In der vorliegenden Arbeit wird die Synthese linearer und zyklischer Aminoketone via nucleophiler aromatischer Substitution von fluorsubstituierten aromatischen Ketonen mit sekundären, aliphatischen Diaminen vorgestellt. Durch eine Adaption der Prozessparameter konnte eine elegante Methode zur Synthese fluorendgruppentragender Oligomere sowie von definierten Makrozyklen entwickelt werden. Die Modifizierung der Oligomere erfolgte sowohl durch Endgruppensubstitution als auch durch Reaktionen an der Carbonylgruppe am Oligomerrückgrat. Als Funktionalisierungsreagenzien wurden Mercaptoessigsäure, LAWESSONs Reagenz und N,N-Dimethylanilin eingesetzt. Die Umsetzung der Makrozyklen mit N,N-Dialkylanilinen ermöglicht die Synthese zyklischer Triphenylmethanfarbstoffe. Die Untersuchung der optischen Eigenschaften dieser zyklischen Kristallviolett-Derivate in Abhängigkeit des pH-Wertes und der Natur des Lösungsmittels sowie der Sensitivität gegenüber Cyanid-Ionen erfolgte mit Hilfe der UV/Vis-Spektroskopie.

This thesis develops various aspects of Algebraic Geometry and its applications in different fields of science. In Chapter 2 we characterise the feasible set of an optimisation problem relevant in chemical process engineering. We consider the polynomial dynamical system associated with mass-action kinetics of a chemical reaction network. Given an initial point, the attainable region of that point is the smallest convex and forward closed set that contains the trajectory. We show that this region is a spectrahedral shadow for a class of linear dynamical systems. As a step towards representing attainable regions we develop algorithms to compute the convex hulls of trajectories. We present an implementation of this algorithm which works in dimensions 2,3 and 4. These algorithms are based on a theory that approximates the boundary of the convex hull of curves by a family of polytopes. If the convex hull is represented as the output of our algorithms we can also check whether it is forward closed or not. Chapter 3 has two parts. In this first part, we do a case study of planar curves of degree 6. It is known that there are 64 rigid isotopy types of these curves. We construct explicit polynomial representatives with integer coefficients for each of these types using different techniques in the literature. We present an algorithm, and its implementation in software Mathematica, for determining the isotopy type of a given sextic. Using the representatives various sextics for each type were sampled. On those samples we explored the number of real bitangents, inflection points and eigenvectors. We also computed the tensor rank of the representatives by numerical methods. We show that the locus of all real lines that do not meet a given sextic is a union of up to 46 convex regions that is bounded by its dual curve. In the second part of Chapter 3 we consider a problem arising in molecular biology. In a system where molecules bind to a target molecule with multiple binding sites, cooperativity measures how the already bound molecules affect the chances of other molecules binding. We address an optimisation problem that arises while quantifying cooperativity. We compute cooperativity for the real data of molecules binding to hemoglobin and its variants. In Chapter 4, given a variety X in n-dimensional projective space we look at its image under the map that takes each point in X to its coordinate-wise r-th power. We compute the degree of the image. We also study their defining equations, particularly for hypersurfaces and linear spaces. We exhibit the set-theoretic equations of the coordinate-wise square of a linear space L of dimension k embedded in a high dimensional ambient space. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with degenerate eigenspectrum.

We develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. We apply our method to the special case where the harmonic map is the Gauss map of a minimal submanifold and the complete manifold is a Grassmannian. With the help of a result by Allard [Allard, W. K. (1972). On the first variation of a varifold. Annals of mathematics, 417-491.], we can study the graph case and have an approach to prove Bernstein-type theorems. This enables us to extend Moser’s Bernstein theorem [Moser, J. (1961). On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics, 14(3), 577-591.] to codimension two, i.e., a minimal p-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a p-plane.

