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Many real-world problems can be modeled by generalized equations. The solution of the latter can be a challenging task, and typically requires the use of some efficient numerical procedures, whose convergence analysis often relies on stability properties of a solution in question, and on a suitable over-estimate for the distance of a given point to the solution set of the problem, called error bound. With this thesis, we aim at a unified approach to first- and second-order conditions for stability properties and error bounds for generalized equations. To this end, we study existing and develop new concepts for generalized first-order derivatives of set-valued mappings, and use them to formulate criteria for Lipschitzian stability properties and Lipschitzian error bound conditions. These criteria can all be regarded as the property that a suitable generalized least singular value of a generalized derivative is nonzero. By considering generalized least singular values as an extended real-valued function that depends on arguments of an underlying mapping, we will be able to obtain second-order conditions arising from generalized derivatives of this function to guarantee non-Lipschitzian stability properties and non-Lipschitzian error bound conditions. This allows us to extend the territory covered by some seminal monographs dealing with stability properties and error bounds for generalized equations under first-order conditions. Furthermore, we discuss some specializations of our findings, and work out relations to existing results. Finally, we also investigate correlations between stability properties and error bounds with respect to different problem-formulations of one and the same generalized equation.

In this thesis, we prove observability inequalities for systems of differential equations in Banach spaces. In particular, we consider non-autonomous systems and systems of elliptic PDE with infinite-dimensional state space. We employ methods from harmonic analysis. This includes a vector-valued version of the Logvinenko-Sereda theorem, generalizing previous work by O. Kovrijkine. Our results are applied to establish null-controllability of control systems in Banach spaces together with precise estimates on the control cost. Furthermore, we consider an inverse problem for the stationary Schrödinger equation in three dimensions. In this setting, we prove that a Kato-class potential is uniquely determined by it's associated Dirichlet-to-Neumann operator. This complements a result by B. Haberman on the Calderón problem for conductivities with unbounded gradient.

Diese Arbeit befasst sich mit der sogenannten Doppelkopie, welche erstmals im Rahmen von Streuamaplituden formuliert wurde und eine Beziehung zwischen Yang-Mills-Theorie und der Gravitation herstellt. Yang-Mills-Streuamplituden tragen sowohl kinematische Informationen als auch Informationen, die mit einer Eigenschaft namens Farbe verbunden sind. Die Doppelkopie besagt, dass die Ersetzung der Farbinformation in Yang-Mills-Amplituden durch eine andere Kopie ihrer kinematischen Information zu Gravitationsamplituden führt, sofern bestimmte algebraische Bedingungen erfüllt sind. Die algebraischen Bedingungen, die für die Doppelkopie erforderlich sind, deuten auf die Existenz einer Algebra hin, die der Kinematik der Yang-Mills-Theorie zugrunde liegt und die kinematische Algebra genannt wird. In den letzten fünfzehn Jahren hat die Doppelkopie die Art und Weise, wie Streuungsberechnungen in der Gravitation durchgeführt werden, revolutioniert, und dennoch bleibt ein grundsätzliches Verständnis der Doppelkopie und der kinematischen Algebra schwer zu fassen. In dieser Arbeit verlassen wir den Rahmen der Streuamplituden und behandeln dieses Problem mit Hilfe von Homotopie-Algebren, den mathematischen Strukturen, die perturbativen Feldtheorien, einschließlich ihrer Off-Shell Struktur, zugrunde liegen. Insbesondere die der Yang-Mills-Theorie zugrunde liegende Algebra lässt sich in algebraische Strukturen für Farbe und Kinematik faktorisieren. Unter Verwendung dieser Faktorisierung konstruieren wir explizit eine kinematische Algebra für die Yang-Mills Theorie bis zur quartischen Ordnung in der Störungstheorie. Dann ersetzen wir die Farbalgebra durch eine zweite Kopie der kinematischen Algebra und erhalten eine Gravitationstheorie bis hinzu und einschließlich der quartischen Wechselwirkungen. Außerdem erklären wir, inwiefern die kinematische Algebra die Struktur ist, die für die Konsistenz der resultierenden Gravitationstheorie verantwortlich ist. Unsere algebraische Herangehenswei, This thesis deals with a relation between Yang-Mills theory and gravity called the double copy, which was first formulated in the framework of scattering amplitudes. Yang-Mills scattering amplitudes carry kinematic information as well as information associated with a property called color. The double copy states that, provided that certain algebraic conditions are met, replacing the color information of Yang-Mills amplitudes with another copy of their kinematic information yields gravitational amplitudes. The algebraic conditions required by the double copy hint at the existence of an algebra underlying the kinematics of Yang-Mills called the kinematic algebra. In the last fifteen years, the double copy has revolutionized the way that scattering computations are performed in gravity, and, yet, a first principle understanding of the double copy and the kinematic algebra remains elusive. In this thesis we divert from scattering amplitudes and address this problem in the framework of homotopy algebras, which are the mathematical structures underlying perturbative field theories, including their off-shell structure. In particular, the algebra underlying Yang-Mills theory factorizes into color and kinematic algebraic structures. Using this factorization, we construct explicitly a kinematic algebra for Yang-Mills theory to quartic order in perturbation theory. Then, following the double copy, we replace the color algebra with a second copy of the kinematic algebra and we obtain gravity up to and including quartic interactions. Moreover, we explain how the kinematic algebra is responsible for the consistency of the resulting gravity theory. Our algebraic approach to the double copy is completely off-shell, gauge independent and local, and provides a novel perspective on the algebraic foundations and origins of the double copy.

Let U be a smooth affine curve over a number field K with a compactification X and let L be a rank 2, geometrically irreducible lisse Ql-sheaf on U with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field E ⊂ Ql, and has bad, infinite reduction at some closed point x of X\U. We show that L occurs as a summand of the cohomology of a family of abelian varieties over U. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when E=Q, then L is isomorphic to the cohomology of an elliptic curve EU -> U., Peer Reviewed

The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.

