A posteriori error estimates for elliptic problems in two and three space dimensions

Let $u \in H$ be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some $\tilde u \in \mathcal{S}$, $\mathcal{S}$ being a suitable finite-element space. Efficient and reliable a posteriors estimates of the error $\| {u - \tilde u} \|$, measuring the (local) quality of $\tilde u$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations

Energy Level Crossings in Molecular Dynamics

Energy level crossings are the landmarks that separate classical from quantum mechanical modeling of molecular systems. They induce non-adiabatic transitions between the otherwise adiabatically decoupled electronic level spaces. This review covers results on the analysis of propagation through level crossings of codimension two, a mathematical justification of surface hopping algorithms, and a spectral study of a linear isotropic system

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlev\'e II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlev\'e II that arises is a double shifted B\"acklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.Comment: 26 pages, 1 Figure, 2 Table

Data and Services at the Integrated Climate Data Center (ICDC) at the University of Hamburg

KlimawandelEarth observation data obtained from remote sensing sensors and in-situ data archives are fundamental for our current understanding of the Earth’s climate system. Such data are an important pre-requisite for Earth System research and should be easy to access and easy to use. In addition such data should be quality assessed and attached with information about uncertainties and long-term stability. If these data sets are stored in a self-explanatory, easy-to-use format, their usefulness and scientific value increase. This is the guideline for the Integrated Climate Data Center (ICDC) at the Center for Earth System Research and Sustainability (CEN), University of Hamburg. ICDC offers a reliable, quick and easy data access along with expert support for users and data providers. The ICDC provides several types of worldwide accessible in situ and satellite Earth observation data of the atmosphere, ocean, land surface, and cryosphere via the web portal http://icdc.zmaw.de. Recently, data from socio-economic sciences have been integrated into ICDC’s data base to enhance interdisciplinary collaboration. On ICDC’s web portal, each data set has its own page. It contains the data access points, a short data description, information about spatiotemporal coverage and resolution, data quality, important reference documents and contacts, and about how to cite the data set. The data are converted into netCDF or ASCII format. Consistency and quality checks are carried out – often in the framework of international collaborations. Literature studies are conducted to learn about potential limitations or preferred application areas of the data offered. The data sets can be accessed through the web page via FTP, HTTP or OPeNDAP. Using the Live Access Server, users can visualize data as maps, along transects and profiles, zoom into key regions, and create time series. In both fields, visualization and data access, ICDC tries to provide fast response times and high reliability

Effective-Range Dependence of Resonantly Interacting Fermions

We extract the leading effective range corrections to the equation of state of the unitary Fermi gas from ab initio fixed-node quantum Monte Carlo (FNQMC) calculations in a periodic box using a density functional theory (DFT), and show them to be universal by considering several two-body interactions. Furthermore, we find that the DFT is consistent with the best available unbiased QMC calculations, analytic results, and experimental measurements of the equation of state. We also discuss the asymptotic effective-range corrections for trapped systems and present the first QMC results with the correct asymptotic scaling.Comment: 11 pages, 5 figures: Updated to match published versio

Smooth adiabatic evolutions with leaky power tails

Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. Adiabatic evolutions without a gap condition do not seem to have this feature in general. This is a known fact for eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the continuous spectrum by considering the Friedrichs model.Comment: Final form, to appear in J. Phys. A; 11 pages, no figure

An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics

An explicit and symplectic integrator called PICKABACK for quantum-classical molecular dynamics is presented. The integration scheme is time reversible and unitary in the quantum part. We use the Lie formalism in order to construct a formal evolution operator which is split by the Strang splitting yielding the symplectic discretization PICKABACK. Finally the new method is compared with a widely used hybrid method in two examples: a collinear collision of a particle with a quantum oscillator and, additionally, a photodissociation process of an ArHCl molecule. It is shown that the PICKABACK algorithm is more stable and accurate at no additional numerical effort

The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class

We explain the exact solution of the 1+1 dimensional Kardar-Parisi-Zhang equation with sharp wedge initial conditions. Thereby it is confirmed that the continuum model belongs to the KPZ universality class, not only as regards to scaling exponents but also as regards to the full probability distribution of the height in the long time limit.Comment: Proceedings StatPhys 2