Despite the fact that each cell in an organism has the same genetic information, it is possible that cells fundamentally differ in their function. The molecular basis for the functional diversity of cells is governed by biochemical processes that regulate the expression of genes. Key to this regulatory process are proteins called transcription factors that recognize and bind specific DNA sequences of a few nucleotides. Here we tackle the problem of identifying the binding sites of a given transcription factor. The prediction of binding preferences from the structure of a transcription factor is still an unsolved problem. For that reason, binding sites are commonly identified by searching for overrepresented sites in a given collection of nucleotide sequences. Such sequences might be known regulatory regions of genes that are assumed to be coregulated, or they are obtained from so-called ChIP-seq experiments that identify approximately the sites that were bound by a given transcription factor. In both cases, the observed nucleotide sequences are much longer than the actual binding sites and computational tools are required to uncover the actual binding preferences of a factor. Aggravated by the fact that transcription factors recognize not only a single nucleotide sequence, the search for overrepresented patterns in a given collection of sequences has proven to be a challenging problem. Most computational methods merely relied on the given set of sequences, but additional information is required in order to make reliable predictions. Here, this information is obtained by looking at the evolution of nucleotide sequences. For that reason, each nucleotide sequence in the observed data is augmented by its orthologs, i.e. sequences from related species where the same transcription factor is present. By constructing multiple sequence alignments of the orthologous sequences it is possible to identify functional regions that are under selective pressure and therefore appear more conserved than others. The processing of the additional information exerted by ortholog sequences relies on a phylogenetic tree equipped with a nucleotide substitution model that not only carries information about the ancestry, but also about the expected similarity of functional sites. As a result, a Bayesian method for the identification of transcription factor binding sites is presented. The method relies on a phylogenetic tree that agrees with the assumptions of the nucleotide substitution process. Therefore, the problem of estimating phylogenetic trees is discussed first. The computation of point estimates relies on recent developments in Hadamard spaces. Second, the statistical model is presented that captures the enrichment and conservation of binding sites and other functional regions in the observed data. The performance of the method is evaluated on ChIP-seq data of transcription factors, where the binding preferences have been estimated in previous studies.

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Die Arbeit liefert einen Betrag zur Objektivierung der Bewertung und Auswertung von waffenspezifischen Spuren auf Geschossen. Durch den Vergleich der Spuren können Geschosse den verwendeten Waffen zugeordnet werden. Es wurde ein Algorithmus entwickelt, der die Auswahl der relevanten Spurenbereiche aus großflächigen Bilder ermöglicht. Ein weiterer Algorithmus kann die Spuren automatisiert um den Drallwinkel korrigieren. Zwei bekannte Vergleichsmodelle wurden an den vorliegenden Spuren angewandt. Ein Vergleichsalgorithmus, basierend auf der Fuzzy-Theorie, wurde zum Spurenvergleich entwickelt. Der Vergleich mittels dieses Algorithmus kann eindimensional oder zweidimensional erfolgen. Eine Gegenüberstellung der Vergleichsansätze zeigt, dass ein Spurenvergleich mit dem Fuzzy-Modell, bei welchen nur die Lage der Riefenspuren einfließt, bessere Ergebnisse liefert als die anderen Modelle und so für die Begründung von Untersuchungsergebnissen im Spurenvergleich verwendet werden kann.

Das in diesem Beitrag beschriebene automatische Verfahren findet bei klassischen geodätischen Berechnungsaufgaben ausgehend von gegebenen Startgrößen (z. B. Festpunktkoordinaten, Messwerte) Berechnungsmöglichkeiten für alle anderen relevanten Größen. Bei redundanten Startgrößen existiert meist eine Vielzahl unterschiedlicher Berechnungsmöglichkeiten, die alle gefunden und deren Ergebnisse berechnet werden. Wenn die Berechnung mehrdeutig ist, aber nur endlich viele Lösungen existieren, werden alle Lösungen gefunden und berechnet. Durch den Vergleich unterschiedlicher Berechnungsergebnisse besteht die Möglichkeit, grobe Fehler in den Startgrößen aufzudecken und ein robustes Endergebnis zu generieren. Das Verfahren arbeitet nicht stochastisch, setzt also kein stochastisches Modell der Messwerte voraus. Die Beschreibung wird mit Beispielen illustriert. Das Verfahren wurde als Webserver-Script realisiert und ist frei im Internet verfügbar. / This contribution describes an automatic method, which can be applied to classical geodetic computation problems. Starting from given input quantities (e. g. coordinates of known points, measurements) computation opportunties for all other relevant quantities are found. For redundant input quantities a multitude of different computation opportunties exists, which are all found automatically, and their results are computed. If the computation is non-unique, but only a finite number of solutions exist, then all solutions are found and computed. By comparison of the different computation results there is the opportunity to detect gross errors in the input quantities and to produce a robust final result. The method does not work stochastically, so no stochastic model of the observations is required. The description of the algorithm is illustrated with the help of examples. The method was implemented as a webserver script and is available for free in the internet.