This thesis explores the application of isoperimetric functions to gain weak functional inequalities involving Dirichlet forms. The connection between such weak functional inequalities and bounds on the convergence speed of the corresponding Markov semi-group is established. Three examples of discrete graphs and the corresponding Dirichlet forms are discussed.

Pattern formation on self deforming materials is playing an increasingly vital role in the study of many biological processes. An example is the cell cortex, a network of interlinked actin filaments connected to the inside of the cell membrane. The stiffness of these filaments makes the cortex rigid and gives the cortex a strong influence in the shape of the cell. Additionally, molecular turnover and motor proteins allow shape changes necessary for the cell to divide and to adapt to its environment. Due to the incredible number of proteins that make up the cell cortex, simulation of the molecular dynamics for the entire cortex is impossible. However, when considering a larger scale, these dynamics can be approximated, making it possible to model the cortex as an active viscoelastic surface. To implement this, we use the surface equivalent of a Maxwell material, additionally distinguishing between shear and dilational stress. The motor proteins are modelled by an advection diffusion equation on the surface combined with a concentration dependent surface tension. Surrounding the surface is a fluid modelled by the Navier-Stokes equations. We study the formation of patterns both analytically and numerically. In the analytical study the 2-dimensional curved surface is reduced to a flat 2-dimensional surface with periodic boundary conditions and no surrounding fluid. We then show, that despite the minimal complexity of this model, spatiotemporal patterns can develop on a viscoelastic surface if the relaxation times are different. For the numerical study the Arbitrary Lagrangian Eulerian method (ALE) is applied. The equations for the surface and bulk are solved separately, but this partial method is numerically unstable for dominantly viscous surfaces. With that in mind, we also implemented a monolithic model for viscous surfaces, solving the bulk and surface equations simultaneously. As a special case the influence of a chiral flow field on a sphere is studied. These chir

The major motivation for this work arose from the problem of simulating diffusion type processes in the human brain network. This thesis addresses numerical methods for parabolic partial differential equations (PDEs) on network structures interpreted as metric spaces (metric graphs). Such domains frequently occur in the context of quantum graphs, where they are studied together with a differential operator and coupling conditions at the vertices of the metric graph. Quantum graphs are popular models for thin, branched structures, and there is a great interest in their studies also from the theoretical point of view. The present work aims to bridge the gap between the theoretical work and the practical usage of quantum graph models by studying arising numerical problems. The main focus is on initial boundary value problems governed by (semilinear) parabolic partial differential equations that involve a second order spatial derivative posed on the edges of the graph. The particularity of these problems are the coupling conditions of the PDEs on their common vertices. The two central methods studied in this thesis are a Galerkin discretization with linear finite elements and a spectral Galerkin discretization with basis functions obtained from an eigenvalue problem on the metric graph. Both approaches follow the method of lines, i.e., Galerkin’s method is applied for the spatial discretization resulting in a system of ordinary differential equations. Spectral accuracy can be obtained with the spectral discretization in space for sufficiently smooth functions that fulfill certain coupling conditions at the vertices. In the finite element approach, the semidiscretization is solved with classical implicit-explicit time stepping methods combined with a graph specific multigrid solver for the arising systems of linear equations in each time step. In the spectral method, the stiffness matrix is diagonal such that exponential integrators can be applied efficiently to solve th

Um das Verständnis für physikalische und biologische Dynamiken zu verbessern, werden oft stellvertretend mathematische Modelle entwickelt, implementiert,validiert und analysiert. Die Entscheidung für oder gegen einen bestimmten Modelltyp, zum Beispiel ob die Auflösung in Raum und Zeit diskret oder kontinuierlich definiert ist, kann erheblichen Einfluss auf die Ergebnisse haben. Insbesondere bei der Untersuchung und Simulation der Dynamiken von biologischen Zellen, die häufig auch als biologische Flüssigkeiten (Biofluids) bezeichnet und in der Literatur oft mit physikalischen Flüssigkeiten verglichen werden, ist die Wahl des geeigneten Modelltyps nicht immer trivial. In diesem Zusammenhang stellt die vorliegende Arbeit drei verschiedene Szenarien vor. Unter Zuhilfenahme von unterschiedlichen mathematischen Modellen werden diese Szenarien dann untersucht. Dabei wird deutlich, dass trotz des ähnlichen Kontextes von physikalischen und biologischen Dynamiken je Szenario unterschiedliche Modelltypen besser geeignet sind und mitunter verschiedene Aussagen liefern. Daher muss für jedes dieser Szenarien die Entscheidung, welches Modell genommen wird und ob dieses in Raum und Zeit diskret oder kontinuierlich ist, neu evaluiert werden. Das erste Szenario befasst sich mit einer rein physikalischen Dynamik und beschreibt das Aufsteigen einer runden Flüssigkeitsblase innerhalb einer anderen Flüssigkeit. In diesem Zusammenhang wird auch häufig von zwei Phasen gesprochen. Dieser Fall dient auch als numerischer Benchmark-Test zur Bewertung der Genauigkeit von Zwei-Phasen-Modellen. Innerhalb dieses Kontextes werden oft Modelle verwendet, die kontinuierlich in Bezug auf Ort und Zeit sind. In der vorliegenden Arbeit wird stellvertretend das Cahn-Hilliard-Navier-Stokes-Modell verwendet. Vor allem wird ein neuer einfacher Diskretisierungsansatz für dieses Modell vorgestellt. Unter Verwendung eines Standard-Benchmark-Tests wird gezeigt, dass die Genauigkeit vergleichbar zu bisherigen Meth, Enhancing the understanding of physical and biological dynamics is crucial, which is why assisting mathematical models are often developed, implemented, validated, and analyzed. The decision for or against a particular model type, for example, whether the resolution in space and time is defined discretely or continuously, can considerably influence the results. Especially when investigating and simulating the dynamics of biological cells, also referred to as biological fluids and in the literature often compared to physical fluids, choosing the appropriate model type is not trivial in every case. This work presents three scenarios, which are further examined with the help of various mathematical models. Despite the similar context, dynamics of physical and biological fluids, some model types are more suited and deliver different results for each scenario. Therefore, the decision should be made new, depending on the scenarios, which model type is optimal, discrete, or continuous in space and time. The first scenario describes pure physical dynamics by the rise of a round fluid bubble within another fluid, which is often referred to as two phases. This setup also serves as a numerical benchmark test to evaluate the accuracy of physical two-phase-models. Within this context, the models used are often continuous regarding space and time. In this work, the Cahn-Hilliard-Navier-Stokes-model is chosen as a representative example. In particular, a new discretization approach for the model is introduced and evaluated by the previous benchmark test, which showcases that the new, more straightforward discretization approach leads to comparably precise results. The second scenario focuses on biological dynamics and describes the untreated growth of a tumor spheroid and further its behavior when exposed to \acl{rt}. These tumor spheroids are, in particular, 3D-assays of in-vitro experiments, which are 3D avascular aggregates of several thousand tumor cells mimicking tumor microa

