Universität Stuttgart

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Subject: search.filters.subject.530Subject: search.filters.subject.510End date: 2022

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Now showing 1 - 9 of 9
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    Chiral metamaterials
    (2016) Eslami, Sahand; Fischer, Peer (Prof. Dr.)
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    Self-adjointness and domain of a class of generalized Nelson models
    (2017) Wünsch, Andreas; Griesemer, Marcel (Prof. Dr.)
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    Optimality principles in human point-to-manifold reaching accounting for muscle dynamics
    (2020) Wochner, Isabell; Driess, Danny; Zimmermann, Heiko; Häufle, Daniel F. B.; Toussaint, Marc; Schmitt, Syn
    Human arm movements are highly stereotypical under a large variety of experimental conditions. This is striking due to the high redundancy of the human musculoskeletal system, which in principle allows many possible trajectories toward a goal. Many researchers hypothesize that through evolution, learning, and adaption, the human system has developed optimal control strategies to select between these possibilities. Various optimality principles were proposed in the literature that reproduce human-like trajectories in certain conditions. However, these studies often focus on a single cost function and use simple torque-driven models of motion generation, which are not consistent with human muscle-actuated motion. The underlying structure of our human system, with the use of muscle dynamics in interaction with the control principles, might have a significant influence on what optimality principles best model human motion. To investigate this hypothesis, we consider a point-to-manifold reaching task that leaves the target underdetermined. Given hypothesized motion objectives, the control input is generated using Bayesian optimization, which is a machine learning based method that trades-off exploitation and exploration. Using numerical simulations with Hill-type muscles, we show that a combination of optimality principles best predicts human point-to-manifold reaching when accounting for the muscle dynamics.
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    Killing and conformal Killing tensors
    (2017) Heil, Konstantin; Semmelmann, Uwe (Prof. Dr.)
    This thesis describes the prolongation connection of Killing tensors in terms of Young symmetrizers. The goal is to give an interpretation to sections of the prolongation bundle for Killing tensors on a manifold as generalized curvature tensors on the cone over that manifold. As a result, this method allows to treat the components of the prolongation bundle as a single object with well-understood symmetries. The developed formalism is then explored in three applications. The first result gives an isomorphism between the symmetric algebra of Killing tensors on a manifold of constant curvature and an algebra generated by parallel two-forms on the cone. That provides a geometric proof of the decomposition of Killing tensors on constant curvature manifolds and the Delong-Takeuchi-Thompson formula, previously obtained by Takeuchi and Thompson. Secondly, this technique, together with some branching rules for holonomy subgroups, yields a new characterization of Sasakian and 3-Sasakian manifolds in terms of Killing tensors satisfying additional curvature conditions. The third application is a new short proof of the result by Dairbekov and Sharafutdinov that the codimension of the zero set of a non-trivial, trace free, conformal Killing tensor is at least two. Throughout this work, special emphasis is placed on the representation theory of the appearing tensor bundles. Therefore, the Killing- and conformal Killing operators are introduced as Stein-Weiss operators. Since Young symmetrizers are a well-established tool in describing tensor representations this approach fits perfectly with the goals of the thesis. A natural consequence of this choice are new, geometric proofs of some established results. Besides those mentioned above these cover: (1) A Weitzenböck formula, which implies that all trace free, conformal Killing tensors on manifolds with non-positive sectional curvature are parallel. (2) The decomposition of occurring representations with respect to the reduced holonomy of a Riemannian product yields that the space of trace free, conformal Killing two-tensors on the product is generated by pullbacks of Killing one- and two-tensors on the factors. Furthermore, this thesis recasts the known examples of Killing tensors on compact Riemannian manifolds in the modern and coordinate free language of differential geometry. It is shown how the example found by Page and Pope generalizes to a construction on all Riemannian submersions with totally geodesic fibres. This technique provides non-parallel symmetric Killing two-tensors on compact Kähler manifolds. That contrasts the fact that on such n-dimensional manifolds do not exist non-parallel Killing forms of degree other than one or n-1. Furthermore, this construction gives a method to compute some eigenvalues of the Lichnerowicz-Laplace operator acting on symmetric two-tensors.
