In the Belgian Namur-Dinant Basin the boundary between the Lustin Formation and the Aisemont Formation (in the Lower rhenana conodont Biozone) corresponds to a fall followed by a rise in sea level, leading to the first recorded late Frasnian coral crisis. The Aisemont Formation records a transgressive-regressive cycle. Prior to the crisis most of the colonial rugose corals were members of the Family Disphyllidae, but these were largely replaced by corals belonging to the Phillipsastraeidae. Among these Frechastraea colonized all environments of the basin and was the main constructor of a biostromal reef in its northernmost proximal area, in the fair-weather wave zone. Corals did not encrust each other and therefore were not firmly attached, but they hug tightly the substrate (a dead coral colony) and rest closely on it to resist to the turbulence of waves. During the Silurian and Devonian, up until the late Frasnian crisis, shallow-water reefs in turbulent water were usually built by encrusting stromatoporoids, whereas rugose corals were restricted to waters of lower energy. Indeed, they were unable to encrust substrates, unlike stromatoporoids and post-Palaeozoic scleractinians, and to live in turbulent habitats. In Belgium argillaceous sedimentation prevented the development of stromatoporoids and provided an opportunity for the corals to colonize empty niches and to construct biostromes in relatively high-energy environments.At the same time Alveolites and stromatoporoids were dominant in a mid-proximal environment below the fair-weather wave base, but within the storm wave zone, where they also constructed biostromes.
In many applications, data and/or parameters are supported on non-Euclidean manifolds. It is important to take into account the geometric structure of manifolds in statistical analysis to avoid misleading results. Although there has been a considerable focus on simple and specific manifolds, there is a lack of general and easy-to-implement statistical methods for density estimation and modeling on manifolds. In this article, we consider a very broad class of manifolds: non-compact Riemannian symmetric spaces. For this class, we provide a very general mathematical result for easily calculating volume changes of the exponential and logarithm map between the tangent space and the manifold. This allows one to define statistical models on the tangent space, push these models forward onto the manifold, and easily calculate induced distributions by Jacobians. To illustrate the statistical utility of this theoretical result, we provide a general method to construct distributions on symmetric spaces. In particular, we define the log-Gaussian distribution as an analogue of the multivariate Gaussian distribution in Euclidean space. With these new kernels on symmetric spaces, we also consider the problem of density estimation. Our proposed approach can use any existing density estimation approach designed for Euclidean spaces and push it forward to the manifold with an easy-to-calculate adjustment. We provide theorems showing that the induced density estimators on the manifold inherit the statistical optimality properties of the parent Euclidean density estimator; this holds for both frequentist and Bayesian nonparametric methods. We illustrate the theory and practical utility of the proposed approach on the space of positive definite matrices.
'Generalist' is a qualifying word that best describes Edouard -'Eddy' -Poty in term of knowledge. He is not just a wellknown palaeontologist. Passionate in all branches of geology, he developed an exceptional background and became one of the very few feeling comfortable out of his area of expertise and capable of seeing the bridges connecting field observations. His research and publications do not necessarily reflect his wide knowledge but those who have or have had the chance to work with him know that Eddy is a living encyclopedia. Moreover, his interest largely crosses over the borders of geology and he can with erudition discuss archaeology, history, architecture, volcanoes, anthropology, literature, cuisine and wine! To understand the man, let us visit his career.Born in 1949 in Seraing, near Liège in Belgium, the young Eddy quickly found his interest in fossils after he discovered brachiopods in the excavation made for the construction of a bridge in Esneux. His first fossils… and the beginning of a lifelong passion for geology. During his childhood and teenage years, he used to search for fossils and minerals while on holiday with his parents, but also around Liège.In 1967, he entered the University of Liège to study geology (at that time 'Geological and Mineralogical Sciences') and learned from Prof. Bellière, Michot and Ubaghs among others. Four years later he presented his master's thesis on Viséan corals. His scientific career was not yet cast because military service was compulsory… but Eddy was not made for the uniform and the discipline which go along with the military. We find him back enrolled in civilian service as a mathematics teacher in Algeria. He spent a year teaching during the week and travelling through the desert during week-ends and holidays to explore the mountains and to collect fossils! Enchanted by Algeria, Eddy would reiterate the experience in the Congo and settled for two more years in Bounia. His life there, if his uncountable anecdotes are true, was epic and amazing.Back in Belgium in 1974, Eddy joined the University of Liège, where for two years he taught the practical classes in geology at the geography department for Prof. Macar. In 1976 he returned to the geology department with an IRSIA research grant to start a PhD under the direction of Prof. Ubaghs. He focused on a group untouched since the 1920's: the Lower Carboniferous rugose corals. Thus started his long-lasting and very friendly collaboration with Raphaël -'Raph' -Conil and his team at the University of Louvain-la-Neuve.
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