Differential K-theory. A survey

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.Comment: 50 pages, report based in particular on work done sponsored the DFG SSP "Globale Differentialgeometrie". v2: final version (only typos corrected), to appear in C. B\"ar et al. (eds.), Global Differential Geometry, Springer Proceedings in Mathematics 17, Springer-Verlag Berlin Heidelberg 201

Classification and repeatability studies of transient electromagnetic measurements with respect to the development of CO2-monitoring techniques

The mitigation of greenhouse gases, like CO2 is a challenging aspect for our society. A strategy to hamper the constant emission of CO2 is utilizing carbon capture and storage technologies. CO2 is sequestrated in subsurface reservoirs. However, these reservoirs harbor the risk of leakage and appropriate geophysical monitoring methods are needed. A crucial aspect of monitoring is the assignment of measured data to certain events occurring. Especially if changes in the measured data are small, suitable statistical methods are needed. In this thesis, a new statistical workflow based on cluster analysis is proposed to detect similar transient electromagnetic signals. The similarity criteria dynamic time warping, the autoregressive distance, and the normalized root-mean-square distance are investigated and evaluated with respect to the classic Euclidean norm. The optimal number of clusters is determined using the gap statistic and visualized with multidimensional scaling. To validate the clustering results, silhouette values are used. The statistical workflow is applied to a synthetic data set, a long-term monitoring data set and a repeat measurement at a pilot CO2-sequestration site in Brooks, Alberta

The Calderon Projection: New Definition and Applications

We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze various ways of writing A|U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well--posed selfadjoint Fredholm extensions under continuous variation of the data.Comment: 60 pages, 4 figures; revised version; index and list of notation added; accepted for publication in J. Geom. Phys; v3 contains a few minor correction

German translation of the Alberta context tool and two measures of research use: methods, challenges and lessons learned

Background: Understanding the relationship between organizational context and research utilization is key to reducing the research-practice gap in health care. This is particularly true in the residential long term care (LTC) setting where relatively little work has examined the influence of context on research implementation. Reliable, valid measures and tools are a prerequisite for studying organizational context and research utilization. Few such tools exist in German. We thus translated three such tools (the Alberta Context Tool and two measures of research use) into German for use in German residential LTC. We point out challenges and strategies for their solution unique to German residential LTC, and demonstrate how resolving specific challenges in the translation of the health care aide instrument version streamlined the translation process of versions for registered nurses, allied health providers, practice specialists, and managers. Methods: Our translation methods were based on best practices and included two independent forward translations, reconciliation of the forward translations, expert panel discussions, two independent back translations, reconciliation of the back translations, back translation review, and cognitive debriefing. Results: We categorized the challenges in this translation process into seven categories: (1) differing professional education of Canadian and German care providers, (2) risk that German translations would become grammatically complex, (3) wordings at risk of being misunderstood, (4) phrases/idioms non-existent in German, (5) lack of corresponding German words, (6) limited comprehensibility of corresponding German words, and (7) target persons’ unfamiliarity with activities detailed in survey items. Examples of each challenge are described with strategies that we used to manage the challenge. Conclusion: Translating an existing instrument is complex and time-consuming, but a rigorous approach is necessary to obtain instrument equivalence. Essential components were (1) involvement of and co-operation with the instrument developers and (2) expert panel discussions, including both target group and content experts. Equivalent translated instruments help researchers from different cultures to find a common language and undertake comparative research. As acceptable psychometric properties are a prerequisite for that, we are currently carrying out a study with that focus

Do self-reported hearing and visual impairments predict longitudinal dementia in older adults?

Background Sensory impairments have been associated with dementia in older adults. However, the contribution of different impairments and how they interact in the development of dementia is not clear. We examined the independent and interaction effects of hearing impairment (HI) and visual impairment (VI) on incident dementia. Design Multi-centric population-based prospective cohort study. Setting Data were taken from the AgeDifferent.de platform, pooling participants aged 75 and older from the German LEILA75+ and AgeCoDe/AgeQualiDe cohorts. Participants Older adults (N = 3497) with mean age 79.8 years, 67.2% female. Measurements Standardized interviews and questionnaires were used to assess self-reported HI and VI at baseline and all-cause dementia in 9 follow-ups, spanning over 20 years. Methods Competing risk regression models were conducted to test the main and interaction effects of HI and VI on dementia incidence, adjusting for established risk factors of dementia and accumulated mortality. Results HI and VI at baseline were reported by 30.3% and 16.6% of individuals, respectively. Adjusting for baseline information on sociodemographics, substance use, cognitive functioning and morbidity, and controlling for accumulated mortality risk, HI (sHR 1.16, 95% CI 1.04–1.30, p = 0.011) but not VI (sHR 1.07, 95% CI 0.90–1.28, p = 0.462) was significantly associated with incident dementia. There was no interaction between HI and VI (sHR 1.09, 95% CI 0.81–1.46, p = 0.567). Conclusions Hearing impairment is associated with an increased incidence of all-cause dementia in older adults. There is no excess risk or risk compensation through the additional presence or absence of visual impairment. Early prevention measures for hearing impairment might help to reduce the long-term risk of dementia

Classification and repeatability studies of transient electromagnetic measurements with respect to the development of CO2-monitoring techniques

The mitigation of greenhouse gases, like CO2 is a challenging aspect for our society. A strategy to hamper the constant emission of CO2 is utilizing carbon capture and storage technologies. CO2 is sequestrated in subsurface reservoirs. However, these reservoirs harbor the risk of leakage and appropriate geophysical monitoring methods are needed. A crucial aspect of monitoring is the assignment of measured data to certain events occurring. Especially if changes in the measured data are small, suitable statistical methods are needed. In this thesis, a new statistical workflow based on cluster analysis is proposed to detect similar transient electromagnetic signals. The similarity criteria dynamic time warping, the autoregressive distance, and the normalized root-mean-square distance are investigated and evaluated with respect to the classic Euclidean norm. The optimal number of clusters is determined using the gap statistic and visualized with multidimensional scaling. To validate the clustering results, silhouette values are used. The statistical workflow is applied to a synthetic data set, a long-term monitoring data set and a repeat measurement at a pilot CO2-sequestration site in Brooks, Alberta

Nichtlineare Dynamik von Metall-Clustern auf Isolatorsubstraten

Die vorliegende Arbeit widmet sich der Untersuchung kleiner Natrium-Cluster, die in Kontakt mit isolierenden Substraten, insbesondere MgO(001) stehen. Im Vordergrund steht dabei die Frage, inwieweit die (geringe) Wechselwirkung zwischen Cluster und Oberfläche die Merkmale des Adsorbats beeinflusst. Während Grundzustandseigenschaften deponierter Cluster durch bereits existierende quantenmechanische Modelle zugänglich sind, gilt das nicht für den Bereich starker Anregungen des Systems. In dieser Arbeit wird nun ein hierarchisches Modell vorgestellt, das diese Lücke schließen soll. Im Rahmen dieses Modells werden die elektronisch aktivsten Komponenten des Systems, die Valenzelektronen des Clusters, quantenmechanisch mit Hilfe der zeitabhängigen Dichtefunktionaltheorie, die elektronisch passiveren Komponenten hingegen durch klassische Molekulardynamik beschrieben. Die Kopplung zwischen Klassik und Quantenmechanik gelingt durch den Einsatz von Pseudopotentialen und Atom-Atom-Potentialen. Sie erfolgt nichtadiabatisch, so dass durch zeitabhängige Rechnungen im Prinzip beliebige Anregungen realisiert werden können. Die Teilchen des Substrats erhalten als zusätzlichen inneren Freiheitsgrad ein Dipolmoment, durch dessen Propagation dynamische Polarisationseffekte berücksichtigt werden können. Das Modell wird zunächst dazu verwendet, um Eigenschaften des Grundzustands, insbesondere die geometrische Struktur der adsorbierten Cluster und deren Anregungsspektrum zu berechnen. Die gefundenen Strukturen dienen als Ausgangspunkt für die weiteren Simulationen. Im nächsten Schritt wird die Dynamik des Depositionsprozesses von Na6 und Na8 auf MgO(001) bei verschiedenen Einschlagenergien untersucht. Die Ergebnisse werden mit analogen Rechnungen mit einem Argonsubstrat verglichen. Desweiteren wird das Modell zur Photoelektronenspektroskopie am adsorbierten Cluster Na8 eingesetzt. Neben der Untersuchung von globalen Größen wie der mittleren Gesamtzahl der emittierten Elektronen, werden auch deren 4-Pi-aufgelöste Winkelverteilungen, sowie die kinetischen Energiespektren berechnet. Schließlich wird in einem abschließenden Kapitel der extrem nichtlineare Prozess von Coulomb-Explosionen eines Na8 Clusters auf der Oberfläche betrachtet. Die Wechselwirkung mit dem Substrat führt zu einer deutlichen Erhöhung des kritischen Ladungszustands für instantane Explosion.The present work is dedicated to the investigation of small sodium clusters which are in contact with an insulating substrate, in most cases MgO(001). The influence of the small interaction between cluster and surface on the properties of the adsorbate is of primary concern. While ground state features of deposited clusters are accessible by existing quantum mechanical models, the regime of strong excitations of the system has so far been out of reach. The aim of this work therefore is to introduce a model which fulfills this task. Within this model the electronically most active components of the system (the valence electrons of the cluster) are treated quantum mechanically in the framework of time-dependent density functional theory, whereas the electronically inert components are described by classical molecular dynamics. The classical part is coupled to the quantum mechanical part by pseudo-potentials and atom-atom potentials. The coupling is done non-adiabatically, so that even strong excitations can be realized by time-resolved calculations. As additional internal degree of freedom each particle of the substrate may acquire a dipole moment. The propagation of the dipole moment allows to incorporate dynamical polarization effects. First of all the model is applied to determine cluster properties close to the ground state, namely the ion geometry of the adsorbed cluster and the optical response. The structures which are determined, are used as starting point for the further calculations. As a next step, the deposition process for Na6 and Na8 on MgO(001) depending on their impact energy is investigated in detail. The results are compared with corresponding calculations using an argon substrate instead. Furthermore the model is used for performing photoelectron spectroscopy of deposited Na8. Aside of global quantities as the average total number of the emitted electrons, their 4-pi-resolved angular distributions as well as their kinetic energy spectra are determined. Finally the last chapter investigates the extremely non-linear process of Coulomb explosions of a Na8 cluster on the surface. The result is a significant increase of the critical charge state for instantaneous explosion due to the interaction with the substrate

Small Eigenvalues on the Conformal Laplacian

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the alpha-genus