Refine
Year of publication
Document Type
- Doctoral Thesis (57)
- Article (29)
- Final Thesis (1)
Has Fulltext
- yes (87)
Is part of the Bibliography
- no (87)
Keywords
- - (22)
- Statistik (5)
- Numerische Mathematik (4)
- Optimale Kontrolle (4)
- fractal (4)
- permutation entropy (4)
- Bioinformatik (3)
- Fraktal (3)
- Optimale Steuerung (3)
- Selbstähnlichkeit (3)
Institute
- Institut für Mathematik und Informatik (87) (remove)
Publisher
- MDPI (14)
- Frontiers Media S.A. (6)
- Springer Nature (3)
- BioMed Central (BMC) (2)
- Oxford University Press (1)
- Wiley (1)
The geometric arena here is a smooth manifold of dimension n equipped with a Riemannian or pseudo-Riemannian metric and an affine connection. Field theories following from a variational principle are considered on this basis. In this context, all invariants which are quadratic in the curvature are determined. The work derives several manifestly covariant formulas for the Euler-Lagrange derivatives or the field equations. Some of these field theories can be interpreted as gravitational theories alternatively to Einstein´s general relativity theory. The work also touches the difficult problem to define and to calculate energy and momentum of a gravitational field.
Tuberculosis (TB) has tremendous public health relevance. It most frequently affects the lung and is characterized by the development of unique tissue lesions, termed granulomas. These lesions encompass various immune populations, with macrophages being most extensively investigated. Myeloid derived suppressor cells (MDSCs) have been recently identified in TB patients, both in the circulation and at the site of infection, however their interactions with Mycobacterium tuberculosis (Mtb) and their impact on granulomas remain undefined. We generated human monocytic MDSCs and observed that their suppressive capacities are retained upon Mtb infection. We employed an in vitro granuloma model, which mimics human TB lesions to some extent, with the aim of analyzing the roles of MDSCs within granulomas. MDSCs altered the structure of and affected bacterial containment within granuloma-like structures. These effects were partly controlled through highly abundant secreted IL-10. Compared to macrophages, MDSCs activated primarily the NF-κB and MAPK pathways and the latter largely contributed to the release of IL-10 and replication of bacteria within in vitro generated granulomas. Moreover, MDSCs upregulated PD-L1 and suppressed proliferation of lymphocytes, albeit with negligible effects on Mtb replication. Further comprehensive characterization of MDSCs in TB will contribute to a better understanding of disease pathogenesis and facilitate the design of novel immune-based interventions for this deadly infection.
Die Arbeit befasst sich mit der Parameterbestimmung in gewöhnlichen Differentialgleichungssystemen aus gegebenen Messdaten. Als Zielfunktion wird die quadratische Abweichungen betrachtet, ebenso wie die Betragssummen- und Tschebyschev-Norm der Differenz von der Lösung der gewöhnlichen Differentialgleichung und des Messwert-Vektors. Zur Anwendung kommen dabei sowohl iterative Optimierungsverfahren als auch direkte Methoden der optimalen Steuerung.
Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic
(2020)
A New Kind of Permutation Entropy Used to Classify Sleep Stages from Invisible EEG Microstructure
(2017)
Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.
Anaplasma phagocytophilum and Anaplasma ovis–Emerging Pathogens in the German Sheep Population
(2021)
Knowledge on the occurrence of pathogenic tick-borne bacteria Anaplasma phagocytophilum and Anaplasma ovis is scarce in sheep from Germany. In 2020, owners from five flocks reported ill thrift lambs and ewes with tick infestation. Out of 67 affected sheep, 55 animals were clinically examined and hematological values, blood chemistry and fecal examinations were performed to investigate the underlying disease causes. Serological tests (cELISA, IFAT) and qPCR were applied to all affected sheep to rule out A. phagocytophilum and A. ovis as a differential diagnosis. Ticks were collected from selected pastures and tested by qPCR. Most animals (n = 43) suffered from selenium deficiency and endoparasites were detected in each flock. Anaplasma spp. antibodies were determined in 59% of examined sheep. Seventeen animals tested positive for A. phagocytophilum by qPCR from all flocks and A. phagocytophilum was also detected in eight pools of Ixodes ricinus. Anaplasma phagocytophilum isolates from sheep and ticks were genotyped using three genes (16S rRNA, msp4 and groEL). Anaplasma ovis DNA was identified in six animals from one flock. Clinical, hematological and biochemical changes were not significantly associated with Anaplasma spp. infection. The 16S rRNA analysis revealed known variants of A. phagocytophilum, whereas the msp4 and groEL showed new genotypes. Further investigations are necessary to evaluate the dissemination and health impact of both pathogens in the German sheep population particularly in case of comorbidities.
Discovering Latent Structure in High-Dimensional Healthcare Data: Toward Improved Interpretability
(2022)
This cumulative thesis describes contributions to the field of interpretable machine learning in the healthcare domain. Three research articles are presented that lie at the intersection of biomedical and machine learning research. They illustrate how incorporating latent structure can provide a valuable compression of the information hidden in complex healthcare data.
Methodologically, this thesis gives an overview of interpretable machine learning and the discovery of latent structure, including clusters, latent factors, graph structure, and hierarchical structure. Different workflows are developed and applied to two main types of complex healthcare data (cohort study data and time-resolved molecular data). The core result builds on Bayesian networks, a type of probabilistic graphical model. On the application side, we provide accurate predictive or discriminative models focusing on relevant medical conditions, related biomarkers, and their interactions.
Background
The alignment of large numbers of protein sequences is a challenging task and its importance grows rapidly along with the size of biological datasets. State-of-the-art algorithms have a tendency to produce less accurate alignments with an increasing number of sequences. This is a fundamental problem since many downstream tasks rely on accurate alignments.
Results
We present learnMSA, a novel statistical learning approach of profile hidden Markov models (pHMMs) based on batch gradient descent. Fundamentally different from popular aligners, we fit a custom recurrent neural network architecture for (p)HMMs to potentially millions of sequences with respect to a maximum a posteriori objective and decode an alignment. We rely on automatic differentiation of the log-likelihood, and thus, our approach is different from existing HMM training algorithms like Baum–Welch. Our method does not involve progressive, regressive, or divide-and-conquer heuristics. We use uniform batch sampling to adapt to large datasets in linear time without the requirement of a tree. When tested on ultra-large protein families with up to 3.5 million sequences, learnMSA is both more accurate and faster than state-of-the-art tools. On the established benchmarks HomFam and BaliFam with smaller sequence sets, it matches state-of-the-art performance. All experiments were done on a standard workstation with a GPU.
Conclusions
Our results show that learnMSA does not share the counterintuitive drawback of many popular heuristic aligners, which can substantially lose accuracy when many additional homologs are input. LearnMSA is a future-proof framework for large alignments with many opportunities for further improvements.