A directed graph (digraph) G can primitively positively construct another digraph H if H is homomorphically equivalent to a first order interpretation of G that only uses primitive positive formulas. In this way, we define the poset of all finite digraphs ordered by primitive positive constructability. This poset, which is the object studied in this dissertation, is important for studying the complexity of constraint satisfaction problems and for studying clones ordered by minion homomorphisms. In this thesis I will introduce the poset and amoung other results I will present the three topmost levels of the poset and I will describe the subposet consisting of digraphs without sources or sinks.

Simulationen elektronischer Strukturen ermöglichen die Bestimmung grundlegender Eigenschaften von Materialien ohne jegliche Experimente. Sie zählen deshalb zu den Standardwerkzeugen, mit denen Fortschritte in materialwissenschaftlichen und chemischen Anwendungen vorangetrieben werden. In den letzten Jahrzehnten hat sich die Dichtefunktionaltheorie (DFT) aufgrund ihrer ausgezeichneten Balance zwischen Genauigkeit und Rechenkosten als die beliebteste Simulationstechnik für elektronische Strukturen etabliert. Jedoch verlangen drängende gesellschaftliche und technologische Herausforderungen nach Lösungen für immer komplexere wissenschaftliche Fragestellungen, sodass selbst die effizientesten DFT-Programme nicht mehr in der Lage sind, Antworten in angemessener Zeit und mit den verfügbaren Rechenressourcen zu liefern. Daher wächst das Interesse an Ansätzen des maschinellen Lernens (ML), die darauf abzielen, Modelle bereitzustellen, die die Vorhersagekraft von DFT-Rechnungen zu vernachlässigbaren Kosten replizieren. In dieser Arbeit wird gezeigt, dass solche ML-DFT Ansätze bisher nicht in der Lage sind, das Vorhersagen der elektronischen Struktur von Materialien auf DFT-Niveau vollständig abzubilden. Davon ausgehend wird in dieser Arbeit ein neuer Ansatz für ML-DFT Modelle vorgestellt. Es wird ein umfassendes Framework für das Training von ML-DFT-Modellen auf Grundlage einer lokalen Darstellung der elektronischen Struktur entwickelt, welcher auch Details wie Strategien zur Datengeneration und Hyperparameteroptimierung beinhaltet. Es werden Ergebnisse vorgestellt, die zeigen, dass mit diesem Framework trainierte Modelle die breite Palette der Vorhersagefähigkeit sowie Genauigkeit von DFT-Simulationen zu drastisch reduzierten Kosten replizieren. Weiterhin wird die allgemeine Nützlichkeit dieses Ansatzes demonstriert, indem Modelle über Längenskalen, Phasengrenzen und Temperaturbereiche hinweg angewendet werden.:List of Tables 10 List of Figures 12 Mathematical notation and a, Electronic structure simulations allow researchers to compute fundamental properties of materials without the need for experimentation. As such, they routinely aid in propelling scientific advancements across materials science and chemical applications. Over the past decades, density functional theory (DFT) has emerged as the most popular technique for electronic structure simulations, due to its excellent balance between accuracy and computational cost. Yet, pressing societal and technological questions demand solutions for problems of ever-increasing complexity. Even the most efficient DFT implementations are no longer capable of providing answers in an adequate amount of time and with available computational resources. Thus, there is a growing interest in machine learning (ML) based approaches within the electronic structure community, aimed at providing models that replicate the predictive power of DFT at negligible cost. Within this work it will be shown that such ML-DFT approaches, up until now, do not succeed in fully encapsulating the level of electronic structure predictions DFT provides. Based on this assessment, a novel approach to ML-DFT models is presented within this thesis. An exhaustive framework for training ML-DFT models based on a local representation of the electronic structure is developed, including minute treatment of technical issues such as data generation techniques and hyperparameter optimization strategies. Models found via this framework recover the wide array of predictive capabilities of DFT simulations at drastically reduced cost, while retaining DFT levels of accuracy. It is further demonstrated how such models can be used across differently sized atomic systems, phase boundaries and temperature ranges, underlining the general usefulness of this approach.:List of Tables 10 List of Figures 12 Mathematical notation and abbreviations 14 1 Introduction 19 2 Background 23 2.1 Density Functional Theory 23 2.2 Sampling of Observables 35 2.3

Via normal Seshadri stratifications, Chirivì, Fang and Littelmann obtained a standard monomial theory (SMT) on the homogeneous coordinate ring of certain embedded projective varieties, that is to say a basis of so called standard monomials. In the case of Schubert varieties such a SMT was already developed combinatorially by Lakshmibai, Musili and Seshadri. We generalize the notion of a Seshadri stratification to closed subvarieties in a product of projective spaces and construct such stratifications on Schubert varieties in every Dynkin type. Using the Littelmann path model, we show that these stratifications provide a geometric explanation of the SMT of Hodge and Young indexed by semistandard Young tableaux, the SMT of Lakshmibai, Musili and Seshadri and more general, of a SMT indexed by sequences of LS-paths.

