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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

Set-based reachability analysis is used extensively in fields such as robotics in the context of, e.g., motion planning and the formal verification of cyber-physical systems. In order to properly model such systems, various set representations have been used throughout the literature. For certain computational tasks, one needs to be able to sample random points within such set representations. This thesis will introduce two approaches, the Ball Walk Algorithm and the Billiard Walk Algorithm, which are both so-called Markov Chain Monte Carlo methods that allow one to generate random samples within a convex set. In addition, the distribution of these samples can be shown to converge to the uniform distribution. The set representation we will focus our efforts on are spectrahedral shadows, which can be seen as the solution sets of positive semi-definite constraints. As a possible application, this thesis will also present two probabilistic solutions to the containment problem for spectrahedral shadows, using uniform random sampling algorithms.

The invasion of the extracellular matrix (ECM) by cancer cells is a highly complex process, but many mathematical models describing it are quite simple. This raises the question how these simpler models differ from the more complex ones regarding their dependence on the input parameters and their qualitative results, and whether simpler models are able to capture and reproduce the results of complex models. To investigate these questions, three cell invasion models are examined: two simple to intermediate partial differential equation (PDE) models with six and eight parameters, respectively, and a complex hybrid model with 17 parameters. The hybrid model was originally developed for angiogenesis, featuring a Cellular Potts model and a finite element formulation, and is extended in this work to describe cell invasion as travelling waves. The models’ parameter sensitivity with respect to initial ECM density, the ECM degradation rate and the cell proliferation probability, is examined using a variance-based method at various points in time. At first, the PDE models exhibit mainly first-order effects from the initial ECM density, and later in time from the ECM degradation rate. At all points in time, the hybrid model is affected the most by the proliferation probability, and by interaction effects between all three parameters. The data fitting abilities are tested by generating data using the hybrid model and estimating the corresponding PDE model parameters. This is done under consideration of various phenomena of the hybrid model, and with different numbers of parameters to be estimated. Using the fitted parameters, the shape approximations and the velocities of the travelling invading waves are compared between the models. The best approximations in shape are performed by the PDE model with eight parameters, the best approximations in wave speed are obtained using the model with six parameters. Overall, the main finding is that even though the model outputs are relat

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

In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which might be a major risk factor for the development of knee osteoarthritis, we develop confidence tubes for curves following a perturbation model on SO(3) using the Gaussian kinematic formula which are equivariant under gait similarities and have precise coverage even for small sample sizes. Applying them to gait curves from eight volunteers undergoing kneeling tasks and adjusting for different walking speeds and marker replacement at different visits, allows us to identify at which phases of the gait cycle the gait pattern changed due to kneeling., Peer Reviewed

The presence of nonignorable missing response variables often leads to complex conditional distribution patterns that cannot be effectively captured through mean regression. In contrast, quantile regression offers valuable insights into the conditional distribution. Consequently, this article places emphasis on the quantile regression approach to address nonrandom missing data. Taking inspiration from fractional imputation, this paper proposes a novel smoothed quantile regression estimation equation based on a sampling importance resampling (SIR) algorithm instead of nonparametric kernel regression methods. Additionally, we present an augmented inverse probability weighting (AIPW) smoothed quantile regression estimation equation to reduce the influence of potential misspecification in a working model. The consistency and asymptotic normality of the empirical likelihood estimators corresponding to the above estimating equations are proven under the assumption of a correctly specified parameter working model. Furthermore, we demonstrate that the AIPW estimation equation converges to an IPW estimation equation when a parameter working model is misspecified, thus illustrating the robustness of the AIPW estimation approach. Through numerical simulations, we examine the finite sample properties of the proposed method when the working models are both correctly specified and misspecified. Furthermore, we apply the proposed method to analyze HIV—CD4 data, thereby exploring variations in treatment effects and the influence of other covariates across different quantiles., Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China, Peer Reviewed

