The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the RayleighTaylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.

Abstract. We give an algorithm to compute N steps of a convolution quadrature approximation to a continuous temporal convolution using only O(N log N ) multiplications and O(log N ) active memory. The method does not require evaluations of the convolution kernel, but instead O(log N ) evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integrodifferential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions.

Laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered. A spectral order, parallelizable method for their numerical inversion is proposed. The method takes into account the available information about the errors arising in the evaluations. Several numerical illustrations are provided.

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Inherited platelet disorders are a heterogeneous group of rare diseases, caused by inherited defects in platelet production and/or function. Their genetic diagnosis would benefit clinical care, prognosis and preventative treatments. Until recently, this diagnosis has usually been performed via Sanger sequencing of a limited number of candidate genes. High-throughput sequencing is revolutionizing the genetic diagnosis of diseases, including bleeding disorders. We have designed a novel high-throughput sequencing platform to investigate the unknown molecular pathology in a cohort of 82 patients with inherited platelet disorders. Thirty-four (41.5%) patients presented with a phenotype strongly indicative of a particular type of platelet disorder. The other patients had clinical bleeding indicative of platelet dysfunction, but with no identifiable features. The high-throughput sequencing test enabled a molecular diagnosis in 70% of these patients. This sensitivity increased to 90% among patients suspected of having a defined platelet disorder. We found 57 different candidate variants in 28 genes, of which 70% had not previously been described. Following consensus guidelines, we qualified 68.4% and 26.3% of the candidate variants as being pathogenic and likely pathogenic, respectively. In addition to establishing definitive diagnoses of well-known inherited platelet disorders, high-throughput sequencing also identified rarer disorders such as sitosterolemia, filamin and actinin deficiencies, and G protein-coupled receptor defects. This included disease-causing variants in DIAPH1 (n=2) and RASGRP2 (n=3). Our study reinforces the feasibility of introducing high-throughput sequencing technology into the mainstream laboratory for the genetic diagnostic practice in inherited platelet disorders.