H.~Herrmann, C.-Y.~Hsiao, G.~Marinescu und der Autor dieser Arbeit bes-chreiben in einem kürzlich erschienen Artikel die vollständige asymptotische Entwicklung des Schwartz-Kerns bestimmter spektraler Projektionen von semi-klassischen Toeplitz-Operatoren auf streng pseudokonvexen, kompakten, einbettbaren CR-Mannigfaltigkeiten. Eine solche asymptotische Expansion führt zu vielen neuen Anwendungen in der CR-Geometrie, und das Ziel dieser Arbeit ist es, dieses Ergebnis zu verallgemeinern. Dazu betrachten wir eine kompakte, nicht-entartete CR Mannigfaltigkeit \((X,T^{1,0}X)\) mit konstanter Signatur \((n_{-},n_{+})\) und einen selbstadjungierten, klassischen Pseudodifferentialoperator \(P\) erster Ordnung, der den Raum der $(0,q)$-Formen auf $X$ auf sich selbst abbildet. Wir definieren und untersuchen dann sogenannte Levi-elliptische Toeplitz-Operatoren für Formen mit niedriger Energie. Genauer betrachten wir für jedes $\lambda>0$ den Operator $ T_{P,\lambda}^{(q)}:=\Pi_\lambda^{(q)}\circ P\circ \Pi_\lambda^{(q)}. $ Dabei bezeichnet $\Pi_\lambda^{(q)}$ - in Analogie zur Szeg\H{o}-Projektion - die orthogonale Projektion auf Formen niedrigerer Energie ist $\mathds 1_{[0,\lambda]}(\Box_b^{(q)})$ bezü-glich des Kohn-Laplace-Operators \(\Box_b^{(q)}\). Im Fall $q=n_-$, nehmen wir an, dass $P$ levi-elliptisch ist, was bedeutet, dass das matrixwertige Hauptsymbole von $P$ bezogen auf die Signatur $(n_-,n_+)$ einer abgeschwächten Elliptizitätsbedingung genügt. Wir betrachten dann den durch die Helffer--Sjöstrand-formel definierten Spektraloperator \(\chi(k^{-1}T^{(q)}_{P,\lambda})\), $q=n_-$, \(k>0\), bezüglich einer glatten Funktion \(\chi\colon \R\to\C\) mit kompaktem Träger in \(\R\setminus\{0\}\). Unter Verwendung mikroanalytischer Methoden von Hsiao--Marinescu und Galasso--Hsiao untersuchen wir das asymptotische Verhalten von \(\chi(k^{-1}T^{(q)}_{P,\lambda})\) für \(k\to+\infty\). Unser Hauptergebnis beschreibt die vollständige asymptotische Entwicklung des Schwartz-Kerns

This thesis explores the application of isoperimetric functions to gain weak functional inequalities involving Dirichlet forms. The connection between such weak functional inequalities and bounds on the convergence speed of the corresponding Markov semi-group is established. Three examples of discrete graphs and the corresponding Dirichlet forms are discussed.

With early works dating back to the 1930’s until today, here is a growing interest in the theory of asymptotic distributions of expected zeros of random polynomials when their degree grows indefinitely. A natural geometric generalization of random polynomials are random sections of a holomorphic line bundle over a complex manifold. In 1999, Shiffman and Zelditch proved that on a compact K¨ahler manifold, the zeros of sections of high tensor powers of a holomorphic line bundle asymptotically equidistribute with respect to the normalized curvature of the line bundle. Their result has numerous applications in mathematical physics and was generalized in many different directions. In this thesis we generalize their result to a semipositively curved holomorphic line bundle over a punctured Riemann surface. To achieve this, we discuss many tools that have proven themselves to represent an appropriate framework to study statistical properties of ensembles of zeros on complex manifolds. We start by proving the existence of spectral gap for the Kodaira Laplacian that is associated to the line bundle. We use this result, together with the technique of analytic localization by Ma and Marinescu, to prove a pointwise global on-diagonal asymptotic expansion of the associated Bergman kernel in our setting. Moreover, we show locally uniform estimates on the Bergman kernel and its derivatives. We use these estimates to prove the locally uniform convergence of the induced Fubini-Study metrics and their potentials to the global curvature and its potential, respectively. We conclude by showing that the expected zeros of holomorphic sections equidistribute with respect to the normalized curvature of the line bundle. Moreover, we apply the theory of meromorphic transforms by Dinh and Sibony estimate the speed of convergence in our equidistribution result.