Influenza A Virus (IAV) infection followed by bacterial pneumonia often leads to hospitalization and death in individuals from high risk groups. Following infection, IAV triggers the process of viral RNA replication which in turn disrupts healthy gut microbial community, while the gut microbiota plays an instrumental role in protecting the host by evolving colonization resistance. Although the underlying mechanisms of IAV infection have been unraveled, the underlying complex mechanisms evolved by gut microbiota in order to induce host immune response following IAV infection remain evasive. In this work, we developed a novel Maximal-Clique based Community Detection algorithm for Weighted undirected Networks (MCCD-WN) and compared its performance with other existing algorithms using three sets of benchmark networks. Moreover, we applied our algorithm to gut microbiome data derived from fecal samples of both healthy and IAV-infected pigs over a sequence of time-points. The results we obtained from the real-life IAV dataset unveil the role of the microbial families Ruminococcaceae, Lachnospiraceae, Spirochaetaceae and Prevotellaceae in the gut microbiome of the IAV-infected cohort. Furthermore, the additional integration of metaproteomic data enabled not only the identification of microbial biomarkers, but also the elucidation of their functional roles in protecting the host following IAV infection. Our network analysis reveals a fast recovery of the infected cohort after the second IAV infection and provides insights into crucial roles of Desulfovibrionaceae and Lactobacillaceae families in combating Influenza A Virus infection. Source code of the community detection algorithm can be downloaded from https://github.com/AniBhar84/MCCD-WN.
Diese Arbeit beschäftigt sich mit der Analyse und Modellierung des Microarrayexperiments. Hierfür wird das gesamte Experiment in fünf Teilprozesse zerlegt, die Reverse Transkription, die Hybridisierung, das Waschen, die Fluoreszenz und die Detektion. Jeder Teilprozess wurde separat modelliert und analysiert. Anschließend wurde die Teilprozesse im Gesamtmodell vereint und dieses für verschiedene Parametersituationen simuliert. Diese Arbeit ermöglicht eine mathematische Handhabung des Microarrayexperiments und deckt seine Abhängigkeit von den einzelnen Schritten des Experiments auf. Dies kann benutzt werden, um Normalisierung und Analyse zu verbessern.
Abstract
The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied.
Neutrophils in Tuberculosis: Cell Biology, Cellular Networking and Multitasking in Host Defense
(2021)
Neutrophils readily infiltrate infection foci, phagocytose and usually destroy microbes. In
tuberculosis (TB), a chronic pulmonary infection caused by Mycobacterium tuberculosis (Mtb),
neutrophils harbor bacilli, are abundant in tissue lesions, and their abundances in blood correlate
with poor disease outcomes in patients. The biology of these innate immune cells in TB is complex.
Neutrophils have been assigned host-beneficial as well as deleterious roles. The short lifespan of
neutrophils purified from blood poses challenges to cell biology studies, leaving intracellular
biological processes and the precise consequences of Mtb–neutrophil interactions ill-defined. The
phenotypic heterogeneity of neutrophils, and their propensity to engage in cellular cross-talk and
to exert various functions during homeostasis and disease, have recently been reported, and such
observations are newly emerging in TB. Here, we review the interactions of neutrophils with Mtb,
including subcellular events and cell fate upon infection, and summarize the cross-talks between
neutrophils and lung-residing and -recruited cells. We highlight the roles of neutrophils in TB
pathophysiology, discussing recent findings from distinct models of pulmonary TB, and emphasize
technical advances that could facilitate the discovery of novel neutrophil-related disease
mechanisms and enrich our knowledge of TB pathogenesis
Interactive Visualization for the Exploration of Aligned Biological Networks and Their Evolution
(2011)
Network Visualization is a widely used tool in biology. The biological networks, as protein-interaction-networks are important for many aspects in life. Today biologists use the comparison of networks of different species (network alignment) to understand the networks in more detail and to understand the underlying evolution. The goal of this work is to develop a visualization software that is able to visualize network alignments and also their evolution. The presented software is the first software for such visualization tasks. It uses 3D graphics and also animations for the dynamic visualization of evolution. This work consists of a review of the Related Work, a chapter about our Graph-based Approach for Interactive Visualization of Evolving Network Alignments, an explanation of the Graph Layout Algorithm and some hints for the Software System.
Trade of cattle between farms forms a complex trade network. We investigate partitions of this network for cattle trade in Germany. These partitions are groups of farms with similar properties and they are inferred directly from the trade pattern between farms. We make use of a rather new method known as stochastic block modeling (SBM) in order to divide the network into smaller units. SBM turns out to outperform the more established community detection method in the context of disease control in terms of trade restriction. Moreover, SBM is also superior to geographical based trade restrictions and could be a promising approach for disease control.
Background
The Earth Biogenome Project has rapidly increased the number of available eukaryotic genomes, but most released genomes continue to lack annotation of protein-coding genes. In addition, no transcriptome data is available for some genomes.
Results
Various gene annotation tools have been developed but each has its limitations. Here, we introduce GALBA, a fully automated pipeline that utilizes miniprot, a rapid protein-to-genome aligner, in combination with AUGUSTUS to predict genes with high accuracy. Accuracy results indicate that GALBA is particularly strong in the annotation of large vertebrate genomes. We also present use cases in insects, vertebrates, and a land plant. GALBA is fully open source and available as a docker image for easy execution with Singularity in high-performance computing environments.
Conclusions
Our pipeline addresses the critical need for accurate gene annotation in newly sequenced genomes, and we believe that GALBA will greatly facilitate genome annotation for diverse organisms.
