Graph-based nonlocal gradient systems and their local limits

In this thesis, we study systems of multiple nonlocally interacting species on a large class of graphs, ranging from graphs to continuous graphs or graphons. The proposed model is based on the theory of metric gradient flows and provides a unified upwind-based framework, including not only concave mobilities but also non-1-homogeneous kinetic relations. Exploiting the system’s gradient flow nature, we prove the existence of measure-valued solutions by establishing a rigorous link to a variational formulation in a quasi-metric Finslerian setting. In addition, the behavior of the arising dynamics is explored in numerical and analytical case studies, displaying phenomena such as the formation of patterns, the aggregation of one species, or the separation of different species. Finally, employing ideas of evolutionary Γ-convergence, we prove for linear mobilities and 1-homogeneous kinetic relations that solutions of nonlocal systems, defined on a suitable family of graphons, converge to solutions of a system of nonlocal interaction equations in Euclidean space. What is more, a similar approximation is possible even after introducing tensor-valued anisotropies into the limiting geometry. Since the upwind-induced prelimit geometries are of non-symmetric Finslerian nature, while the limiting geometry is of Riemannian type, our graphon-to-local limit can also be viewed as a non-symmetric-to-symmetric limit of gradient structures

An overview of the objectives of and the approaches to propensity score analyses

The assessment of treatment effects from observational studies may be biased with patients not randomly allocated to the experimental or control group. One way to overcome this conceptual shortcoming in the design of such studies is the use of propensity scores to adjust for differences of the characteristics between patients treated with experimental and control interventions. The propensity score is defined as the probability that a patient received the experimental intervention conditional on pre-treatment characteristics at baseline. Here, we review how propensity scores are estimated and how they can help in adjusting the treatment effect for baseline imbalances. We further discuss how to evaluate adequate overlap of baseline characteristics between patient groups, provide guidelines for variable selection and model building in modelling the propensity score, and review different methods of propensity score adjustments. We conclude that propensity analyses may help in evaluating the comparability of patients in observational studies, and may account for more potential confounding factors than conventional covariate adjustment approaches. However, bias due to unmeasured confounding cannot be corrected fo

Non-classical correlations between a C-band telecom photon and a stored spin-wave

Future ground-based quantum information networks will likely use single photons transmitted through optical fibers to entangle individual network nodes. To extend communication distances and overcome limitations due to photon absorption in fibers the concept of quantum repeaters has been proposed. For that purpose, it is required to achieve quantum correlations between the material nodes and photons at telecom wavelengths which can be sent over long distances in optical fibers. Here we demonstrate non-classical correlation between a frequency converted telecom C-band photon and a spin-wave stored in an atomic ensemble quantum memory. The photons emitted from the ensemble and heralding the spin-waves are converted from 780 nm to 1552 nm by means of an all-solid-state integrated waveguide non-linear device. We show ultra-low noise operation of the device enabling a high signal to noise ratio of the converted single photon, leading to a high spin-wave heralding efficiency. The presented work is an enabling step towards the practical entanglement of remote quantum memories and the entanglement of quantum systems operating at different wavelengths.Comment: 9 pages, 5 figure

Graph-to-local limit for the nonlocal interaction equation

We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.Comment: 48 pages. Comments welcom

Untersuchung von Oxidationsprozessen an Siliziumnanodrähten mittels Molekulardynamik