Characterizing the set of correlations that can arise from performing measurements in a quantum description of Nature is a relevant, but challenging task. Such a characterization, known as the quantum causal compatibility problem, provides us with insights in the advantages of quantum theory over a classical theory, but can also show its limitations. This problem becomes particularly challenging when the quantum states and measurements are required to be compatible with a given causal structure. A causal structure dictates the causal dependencies of the parties and systems involved in the experiment. One can think of the Bell scenario as one of the simplest causal structures, in which two spatially separated parties, Alice and Bob, are assumed to perform a measurement on a shared source. In more general causal structures we might have more parties, and more sources that are independent of each other. Recently, a systematic way of analyzing the correlations in classical and quantum causal structures was proposed in the form of the inflation technique. In the inflation technique the causal dependencies, which are difficult to encode algorithmically, are relaxed to easy-to-encode symmetry constraints on a larger number of parties. For the classical case, this provides a converging hierarchy of linear programs for the causal compatibility problem. For the quantum case, it instead yields a hierarchy of increasingly restrictive semidefinite programming relaxations. It is, however, unknown whether this hierarchy is also complete. One of the main results of this thesis is to show that a modified version of the quantum inflation technique is convergent for the quantum causal compatibility problem. This modified hierarchy introduces an additional parameter, r, that restricts the Schmidt rank of the observables. For each value of r we provide a hierarchy of compatibility tests that is complete in the sense that it will detect, at some finite level, any probability distribution

Seit Beginn der Pandemie Anfang des Jahres 2020 werden statistische Daten erhoben mit dem Ziel, die Ausbreitung von COVID-19 zu charakterisieren. Grundlage der statistischen Analysen bilden die Zeitreihen der täglich erfassten Anzahl von Neuinfektionen. Die Dynamik der Pandemie lässt sich als Trajektorie in einem Phasenraum visualisieren. Dieser Zugang und ein Vergleich mit dem mathematischen Modell des logistischen Wachstums ermöglicht eine Analyse der Wirksamkeit von Maßnahmen und liefert Hinweise für eine Optimierung von Strategien zur Eindämmung der Virusausbreitung.:1. Einleitung 2. Zeitliche Entwicklung der Infektionszahlen und logistisches Wachstum 3. Pandemie im Phasenraum und log-log-Darstellung 3.1 Dynamik der Pandemie bis 02.03.2020 bis 27.06.2021 3.2 Dynamik der Pandemie von 28.06.2021 bis 02.06.2023 4. Diskussion 4.1 Datenerhebung und statistische Auswertung 4.2 Zeitliche, räumliche und sachliche Analyse der Daten 4.3 Schlussfolgerungen und Ausblick 5. Anhang 6. Literatur, Since the start of the pandemic at the beginning of 2020, statistical data have been collected with the aim of characterizing the spread of COVID-19. The basis of the statistical analyzes is the time series of the number of new infections recorded daily. The dynamics of the pandemic can be visualized as a trajectory in a phase space. This approach and a comparison with the mathematical model of logistic growth enables us an analysis of the effectiveness of measures and provides evidence for optimizing strategies for containment of the virus.:1. Einleitung 2. Zeitliche Entwicklung der Infektionszahlen und logistisches Wachstum 3. Pandemie im Phasenraum und log-log-Darstellung 3.1 Dynamik der Pandemie bis 02.03.2020 bis 27.06.2021 3.2 Dynamik der Pandemie von 28.06.2021 bis 02.06.2023 4. Diskussion 4.1 Datenerhebung und statistische Auswertung 4.2 Zeitliche, räumliche und sachliche Analyse der Daten 4.3 Schlussfolgerungen und Ausblick 5. Anhang 6. Literatur

We study the geometry of linear submanifolds and strata of k-differentials. Moreover, we give applications in the theory of tropical geometry and the theory of Higgs bundles.

We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in Mg. Our results regarding quadratic differentials imply that the quadratic strata in genus 6 are uniruled.

For genus g=r(r+1)2+1, we prove that via the forgetful map, the universal Prym-Brill-Noether locus Rrg has a unique irreducible component dominating the moduli space Rg of Prym curves.

The hierarchical feature regression (HFR) is a novel graph-based regularized regression estimator, which mobilizes insights from the domains of machine learning and graph theory to estimate robust parameters for a linear regression. The estimator constructs a supervised feature graph that decomposes parameters along its edges, adjusting first for common variation and successively incorporating idiosyncratic patterns into the fitting process. The graph structure has the effect of shrinking parameters towards group targets, where the extent of shrinkage is governed by a hyperparameter, and group compositions as well as shrinkage targets are determined endogenously. The method offers rich resources for the visual exploration of the latent effect structure in the data, and demonstrates good predictive accuracy and versatility when compared to a panel of commonly used regularization techniques across a range of empirical and simulated regression tasks.

Matroids are combinatorial objects that generalize linear independence. A matroid can be represented geometrically by its Bergman fan and we compare the symmetries of these two objects. Sometimes, the Bergman fan has additional automorphisms, which are related to Cremona transformations in projective space. Their existence depends on a combinatorial property of the matroid, as has been shown by Shaw and Werner, and we study the consequences for the structure of such matroids. This allows us to gain a better understanding of the so-called Cremona group of a matroid and we apply our results to root system matroids., Matroide sind kombinatorische Objekte, die lineare Abhängigkeit verallgemeinern. Ein Matroid kann durch seinen Bergmanfächer geometrisch dargestellt werden und wir vergleichen die Symmetrien dieser beiden Objekte. Manchmal besitzt der Bergmanfächer zusätzliche Automorphismen, die mit Cremonatransformationen im projektiven Raum zusammenhängen. Deren Existenz hängt von einer kombinatorischen Eigenschaft des Matroids ab, wie Shaw und Werner gezeigt haben, und wir untersuchen die Konsequenzen für die Struktur solcher Matroide. Auf diese Weise können wir die sogenannte Cremonagruppe eines Matroids besser verstehen und wir wenden unsere Resultate auf Wurzelsystemmatroide an.