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    Effective transport coefficients of anisotropic disordered materials
    (2022) Hilfer, R.; Hauskrecht, J.
    A novel effective medium theory for homogenized transport coefficients of anisotropic mixtures of possibly anisotropic materials is developed. Existing theories for isotropic systems cannot be easily extended, because that would require geometric characterizations of anisotropic connectivity. In this work anisotropic connectivity is characterized by introducing a tensor that is constructed from a histogram of local percolating directions. The construction is inspired by local porosity theory. A large number of known and unknown generalized effective medium approximations for anisotropic media are obtained as limiting special cases from the new theory. Among these limiting cases the limit of strong cylindrical anisotropy is of particular interest. The parameter space of the generalized theory is explored, and the advanced results are applied to experiment.
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    Formen und Kräfte : ein mathematisch-physikalischer Gang zur Kunst auf dem Campus Vaihingen
    (Stuttgart : Fakultät 8 - Mathematik und Physik, Universität Stuttgart, 2022) Stroppel, Markus; Scheffler, Marc; Engstler, Katja Stefanie; Engstler, Katja Stefanie (Konzept und Gestaltung)
    Der Rundgang erläutert und interpretiert einzelne Objekte und künstlerische Elemente der Lernstraße auf dem Campus Vaihingen aus mathematischer und physikalischer Sicht für die interessierte Allgemeinheit, aber auch für Schüler­innen und Schüler und für Studierende.
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    Simulating stochastic processes with variational quantum circuits
    (2022) Fink, Daniel
    Simulating future outcomes based on past observations is a key task in predictive modeling and has found application in many areas ranging from neuroscience to the modeling of financial markets. The classical provably optimal models for stationary stochastic processes are so-called ϵ-machines, which have the structure of a unifilar hidden Markov model and offer a minimal set of internal states. However, these models are not optimal in the quantum setting, i.e., when the models have access to quantum devices. The methods proposed so far for quantum predictive models rely either on the knowledge of an ϵ-machine, or on learning a classical representation thereof, which is memory inefficient since it requires exponentially many resources in the Markov order. Meanwhile, variational quantum algorithms (VQAs) are a promising approach for using near-term quantum devices to tackle problems arising from many different areas in science and technology. Within this work, we propose a VQA for learning quantum predictive models directly from data on a quantum computer. The learning algorithm is inspired by recent developments in the area of implicit generative modeling, where a kernel-based two-sample-test, called maximum mean discrepancy (MMD), is used as a cost function. A major challenge of learning predictive models is to ensure that arbitrarily many time steps can be simulated accurately. For this purpose, we propose a quantum post-processing step that yields a regularization term for the cost function and penalizes models with a large set of internal states. As a proof of concept, we apply the algorithm to a stationary stochastic process and show that the regularization leads to a small set of internal states and a constantly good simulation performance over multiple future time steps, measured in the Kullback-Leibler divergence and the total variation distance.
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    Topology optimization of metalization grid patterns to improve the Power conversion efficiency of thin-film solar cells
    (2021) Braun, Benedikt
    Der metallische Leiter, welcher in Form eines Gitters auf der Oberfläche einer Solarzelle angebracht ist, heißt Grid. Die Funktion dieses Grids ist es, den in der Absorberschicht einer Solarzelle erzeugten Strom, ohne große Verluste, an der Oberfläche zum externen Abgreifpunkt zu leiten. Durch die sehr gute Leitfähigkeit des Grids wird ein verlustarmer Ladungstransport ermöglicht. Allerdings bewirkt das für Lichtstrahlen undurchdringbare Grid eine Abschattung der Absoberschicht und verhindert, dass an dieser Stelle Strom erzeugt werden kann. Wenn kein Grid angebracht ist, fließt die Ladung durch die oberste Schicht einer Solarzelle. Diese besteht aus transparenten leitfähigen Oxiden (engl. transparent conducting oxides (TCO)). Das TCO lässt Lichtstrahlen durch und dadurch kann Strom erzeugt werden. Obwohl die Schicht den Strom leiten kann, besitzt sie denoch einen sehr hohen elektrischen Widerstand. Das bedeutet, eine geeignete Wahl des Gridmusters verschattet möglichst wenig Fläche der Solarzelle und bietet trotzdem einen flächendeckenden, verlustarmen Ladungsabtransport. Ein Gridmuster, welches beide Anforderungen bestens erfüllt, soll in dieser Bachelorarbeit mithilfe von Topologie-Optimierung gefunden werden. Topologie-Optimierung ist eine mathematische Optimierungsmethode, mit der, innerhalb eines Gebietes, eine optimale Materialverteilung gefunden werden kann, um eine hohe, strukturbedingte Leistung zu erzielen. Im Zuge dieser Arbeit ist dieses Gebiet die Oberfläche einer Solarzelle und das Material, welches auf der Oberfläche verteilt werden soll, ist das Metall, welches das Gridmuster bildet. Die Leistung einer Solarzelle wird mit dem Wirkungsgrad angegeben. Der Wirkungsgrad ist die Effizienz, mit der die Solarenergie in elektrische Energie umgewandelt werden kann. Zur Berechnung des Wirkungsgrades wird das Gebiet mit einem Voronoi-Diagramm in Simplizes unterteilt. Basierend auf der Poisson-Gleichung für elektrische Leitfähigkeit, kann die Ladung, die durch ein Simplex fließt, mit einer Finite-Elemente-Methode berechnet werden. Aus den einzelnen generierten Strömen lässt sich ein Gesamtstrom berechnen, mit welchem die erzeugte, elektrische Energie berechnet werden kann. Der einzige Parameter, welcher zur Berechnung der Effizienz einer Solarzelle benötigt und in dieser Arbeit variiert wird, ist das Gridmuster. Die Komponenten des Dichtevektors geben dabei die Metalldichte eines jeden Simplexes an. Zur Optimierung dieses Dichtevektors werden in dieser Arbeit Optimierungsverfahren verglichen, die in Richtung des steilsten Abstiegs optimieren. Mit einem dieser Ver- fahren werden weitere Modifizierungen des Dichtevektors getestet. Eine der Modifizierungen betrifft dabei die Umgebung des externen Abgreifpunktes. Die aufgebrachte Gridfläche muss an dieser Stelle groß genug sein, damit ein externer Kontakt ohne Probleme angebracht werden kann. Die nächste Modifizierung, die verwendet wird, ist eine Methode zur lokalen Optimierung. Dabei werden die durch die Diskretisierung entstandenen Simplizes zufällig in mehrere lokalen Teilgebiete eingeteilt und der Reihe nach optimiert. Besitzt eine Komponente des Dichtevektors einen Wert von 0 steht dies für kein Grid, während ein Wert von 1 für das vorhanden sein von Grid steht. Die Komponenten des Dichtevektors repräsentieren dabei jeweils ein Simplex und damit eine Teilfläche der Solarzelle. Eine Modifizierung ermöglicht außer den Werten 0 (kein Grid, schlecht leitend, Strom wird erzeugt) und 1 (Grid, gut leitend, kein Strom wird erzeugt) Zwischenwerte. Mit diesen Zwischenwerten kann eine kontinuierliche Optimierung durchgeführt werden. Die Leitfähigkeit bzw. die Möglichkeit Strom zu generieren, wird dabei für Zwischenwerte interpoliert. Je nach Wahl der Interpolationsfunktion, kann der Wert der Leitfähigkeit für Zwischenwerte gut oder schlecht sein. Ebenso für die Menge an generiertem Strom. Sowohl niedrige als auch hohe Werte kommen mit Vorteilen, weshalb eine geschickte Kombination zu einem verbesserten Optimierungsverhalten führen kann. Die letzte Modifizierung, die eine Rolle spielt, ist das Gridmuster, von welchem ausgehend optimiert wird. Dabei wird, unter anderem, das im Labor vom Zentrum für Sonnenenergie- und Wasserstoffforschung Baden-Württemberg (ZSW) verwendete Gridmuster optimiert. Das Ziel dieser Arbeit ist es, mit den kombinierten Methoden und den Ergebnissen der damit durchgeführten Optimierungen ein neues Gridmuster zu konstruieren, welches dem bisher verwendeten Gridmuster überlegen ist.
