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Quantitative estimates for stress concentration of the Stokes flow between adjacent circular cylinders
(2020)SAM Research ReportWhen two inclusions with high contrast material properties are located close to each other in a homogeneous medium, stress may become arbitrarily large in the narrow region between them. In this paper, we investigate such stress concentration in the two-dimensional Stokes flow when inclusions are the two-dimensional cross sections of circular cylinders of the same radii and the background velocity field is linear. We construct two ...Report -
Exceptional points in parity-time-symmetric subwavelength metamaterials
(2020)SAM Research ReportWhen sources of energy gain and loss are introduced to a wave scattering system, the underlying mathematical formulation will be non-Hermitian. This paves the way for the existence of exceptional points, whereby eigenmodes are linearly dependent. The main goal of this work is to study the existence and consequences of exceptional points in the setting of high-contrast subwavelength metamaterials. We begin by studying a system of two ...Report -
Die Schweizer Selbständigerwerbenden in der Covid19-Pandemie
(2020)Der Bundesrat hat am 13. März einen partiellen Lockdown der Schweiz beschlossen, der viele Betriebe stark betroffen hat. In der Woche nach Ostern (14.-20. April) haben wir die Inhaberinnen und Inhaber von 1'011 kleinen und mittleren Unternehmen (KMU) zu den Auswirkungen der Krise auf das Unternehmen und ihre persönliche Situation befragt. 409 der befragten Selbständigen führen einen Betrieb, der in dieser Zeit geschlossen war. Es zeigt ...Report -
Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²
(2020)SAM Research ReportIn a polygon Ω ⊂ R2, we consider mixed hp-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and hp spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in Ω, we prove exponential rates of convergence of the mixed ...Report -
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Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws
(2020)SAM Research ReportWe present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the ”nonlocal pair-interaction model” which was recently introduced by Du, Huang, and LeFloch. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and ...Report -
Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies
(2020)SAM Research ReportCircuits of biological neurons, such as in the functional parts of the brain can be modeled as networks of coupled oscillators. Inspired by the ability of these systems to express a rich set of outputs while keeping (gradients of) state variables bounded, we propose a novel architecture for recurrent neural networks. Our proposed RNN is based on a time-discretization of a system of second-order ordinary differential equations, modeling ...Report