• Zahlen und Maßsysteme sind bereits aus dem antiken Ägypten und aus Mesopotamien belegt. Im 4. Jahrtausend vor unserer Zeitrechnung haben sich mit der hierarchisierten Gesellschaft auch Zahl- und Schriftzeichen entwickelt. Sie dienten vor allem der Zuteilung von Ressourcen. • Die 13 Bücher der »Elemente« von Euklid (3. Jahrhundert vor unserer Zeit) sind die früheste erhaltene axiomatisch-deduktiv aufgebaute Sammlung mathematischen Wissens. Die Begeisterung für dieses Werk hielt über Jahrhunderte an. • Die Ordnung von Beobachtungen durch die mathematische Erfassung und Auswertung von Daten ist in Wissenschaft und Alltag selbstverständlich. Mathematische Techniken der Statistik und Kartierung halfen Dr. John Snow im 19. Jahrhundert, die Ausbreitung der Cholera zu erforschen und zu bekämpfen. • Trotzdem stößt die Mathematik bei der Schaffung von Ordnungen auch an Grenzen: Denn es gibt weder eine Garantie noch eine Anleitung für deren bestmögliche Nutzung. Dies zeigen nicht zuletzt Krisen wie die Coronapandemie oder die Klimakrise.

Highlights • We study dormancy in the ‘rare mutation’ regime of stochastic adaptive dynamics. • We first derive the polymorphic evolution sequence, based on prior work. • Our evolutionary branching criterion extends a result by Champagnat and Méléard. • In a classical model dormancy can favour evolutionary branching. • Dormancy also affects several more population characteristics. Abstract In this paper, we investigate the consequences of dormancy in the ‘rare mutation’ and ‘large population’ regime of stochastic adaptive dynamics. Starting from an individual-based micro-model, we first derive the Polymorphic Evolution Sequence of the population, based on a previous work by Baar and Bovier (2018). After passing to a second ‘small mutations’ limit, we arrive at the Canonical Equation of Adaptive Dynamics, and state a corresponding criterion for evolutionary branching, extending a previous result of Champagnat and Méléard (2011). The criterion allows a quantitative and qualitative analysis of the effects of dormancy in the well-known model of Dieckmann and Doebeli (1999) for sympatric speciation. In fact, quite an intuitive picture emerges: Dormancy enlarges the parameter range for evolutionary branching, increases the carrying capacity and niche width of the post-branching sub-populations, and, depending on the model parameters, can either increase or decrease the ‘speed of adaptation’ of populations. Finally, dormancy increases diversity by increasing the genetic distance between subpopulations.

Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra. In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset. We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distin

In this thesis we consider the application of Fenchel's duality theory and gradient-based methods for the training and hyperparameter optimization of Support Vector Machines. We show that the dualization of convex training problems is possible theoretically in a rather general formulation. For training problems following a special structure (for instance, standard training problems) we find that the resulting optimality conditions can be interpreted concretely. This approach immediately leads to the well-known notion of support vectors and a formulation of the Representer Theorem. The proposed theory is applied to several examples such that dual formulations of training problems and associated optimality conditions can be derived straightforwardly. Furthermore, we consider different formulations of the primal training problem which are equivalent under certain conditions. We also argue that the relation of the corresponding solutions to the solution of the dual training problem is not always intuitive. Based on the previous findings, we consider the application of customized optimization methods to the primal and dual training problems. A particular realization of Newton's method is derived which could be used to solve the primal training problem accurately. Moreover, we introduce a general convergence framework covering different types of decomposition methods for the solution of the dual training problem. In doing so, we are able to generalize well-known convergence results for the SMO method. Additionally, a discussion of the complexity of the SMO method and a motivation for a shrinking strategy reducing the computational effort is provided. In a last theoretical part, we consider the problem of hyperparameter optimization. We argue that this problem can be handled efficiently by means of gradient-based methods if the training problems are formulated appropriately. Finally, we evaluate the theoretical results concerning the training and hyperparameter optimizatio

This thesis studies regression discontinuity designs with the use of additional covariates for estimation of the average treatment effect. We prove asymptotic normality of the covariate-adjusted estimator under sufficient regularity conditions. In the case of a high-dimensional setting with a large number of covariates depending on the number of observations, we discuss a Lasso-based selection approach as well as alternatives based on calculated correlation thresholds. We present simulation results on those alternative selection strategies.:1. Introduction 2. Preliminaries 3. Regression Discontinuity Designs 4. Setup and Notation 5. Computing the Bias 6. Asymptotic Behavior 7. Asymptotic Normality of the Estimator 8. Including Potentially Many Covariates 9. Simulations 10. Conclusion

The development of mathematical reasoning skills has increasingly been of focus for the teaching and learning of mathematics. This research utilizes a teaching simulation using the methodology of scriptwriting, in which prospective teachers are asked to complete a script of a dialogue from a classroom simulation involving fraction multiplication and division with justification, assisting fictional students to work through their difficulties and helping them to justify their reasoning. Such tasks allow for the examination of the prospective teacher moves to support student reasoning through their imagined action and choice of words. Scripts from forty-one prospective primary teachers were examined for the study, and five clusters based on the type of teacher move for supporting student reasoning were found. Overall, the prospective teachers emphasized the elicitation and facilitation of students’ ideas. The cluster analysis, however, provided a nuanced examination of the cohort’s teacher moves. While cluster one saw the highest incident of eliciting teacher moves, albeit only in the low potential category, clusters two and three mostly used facilitating teacher moves, but varied in their use of high and low potential moves. Cluster four concentrated moves on facilitating, eliciting, and responding to student reasoning. Cluster five employed teacher moves from all main categories, with some instances of high potential moves in all categories except extending student reasoning, which can better support reasoning. The prospective mathematics teachers’ scripts and the five clusters that were found during analysis are discussed with implications for future teacher education and the support of building mathematical reasoning., Peer Reviewed