We propose three novel consistent specification tests for quantile regression models which generalize former tests in three ways. First, we allow the covariate effects to be quantile-dependent and nonlinear. Second, we allow parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. We are thereby able to test for general functional forms, while retaining linear effects as special cases. In both cases, the induced class of conditional distribution functions is tested with a Cramér–von Mises type test statistic for which we derive the theoretical limit distribution and propose a bootstrap method. Third, a modified test statistic is derived to increase the power of the tests. We highlight the merits of our tests in a detailed MC study and two real data examples. Our first application to conditional income distributions in Germany indicates that there are not only still significant differences between East and West but also across the quantiles of the conditional income distributions, when conditioning on age and year. The second application to data from the Australian national electricity market reveals the importance of using interaction effects for modeling the highly skewed and heavy-tailed distributions of energy prices conditional on day, time of day and demand., Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659, Peer Reviewed

The notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes t, Peer Reviewed

The use of mathematical models in the field of tumor-immune interactions offers an analytical framework, which allows the study of specific questions related to tumor-immune dynamics and potential approaches for tumor treatment. This thesis deals with various mathematical approaches to describe tumor-immune interactions using systems of ordinary and delay differential equations.

Moral hazard risk in insurance describes the possibility of fraudulent policy holder behaviour, resulting in losses which are hard to predict and quantify. For disability income insurance (DII), the “problem child” of many insurance companies, moral hazard is one possible explanation of its poor performance. Deception in DII is often-times motivated by financial need or by the fear to lose employment in worsening economic conditions. In this thesis, we investigate if the labour market influences DII claims experience and develop a framework for insurance companies to hedge moral hazard risk in DII portfolios. First, we use regression models on UK data from 2004 to 2016 to assess the influence of unemployment rate on DII claims and recoveries. Our analysis shows that in times of rising unemployment in the UK, the number of new claims increases and the number of DII recoveries falls. At the same time, we account for other potentially influential economic indicators and find that besides unemployment, the business and consumer confidence index have a significant effect on DII inceptions. Second, we build economic tracking portfolios to mimic the most important economic indicators explaining DII claims experience according to our regression models. Third, we use the results of our statistical models and portfolio constructions to develop a partial hedge for DII loss. The hedging strategy aims to offset the part of DII loss which is attributed to moral hazard risk. As a final step, we illustrate potential benefits of our hedging strategy by means of a simulation study. We find that adjusting for economic conditions via employing the hedging portfolio can improve accuracy of expected loss and reduce the overall loss reserve.

Die Anwenung von Operator-Splitting-Methoden auf gewöhnliche Differentialgleichungen ist gut etabliert. Für Differential-algebraische Gleichungen und partielle Differential-algebraische Gleichungen unterliegt sie jedoch vielen Einschränkungen aufgrund des Vorhandenseins von Nebenbedingungen. Die räumliche Diskretisierung reduziert PDAEs und lenkt unseren Fokus auf das Konzept der DAEs. Um eine reibungslose Übertragung des Operator-Splittings von ODEs auf DAEs durchzuführen, ist es wichtig, eine geeignete entkoppelte Struktur für das gewünschte Differential-algebraische System zu haben. In dieser Arbeit betrachten wir ein Modell, das partielle Differentialgleichungen für elektromagnetische Bauelemente - modelliert durch die Maxwell-Gleichungen - mit Differential-algebraischen Gleichungen koppelt, die die elementaren Schaltungselemente beschreiben. Nach der räumlichen Diskretisierung der klassischen Formulierung der Maxwell-Gleichungen mit Hilfe der finiten Integrationstechnik formulieren wir das resultierende gekoppelte System als Differential-algebraische Gleichung. Um eine geeignete Entkopplung zu bekommen, verwenden wir den zweigorientierten Loop-Cutset-Ansatz für die Schaltungsmodellierung. Daraus folgt, dass wir in der Lage sind, eine geeignete Operatorzerlegung so zu konstruieren, dass wir eine natürliche topologisch entkoppelte Port-Hamiltonsche DAE-Struktur erhalten. Wir schlagen einen Operator-Splitting-Ansatz für die Schaltungs-DAEs und gekoppelten Feld-Schaltungs-DAEs in entkoppelter Form vor und analysieren seine numerischen Eigenschaften. Darüber hinaus nutzen wir das Hamiltonsche Verhalten der inhärenten gewöhnlichen Differentialgleichung durch die Verwendung expliziter und energieerhaltender Zeitintegrations-methoden. Schließlich führen wir numerische Tests, um das mathematische Modell zu illustrieren und die Konvergenzergebnisse für das vorgeschlagene DAE-Operator-Splitting zu demonstrieren., Le equazioni algebriche differenziali e algebriche alle derivate parziali hanno avuto un enorme successo come modelli di sistemi dinamici vincolati. Nella modellazione matem- atica, spesso si desidera catturare diversi aspetti di una situazione come le leggi di conservazione della fisica, il trasporto convettivo o la diffusione. Queste aspetti si riflettono nel sistema di equazioni del modello come operatori diversi. La tecnica dell’Operator Splitting si è rivelata una strategia di successo per affrontare problemi così complicati. L’applicazione dei metodi di Operator Splitting alle equazioni differenziali ordinarie (ODE) è ormai una tecnologia ben consolidata. Tuttavia, per equazioni algebriche differenziali (DAE) e algebriche differenziali parziali (PDAE), l’approccio è soggetto a molte restrizioni dovute alla presenza di vincoli e alla proprietà di indice. La discretizzazione spaziale riduce le PDAE e indirizza la nostra attenzione al concetto di DAE. Le DAE emergono in problemi dinamici vincolati come circuiti elettrici o reti di trasporto di energia. Al fine di generalizzare agevolmente la tecnica dell’Operator Splitting dalle ODE alle DAE, è importante avere una struttura disaccoppiata adeguata per il sistema algebrico differenziale desiderato. In questa tesi, consideriamo un modello che accoppia equazioni differenziali alle derivate parziali per dispositivi elettromagnetici -modellati dalle equazioni di Maxwell- con equazioni algebriche differenziali che descrivono gli elementi base del circuito. Dopo aver discretizzato spazialmente la formulazione classica delle equazioni di Maxwell usando la tecnica di integrazione finita, formuliamo il sistema accoppiato risultante come una equazione algebrica differenziale. Interpretando il dispositivo elettromagnetico come un elemento capacitivo, l’indice dell’intero sistema di circuito e campo accoppiato può essere specificato utilizzando le proprietà topologiche del circuito e non supera il valore di due. Per eseguire, The application of operator splitting methods to ordinary differential equations (ODEs) is well established. However, for differential-algebraic equations (DAEs) and partial differential-algebraic equations (PDAEs), it is subjected to many restrictions due to the presence of constraints. In constrained dynamical problems as electrical circuits or energy transport networks, DAEs arise. In order to perform a smooth transfer of the operator splitting from ODEs to DAEs, it is important to have a suitable decoupled structure for the desired differential-algebraic system. In this thesis, we consider a model which couples partial differential equations for electro- magnetic devices -modeled by Maxwell’s equations- with differential-algebraic equations describing the basic circuit elements. After spatially discretizing the classical formulation of Maxwell’s equations using the finite integration technique, we formulate the resulting coupled system as a differential-algebraic equation. To perform an appropriate decoupling, we use the branch oriented loop-cutset approach for circuit modeling. It follows that we are able to construct a suitable operator decomposition such that we obtain a natural topologically decoupled port-Hamiltonian DAE structure. We propose an operator splitting approach for the decoupled circuit and coupled field- circuit DAEs using the Lie-Trotter and Strang splitting algorithms and analyze its numerical properties. Furthermore, we exploit the Hamiltonian behavior of the system’s inherent ordinary differential equation by the utilization of explicit and energy-preserving time integration methods. Based on the convergence analysis of the ODE operator splitting method, we derive convergence results for the proposed approach that depends on the index of the system and thus on its topological structure. Finally, we perform numerical tests, to underline the mathematical model and to demonstrate the convergence results for the proposed DAE operator splitting.

We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser’s and Milnor’s proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces., Peer Reviewed

In der vorliegenden Arbeit untersuchen wir unendliche Matrizen, die lineare Operatoren in unendlichdimensionalen Räumen charakterisieren. Dabei betrachten wir insbesondere den Hilbertraum ℓ2(ℕ) der quadratsummierbaren Folgen (reeller oder) komplexer Zahlen. Das Kernziel ist herauszufinden, unter welchen Bedingungen an die Einträge einer unendlichen Matrix M der durch diese repräsentierte lineare Operator wohldefiniert ist. Anders formuliert fragen wir uns, wann das Matrix-Vektor-Produkt Mx einer unendlichen Matrix M und einer Folge x ∈ ℓ2(ℕ), die wir als Vektor mit unendlich vielen Zeilen auffassen, wieder eine quadratsummierbare Folge ist. Darüber hinaus wollen wir die Stetigkeit solcher Operatoren untersuchen. Eine allgemeine Aussage kann man darüber nicht treffen, man kann allerdings eine Reihe von Beispielen finden. Dafür erinnern wir zunächst an einige mathematische Grundlagen, etwa aus der Hilbertraumtheorie. Abschließend übertragen wir ausgewählte Ergebnisse auf den Hardy-Raum auf der offenen Einheitskreisscheibe H2(), der isometrisch isomorph zum Folgenraum ℓ2(ℕ0) ist.

The proportional subdistribution hazards (PSH) model is popularly used to deal with competing risks data. Censored quantile regression provides an important supplement as well as variable selection methods due to large numbers of irrelevant covariates in practice. In this paper, we study variable selection procedures based on penalized weighted quantile regression for competing risks models, which is conveniently applied by researchers. Asymptotic properties of the proposed estimators, including consistency and asymptotic normality of non-penalized estimator and consistency of variable selection, are established. Monte Carlo simulation studies are conducted, showing that the proposed methods are considerably stable and efficient. Real data about bone marrow transplant (BMT) are also analyzed to illustrate the application of the proposed procedure., National Natural Science Foundation of China, Scientific Research Foundation of North China University of Technology, Fundamental Research Funds for Beijing Universities, NCUT, National Natural Science Foundation of China, China Statistical Research Project, Peer Reviewed

In random-effects models, hierarchical linear models, or multilevel models, it is typically assumed that the variances within higher-level units are homoscedastic, meaning that they are equal across these units. However, this assumption is often violated in research. Depending on the degree of violation, this can lead to biased standard errors of higher-level parameters and thus to incorrect inferences. In this article, we describe a resampling technique for obtaining standard errors—Zitzmann’s jackknife. We conducted a Monte Carlo simulation study to compare the technique with the commonly used delete-1 jackknife, the robust standard error in Mplus, and a modified version of the commonly used delete-1 jackknife. Findings revealed that the resampling techniques clearly outperformed the robust standard error in rather small samples with high levels of heteroscedasticity. Moreover, Zitzmann’s jackknife tended to perform somewhat better than the two versions of the delete-1 jackknife and was much faster., Peer Reviewed

Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers., Peer Reviewed

The Schönhage-Strassen-Algorithm multiplies two integers of length n in O(n log n log log n) steps on a multitape Turing machine. The main goal of this thesis is to give a verified implementation of the algorithm as well as verified runtime bounds. Integers are represented as LSBF (least significant bit first) boolean lists, on which simple algorithms for addition, subtraction and grid multiplication are given as well as a verified implementation of the Karatsuba-Multiplication with runtime O(nlog23). The already existing AFP-entry for Number Theoretic Transforms is adapted to a more general setting, which allows its application to certain quotient rings used in the Schönhage-Strassen-Algorithm. After some final preparations, an implementation of the Schönhage-Strassen-Algorithm with runtime O(n log n log log n) based on the original paper is given.

Organische Halbleiterbauelemente sind eine vielversprechende Technologie, die das Spektrum der optoelektronischen Halbleiterbauelemente erweitert und etablierte Technologien basierend auf anorganischen Halbleitermaterialien ersetzen kann. Für Display- und Beleuchtungsanwendungen werden sie z. B. als organische Leuchtdioden oder Transistoren verwendet. Eine entscheidende Eigenschaft organischer Halbleitermaterialien ist, dass die Ladungstransporteigenschaften stark von der Temperatur im Bauelement beeinflusst werden. Insbesondere nimmt die elektrische Leitfähigkeit mit der Temperatur zu, so dass Selbsterhitzungseffekte, einen großen Einfluss auf die Leistung der Bauelemente haben. Mit steigender Temperatur nimmt die elektrische Leitfähigkeit zu, was wiederum zu größeren Strömen führt. Dies führt jedoch zu noch höheren Temperaturen aufgrund von Joulescher Wärme oder Rekombinationswärme. Eine positive Rückkopplung liegt vor. Im schlimmsten Fall führt dieses Verhalten zum thermischen Durchgehen und zur Zerstörung des Bauteils. Aber auch ohne thermisches Durchgehen führen Selbsterhitzungseffekte zu interessanten nichtlinearen Phänomenen in organischen Bauelementen, wie z. B. die S-förmige Beziehung zwischen Strom und Spannung. In Regionen mit negativem differentiellen Widerstand führt eine Verringerung der Spannung über dem Bauelement zu einem Anstieg des Stroms durch das Bauelement. Diese Arbeit soll einen Beitrag zur mathematischen Modellierung, Analysis und numerischen Simulation von organischen Bauteilen leisten. Insbesondere wird das komplizierte Zusammenspiel zwischen dem Fluss von Ladungsträgern (Elektronen und Löchern) und Wärme diskutiert. Die zugrundeliegenden Modellgleichungen sind Thermistor- und Energie-Drift-Diffusion-Systeme. Die numerische Diskretisierung mit robusten hybriden Finite-Elemente-/Finite-Volumen-Methoden und Pfadverfolgungstechniken zur Erfassung der in Experimenten beobachteten S-förmigen Strom-Spannungs-Charakteristiken wird vorgestellt., Organic semiconductor devices are a promising technology to extend the range of optoelectronic semiconductor devices and to some extent replace established technologies based on inorganic semiconductor materials. For display and lighting applications, they are used as organic light-emitting diodes (OLEDs) or transistors. One crucial property of organic semiconductor materials is that charge-transport properties are heavily influenced by the temperature in the device. In particular, the electrical conductivity increases with temperature, such that self-heating effects caused by the high electric fields and strong recombination have a potent impact on the performance of devices. With increasing temperature, the electrical conductivity rises, which in turn leads to larger currents. This, however, results in even higher temperatures due to Joule or recombination heat, leading to a feedback loop. In the worst case, this loop leads to thermal runaway and the complete destruction of the device. However, even without thermal runaway, self-heating effects give rise to interesting nonlinear phenomena in organic devices, like the S-shaped relation between current and voltage resulting in regions where a decrease in voltage across the device results in an increase in current through it, commonly denoted as regions of negative differential resistance. This thesis aims to contribute to the mathematical modeling, analysis, and numerical simulation of organic semiconductor devices. In particular, the complicated interplay between the flow of charge carriers (electrons and holes) and heat is discussed. The underlying model equations are of thermistor and energy-drift-diffusion type. Moreover, the numerical approximation using robust hybrid finite-element/finite-volume methods and path-following techniques for capturing the S-shaped current-voltage characteristics observed in experiments are discussed.

Diese Dissertation behandelt arithmetische Eigenschaften von Familien algebraischer Varietäten und deren speziellen Untervarietäten. Im ersten Kapitel definieren wir sogenannte absolut spezielle Untervarietäten mithilfe von Delignes Begriff der absoluten Hodgeklassen. Ausgehend von der Vermutung, dass alle Hodgeklassen absolute Hodgeklassen sind, erwarten wir, dass alle speziellen Untervarietäten absolut speziell sind. Wir beweisen diese Erwartung für Untervarietäten, die eine bestimmte Monodromiebedingung erfüllen. Das zweite Kapitel führt eine l-adische Version von speziellen Untervarietäten ein, die wir l-Galois spezielle Untervarietäten nennen. Wir studieren bewiesene und vermutete Eigenschaften dieser Untervarietäten und deren Zusammenhang zur Struktur des l-Galois exzeptionellen Locus und zur Mumford-Tate Vermutung. Im dritten Kapitel beweisen wir eine Rapoport-Zink Uniformisierung für den Modulraum der primitiv polarisierten K3 Flächen und kubischen Vierfaltigkeiten mit supersingulärer Reduktion. In beiden Fällen ist der Modulraum uniformisiert von einer explizit definierten rigid analytischen Untervarietät einer lokalen Shimura-Varietät von orthogonalem Typ., This thesis studies arithmetic aspects of families of algebraic varieties and their special subvarieties. In the first part, we use Deligne's framework of absolute Hodge classes to define a notion of absolutely special subvarieties. The conjecture that all Hodge classes are absolute Hodge predicts that every special subvariety is absolutely special. We prove this prediction for subvarieties satisfying a certain monodromy condition. The second part introduces an l-adic analog of special subvarieties that we call l-Galois special subvarieties. We study the properties of these subvarieties and discuss how known and unknown properties of l-Galois special subvarieties are related to the structure of the l-Galois exceptional locus and to the Mumford-Tate conjecture. In the third chapter, we prove a Rapoport-Zink type uniformization result for the moduli space of polarized K3 surfaces and cubic fourfolds. We show that in both cases, the tube over the supersingular locus of the moduli space is uniformized by an explicitly described rigid analytic open subvariety of a local Shimura variety of orthogonal type.

Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is NP-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be NP-hard, which settles an open question., Peer Reviewed

We consider three examples of functors from Lorentzian categories and their applications in finiteness results, singularity theorems and boundary constructions. The third example is a novel functor from the category of ordered measure spaces to the category of Lorentzian pre-length spaces in the sense of Kunzinger–Sämann., Peer Reviewed

In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we construct a generalized Weil operator from the complex conjugate of the cohomology of a semisimple local system to the cohomology of its dual local system, which is functorial with respect to smooth restrictions. This operator allows us to study the Poincaré pairing, usually not positive definite, in terms of a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems. As an application, we give a new proof of Sabbah’s Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, by adapting the method of de Cataldo-Migliorini, without using the category of polarizable twistor modules. This answers a question of Sabbah., Peer Reviewed

Limit-Orderbücher sind das Standardinstrument der Preisbildung in modernen Finanzmärkten. Während Strom traditionell in Auktionen gehandelt wird, gibt es Intraday Strommärkte wie beispielsweise den SIDC-Markt, in welchem Käufer und Verkäufer über Limit-Orderbücher zusammentreffen. In dieser Arbeit werden wir stochastische Modelle von Limit-Orderbüchern auf der Grundlage der zugrundeliegenden Marktmikrostruktur entwickeln. Einen besonderen Schwerpunkt legen wir dabei auf die Berücksichtigung besonderer Merkmale der Intraday-Strommärkte, die sich zum Teil deutlich von denen der Finanzmärkte unterscheiden. Die in dieser Arbeit entwickelten Modelle beginnen mit einer realistischen und mikroskopischen Beschreibung der Marktdynamik. Große Preisänderungen über kurze Zeiträume werden ebenso berücksichtigt wie begrenzte grenzüberschreitende Aktivitäten. Diese mikroskopischen Modelle sind im Allgemeinen zu rechenintensiv für praktische Anwendungen. Das Hauptziel dieser Arbeit ist es daher, geeignete Approximationen dieser mikroskopischen Modelle durch sogenannte Skalierungsgrenzprozesse herzuleiten. Zu diesem Zweck werden sorgfältig Skalierungsannahmen formuliert und in die mikroskopischen Modelle eingebaut. Diese Annahmen ermöglichen es uns, ihr Hochfrequenzverhalten zu untersuchen, vorausgesetzt, dass die Größe eines einzelnen Auftrags gegen Null konvergiert, während die Auftragseingangsrate gegen unendlich tendiert. Die Kalibrierung mathematischer Modelle ist aus Anwendersicht eines der Hauptanliegen. Dabei ist bekannt, dass Änderungspunkte (abrupte Schwankungen) in hochfrequenten Finanzdaten vorhanden sind. Falls sie durch endogene Effekte verursacht wurden, muss bei der Schätzung solcher Änderungspunkte die Abhängigkeit von den zugrundeliegenden Daten berücksichtigt werden. Daher erweitern wir im letzten Teil dieser Arbeit die bestehende Literatur zur Erkennung von Änderungspunkten, so dass auch zufällige, von den Daten abhängige Änderungspunkte gehandhabt werden können., Limit order books are the standard instrument for price formation in modern financial markets. While electricity has traditionally been traded through auctions, there are intraday electricity markets, such as the SIDC market, in which buyers and sellers meet via limit order books. In this thesis, stochastic models of limit order books are developed based on the underlying market microstructure. A particular focus is set on incorporating unique characteristics of intraday electricity markets, some of which are quite different from those of financial markets. The developed models in this thesis start with a realistic and microscopic description of the market dynamics. Large price changes over short time periods are considered, as well as limited cross-border activities. These microscopic models are generally computationally too intensive for practical applications. The main goal of this thesis is therefore to derive suitable approximations of these microscopic models by so-called scaling limits. For this purpose, appropriate scaling assumptions are carefully formulated and incorporated into the microscopic models which allow us to study their high-frequency behavior when the size of an individual order converges to zero while the order arrival rate tends to infinity. Calibration of mathematical models is one of the main concerns from a practitioner’s point of view. It is well known that change points (abrupt variations) are present in high-frequency financial data. If they are caused by endogenous effects, the dependence on the underlying data must be considered when estimating such change points. In the final part of this thesis, we extend the existing literature on change point detection so that random change points depending on the data can also be handled.

This article constructs examples of associative submanifolds in G2–manifolds obtained by resolving G2–orbifolds using Joyce’s generalised Kummer construction. As the G2–manifolds approach the G2G2–orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison., Peer Reviewed

We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus ≤ 5 and polarized K3 surfaces of degree ≤ 8., Peer Reviewed

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D Navier-Stokes equations and the von Kármán equations with Morley, discontinuous Galerkin, C0 interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms., Peer Reviewed

We consider multi-edge or banana graphs bn on n internal edges ei with different masses mi . We focus on the cut banana graphs ( R(bn)) from which the full result R(bn) can be derived through dispersion. We give a recursive definition of ( R(bn)) through iterated integrals. We discuss the structure of this iterated integral in detail. A discussion of accompanying differential equations, of monodromy and of a basis of master integrals is included., Peer Reviewed

Piezoelektrika haben ein breit gefächertes Anwendungsspektrum in Industrie, Alltag und Forschung. Dies erfordert ein genaues Wissen über das Materialverhalten der betrachteten piezoelektrischen Elemente, was mit dem Lösen von simulationsgestützten inversen Parameteridentifikationsproblemen einhergeht. Die vorliegende Arbeit befasst sich mit der optimalen Versuchsplanung (OED) für dieses Problem. Piezoelektrische Materialien weisen die Eigenschaft auf, sich als Reaktion auf angelegte Potentiale oder Kräfte mechanisch oder elektrisch zu verändern (direkter und indirekter piezoelektrischer Effekt). Um eine Spannung anzulegen und den indirekten piezoelektrischen Effekt auszunutzen, werden Elektroden aufgebracht, deren Konfiguration einen erheblichen Einfluss auf mögliche Systemantworten hat. Daher werden das Potential, die Anzahl und die Größe der Elektroden zunächst im zweidimensionalen Fall optimiert. Das piezoelektrische Verhalten basiert im betrachteten Kleinsignalbereich auf zeitabhängigen, linearen partiellen Differentialgleichungen. Die Herleitung sowie Existenz und Eindeutigkeit der Lösungen werden gezeigt. Zur Berechnung der elektrischen Ladung und der Impedanz, die für das Materialidentifikationsproblem und damit für die Versuchsplanung relevant sind, werden zeit- und frequenzabhängige Simulationen auf Basis der Finite Elemente Methode (FEM) mit dem FEM Simulationstool FEniCS durchgeführt. Es wird auf Nachteile bei der Berechnung der Ableitungen eingegangen und erste adjungierte Gleichungen formuliert. Die Modellierung des Problems der optimalen Versuchsplanung erfolgt hauptsächlich durch die Kontrolle des Potentials der Dirichlet Randbedingungen des Randwertproblems. Anhand mehrerer numerischer Beispiele werden die resultierenden Konfigurationen gezeigt. Weitere Ansätze zur Elektrodenmodellierung, z.B. durch Kontrolle der Materialeigenschaften, werden ebenfalls vorgestellt. Schließlich wird auf mögliche Erweiterungen des vorgestellten OED Problems hingewiesen, Piezoelectrics have a wide range of applications in industry, everyday life and research. This requires an accurate knowledge of the material behavior, which implies the solution of simulation-based inverse identification problems. This thesis focuses on the optimal design of experiments addressing this problem. Piezoelectric materials exhibit the property of mechanical or electrical changes in response to applied potentials or forces (direct and indirect piezoelectric effect). To apply voltage and to exploit the indirect piezoelectric effect, electrodes are attached whose configura- tion have a significant influence on possible system responses. Therefore, the potential, the number and the size of the electrodes are initially optimized in the two-dimensional case. The piezoelectric behavior in the considered small signal range is based on a time dependent linear partial differential equation system. The derivation as well as the exis- tence, uniqueness and regularity of the solutions of the equations are shown. Time- and frequency-dependent simulations based on the finite element method (FEM) with the FEM simulation tool FEniCS are performed to calculate the electric charge and the impedance, which are relevant for the material identification problem and thus for the experimental design. Drawbacks in the derivative calculations are pointed out and a first set of adjoint equations is formulated. The modeling of the optimal experimental design (OED) prob- lem is done mainly by controlling the potential of the Dirichlet boundary conditions of the boundary value problem. Several numerical examples are used to show the resulting configurations and to address the difficulties encountered. Further electrode modeling ap- proaches for example by controlling the material properties are then discussed. Finally, possible extensions of the presented OED problem are pointed out.