Maligne Erkrankungen zeigen oft charakteristische genetische Veränderungen. Das Auffinden derartiger Veränderungen wurde in den letzten Jahren durch verfeinerte molekulare Techniken erleichtert. Viele genetische Ereignisse in den maligne transformierten Zellen sind jedoch noch ungeklärt. Die präzise Bestimmung der Bruchpunktregionen chromosomaler Veränderungen bei T-Zell akuten lymphatischen Leukämien ist Inhalt dieser Arbeit. Hierzu wurde die „Fine Tiling-Comparative Genomhybridisierung“ (FT-CGH) mit der „Ligation mediated-PCR“ (LM-PCR) kombiniert. Diese Methoden wurden zunächst an Zelllinien etabliert und anschließend in verschiedenen Leukämieproben eingesetzt. Chromosomale Aberrationen gehen häufig mit Verlust oder Gewinn von genetischem Material einher. Diese unbalancierten Anomalien lassen sich durch die Comparative Genomhybridisierung (CGH) ermitteln. Dieses Verfahren ermöglicht Differenzen der DNA-Menge einer zu untersuchenden Probe bezogen auf eine interne Kontrollprobe zu detektieren. Bei der Fine Tiling-CGH werden gezielt chromosomale Abschnitte hochauflösend auf eventuelle Abweichungen des DNA-Gehaltes analysiert. Anschließend werden die detektierten Bruchpunktregionen der DNA Schwankungen mittels der LM-PCR untersucht. Ein Abgleich mit einer internen Kontrollzelllinie HEK 293-T lässt atypische PCR-Fragmente bei der untersuchten Probe aufspüren. Der anschließende Sequenzabgleich unter der Verwendung des BLASTn Suchprogramms (National Center for Biotechnology Information) führte in den untersuchten Zelllinien, wie auch in den T-Zell akuten lymphatischen Leukämieproben zur Identifizierung verschiedener genomischer Veränderungen. Neben einfachen Deletionen wurden auch bisher ungeklärte komplexere chromosomale Translokationen nachgewiesen. So konnte unter anderem bei einer lymphoblastischen T-Zell-Leukämie die Translokation t(12;14)(q23;q11.2) auf genomischer Ebene geklärt werden. Hierbei fand im Abschnitt 14q11 innerhalb des TRA/D Locus eine Deletion von 89 Kilobasen statt. Die Bruchenden wurden mit der Sequenz des open reading frames C12orf42, welches im 12q23 Chromosomenabschnitt lokalisiert ist, zusammengelagert. Bei dieser chromosomalen Aberration wurde die C12orf42 Sequenz zerstört und 1,3 Kilobasen deletiert. Des Weiteren konnte bei einer akuten lymphoblastischen T-Zell-Leukämie die Inversion inv(14)(q11q32) mit involvierten TRA/D und IGH Locus auf Sequenzebene geklärt werden. Der Bruch des 14q11 Bereiches fand zwischen dem Genabschnitt der konstanten Region (TRAC) des TRA/D Locus und dem DAD1 (defender against cell death 1) Gens statt, wobei im beteiligten genetischen Abschnitt keine Rekombinasesignalsequenz (RSS) zu finden ist. Dieses belegt, dass fehlerhafte Umlagerungen innerhalb des Genoms nicht ausschließlich auf die Rekombinase zurückzuführen sind. Die vorliegende Arbeit zeigt, dass die Kombination aus FT-CGH und LM-PCR eine präzise Bruchpunktanalyse unbekannter chromosomaler Aberrationen, welche mit Imbalancen einhergehen, ermöglicht. Diese genaue Analyse dient der Identifizierung von Genen, welche direkt und indirekt durch diese genomischen Umlagerungen betroffen sind. Das Wissen über diese Veränderungen kann für das Verständnis der Pathogenese, für diagnostische Zwecke und zum Nachweis der minimalen Resterkrankung eingesetzt werden. Eine Klärung beteiligter Gene und Signalwege wird es erlauben, zielgerichtete und individualisierte Therapiestrategien zu entwickeln.
Self-similar sets are a class of fractals which can be rigorously defined and treated by mathematical methods. Their theory has been developed in n-dimensional space, but we have just a few good examples of self-similar sets in three-dimensional space. This thesis has two different aims. First, to extend fractal constructions from two-dimensional space to three-dimensional space. Second, to study some of the properties of these fractals such as finite type, disk-likeness, ball-likeness, and the Hausdorff dimension of boundaries. We will use the neighbor graph tool for creating new fractals, and studying their properties.
Background
An important initial phase of arguably most homology search and alignment methods such as required for genome alignments is seed finding. The seed finding step is crucial to curb the runtime as potential alignments are restricted to and anchored at the sequence position pairs that constitute the seed. To identify seeds, it is good practice to use sets of spaced seed patterns, a method that locally compares two sequences and requires exact matches at certain positions only.
Results
We introduce a new method for filtering alignment seeds that we call geometric hashing. Geometric hashing achieves a high specificity by combining non-local information from different seeds using a simple hash function that only requires a constant and small amount of additional time per spaced seed. Geometric hashing was tested on the task of finding homologous positions in the coding regions of human and mouse genome sequences. Thereby, the number of false positives was decreased about million-fold over sets of spaced seeds while maintaining a very high sensitivity.
Conclusions
An additional geometric hashing filtering phase could improve the run-time, accuracy or both of programs for various homology-search-and-align tasks.
Phylogenetic (i.e., leaf-labeled) trees play a fundamental role in evolutionary research. A typical problem is to reconstruct such trees from data like DNA alignments (whose columns are often referred to as characters), and a simple optimization criterion for such reconstructions is maximum parsimony. It is generally assumed that this criterion works well for data in which state changes are rare. In the present manuscript, we prove that each binary phylogenetic tree T with n ≥ 20k leaves is uniquely defined by the set Ak (T), which consists of all characters with parsimony score k on T. This can be considered as a promising first step toward showing that maximum parsimony as a tree reconstruction criterion is justified when the number of changes in the data is relatively small.
In dieser Arbeit beschäftigen wir uns mit symplektischen Lie-Algebren und metrischen, symplektischen Lie-Algebren. Wir erweitern ein bestehendes Klassifikationsschema mittels quadratischer Erweiterungen für metrische Lie-Algebren mit halbeinfachen, schiefsymmetrischen Derivationen so, dass damit sämtliche metrischen, symplektischen Lie-Algebren auf Isomorphie untersucht werden können. Damit bestimmen wir die Isomorphieklassen aller nichtabelschen, metrischen, symplektischen Lie-Algebren, der Dimension kleiner als zehn, sowie alle mit einem Index von kleiner oder gleich drei. Anschließend wird in Analogie zur Herangehensweise für metrische Lie-Algebren ein Klassifikationsschema für symplektische Lie-Algebren mit ausgeartetem Zentrum mittels quadratischer Erweiterungen aufgebaut, was uns zudem ein Klassifikationsschema für nilpotente, symplektische Lie-Algebren liefert. Abschließend berechnen wir konkret ein Repräsentantensystem der Isomorphieklassen aller sechsdimensionalen, nilpotenten, symplektischen Lie-Algebren.
Objektive Eingruppierung sequenzierter Tollwutisolate mithilfe des Affinity Propagation Clusterings.
