Grassmannians and Koszul duality

Let \(X\) be a partial flag variety, stratified by orbits of the Borel. We give a criterion for the category of modular perverse sheaves to be equivalent to modules over a Koszul ring. This implies that modular category \(\mathcal O\) is governed by a Koszul-algebra in small examples.     

Determination of acoustic parameters in flow fields using OpenFOAM

In the present paper a Computational Aero Acoustics approach based on Lighthill's and Curle's Acoustic Analogies was implemented in OpenFOAM 2.1.1. The novel developed OpenFOAM application solvers acousticFoam and acousticRhoFoam can be used for transient incompressible and compressible simulations respectively. The CAA approach takes High Performance Computing environment into account and realize the computation of flow and acoustic fields within the acoustical near field on one mesh only. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Approach for Martensitic Transformations and Crystal Plasticity

This work is motivated by cryogenic turning which allows end shape machining and simultaneously attaining a hardened surface due to deformation induced martensitic transformations. To study the process on the microscale, a multivariant phase field model for martensitic transformations in conjunction with a crystal plastic material model is introduced. The evolution of microstructure is assumed to follow a time-dependent Ginzburg-Landau equation. To solve the field equations the finite element method is used. Time integration is performed with Euler backward schemes, on the global level for the evolution equation of the phase field, and on the element level for the crystal plastic material law. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sensitivities for topological graph changes in finite element meshes and application to adaptive refinement

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topological mesh derivative for a given objective function that depends on the discrete solution of the underlying PDE. These sensitivities may in turn be used as refinement indicators within an adaptive algorithm. For their calculation, we rely on the first-order asymptotic expansion of the Galerkin solution with respect to the topological mesh change. As a proof of concept, we consider the total potential energy of a linear symmetric second-order elliptic PDE, minimization of which is known to decrease the approximation error in the energy norm. In this case, our approach yields local sensitivities that are closely related to the reduction of the energy error upon refinement and may therefore be used as refinement indicators in an adaptive algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective semi-analytic integration for hypersingular Galerkin boundary integral equations for the Helmholtz equation in 3D

We deal with the Galerkin discretization of the boundary integral equations corresponding to problems with the Helmholtz equation in 3D. Our main result is the semi-analytic integration for the bilinear form induced by the hypersingular operator. Such computations have already been proposed for the bilinear forms induced by the single-layer and the double-layer potential operators in the monograph The Fast Solution of Boundary Integral Equations by O. Steinbach and S. Rjasanow and we base our computations on these results.     

Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.     

Estimating summary functionals in multistate models with an application to hospital infection data

Inhomogeneous Markov chains are a relevant framework for analysing event histories. The fundamental estimator in the presence of incompletely observed data is the Nelson-Aalen estimator of the multivariate cumulative hazards. It may be summarised in terms of probability estimates via the empirical transition matrix. The empirical transition matrix has only slowly entered applications, one reason being previous lack of software. In a number of applications, further summary measures are desired. We illustrate how they may be computed from the empirical transition matrix and why bootstrapping their variance works. In contrast, computing such summaries outside the present framework has typically led to biased results. As an example, we consider in more detail hospital stay following infectious complication. This summary quantity is often considered by clinical decision makers, but reliable estimates require modelling the timing of infection as in the present set-up. In this context, we also derive new summary measures that further distinguish between patients discharged and patients deceased.