Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let be an open and closed subset of the boundary of M. We say that has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance from a point to is bounded uniformly in x (and hence, in particular, intersects all connected components of M). For manifolds with finite width, we prove a Poincaré inequality for functions vanishing on , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces , .     

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.     

Regularity for Eigenfunctions of Schr?dinger Operators

We prove a regularity result in weighted Sobolev (or Babu?ka?CKondratiev) spaces for the eigenfunctions of certain Schr?dinger-type operators. Our results apply, in particular, to a non-relativistic Schr?dinger operator of an N-electron atom in the fixed nucleus approximation. More precisely, let ${\mathcal{K}_{a}^{m}(\mathbb{R}^{3N},r_S)}$ be the weighted Sobolev space obtained by blowing up the set of singular points of the potential ${V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}}$ , ${x \in \mathbb{R}^{3N}}$ , ${b_j, c_{ij} \in \mathbb{R}}$ . If ${u \in L^2(\mathbb{R}^{3N})}$ satisfies ${(-\Delta + V) u = \lambda u}$ in distribution sense, then ${u \in \mathcal{K}_{a}^{m}}$ for all ${m \in \mathbb{Z}_+}$ and all a ?? 0. Our result extends to the case when b _j and c _ij are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a?<?3/2.     

Die Siliciumbestimmung im Ferrosilicium

Ohne Zusammenfassung     

Manifolds with small Dirac eigenvalues are nilmanifolds

Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded diameter, and almost non-negative scalar curvature. Let r = 1 if n = 2,3 and r = 2^[n/2]-1 + 1 if n ≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost non-negative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.      

LC-MS/MS determination of potential endocrine disruptors of cortico signalling in rivers and wastewaters

A targeted analytical method was established to determine a large number of chemicals known to interfere with the gluco- and mineralocorticoid signalling pathway. The analytes comprise 30 glucocorticoids and 9 mineralocorticoids. Ten out of these corticosteroids were primary metabolites. Additionally, 14 nonsteroids were included. These analytes represent a broader range of possible adverse modes of action than previously reported. For the simultaneous determination of these structurally diverse compounds, a single-step multimode solid-phase extraction and pre-concentration was applied. Extracts were separated by a short linear HPLC gradient (20 min) on a core shell RP column (2.7 μm particle size) and compounds identified and quantified by LC-MS/MS. The method provided excellent retention time reproducibility and detection limits in the low nanograms per litre range. Untreated hospital wastewater, wastewater treatment plant influent, treated effluent and river waters were analysed to demonstrate the applicability of the method. The results show that not all compounds were sufficiently eliminated by the wastewater treatment, resulting in the presence of several steroids (～20 ng/L) and nonsteroids in the final effluent, some of them at high concentrations up to 200 ng/L. Most of the detected mono-hydroxylated steroidal transformation products were found at significantly higher concentrations than their parent compounds. We therefore recommend to include these potentially bioactive metabolites in environmental toxicity assessment.     

Acylthioketene-thioacylketene-thiet-2-one rearrangements

Flash vacuum thermolysis (FVT) of 6-aryl-1,3-dioxine-4-thiones 9 leads to the formation of acylthioketenes 10, which are characterized by Ar matrix IR spectroscopy as well as on-line tandem mass spectrometry. The thioketenes 10 undergo a 1,3-shift of the aryl group to generate thioacylketenes 11. Ketenes 11 cyclize to 3-aryl-thiet-2-ones 12, which are also characterized by matrix IR spectroscopy and tandem mass spectrometry. The thiet-2-ones 12 undergo two kinds of reaction under the FVT conditions: (i) cheletropic CO extrusion with formation of arylthioketenes 13, and (ii) cycloreversion to COS and arylacetylene.     

Inductively coupled plasma mass spectrometry (ICP MS): a versatile tool

Inductively coupled plasma (ICP) mass spectrometry (MS) is routinely used in many diverse research fields such as earth, environmental, life and forensic sciences and in food, material, chemical, semiconductor and nuclear industries. The high ion density and the high temperature in a plasma provide an ideal atomizer and element ionizer for all types of samples and matrices introduced by a variety of specialized devices. Outstanding properties such as high sensitivity (ppt-ppq), relative salt tolerance, compound-independent element response and highest quantitation accuracy lead to the unchallenged performance of ICP MS in efficiently detecting, identifying and reliably quantifying trace elements. The increasing availability of relevant reference compounds and high separation selectivity extend the molecular identification capability of ICP MS hyphenated to species-specific separation techniques. While molecular ion source MS is specialized in determining the structure of unknown molecules, ICP MS is an efficient and highly sensitive tool for target-element orientated discoveries of relevant and unknown compounds. This special-feature, tutorial article presents the principle and advantages of ICP MS, highlighting these using examples from recently published investigations.     

Bromine enrichment in the near-surface region of Br-doped NaCl single crystals diagnosed by Rutherford backscattering spectrometry

Bromine released from sea-salt aerosols and seawater ice is known for its high chemical reactivity. Previous studies have suggested that its availability to the gas-phase could be enhanced by segregation processes increasing Br concentration on the aerosol surface as compared to the bulk. However, little is known about the composition within the near-surface region, that is, the outermost approximately 100 monolayers. We used Rutherford backscattering spectrometry (RBS) to measure Br concentration profiles to a depth of about 750 nm of Br-doped NaCl single crystals to characterize the thermodynamics and kinetics of Br segregation to the near-surface region in moist air. These experiments were carried out on cleavage planes of melt-grown and of annealed solution-grown crystals at room temperature and relative humidities (RH) too low for formation of a stable liquid phase. Segregation of Br was below the detection limit on melt-grown crystals with Br/Cl = 0.01. In the case of annealed solution-grown crystals with Br/Cl = 0.002, average segregations of (0.24 +/- 0.11) x 10(15) and (0.42 +/- 0.12) x 10(15) Br atoms cm-2 were observed at 50% and 65% RH, respectively. No segregation was found at 20% RH. The observed Br segregation can be explained by the formation of an adsorbed liquid layer (depending on crystal surface properties and relative humidity) and preferential, diffusion-limited dissolution of Br into this layer according to the partition coefficient of Br between aqueous and solid NaCl. The thickness of the adsorbed liquid layer, which depends on crystal surface geometry and on relative humidity, can be estimated to range from 4 to at most 59 nm on the basis of measured Br concentrations and partition coefficients. Applying this concept of partitioning to natural sea salt suggests a Br/Cl molar ratio of up to 0.2 in adsorbed surface water of crystallized natural aerosol particles compared to about 0.0015 in seawater. This would have a major impact on heterogeneous reactions on sea-salt particles under dry conditions such as in the freeze-dried Arctic boundary layer.