(2018)
Das International Committee on Taxonomy of Viruses (ICTV) reguliert die Nomenklatur von Viren sowie die Entstehung neuer Taxa (dazu gehören: Ordnung, Familie, Unterfamilie, Gattung und Art/Spezies). Dank dieser Anstrengungen ist die Einteilung für verschiedenste Viren in diese Kategorien klar und transparent nachvollziehbar. In den vergangenen Jahrzehnten sind insgesamt mehr als 21.000 Datensätze der Spezies „rabies lyssavirus“ (RABV) sequenziert worden. Eine weiterführende Unterteilung der sequenzierten Virusisolate dieser Spezies ist bislang jedoch nicht einheitlich vorgeschlagen. Die große Anzahl an sequenzierten Isolaten führte auf Basis von phylogenetischen Bäumen zu uneindeutigen Ergebnissen bei der Einteilung in Cluster. Inhalt meiner Dissertation ist daher ein Vorschlag, diese Problematik mit der Anwendung einer partitionierenden Clusteringmethode zu lösen. Dazu habe ich erstmals die Methodik des affinity propagation clustering (AP) für solche Fragestellungen eingesetzt. Als Datensatz wurden alle verfügbaren sequenzierten Vollgenomisolate der Spezies RABV analysiert. Die Analysen des Datensatzes ergaben vier Hauptcluster, die sich geographisch separieren ließen und entsprechend als „Arctic“, „Cosmopolitain“, „Asian“ und „New World“ bezeichnet wurden. Weiterführende Analysen erlaubten auch eine weitere Aufteilung dieser Hauptcluster in 12-13 Untercluster. Zusätzlich konnte ich einen Workflow generieren, der die Möglichkeit bietet, die mittels AP definierten Cluster mit den Ergebnissen der phylogenetischen Auswertungen zu kombinieren. Somit lassen sich sowohl Verwandtschaftsverhältnisse erkennen als auch eine objektive Clustereinteilung vornehmen. Dies könnte auch ein möglicher Analyseweg für weitere Virusspezies oder andere vergleichende Sequenzanalysen sein.
Numerische Lösung von Optimalsteuerungsaufgaben unter Nebenbedingungen mit biologischen Anwendungen
(2010)
In dieser Dissertation wird ein Verfahren zur Lösung von Optimalsteuerungsaufgaben mit Steuer-Zustandsbeschränkungen vorgestellt. Dazu werden die notwendigen Bedingungen an eine optimale Lösung benutzt, die ein System aus algebraischen Gleichungen, Ungleichungen und Differentialgleichungen erzeugen. Dieses System wird mit einem Newton-ähnlichen Ansatz gelöst. Außerdem wird die Erweiterung auf Problemen mit reinen Zustandsbeschränkungen vorgeführt. Eine deutliche Verbesserung der Konvergenzergebnisse kann durch die Anwendung der Fisher-Burmeister-Funktion auf die Komplementaritätsbedingungen erzielt werden. Die Iterationsverfahren werden auf eine Reihe von restringierten Optimalsteuerungsaufgaben (Aufgaben mit reinen Steuerbeschränkungen, gemischten Steuer-Zustandbeschränkungen und reinen Zustandsbeschränkungen für einzelne Zeitpunkte und für das gesamte Optimierungsintervall) angewendet, um ihr Verhalten bei verschiedenen Startwerten sowie unterschiedlichen Schrittweitenansätzen zu untersuchen. Dazu werden zum einen zwei aus der Literatur bekannte Aufgaben (das Rayleigh-Problem und das Minimum-Ernergy-Problem) gelöst und zum anderen werden zwei Probleme mit biologischem Hintergrund untersucht. So wird eine Optimalsteuerungsaufgabe aus der Fischerei um geeignete Einnahmenbedingungen erweitert, die absichern sollen, dass die Fischer keine längeren Phasen ohne Kapitalzuwachs haben. Dazu wird zwischen einer globalen Bedingung und einer Bedingung für endlich viele Zeitpunkte unterschieden. Desweiteren wird ein Modell einer HIV-Erkrankung untersucht, bei dem die numerischen Verfahren, die die notwendigen Bedingungen an eine optimale Lösung benutzen, nur für geringe Behandlungszeiten (bis zu 50 Tage) das Problem lösen. Es zeigt sich, dass die Stabilität dieser Verfahren deutlich verbessert werden kann, wenn das Modell um eine Obergrenze für die T-Zellen erweitert wird. Den Abschluss der Dissertation bildet ein Kapitel zur Konvergenzuntersuchung, in dem sich zeigt, dass die verwendeten Iterationsverfahren teilweise von sehr schlechter Konvergenzordnung sind, da die Bedingung für eine lineare Konvergenz nicht erfüllt wird.
The constructions of Lévy processes from convolution semigroups and of product systems from subproduct systems respectively, are formally quite similar. Since there are many more comparable situations in quantum stochastics, we formulate a general categorial concept (comonoidal systems), construct corresponding inductive systems and show under suitable assumptions general properties of the corresponding inductive limits. Comonoidal systems in different tensor categories play a role in all chapters of the thesis. Additive deformations are certain comonoidal systems of algebras. These are obtained by deformation of the algebra structure of a bialgebra. If the bialgebra is even a Hopf algebra, then compatibility with the antipode automatically follows. This remains true also in the case of braided Hopf algebras. Subproduct systems are comonoidal systems of Hilbert spaces. In the thesis we deal with the question, what are the possible dimensions of finite-dimensional subproduct systems. In discrete time, this can be reduced to the combinatorial problem of determining the complexities of factorial languages. We also discuss the rational and continuous time case. A further source for comonoidal systems are universal products, which are used in quantum probability to model independence. For the (r,s)-products, which were recently introduced by S. Lachs, we determine the corresponding product of representations by use of a generalized GNS-construction.
Statistical Methods and Applications for Biomarker Discovery Using Large Scale Omics Data Set
(2023)
This thesis focuses on identifying genetic factors associated with human kidney disease progression, with three articles presented. Article I describes the identification of loci associated with UACR through trans-ethnic, European-ancestry-specific, and diabetes-specific meta-analyses. An approximate conditional analysis was performed to identify additional independent UACR-associated variants within identified loci. The genome-wide significance level of 𝛼=5×10−8 is used for both primary GWAS association and conditional analyses. However, unlike primary association tests, conditional tests are limited to specific genomic regions surrounding primary GWAS index signals rather than being applied on a genome-wide scale.
In article II, we hypothesized that the application of 𝛼=5×10−8 is overly strict and results in a loss of power. To address this issue, we developed a quasi-adaptive method within a weighted hypothesis testing framework. This method exploits the type I error (𝛼=0.05) by providing less conservative SNP specific 𝛼-thresholds to select secondary signals in conditional analysis. Through simulation studies and power analyses, we demonstrate that the quasi-adaptive method outperforms the established criterion 𝛼=5×10−8 as well as the equal weighting scheme (the Sidak-correction). Furthermore, our method performs well when applied to real datasets and can potentially reveal previously undetected secondary signals in existing data.
In article III, we extended our quasi-adaptive method to identify plausible multiple independent signals at each locus (a secondary signal, a tertiary signal, a signal of 4th, and beyond) and applied it to the publically available GWAS meta-analysis to detect additional multiple independent eGFR-associated signals. The improved quasi-adaptive method successfully identified additional novel replicated independent SNPs that would have gone undetected by applying too conservative genome-wide significance level of 𝛼=5× 10−8. Colocalization analysis based on the novel independent signals identified potentially functional genes across the kidney and other tissues.
