for detail

Cranial anatomy and taxonomy of the erythrosuchid archosauriform ‘Vjushkovia triplicostata’ Huene, 1960, from the Early Triassic of European Russia

Erythrosuchidae are a globally distributed and important group of apex predators that occupied Early and Middle Triassic terrestrial ecosystems following the Permo-Triassic mass extinction. The stratigraphically oldest known genus of Erythrosuchidae is Garjainia Ochev, 1958, which is known from the late Early Triassic (late Olenekian) of European Russia and South Africa. Two species of Garjainia have been reported from Russia: the type species, Garjainia prima Ochev, 1958, and ‘Vjushkovia triplicostata’ von Huene, 1960, which has been referred to Garjainia as either congeneric (Garjainiatriplicostata) or conspecific (G. prima). The holotype of G. prima has received relatively extensive study, but little work has been conducted on type or referred material attributed to ‘V. triplicostata’. However, this material includes well-preserved fossils representing all parts of the skeleton and comprises seven individuals. Here, we provide a comprehensive description and review of the cranial anatomy of material attributed to ‘V. triplicostata’, and draw comparisons with G. prima. We conclude that the two Russian taxa are indeed conspecific, and that minor differences between them result from a combination of preservation or intraspecific variation. Our reassessment therefore provides additional information on the cranial anatomy of G. prima. Moreover, we quantify relative head size in erythrosuchids and other early archosauromorphs in an explicit phylogenetic context for the first time. Our results show that erythrosuchids do indeed appear to have disproportionately large skulls, but that this is also true for other early archosauriforms (i.e. proterosuchids), and may reflect the invasion of hypercarnivorous niches by these groups following the Permo-Triassic extinction.© 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited