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We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Holder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm-Liouville problems. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm-Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.
LetH be a Schrodinger operator defined on a noncompact Riemannianmanifold Omega, and let W is an element of L-infinity (Omega; R). Suppose that the operator H + W is critical in Omega, and let phi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction ofH satisfying vertical bar u vertical bar <= C-phi in Omega for some constant C > 0, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K is an element of Omega the operator H admits a positive solution in (Omega) over bar = Omega \ K, and vertical bar u vertical bar <= C psi in (Omega) over bar for some constant C > 0, where psi is a positive solution of minimal growth in a neighborhood of infinity in Omega. Under natural assumptions, this result holds also in the context of infinite graphs, and Dirichlet forms.
We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.
In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.
Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary
(2023)
Non-local boundary conditions – for example the Atiyah–Patodi–Singer (APS) conditions – for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, Bär and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While Bär and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions.
In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed.
We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems.
Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions – the analogue of the Atiyah–Patodi–Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.
Several numerical tools designed to overcome the challenges of smoothing in a non-linear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter, and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.
Das Eigene und das Fremde
(2023)
Die vorliegende Arbeit stellt eine Untersuchung des Fremdverstehens von Lehrkräften im Mathematikunterricht dar. Mit ‚Fremdverstehen‘ soll dabei – in Anlehnung an den Soziologen Alfred Schütz – der Prozess bezeichnet werden, in welchem eine Lehrkraft versucht, das Verhalten einer Schülerin oder eines Schülers zu verstehen, indem sie dieses Verhalten auf ein Erleben zurückführt, das ihm zugrunde gelegen haben könnte. Als ein wesentliches Merkmal des Prozesses stellt Schütz in seiner Theorie des Fremdverstehens heraus, dass das Fremdverstehen eines Menschen immer auch auf seinen eigenen Erlebnissen basiert. Aus diesem Grund wird in der Arbeit ein methodischer Zweischritt vorgenommen: Es werden zunächst die mathematikbezogenen Erlebnisse zweier Lehrkräfte nachgezeichnet, bevor dann ihr Fremdverstehen in konkreten Situationen im Mathematikunterricht rekonstruiert wird. In der ersten Teiluntersuchung (= der Rekonstruktion eigener Erlebnisse der untersuchten Lehrkräfte) erfolgt die Datenerhebung mit Hilfe biographisch-narrativer Interviews, in denen die untersuchten Lehrkräfte angeregt werden, ihre mathematikbezogene Lebensgeschichte zu erzählen. Die Analyse dieser Interviews wird im Sinne der rekonstruktiven Fallanalyse vorgenommen. Insgesamt führt die erste Teiluntersuchung zu textlichen Darstellungen der rekonstruierten mathematikbezogenen Lebensgeschichte der untersuchten Mathematiklehrkräfte. In der zweiten Teiluntersuchung (= der Rekonstruktion des Fremdverstehens der untersuchten Lehrkräfte) werden dann narrative Interviews geführt, in denen die untersuchten Lehrkräfte von ihrem Fremdverstehen in konkreten Situationen im Mathematikunterricht erzählen. Die Analyse dieser Interviews erfolgt mit Hilfe eines dreischrittigen Analyseverfahrens, welches die Autorin eigens zum Zweck der Rekonstruktion von Fremdverstehen entwickelte. Am Ende dieser zweiten Teiluntersuchung werden sowohl das rekonstruierte Fremdverstehen der Lehrkräfte in verschiedenen Unterrichtssituationen dargestellt als auch Strukturen, die sich in ihrem Fremdverstehen abzeichnen. Mit Hilfe einer theoretischen Verallgemeinerung werden schließlich – auf Basis der Ergebnisse der zweiten Teiluntersuchung – Aussagen über fünf Merkmale des Fremdverstehens von Lehrkräften im Mathematikunterricht im Allgemeinen gewonnen. Mit diesen Aussagen vermag die Arbeit eine erste Beschreibung davon hervorzubringen, wie sich das Phänomen des Fremdverstehens von Lehrkräften im Mathematikunterricht ausgestalten kann.
We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.