Overall, these articles contribute to the understanding of genetic factors associated with human kidney disease progression and provide novel methods for identifying secondary and multiple independent signals in conditional GWAS analyses.
Liver diseases are important causes of morbidity and mortality worldwide. The aim of
this study was to identify differentially expressed microRNAs (miRNAs), target genes, and key
pathways as innovative diagnostic biomarkers in liver patients with different pathology and functional
state. We determined, using RT-qPCR, the expression of 472 miRNAs in 125 explanted livers from
subjects with six different liver pathologies and from control livers. ANOVA was employed to
obtain differentially expressed miRNAs (DEMs), and miRDB (MicroRNA target prediction database)
was used to predict target genes. A miRNA–gene differential regulatory (MGDR) network was
constructed for each condition. Key miRNAs were detected using topological analysis. Enrichment
analysis for DEMs was performed using the Database for Annotation, Visualization, and Integrated
Discovery (DAVID). We identified important DEMs common and specific to the different patient
groups and disease progression stages. hsa-miR-1275 was universally downregulated regardless
the disease etiology and stage, while hsa-let-7a*, hsa-miR-195, hsa-miR-374, and hsa-miR-378 were
deregulated. The most significantly enriched pathways of target genes controlled by these miRNAs
comprise p53 tumor suppressor protein (TP53)-regulated metabolic genes, and those involved in
regulation of methyl-CpG-binding protein 2 (MECP2) expression, phosphatase and tensin homolog
(PTEN) messenger RNA (mRNA) translation and copper homeostasis. Our findings show a novel
panel of deregulated miRNAs in the liver tissue from patients with different liver pathologies. These
miRNAs hold potential as biomarkers for diagnosis and staging of liver diseases.
The goal of this doctoral thesis is to create and to implement methods for fully automatic segmentation applications in magnetic resonance images and datasets. The work introduces into technical and physical backgrounds of magnetic resonance imaging (MRI) and summarizes essential segmentation challenges in MRI data including technical malfunctions and ill-posedness of inverse segmentation problems. Theoretical background knowledge of all the used methods that are adapted and extended to combine them for problem-specific segmentation applications are explained in more detail. The first application for the implemented solutions in this work deals with two-dimensional tissue segmentation of atherosclerotic plaques in cardiological MRI data. The main part of segmentation solutions is designed for fully automatic liver and kidney parenchyma segmentation in three-dimensional MRI datasets to ensure computer-assisted organ volumetry in epidemiological studies. The results for every application are listed, described and discussed before important conclusions are drawn. Among several applied methods, the level set method is the main focus of this work and is used as central segmentation concept in the most applications. Thus, its possibilities and limitations for MRI data segmentation are analyzed. The level set method is extended by several new ideas to overcome possible limitations and it is combined as important part of modularized frameworks. Additionally, a new approach for probability map generation is presented in this thesis, which reduces data dimensionality of multiple MR-weightings and incorporates organ position probabilities in a probabilistic framework. It is shown, that essential organ features (i.e. MR-intensity distributions, locations) can be well represented in the calculated probability maps. Since MRI data are produced by using multiple MR- weightings, the used dimensionality reduction technique is very helpful to generate a single probability map, which can be used for further segmentation steps in a modularized framework.
A lot of research data has become available since the outbreak of the COVID-19
pandemic in 2019. Connecting this data is essential for the understanding of the
SARS-CoV-2 virus and the fight against the pandemic.
Amongst biological and biomedical research data, computational models targeting
COVID-19 have been emerging and their number is growing constantly. They are a
central part of the field of Systems Biology, which aims to understand the mechanisms
and behaviour of biological systems. Model predictions help to understand the
mechanisms of the novel coronavirus and the life-threatening disease it is causing.
Both biomedical research data and modelling data regarding COVID-19 have
previously been stored in separated domain-specific graph databases. MaSyMoS,
short for Management System for Models and Simulations, is a graph database for
storing simulation studies of biological and biochemical systems. The CovidGraph
project integrates research data regarding COVID-19 and the coronavirus family
from various data resources in a knowledge graph.
In this thesis, we integrate simulation models from MaSyMoS, including models
targeting COVID-19, into the CovidGraph. Therefore, we present a concept for
the integration of simulation studies and the linkage through ontology terms and
reference publications in the CovidGraph. Ultimately, we connect data from the field
of systems biology and biomedical research data in a graph database.
Phytopathogenic Verticillia cause Verticillium wilt on numerous economically important crops. Plant infection begins at the roots, where the fungus is confronted with rhizosphere inhabiting bacteria. The effects of different fluorescent pseudomonads, including some known biocontrol agents of other plant pathogens, on fungal growth of the haploid Verticillium dahliae and/or the amphidiploid Verticillium longisporum were compared on pectin-rich medium, in microfluidic interaction channels, allowing visualization of single hyphae, or on Arabidopsis thaliana roots. We found that the potential for formation of bacterial lipopeptide syringomycin resulted in stronger growth reduction effects on saprophytic Aspergillus nidulans compared to Verticillium spp. A more detailed analyses on bacterial-fungal co-cultivation in narrow interaction channels of microfluidic devices revealed that the strongest inhibitory potential was found for Pseudomonas protegens CHA0, with its inhibitory potential depending on the presence of the GacS/GacA system controlling several bacterial metabolites. Hyphal tip polarity was altered when V. longisporum was confronted with pseudomonads in narrow interaction channels, resulting in a curly morphology instead of straight hyphal tip growth. These results support the hypothesis that the fungus attempts to evade the bacterial confrontation. Alterations due to co-cultivation with bacteria could not only be observed in fungal morphology but also in fungal transcriptome. P. protegens CHA0 alters transcriptional profiles of V. longisporum during 2 h liquid media co-cultivation in pectin-rich medium. Genes required for degradation of and growth on the carbon source pectin were down-regulated, whereas transcripts involved in redox processes were up-regulated. Thus, the secondary metabolite mediated effect of Pseudomonas isolates on Verticillium species results in a complex transcriptional response, leading to decreased growth with precautions for self-protection combined with the initiation of a change in fungal growth direction. This interplay of bacterial effects on the pathogen can be beneficial to protect plants from infection, as shown with A. thaliana root experiments. Treatment of the roots with bacteria prior to infection with V. dahliae resulted in a significant reduction of fungal root colonization. Taken together we demonstrate how pseudomonads interfere with the growth of Verticillium spp. and show that these bacteria could serve in plant protection.
In phylogenetics, evolutionary relationships of different species are represented by phylogenetic trees.