Siliziumnanodrähte (SiNWs) bieten eine aussichtsreiche Grundlage zur Entwicklung neuartiger nanoelektronischer Bauelemente, wie Feldeffekttransistoren oder Sensoren. Dabei ist insbesondere die Oxidation der Drähte interessant, weil diese weitreichenden Einfluss auf die elektronischen Eigenschaften der Bauelemente hat, die aus den SiNWs gefertigt werden. Die Größe der untersuchten Strukturen erfordert eine atomistische Analyse des Oxidationsprozesses. In der vorliegenden Arbeit wird der bisher wenig verstandene Beginn der Oxidation dünner Drähte molekulardynamisch simuliert, wobei als Potential ein reaktives Kraftfeld dient. Dabei wird sich intensiv mit dem Transfer elektrischer Ladungen zwischen Atomen unterschiedlicher Elektronegativitäten während der Simulationen auseinandergesetzt. Desweiteren werden Strukturen, die während der Oxidation von SiNWs der Orientierungen und bei Temperaturen von 300 K und 1200 K entstehen, untersucht. Ein Fokuspunkt dieser Untersuchungen ist die Analyse der Anzahl am Draht adsorbierter Sauerstoffatome während der frühen Oxidationsphase. Darüber hinaus wird die Dichte der entstehenden Strukturen beleuchtet. Dies geschieht mit einer hohen radialen Auflösung und erstmalig während der gesamten Simulation. Hierbei zeigt sich, dass während des Übergangs von kristallinem Silizium zu amorphem Siliziumdioxid zwischen den Siliziumatomen Sauerstoff eingelagert wird, die Kristallstruktur des Siliziums sich zunächst jedoch noch nicht auflöst. Dadurch entsteht ein charakteristisches Muster hoher und niedriger Dichten, das von der ursprünglichen Kristallstruktur des SiNW abhängt.:Abbildungsverzeichnis Abkürzungsverzeichnis Symbolverzeichnis 1 Einleitung 2 Einführung zu Siliziumnanodrähten 2.1 Kristallstuktur von Silizium 2.2 Ideale Siliziumnanodrähte 2.3 Herstellung von Siliziumnanodrähten 3 Grundlagen der Molekulardynamik 3.1 Newtonsche Axiome 3.2 Einige grundlegende Begriffe der statistischen Physik 3.3 Molekulardynamik 3.4 Reaktives Kraftfeld 3.5 Methoden zur Beschreibung des Ladungstransfers 3.6 Thermostat und Barostat 3.7 Large-scale Atomic/Molecular Massively Parallel Simulator 4 Entwicklung des Modellsystems 4.1 Ausgangsstruktur 4.2 Vorrelaxation 4.3 Ablauf der Oxidation 4.4 Verwendeter ReaxFF-Parametersatz 4.5 Optimierung der Zeitschrittweite 4.5.1 Modellsystem, Relaxation und Oxidation 4.5.2 Festlegung der Zeitschrittweite 4.6 Optimierung der Systemlänge 4.6.1 Modellsystem, Relaxation und Oxidation 4.6.2 Festlegung der Systemlänge 4.7 Einfluss des globalen, instantanen Ladungstransfers auf die Simulation 4.7.1 Festlegung des Einsetzabstands 4.7.2 Vergleich mit Daten von Khalilov et al. 5 Variation von System- und Einsetztemperatur sowie Drahtorientierung 5.1 Variation von System- und Einsetztemperatur 5.1.1 Untersuchung des Oxidationsgrads 5.1.2 Untersuchung von Dichten und Grenzflächenpositionen 5.2 Variation der Drahtorientierung 5.2.1 Untersuchung des Oxidationsgrads 5.2.2 Untersuchung von Dichten und Grenzflächenpositionen 6 Zusammenfassung und Ausblick 6.1 Zusammenfassung 6.2 Ausblick Literaturverzeichni

Global, Parameterwise and Joint Shrinkage Factor Estimation

The predictive value of a statistical model can often be improved by applying shrinkage methods. This can be achieved, e.g., by regularized regression or empirical Bayes approaches. Various types of shrinkage factors can also be estimated after a maximum likelihood fit has been obtained: while global shrinkage modifies all regression coefficients by the same factor, parameterwise shrinkage factors differ between regression coefficients. The latter ones have been proposed especially in the context of variable selection. With variables which are either highly correlated or associated with regard to contents, such as dummy variables coding a categorical variable, or several parameters describing a nonlinear effect, parameterwise shrinkage factors may not be the best choice. For such cases, we extend the present methodology by so-called 'joint shrinkage factors', a compromise between global and parameterwise shrinkage. Shrinkage factors are often estimated using leave-one-out resampling. We also discuss a computationally simple and much faster approximation to resampling-based shrinkage factor estimation, can be easily obtained in most standard software packages for regression analyses. This alternative may be relevant for simulation studies and other computerintensive investigations. Furthermore, we provide an R package shrink implementing the mentioned shrinkage methods for models fitted by linear, generalized linear, or Cox regression, even if these models involve fractional polynomials or restricted cubic splines to estimate the influence of a continuous variable by a nonlinear function. The approaches and usage of the package shrink are illustrated by means of two examples

Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics

We explore the dynamical behavior and energetic properties of a model of two species that interact nonlocally on finite graphs. The authors recently introduced the model in the context of nonquadratic Finslerian gradient flows on generalized graphs featuring nonlinear mobilities. In a continuous and local setting, this class of systems exhibits a wide variety of patterns, including mixing of the two species, partial engulfment, or phase separation. This work showcases how this rich behavior carries over to the graph structure. We present analytical and numerical evidence thereof.Comment: arXiv admin note: substantial text overlap with arXiv:2107.1128

Storing single photons emitted by a quantum memory on a highly excited Rydberg state

Strong interaction between two single photons is a long standing and important goal in quantum photonics. This would enable a new regime of nonlinear optics and unlock several applications in quantum information science, including photonic quantum gates and deterministic Bell-state measurements. In the context of quantum networks, it would be important to achieve interactions between single photons from independent photon pairs storable in quantum memories. So far, most experiments showing nonlinearities at the single-photon level have used weak classical input light. Here we demonstrate the storage and retrieval of a paired single photon emitted by an ensemble quantum memory in a strongly nonlinear medium based on highly excited Rydberg atoms. We show that nonclassical correlations between the two photons persist after retrieval from the Rydberg ensemble. Our result is an important step towards deterministic photon–photon interactions, and may enable deterministic Bell-state measurements with multimode quantum memories.Peer ReviewedPostprint (published version