The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

Temporal logics are a very popular family of logical languages, used to specify properties of abstracted systems. In the last few years, many extensions of temporal logics have been proposed, in order to address the need to express more than just abstract properties. In our work we study temporal logics extended by local constraints, which allow to express quantitative properties on data values from an arbitrary relational structure called the concrete domain. An example of concrete domain can be (Z, <, =), where the integers are considered as a relational structure over the binary order relation and the equality relation. Formulas of temporal logics with constraints are evaluated on data-words or data-trees, in which each node or position is labeled by a vector of data from the concrete domain. We call the constraints local because they can only compare values at a fixed distance inside such models. Several positive results regarding the satisfiability of LTL (linear temporal logic) with constraints over the integers have been established in the past years, while the corresponding results for branching time logics were only partial. In this work we prove that satisfiability of CTL* (computation tree logic) with constraints over the integers is decidable and also lift this result to ECTL*, a proper extension of CTL*. We also consider other classes of concrete domains, particularly ones that are \"tree-like\". We consider semi-linear orders, ordinal trees and trees of a fixed height, and prove decidability in this framework as well. At the same time we prove that our method cannot be applied in the case of the infinite binary tree or the infinitely branching infinite tree. We also look into extending the expressiveness of our logic adding non-local constraints, and find that this leads to undecidability of the satisfiability problem, even on very simple domains like (Z, <, =). We then find a way to restrict the power of the non-local constraints to regain decidability.

Although the human brain has been studied over past decades at morphological and histological levels, much remains unknown about its molecular and genetic mechanisms. Furthermore, when compared with our closest relative the chimpanzee, the human brain strikingly shows great morphological changes that have been often associated with our cognitive specializations and skills. Nevertheless, such drastic changes in the human brain may have arisen not only through morphological changes but also through changes in the expression levels of genes and transcripts. Gene regulatory networks are complex and large-scale sets of protein interactions that play a fundamental role at the core of cellular and tissue functions. Among the most important players of such regulatory networks are transcription factors (TFs) and the transcriptional circuitries in which TFs are the central nodes. Over past decades, several studies have focused on the functional characterization of brain-specific TFs, highlighting their pathways, interactions, and target genes implicated in brain development and often disorders. However, one of the main limitations of such studies is the data collection which is generally based on an individual experiment using a single TF. To understand how TFs might contribute to such human-specific cognitive abilities, it is necessary to integrate the TFs into a system level network to emphasize their potential pathways and circuitry. This thesis proceeds with a novel systems biology approach to infer the evolution of these networks. Using human, chimpanzee, and rhesus macaque, we spanned circa 35 million years of evolution to infer ancestral TF networks and the TF-TF interactions that are conserved or shared in important brain regions. Additionally, we developed a novel method to integrate multiple TF networks derived from human frontal lobe next-generation sequencing data into a high confidence consensus network. In this study, we also integrated a manually curated list of TFs important for brain function and disorders. Interestingly, such “Brain-TFs” are important hubs of the consensus network, emphasizing their biological role in TF circuitry in the human frontal lobe. This thesis describes two major studies in which DNA microarray and RNA-sequencing (RNA-seq) datasets have been mined, directing the TFs and their potential target genes into co-expression networks in human and non-human primate brain genome-wide expression datasets. In a third study we functionally characterized ZEB2, a TF implicated in brain development and linked with Mowat-Wilson syndrome, using human, chimpanzee, and orangutan cell lines. This work introduces not only an accurate analysis of ZEB2 targets, but also an analysis of the evolution of ZEB2 binding sites and the regulatory network controlled by ZEB2 in great apes, spanning circa 16 million years of evolution. In summary, those studies demonstrated the critical role of TFs on the gene regulatory networks of human frontal lobe evolution and functions, emphasizing the potential relationships between TF circuitries and such cognitive skills that make humans unique.

For the modelling of chemistry we use undirected, labelled graphs as explicit models of molecules and graph transformation rules for modelling generalised chemical reactions. This is used to define artificial chemistries on the level of individual bonds and atoms, where formal graph grammars implicitly represent large spaces of chemical compounds. We use a graph rewriting formalism, rooted in category theory, called the Double Pushout approach, which directly expresses the transition state of chemical reactions. Using concurrency theory for transformation rules, we define algorithms for the composition of rewrite rules in a chemically intuitive manner that enable automatic abstraction of the level of detail in chemical pathways. Based on this rule composition we define an algorithmic framework for generation of vast reaction networks for specific spaces of a given chemistry, while still maintaining the level of detail of the model down to the atomic level. The framework also allows for computation with graphs and graph grammars, which is utilised to model non-trivial chemical systems. The graph generation relies on graph isomorphism testing, and we review the general individualisation-refinement paradigm used in the state-of-the-art algorithms for graph canonicalisation, isomorphism testing, and automorphism discovery. We present a model for chemical pathways based on a generalisation of network flows from ordinary directed graphs to directed hypergraphs. The model allows for reasoning about the flow of individual molecules in general pathways, and the introduction of chemically motivated routing constraints. It further provides the foundation for defining specialised pathway motifs, which is illustrated by defining necessary topological constraints for both catalytic and autocatalytic pathways. We also prove that central types of pathway questions are NP-complete, even for restricted classes of reaction networks. The complete pathway model, including constraints for catalytic and autocatalytic pathways, is implemented using integer linear programming. This implementation is used in a tree search method to enumerate both optimal and near-optimal pathway solutions. The formal methods are applied to multiple chemical systems: the enzyme catalysed beta-lactamase reaction, variations of the glycolysis pathway, and the formose process. In each of these systems we use rule composition to abstract pathways and calculate traces for isotope labelled carbon atoms. The pathway model is used to automatically enumerate alternative non-oxidative glycolysis pathways, and enumerate thousands of candidates for autocatalytic pathways in the formose process.

We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.

In der vorliegenden Dissertation behandeln wir spezielle matrizielle Potenzmomentenprobleme: Das rechtsseitige bzw. linksseitige alpha-Stieltjes Momentenproblem für beliebige reelle alpha. Wir widmen uns in dieser Arbeit vorwiegend dem vollständig nichtdegenerierten Fall. Im Verlauf der Arbeit werden wir mehrere Parametrisierungen der vorgegebenen Momentenfolge einführen, untersuchen und untereinander in Verbindung bringen. Speziell befassen wir uns mit der alpha-Stieltjes-Parametrisierung, der kanonischen Hankel-Parametrisierung, dem Favard-Paar, der alpha-Dyukarev-Stieltjes-Parametrisierung, dem alpha-Dyukarev-Quadrupel und dem alpha-Stieltjes-Quadrupel. Wir werden das betrachtete Potenzmomentenproblem mithilfe der sogenannten Stieltjes-Transformation in ein äquivalentes Interpolationsproblem überführen. Durch die Heranziehung des Systems von Potapovschen fundamentalen Matrixungleichungen können wir eine vollständige Beschreibung der Lösungsmenge in Form einer gebrochen linearen Transformation vornehmen, deren erzeugende Matrixfunktion ein aus dem alpha-Dyukarev-Quadrupel gebildetes 2qx2q-Matrixpolynom, auch Resolventenmatrix genannt, ist und als deren Parametermenge eine spezielle Klasse von geordneten Paaren von meromorphen qxq-Matrixfunktionen, den sogenannten Stieltjes-Paaren, fungiert. Hierbei spielt die Signaturmatrix Jq-Schlange eine wichtige Schlüsselrolle. Weiterhin erfolgt eine ausführliche Diskussion zweier in gewissem Sinne extremaler Lösungen des umformulierten Potenzmomentenproblems, eine multiplikative Zerlegung der Resolventenmatrix in lineare Matrixpolynome, wodurch eine Verbindung zu einem möglichen Algorithmus vom Schur-Typ geschaffen wird, eine alternative Beschreibung der Lösungsmenge mit einer Teilmenge von qxq-Schurfunktionen als Parametermenge und eine Betrachtung einiger Zusammenhänge zum Hausdorffschen Momentenproblem.:0 Einleitung 1 Erste Beobachtungen zu matriziellen alpha-Stieltjes Momentenproblemen 2 Über einige zu Matrizenfolgen gehörige Parametrisierungen und Matrixpolynome 3 Die alpha-Dyukarev-Stieltjes-Parametrisierung von alpha-Stieltjes-positiv definiten Folgen 4 Konstruktion einer Resolventenmatrix für vollständig nichtdegenerierte matrizielle alpha-Stieltjes Momentenprobleme 5 Die multiplikative Struktur der Folge von 2qx2q-alpha-Dyukarev-Matrixpolynomen bezüglich alpha-Stieltjes-positiv definiter Folgen 6 Eine alternative Beschreibung der Lösungsmenge für vollständig nichtdegenerierte matrizielle alpha-Stieltjes Momentenprobleme 7 Das alpha-Stieltjes-Quadrupel bezüglich alpha-Stieltjes-positiv definiter Folgen 8 Weitere Zusammenhänge zwischen einigen Parametrisierungen alpha-Stieltjes-positiv definiter Folgen 9 Einige Zusammenhänge zum matriziellen Hausdorffschen Momentenproblem A Einige Aussagen zur Integrationstheorie nichtnegativ hermitescher Maße B Über die Stieltjes-Transformation von nichtnegativ hermiteschen Maßen C Einige Aussagen der Matrizentheorie D Einige Aussagen der J-Theorie E Einige Aussagen über ganze Funktionen aus J-Potapov-Klassen bezüglich Halbebenen F Einige Aussagen über Stieltjes-Paare von meromorphen Matrixfunktionen G Einige Aussagen über Teilklassen von Schur-Funktionen auf Halbebenen

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Many human neurological and psychiatric disorders are substantially heritable and there is growing inter-est in searching for genetic variants explaining variability in disease-induced alterations of brain anatomy and function, as measured using neuroimaging techniques. The standard analysis approach in genetic neuroimaging is the mass-univariate linear modeling approach, which is disadvantageous, since it cannot account for dependencies among collinear variables and has to be corrected for multiple testing. In con-trast, multivariate methods include combined information from multiple variants simultaneously into the analysis, and can therefore account for the correlation structure in both the neuroimaging and the genetic data. Partial Least Squares Analysis and Canonical Correlation Analysis are common multivariate ap-proaches and different variants have been established for genetic neuroimaging. However, a compre-hensive comparison with respect to data characteristics and strengths and weaknesses of these methods was missing to date. This thesis elaborately compared three multivariate techniques, Sparse Canonical Correlation Analysis (Sparse CCA), Bayesian Inter-Battery Factor Analysis (Bayesian IBFA) and Partial Least Squares Corre-lation (PLSC) in order to express a clear statement on which method in to choose for analysis in genetic neuroimaging. It was shown that for highly collinear neuroimaging data, Bayesian IBFA could not be recommended, since additional post-processing steps were required to differentiate between causal and non-informative components. In contrast, Sparse CCA and PLSC were suitable for genetic neuroimaging data. Among the two, the use of Sparse CCA was recommended in situations with relatively low-dimensional neuroimaging and genetic data, since its predictive power was higher when data dimension-ality was below 400 times sample size. For higher dimensionalities, the predictive power of PLSC ex-ceeded that of Sparse CCA. Thus, for multivariate modeling of high-dimensional neuroimaging-genetics-associations, a preference for the usage of PLSC was indicated. The remainder of this thesis dealt with the improvement of the computational efficiency of multivariate statistics in genetic neuroimaging, since it can be expected that there will be a growth in cost- and time-efficient DNA sequencing as well as neuroimaging techniques in the coming years, which will result in excessively long computation times due to increasing data dimensionality. To accommodate this large number of variables, a new and computational efficient statistical approach named PLSC-RP was pro-posed, which incorporates a method for dimensionality reduction named Random projection (RP) into traditional PLSC in order to represent the originally high-dimensional data in lower dimensional spaces. Subsequently, PLSC is used for multivariate analysis of compressed data sets. Finally, the results are transformed back to the original spaces to enable the interpretation of original variables. It was demon-strated that the usage of PLSC-RP reduced computation times from hours to seconds compared to its state-of-the-art counterpart PLSC. Nonetheless, the accuracy of the results was not impaired, since the results of PLSC-RP and PLSC were statistically equivalent. Furthermore, PLSC-RP could be used for inte-grative analysis of data sets containing high-dimensional neuroimaging data, high-dimensional genetic data or both, and was therefore shown to be independent of the statistical data type. Thus, PLSC-RP opens up a wide range of possible applications.

This thesis is devoted to the rigorous study of approximations for (multi-phase) mean curvature flow and related equations. We establish convergence towards weak solutions of the according geometric evolution equations in the BV-setting of finite perimeter sets. Our proofs are of variational nature in the sense that we use the gradient flow structure of (multi-phase) mean curvature flow. We study two classes of schemes, namely phase-field models and thresholding schemes. The starting point of our investigation is the fact that both, the Allen-Cahn Equation and the thresholding scheme, preserve this gradient flow structure. The Allen-Cahn Equation is a gradient flow itself, while the thresholding scheme is a minimizing movements scheme for an energy that Γ-converges to the total interfacial energy. In both cases we can incorporate external forces or a volume-constraint. In the spirit of the work of Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2):253–271, 1995), our results are conditional in the sense that we assume the time-integrated energies to converge to those of the limit. Although this assumption is natural, it is not guaranteed by the a priori estimates at hand.

In this dissertation we considered a nonlinear sigma model with gravitino field. This is a supersymmetric extension of the nonlinear sigma model in the string theory, and we set up the geometric model using commuting variables, such that we could analyze it using the tools from calculus of variations. We introduced an action functional which corresponds to the super harmonic map functional, which has four arguments: a map between Riemannian manifolds, a vector spinor, a Riemannian metric and a gravitino. After getting the total variation formula, we considered the symmetries that the action functional possesses. By Noether's principle these families of symmetries induces conservation laws, which help to interpret the energy-momentum tensor and the supercurrent as holomorphic sections of some complex bundle. We also discussed the supersymmetry of our model. It turns out that the supersymmetry only remains in some particular cases, which is still useful in the analysis. Then we defined the weak solution in the distributional sense, and using Riesz potential estimates and Riviere regularity theory, we could improve the regularity of the weak solutions. More precisely, when the Riemannian metric and the gravitino are smooth, then any weak solution is actually smooth; and when the gravitino are coarse but subcritical, we can still show that the weak solutions are Holder continuous. Next we considered the compactness of solutions with bounded energies. We showed the small energy regularity on local domains and gap properties on the global surface. We also established the Pohozaev identities and thus showed the removable singularity theorem. Finally, for a sequence of solutions of uniformly bounded energies with respect to a converging sequence of gravitino fields, we showed that they converges weakly. Actually away from finite points, the convergence is strong and at those points, the energies concentrate. After a rescaling, each of these points corresponds to finitely some Dirac-harmonic maps with curvature terms defined on the Riemann sphere. Moreover, we established the energy identities along the weakly convergent sequences modulo these bubbles.