In this thesis, we are mainly concerned with the reconstruction of ancestral sequences and the accuracy of this reconstruction given a rooted binary phylogenetic tree.
For example, we wish to estimate the DNA sequences of the ancestors given the observed DNA sequences of today living species.
In particular, we are interested in reconstructing the DNA sequence of the last common ancestor of all species under consideration. Note that this last common ancestor corresponds to the root of the tree.
There exist various methods for the reconstruction of ancestral sequences.
A widely used principle for ancestral sequence reconstruction is the principle of parsimony (Maximum Parsimony).
This principle means that the simplest explanation it the best.
Applied to the reconstruction of ancestral sequences this means that a sequence which requires the fewest evolutionary changes along the tree is reconstructed.
Thus, the number of changes is minimized, which explains the name of Maximum Parsimony.
Instead of estimating a whole DNA sequence, Maximum Parsimony considers each position in the sequence separately. Thus in the following, each sequence position is regarded separately, and we call a single position in a sequence state.
It can happen that the state of the last common ancestor is reconstructed unambiguously, for example as A. On the other hand, Maximum Parsimony might be indecisive between two DNA nucleotides, say for example A and C.
In this case, the last common ancestor will be reconstructed as {A,C}.
Therefore we consider, after an introduction and some preliminary definitions, the following question in Section 3: how many present-day species need to be in a certain state, for example A, such that the Maximum Parsimony estimate of the last common ancestor is also {A}?
The answer of this question depends on the tree topology as well as on the number of different states.
In Section 4, we provide a sufficient condition for Maximum Parsimony to recover the ancestral state at the root correctly from the observed states at the leaves.
The so-called reconstruction accuracy for the reconstruction of ancestral states is introduced in Section 5. The reconstruction accuracy is the probability that the true root state is indeed reconstructed and always takes two processes into account: on the one hand the approach to reconstruct ancestral states, and on the other hand the way how the states evolve along the edges of the tree. The latter is given by an evolutionary model.
In the present thesis, we focus on a simple symmetric model, the Neyman model.
The symmetry of the model means for example that a change from A to C is equally likely than a change from C to A.
Intuitively, one could expect that the reconstruction accuracy it the highest when all present-day species are taken into account. However, it has long been known that the reconstruction accuracy improves when some taxa are disregarded for the estimation.
Therefore, the question if there exits at least a lower bound for the reconstruction accuracy arises, i.e. if it is best to consider all today living species instead of just one for the reconstruction.
This is bad news for Maximum Parsimony as a criterion for ancestral state reconstruction, and therefore the question if there exists at least a lower bound for the reconstruction accuracy arises.
In Section 5, we start with considering ultrametric trees, which are trees where the expected number of substitutions from the root to each leaf is the same.
For such trees, we investigate a lower bound for the reconstruction accuracy, when the number of different states at the leaves of the tree is 3 or 4.
Subsequently in Section 6, in order to generalize this result, we introduce a new method for ancestral state reconstruction: the coin-toss method.
We obtain new results for the reconstruction accuracy of Maximum Parsimony by relating Maximum Parsimony to the coin-toss method.
Some of these results do not require the underlying tree to be ultrametric.
Then, in Section 7 we investigate the influence of specific tree topologies on the reconstruction accuracy of Maximum Parsimony. In particular, we consider balanced and imbalanced trees as the balance of a tree may have an influence on the reconstruction accuracy.
We end by introducing the Colless index in Section 8, an index which measures the degree of balance a rooted binary tree can have, and analyze its extremal properties.
Parsimonious Histograms
(2010)
The dissertation is concerned with the construction of data driven histograms. Histograms are the most elementary density estimators at all. However, they require the specification of the number and width of the bins. This thesis provides two new construction methods delivering adaptive histograms where the required parameters are determined automatically. Both methods follow the principle of parsimony, i.e. the histograms are solutions of predetermined optimization problems. In both cases, but under different aspects, the number of bins is minimized. The dissertation presents the algorithms that solve the optimization problems and illustrates them by a number of numerical experiments. Important properties of the estimators are shown. Finally, the new developed methods are compared with standard methods by an extensive simulation study. By means of synthetic samples of different size and distribution the histograms are evaluated by special performance criteria. As one main result, the proposed methods yield histograms with considerably fewer bins and with an excellent ability of peak detection.
Tafazzin is an acyltransferase with key functions in remodeling of the mitochondrial phospholipid cardiolipin (CL) by exchanging single fatty acids species in CL. Tafazzin-mediated CL remodeling determines the actual CL compositions and has been implicated in mitochondrial morphology and function. Thus, any deficiency of tafazzin leads to altered fatty acid composition of CL which is directly associated with impaired mitochondrial respiration and ATP production. Mutations in the tafazzin encoding gene TAZ, are the cause of the severe X-linked genetic disease, BARTH syndrome (BTHS).
Previous work provided first hints on a linkage of CL composition and subsequent limitations in the cellular ATP levels which may contribute to the restriction of growth. However, in C6 cells ATP levels remained unaltered due to compensatory activation of glycolysis. Moreover, it has been demonstrated that the substantial changes in CL composition are similarly resulting from knocking down either cardiolipin synthase (CRLS) or TAZ. This has also been shown in C6 glioma cells. Most notably only the knock down of TAZ, but not that of CRLS, compromised proliferation of C6 glioma cells. Therefore, a CL- independent role of TAZ in regulating cell proliferation is postulated.
In this study, any linkage of the lack of tafazzin to cellular proliferation should be investigated in more detail to allow first insight into underlying mechanisms.
The results of the current study demonstrate that the tafazzin knockout in C6 glioma cells show changes in global gene expression by applying transcriptome analysis using the- microarray Clarion S rat Affymetrix array. Out of 22,076 total number of genes detected, 1,099 genes were differentially expressed in C6 knockout cells which were either ≥2 and ≥4 fold up or down regulated genes. Furthermore, expression of selected target genes was validated using RT-qPCR. We have hypothesised that the changes in TAZ dependent gene expression is via PPAR transcription factor. According to eukaryotic promoter database (EPD) for selected target genes, exhibited at least one putative binding site for PPARG and PPARA transcription factors. However, pioglitazone and LG100268, synthetic ligands of PPARG and RXR, could not show any effect on changes in gene expression in C6 TAZ cells. Another class of cellular lipids, oxylipins were found to occur in significantly higher amounts in C6 TAZ cells compared to C6 cells which makes them candidates for mediating cellular effects and regulating gene expression via PPARs. A computational tool CiiiDER was used to for the prediction of transcription factor binding site. The transcription factors enriched in TAZ- regulated genes were found to be HOXA5 and PAX2, binding sites of which could be detected in 100 % of TAZ- regulated genes (>2-fold). By applying IPA to the differentially expressed genes we could identify lipid metabolism, and cholesterol superpathway in particular as the most affected pathway in C6 TAZ cells. This pathway consists of 20 genes, of which all (20/20) appeared to be differentially regulated in C6 TAZ cells. Of all the 20 genes, 4 of the differentially expressed genes were selected for further validation by RT-qPCR. By IPA it was possible to identify the upstream regulators that might be responsible for the differential expression of genes in C6 deficient cells. Some of the genes ACACA, HMGCR, FASN, ACSL1, 3 and, 5 identified was decreased by predicted activation and inhibition of the regulators. Further we have analysed the levels of cellular cholesterol content in C6 and C6 TAZ (w/o Δ5 and FL) cells. In C6 cells cholesterol is present more in its free form. C6 TAZ cells have increased amount of cholesterol compared to C6 cells. However, Δ5 and FL expressed C6 TAZ cells showed less amount of cholesterol.