This thesis is concerned with two problems in the Calculus of Variations touching on two central aspects of Materials Science: the structure of solid matter and its dynamic behavior. The problem pertaining to the first aspect is the analysis of the rigidity properties of possibly branched microstructures formed by shape memory alloys undergoing cubic-to-tetragonal transformations. On the basis of a variational model in the framework of linearized elasticity, we derive a non-convex and non-discrete valued differential inclusion describing the local volume fractions of such structures. Our main result shows the inclusion to be rigid without additional regularity assumptions and provides a list of all possible solutions. We give constructions ensuring that the various types of solutions indeed arise from the variational model and quantitatively describe their rigidity via H-measures. Our contribution to the second aspect is a conditional result on the convergence of the Allen-Cahn Equations to multi-phase mean curvature flow, which is a popular model for grain growth in polychrystalline metals. The proof relies on the gradient flow structure of both models and borrows ideas from certain convergence proofs for minimizing movement schemes.:1 Introduction 1.1 Shape memory alloys 1.2 Multi-phase mean curvature flow 2 Branching microstructures in shape memory alloys: Rigidity due to macroscopic compatibility 2.1 The main rigidity theorem 2.2 Outline of the proof 2.3 Proofs 3 Branching microstructures in shape memory alloys: Constructions 3.1 Outline and setup 3.2 Branching in two linearly independent directions 3.3 Combining all mechanisms for varying the volume fractions 4 Branching microstructures in shape memory alloys: Quantitative aspects via H-measures 4.1 Preliminary considerations 4.2 Structure of the H-measures 4.3 The transport property and accuracy of the approximation 4.4 Applications of the transport property 5 Convergence of the Allen-Cahn Equation to multi-phase mean curvature flow 5.1 Main results 5.2 Compactness 5.3 Convergence 5.4 Forces and volume constraints

Eine Verallgemeinerung der klassischen Resultate von Von Neumann und Birkhoff ist die Frage nach gewichteten Versionen ihrer Theoreme. Eine zentrale Antwort auf diese Fragestellung lieferte Jean Bourgain 1988 mit seinem Rückkehrzeitentheorem. Aufbauend auf dem Beweis von Bourgain, Furstenberg, Katznelson und Ornstein aus dem Jahr 1989 sowie dem Buch von Assani präsentieren wir einen ausführlichen und vollständigen Beweis und besprechen insbesondere den Fall von dynamischen Systemen mit rein atomarer invarianter sigma-Algebra. / In this diploma thesis we present a detailed proof of Bourgain's Return Times Theorem due to Bourgain, Furstenberg, Katznelson and Ornstein following their paper as well as the book by Assani. In particular, we discuss the case of systems with a purely atomic invariant sigma-algebra in all details.

This work investigate two different approaches for the parametrization of a special moment problem of Stieltjes-type. On the one hand we deal with systems of Potapov's fundamental matrix inequalities. Thereby, we examine certain invariant subspaces, so-called Dubovoj subspaces, and special matrix polynomials as wells as their associated J- forms. On the other hand we consider a Schur-analytic approach and present a special one-step algorithm. Moreover, considerations on linear fractional transformations of matrices serve as an important tool for the development of the algorithm. Both representations aim at a description of the solution in the non-degenerate case as well as in the different degenerate cases.