There is widespread agreement, that today’s students must develop competencies in the efficient use of information and communication technology (ICT) to cope with the demands of the 21st century. To meet this requirement, teachers must integrate ICT into their classroom activities on a regular basis. Studies have shown that the use of ICT in the classroom correlates with the level of professional knowledge and with affective-motivational dispositions (such as emotions and self-efficacy) of teachers. However, the relations between these dispositions and the extent to which these relations differ between pre- and in-service teachers have not yet been investigated. Hence, the present study examines the dispositions of 148 German pre-service and 132 German in-service primary school teachers to use ICT in geometry classes and tests for differences between these groups. To this end, a series of path models have been investigated on the basis of control-value theory in a quantitative study. Results of the invariance testing revealed only minor differences in the relations between the investigated dispositions: For in-service teachers a negative correlation between the assumed value of ICT for teaching geometry and the professional knowledge regarding ICT was found. The same does not hold true for pre-service teachers. Apart from this difference, however, the two groups were very similar. It can therefore be concluded that learning opportunities regarding the use of ICT in geometry classes do not need to differ greatly for the pre-service and in-service teachers., Peer Reviewed

We consider mixed-integer optimal control problems, whose optimality conditions involve global combinatorial optimization aspects for the corresponding Hamiltonian pointwise in time. We propose a time-domain decomposition, which makes this problem class accessible for mixed-integer programming using parallel-in-time direct discretizations. The approach is based on a decomposition of the optimality system and the interpretation of the resulting subproblems as suitably chosen mixed-integer optimal control problems on subintervals in time. An iterative procedure then ensures continuity of the states at the boundaries of the subintervals via co-state information encoded in virtual controls. We prove convergence of this iterative scheme for discrete-continuous linear-quadratic problems and present numerical results both for linear-quadratic as well as nonlinear problems., Peer Reviewed

We compute the integral Picard group of the moduli stack of polarized K3 surfaces of fixed degree whose singularities are at most rational double points, and of its coarse moduli space. We also compute the integral Picard group of the stack of quasi‐polarized K3 surfaces, and of the stacky period domain., Peer Reviewed

We study needle formation at martensite/martensite macro interfaces in shape-memory alloys. We characterize the scaling of the energy in terms of the needle tapering length and the transformation strain, both in geometrically linear and in finite elasticity. We find that linearized elasticity is unable to predict the value of the tapering length, as the energy tends to zero with needle length tending to infinity. Finite elasticity shows that the optimal tapering length is inversely proportional to the order parameter, in agreement with previous numerical simulations. The upper bound in the scaling law is obtained by explicit constructions. The lower bound is obtained using rigidity arguments, and as an important intermediate step we show that the Friesecke–James–Müller geometric rigidity estimate holds with a uniform constant for uniformly Lipschitz domains., Peer Reviewed

The known a posteriori error analysis of hybrid high-order methods treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks., Peer Reviewed

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart (m = 1) or Morley (m = 2) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated L2 error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability., Peer Reviewed