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    Energy estimates for the two-dimensional Fermi polaron
    (2017) Linden, Ulrich; Griesemer, Marcel (Prof. Dr.)
    This thesis is concerned with the quantum mechanical system of a single particle interacting with an ideal gas of identical fermions by point interaction. In the physics literature this system is often referred to as Fermi polaron. We investigate the two-dimensional Fermi polaron. Unlike the one-dimensional case, point interactions in two or three dimensions cannot be implemented as perturbation of the quadratic form of the Laplacian. Either they are obtained as self-adjoint extensions of the Laplacian restricted to functions that vanish when the coordinates of two particles coincide, or they are constructed by a suitable limiting process. Choosing the second approach, a many-body operator with two-particle point interaction has firstly been rigorously defined by Dell'Antonio, Figari and Teta. We consider the Fermi polaron confined to a box with periodic boundary conditions and we identify a broad class of regularization schemes that approximate the Hamiltonian of the Fermi polaron as limit operator in the strong resolvent sense. The Hamiltonian is not given by a closed form, which could be conveniently used in standard variational principles. We establish a novel variational principle that characterizes all bound states, i.e. all energy eigenstates below the bottom of the spectrum of the kinetic energy. This variational principle turns out to be very useful for the following purposes. The ground state of the Fermi polaron is expected to be well approximated by the polaron and the molecule ansatz in the regime of weak and strong coupling between the impurity and the Fermi gas, respectively. In the physics literature, these two classes of trial states are used for variational computations with the (ultraviolet) regularized Hamiltonian. Although the implicit expressions for the minimal energy of both classes allow for the removal of the ultraviolet cutoff, it remains unclear whether the results are upper bounds to the ground state energy of the Fermi polaron. We show that the minimization of energy over polaron and molecule trial states can be reformulated in a natural way with the help of our variational principle. By doing so, the classes of trial states simplify considerably, and since there is no reference to regularized quantities, we can prove that the expressions for the polaron and the molecule energy in the physics literature are indeed upper bounds to the ground state energy of the Fermi polaron. As a further application of the variational principle, we prove analytically that to first order in a particle-hole expansion the molecule ansatz yields a better approximation to the ground state energy than the polaron ansatz if the coupling between the impurity and the Fermi gas is strong enough. So far, this had only been done numerically. The concluding chapter is devoted to the derivation of a lower bound to the ground state energy of the Fermi polaron in two-dimensional space. We show that the ground state energy can be bounded from below by a quantity that does not depend on the number of fermions in the Fermi gas. This result is correct under the assumption that the ratio of the mass of the impurity and the mass of a fermion exceeds 1.225. We also present a method which might yield a similar result for lower mass ratios. This method gives an estimate for the quadratic form of the regularized Hamiltonian in position space representation. In this connection, we present an inequality that bounds a singular potential of a two-dimensional Fermi gas depending only on the minimal distance between two fermions by the kinetic energy of the Fermi gas uniformly in the number of fermions. This inequality also applies to a potential with singularity 1/r^2, for which the Hardy inequality does not hold in two dimensions. Therefore, the full antisymmetry of the wave function has to be taken into account.