Gegenstand dieser Dissertation ist die non-parametrische Schätzung der Kovarianz in multi-dimensionalen Lévy-Prozessen auf der Basis von Hochfrequenzbeobachtungen. Im ersten Teil der Arbeit wird eine modifizierte Version der von Jacod und Reiß vorgeschlagenen Methode der Hochfrequenzbeobachtung für die Ermittlung der Kovarianz multi-dimensionaler Lévy-Prozesse gegeben. Es wird gezeigt, dass der Kovarianzschätzer optimal im Minimaxsinn ist. Darüber hinaus demonstrieren wir, dass die Indexaktivität der co-jumps durch das harmonische Mittel der Sprungaktivitätsinzidenzen der Komponenten von unten beschränkt wird. Der zweite Teil behandelt das Problem der adaptiven Schätzung. Ausgehend von einer Familie asymptotischer Minimax-Schätzer der Kovarianz, erhalten wir einen datenbasierten Schätzer. Wir wenden Lepskii’s Methode an, um die Kovarianz an die unbekannte Aktivität des co-jumps Indexes des Sprungteils anzupassen. Da wir es mit einem Adaptierungsproblem zu tun haben, müssen wir eine Schätzung der charakteristischen Funktion des multi-dimensionalen Lévy-Prozesses konstruieren, damit die charakteristische Funktion weder von einer semiparametrischen Annahme abhängt noch schnell abfällt. Aus diesem Grund wird auf Basis von Neumanns Methode ein trunkierter Schätzer für die empirische charakteristische Funktion konstruiert. Die Anwesenheit der trunkierten, empirischen charakteristischen Funktion im Zähler führt jedoch zu einer Situation, die auch bei der Deconvolution auftritt, d.h. einem irregulären Verhalten des stochastischen Fehlers. Dieser U-förmige stochastische Fehler verhindert die Anwendung von Lepskii’s Grundsatz. Um diesem Problem, entgegenzuwirken, entwickeln wir eine Strategie, welche zu einem Orakelstart von Lepskii's Methode führt, mit deren Hilfe ein monoton steigender stochastischer Fehler konstruiert wird. Dies erlaubt uns, ein Balancing Principle einzuführen und einen adaptiven Schätzer für die Kovarianz zu erhalten, der fast-optimale Raten erzeugt., In this thesis, we consider the problem of nonparametric estimation for the continuous part of the covariation of a multi-dimensional Lévy process from high-frequency observations. This continuous part of covariation is also called covariance. The first part modifies the high-frequency estimation method, proposed by Jacod and Reiss, to cover estimation of the covariance of multi-dimensional Lévy processes. The covariance estimator is shown to be optimal in the minimax-sense. Moreover, the co-jump index activity is proved to be bounded from below by the harmonic mean of the jump activity indices of the components. In the second part, we address the problem of the adaptive estimation. Starting from an asymptotically minimax family of estimators for the covariance, we derive a data-driven estimator. Lepskii's method is applied to adapt the covariance to the unknown co-jump index activity of the jump part. Faced with an adaptation problem, we need to secure an estimation for the characteristic function of the multi-dimensional Lévy process so that it does not depend on a semiparametric assumption and, at the same time, does not decay fast. For this reason, a truncated estimator for the empirical characteristic function is constructed based on Neumann's method. The presence of the truncated empirical characteristic function in the denominator leads to a situation similar to the deconvolution problem, i.e., an irregular behavior of the stochastic error. This U-shaped stochastic error does not permit us to apply Lepskii's principle. To counteract this problem, we establish a strategy to obtain an oracle start of Lepskii's method, according to which a monotonically increasing stochastic error is constructed. This enables us to apply a balancing principle and build an adaptive estimator for the covariance which obtains near-optimal rates.

Die hier dargelegte Dissertation ist motiviert durch die Vorschläge von Joyce, Doan und Walpuski zur Definitionen enumerativer Invarianten für G2-Mannigfaltigkeit, durch das Zählen gewisser kalibrierter Untermannigfaltigkeiten, sogenannter assoziativen Untermannigfaltigkeiten. In Kapitel 1, werde ich Definitionen und grundlegende Fakten über G2-Mannigfaltigkeit und deren assoziative Untermannigfaltigkeit wiederholen. Darüber hinaus erläutere ich die Konstruktion von G2-Mannigfaltigkeit als verdrehte verbundener Summe. Kapitel 2 schafft die nötige Grundlage für das darauf folgende dritte Kapitel. Hier definiere ich den Modul-Raum der asymptotisch zylindrischen assoziativen Untermannigfaltigkeiten zusammen mit seiner natürlichen Topologie und zeige, dass der Modul-Raum lokal homeomorph zur Urbild-Menge der Null einer glatten Abbildung zwischen zwei endlich-dimensionalen Räu- men ist. In besonderen Fällen ist dieser Modul-Raum eine Lagrangesche Untermannigfaltigkeit des Modul Raums der holomorphen Kurven einer asymptotisch zylindrischen Calabi-Yau Man- nigfaltigkeit. In Kapitel 3 beweise ich ein Klebe-Theorem für ein Paar von asymptotisch zylindrischen as- soziativen Untermannigfaltigkeiten in einem zusammenpassenden Paar von asymptotisch zylin- drischen G2-Mannigfaltigkeiten. Hiermit konstruiere ich neue geschlossene und starre (rigid) assoziative Untermannigfaltigkeiten in verdrehten verbundenen Summe G2-Mannigfaltigkeiten. In Kapitel 4 untersuche ich den Modul-Raum der konisch singulären assoziativen Un- termannigfaltigkeiten in G2-Mannigfaltigkeiten. Durch das Umformulieren des Indexes des Operators, der die Deformationstheorie kontrolliert, in bestimmte Stabilität-Indizes des zu- grundeliegenden assoziativen Kegels begründe ich, dass in einem generischen Pfad in dem Raum der ko-geschlossenen G2-Strukturen keine asymptotisch konischen assoziative Unter- mannigfaltigkeiten existieren, die mindestens eine Singularität besitzen, die auf einem Kegel mit Stabiltätsindex, The dissertation presented here is motivated from the proposals made by Joyce, Doan and Walpuski to define enumerative invariants of G2-manifolds by counting certain calibrated submanifolds, called associative submanifolds. In Chapter 1, I review the definitions and basic facts of G2-manifolds and associative submanifolds. Moreover, I explain the construction of G2-manifolds as twisted connected sums. Chapter 2 serves as a necessary groundwork for Chapter 3. Here, I define the moduli space of asymptotically cylindrical associative submanifolds with its natural topology and prove that the moduli space is locally homeomorphic to the zero set of a smooth map between two finite-dimensional spaces. In the best scenario, this moduli space is a Lagrangian submanifold of the moduli space of holomorphic curves in the asymptotic Calabi-Yau 3-fold. In Chapter 3, I prove a gluing theorem for a pair of asymptotically cylindrical associative submanifolds in a matching pair of asymptotically cylindrical G2-manifolds. Using this I construct new closed and rigid associative submanifolds of twisted connected sum G2-manifolds. In Chapter 4, I study the moduli space of conically singular associative submanifolds in G2-manifolds. By reformulating the index of the operator that controls the deformation theory in terms of certain stability-index of the associative cones, I establish that in a generic path of co-closed G2-structures there are no conically singular associative submanifolds that have at least one singularity modeled on a cone of stability-index greater than one. This result applies to all special Lagrangian cones, except the Harvey-Lawson T2-cone and a union of two special Lagrangian planes. Additionally, it applies to all associative cones whose links are null-torsion holomorphic curves in S6. Furthermore, parts of Chapter 4 also serve as a necessary groundwork for Chapter 5. The naive counting of associative submanifolds does not lead to an invariant due to several transit