Previous work established that knockout of tafazzin in C6 cells showed decreased cell proliferation in the absence of any changes in ATP content. To understand this phenomenon cellular senescence associated β-galactosidase in C6 and C6 TAZ cells was performed. C6 TAZ cells showed increased percentage of β-gal positive cells compared to C6 cells. Moreover, senescent associated secretory phenotype (SASP) represented by e.g. CXCL1, IL6, and IL1α was determined using RT-qPCR. Gene expression of these SASP factors was significantly upregulated in C6 TAZ cells.
Several human tafazzin isoforms exists due to alternate splicing. However, whether these isoforms differ in function and in CL remodelling activity or specificity, in particular, is unknown. The purpose of this work was to determine if specific isoforms, such as human isoform lacking exon 5 (Δ5), rat full length tafazzin (FL) and enzymatically dead full length tafazzin (H69L), can restore the wild type phenotype in terms of CL composition, cellular proliferation, and gene expression profile. Therefore, in the second part, it was demonstrated that expression of Δ5 to some extent and rat full length tafazzin can completely restore CL composition, in C6 TAZ cells which is naturally linked to the restoration of mitochondrial respiration. As expected, a comparable restoration of CL composition could not be seen after re-expressing an enzymatically dead full-length rat TAZ, (H69L; TAZ Mut). Furthermore, re-expression of the TAZ Mut largely failed to reverse the alterations in gene expression, in contrast re-expression of the TAZ FL and the Δ5 isoforms reversed gene expression to a larger extent. Moreover, only rat full length TAZ was able to reverse proliferation rate. Surprisingly, the expression of Δ5 in C6 TAZ cells did not promote proliferation of the wild type. Different effects of Δ5 and FL on CL composition and cell proliferation points to the specific and in part non-enzymatic functions of tafazzin isoforms, but this certainly requires further analysis.
Gram-negative bacteria secrete lipopolysaccharides (LPS), leading to a host immune
response of proinflammatory cytokine secretion. Those proinflammatory cytokines are
TNF-α and IFN-γ, which induce the production of indoleamine 2,3-dioxygenase (IDO). IDO production is increased during severe sepsis, and septic shock. High IDO
levels are associated with increased mortality. This enzyme catalyzes the degradation of tryptophan (TRP) to kynurenine (KYN) along the kynurenine pathway (KP).
KYN is further degraded to kynurenic acid (KYNA). Increased IDO levels accompany
with increased levels of KYNA, which is associated with immunoparalysis.
Due to its central role, the KP is a potential target of therapeutic intervention.
The degradation of TRP to KYN by IDO was intervened by 1-Methyltryptophan (1-
MT), which is assumed to inhibit IDO. By administering 1-MT, the survival of
1-MT-treated mice suffering from sepsis increased compared to mice not treated with
1-MT. The levels of downstream metabolites such as KYN and KYNA were
expected to be decreased. Surprisingly, in healthy mice and pigs, an increase in KYNA
after 1-MT administration was reported. Those unexpected metabolite alterations after 1-MT administration, and the mode of action, were not the focus of recent
research. Hence, there is no explanation for KYNA increase, while KYN did not change.
This thesis aims to postulate a possible degradation pathway of 1-MT along the KP
with the help of ordinary differential equation (ODE) systems.
Moreover, the developed ODE models were used to determine the ability of 1-MT to
inhibit IDO in vivo. Therefore, a multiplicity of ODE models were developed, including
a model of the KP, an extension by lipopolysaccharide (LPS) administration, and 1-MT
administration.
Moreover, seven ODE models were developed, all considering possible degradation pathways of 1-MT. The most likely degradation pathway was combined with the ODE model
of LPS administration, including the inhibitory effects of 1-MT.
Those models consist of several dependent equations describing the dynamics of the KP.
For each component of the KP, one equation describes the alterations over time. Equations for TRP, KYN, KYNA, and quinolinic acid (QUIN) were developed.
Moreover, the alterations of serotonin (SER) were also included. All together belong
to the TRP metabolism. They include the degradation of TRP to SER and to KYN,
which is further degraded to KYNA and QUIN. Every degradation is catalyzed by an enzyme. Therefore, Michaelis-Menten (MM) equations were used employing the substrate
constant Km and the maximal degradation velocity Vmax. To reduce the complexity of
parameter calculation, Km values of the different enzymes were fixed to literature values.
The remaining parameters of the equations were determined so that the trajectories of
the calculated metabolite levels correspond to data. The parameters of different models were determined. To propose a degradation pathway of 1-MT leading to increased
KYNA levels, seven models were developed and compared. The most likely model was
extended to test whether the inhibitory effects of 1-MT on IDO can be determined.
Three different approaches determined the ODE model parameters of the different hypothesis of 1-MT degradation. In the first approach, ODE model parameters were fixed
to values fitted to an independent data set. In the second approach, parameters were
fitted to a subset of the data set, which was used for simulations of the different hypotheses. The third approach calculated ODE model parameters 100 times without
fixed parameters. The parameter set ending up in trajectories of the TRP metabolites,
which have the smallest distance to the data, was assumed to be the most likely. The
ODE model parameters were fitted to data measured in pigs. Two different
experimental models delivered data used in this thesis. The first experimental model
activates IDO by LPS administration in pigs. The second one combines the IDO
activation by LPS with the administration of 1-MT in pigs.
The most likely hypothesis, according to approach 1 was the degradation of 1-MT to
KYNA and TRP. For the second data set the most likely one was the direct degradation of 1-MT to KYNA. With approach 2 the most likely degradation pathways were
the combination of all degradation pathways and the degradation of 1-MT to TRP and
TRP to KYNA. With approach 3 the most likely way of KYNA increase was given by
the direct degradation of 1-MT to KYNA. In summary, the three approaches revealed
hypothesis 2, the direct degradation of 1-MT to KYNA most frequently. A cell-free
assay validated this result. This experiment combined 1-MT or TRP with or without
the enzyme kynurenine aminotransferase (KAT). KAT was already shown to degrade
TRP directly to KYNA. The levels of TRP, KYN and KYNA were measured. The
highest KYNA levels were yielded with an assay adding KAT to 1-MT, corresponding
to hypothesis 2. The models describing the inhibitory effects of 1-MT revealed that
the model without inhibitory effects of 1-MT on IDO was more likely for all three approaches.