We propose a fully-corrective generalized conditional gradient method (FC-GCG) for the minimization of the sum of a smooth, convex loss function and a convex one-homogeneous regularizer over a Banach space. The algorithm relies on the mutual update of a finite set Ak of extremal points of the unit ball of the regularizer and of an iterate uk cone(Ak). Each iteration requires the solution of one linear problem to update Ak and of one finite dimensional convex minimization problem to update the iterate. Under standard hypotheses on the minimization problem we show that the algorithm converges sublinearly to a solution. Subsequently, imposing additional assumptions on the associated dual variables, this is improved to a linear rate of convergence. The proof of both results relies on two key observations: First, we prove the equivalence of the considered problem to the minimization of a lifted functional over a particular space of Radon measures using Choquet’s theorem. Second, the FC-GCG algorithm is connected to a Primal-Dual-Active-Point method on the lifted problem for which we finally derive the desired convergence rates., Peer Reviewed

We investigate the proximal map for the weighted mean absolute error function. An algorithm for its efficient and vectorized evaluation is presented. As a demonstration, this algorithm is applied to a nonsmooth energy minimization problem., DFG http://dx.doi.org/10.13039/501100001659, Peer Reviewed

Regularized structural equation models have gained considerable traction in the social sciences. They promise to reduce overfitting by focusing on out-of-sample predictions and sparsity. To this end, a set of increasingly constrained models is fitted to the data. Subsequently, one of the models is selected, usually by means of information criteria. Current implementations of regularized structural equation models differ in their optimizers: Some use general purpose optimizers whereas others use specialized optimization routines. While both approaches often perform similarly, we show that they can produce very different results. We argue that in particular, the interaction between optimizer and selection criterion (e.g., BIC) contributes to these differences. We substantiate our arguments with an empirical demonstration and a simulation study. Based on these findings, we conclude that researchers should consider specialized optimizers whenever possible. To facilitate the implementation of such optimizers, we provide the R package lessSEM., Peer Reviewed

We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability., Peer Reviewed

Given a polarizable ℤ-variation of Hodge structures \(\mathbb{V}\) over a complex smooth quasi-projective base \(S\), a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of \(S\), called the special subvarieties for \(\mathbb{V}\). Our main result in this paper is that, if the level of \({\mathbb{V}}\) is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree \(d\) smooth hypersurfaces in \(\mathbf{P}^{n+1}_{\mathbb{C}}\), \(n\geq 3\), \(d\geq 5\) and \((n,d)\neq (4,5)\), is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in \(S^{\mbox{an}}\) as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in \(S\) of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension., Peer Reviewed

Planning longitudinal studies can be challenging as various design decisions need to be made. Often, researchers are in search for the optimal design that maximizes statistical power to test certain parameters of the employed model. We provide a user-friendly Shiny app OptDynMo available at https://shiny.psychologie.hu-berlin.de/optdynmo that helps to find the optimal number of persons (N) and the optimal number of time points (T) for which the power of the likelihood ratio test (LRT) for a model parameter is maximal given a fixed budget for conducting the study. The total cost of the study is computed from two components: the cost to include one person in the study and the cost for measuring one person at one time point. Currently supported models are the cross-lagged panel model (CLPM), factor CLPM, random intercepts cross-lagged panel model (RI-CLPM), stable trait autoregressive trait and state model (STARTS), latent curve model with structured residuals (LCM-SR), autoregressive latent trajectory model (ALT), and the latent change score model (LCS)., Peer Reviewed

The “shape” of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation, and scaling, its “form” its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian L2-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even nonlinear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of (a) astragalus shapes of wild and domesticated sheep and (b) cell forms generated in a biophysical model, as well as (c) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines. Supplementary materials for this article are available online., Peer Reviewed

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We investigate the maximum principle for the weak solutions to the Cauchy problem for the hyperbolic fourth-order linear equations with constant complex coefficients in the plane bounded domain., Peer Reviewed