The correctness of hypothesis 2 has to be confirmed by further in vitro experiments. It
also has to be investigated which reactions promote the degradation of 1-MT to KYNA.
The missing inhibitory properties of 1-MT on IDO, determined by the in silico ODE
models, align with previous research. It was shown that the saturation of 1-MT was too
low, e.g. in pigs, to inhibit IDO efficiently.
In this study, the first possible degradation pathway of 1-MT along the KP is proposed.
The reliability of the results depends on the quality of the experimental data, and the
season, when data were measured. Moreover, the results vary between the different
approaches of parameter fitting. Different approaches of parameter fitting have to be
included in the analysis to get more evidence for the correctness of the results.
Jump penalized L1-Regression
(2012)
Die vorgelegte Arbeit beschäftigt sich mit Kurvenschätzung in einem Regressionsmodell für eindimensionale verrauschte Daten, welche die Ausreißer enthalten können. Dabei ist die Regression Funktion, also Funktion welche a priori unbekannt ist und welche geschätzt werden soll, eine beliebige absolut-integrierbare Funktion auf dem Intervall [0, 1) und Regression Schätzer eine Stückweise-konstante Funktion auf dem Intervall [0, 1). Die von uns betrachtende Schätzer sind stückweise-konstante Funktionen, welche die L1-Version den sogenannten Potts Funktional minimieren (s. [8]). Das L1 Potts Funktional ist so gewählt, dass einerseits die Komplexität des Schätzers in Form der Anzahl ihrer Sprünge beachtet wird und anderseits die absolute Abweichungen von den Daten betrachtet werden. Die Stufen des Minimierers vom L1 Potts Funktional entsprechen den lokalen Medianen von verrauschten Daten, im Gegensatz dazu entsprechen die Stufen des Minimierers von dem klassischen Potts Funktional (L2-Fall) den lokalen Mittelwerten von den Daten. Der Vorteil der L1-Version gegenüber L2-Version des Potts Funktionals kann dadurch erklärt werden, dass die Mediane bekannterweise viel robuster gegen Ausreißer als Mittelwerte sind. In der vorgelegten Arbeit wurden die asymptotischen Eigenschaften sowohl von der L1 Potts Funktionals als auch von seinen Minimierer studiert. Unter anderem, es konnte die Konsistenz des Schätzers für den Fall, dass die Originalfunktion f selbst eine Stufenfunktion ist, gezeigt werden. Dies stellt das Hauptergebnis der Arbeit dar. Konsistenz heißt hier, dass unter bestimmten Bedingungen die Minimierer vom L1 Potts Funktional gegen die Originalfunktion f konvergieren.
Given a manifold with a string structure, we construct a spinor bundle on its loop space. Our construction is in analogy with the usual construction of a spinor bundle on a spin manifold, but necessarily makes use of tools from infinite dimensional geometry. We equip this spinor bundle on loop space with an action of a bundle of Clifford algebras. Given two smooth loops in our string manifold that share a segment, we can construct a third loop by deleting this segment. If this third loop is smooth, then we say that the original pair of loops is a pair of compatible loops. It is well-known that this operation of fusing compatible loops is important if one wants to understand the geometry of a manifold through its loop space. In this work, we explain in detail how the spinor bundle on loop space behaves with respect to fusion of compatible loops. To wit, we construct a family of fusion isomorphisms indexed by pairs of compatible loops in our string manifold. Each of these fusion isomorphisms is an isomorphism from the relative tensor product of the fibres of the spinor bundle over its index pair of compatible loops to the fibre over the loop that is the result of fusing the index pair. The construction of a spinor bundle on loop space equipped with a fusion product as above was proposed by Stolz and Teichner with the goal of studying the Dirac operator on loop space". Our construction combines facets of the theory of bimodules for von Neumann algebras, infinite dimensional manifolds, and Lie groups and their representations. We moreover place our spinor bundle on loop space in the context of bundle gerbes and bundle gerbe modules.
Geometric T-Duality
(2022)
From a physicists point of view T-duality is a relation connecting string
theories on different spacetimes. Mathematically speaking, T-duality should be a symmetric relation on
the space of toroidal string backgrounds. Such a background consists of: a smooth manifold M; a torus bundle E over M - the total space modelling spacetime; a Riemannian metric g on E - modelling the field of gravity; a U(1)-bundle gerbe G with connection over E - modelling the Kalb-
Ramond field.
But as of now no complete model for T-duality exists. The three most notable
approaches for T-duality are given by the differential approaches by Buscher in the form of the Buscher rules and by Bouwknegt, Evslin and Mathai in the form of T-duality with H-flux on the one hand, and by the topological approach given by Bunke, Rumpf and Schick which is known as topological T-duality. In this thesis we combine these different approaches to form the first model for T-duality over complete geometric toroidal string backgrounds and we will introduce an example for this geometric T-duality inspired by the Hopf bundle.
Abstract
With the advent of molecular genetic methods, an increasing number of morphologically cryptic taxa has been discovered. The majority of them, however, remains formally undescribed and without a proper name although their importance in ecology and evolution is increasingly being acknowledged. Despite suggestions to complement traditional descriptions with genetic characters, the taxonomic community appears to be reluctant to adopt this proposition. As an incentive, we introduce QUIDDICH, a tool for the QUick IDentification of DIgnostic CHaracters, which automatically scans a DNA or amino acid alignment for those columns that allow to distinguish taxa and classifies them into four different types of diagnostic characters. QUIDDICH is a system‐independent, fast and user‐friendly tool that requires few manual steps and provides a comprehensive output, which can be included in formal taxonomic descriptions. Thus, cryptic taxa do not have to remain in taxonomic crypsis and, bearing a proper name, can readily be included in biodiversity assessments and ecological and evolutionary analyses. QUIDDICH can be obtained from the comprehensive R archive network (CRAN, https://cran.r-project.org/package=quiddich).
This thesis revolves around a new concept of independence of algebras. The independence nicely fits into the framework of universal products, which have been introduced to classify independence relations in quantum probability theory; the associated product is called (r,s)-product and depends on two complex parameters r and s. Based on this product, we develop a theory which works without using involutive algebras or states. The following aspects are considered: 1. Classification: Universal products are defined on the free product of algebras (the coproduct in the category of algebras) and model notions of independence in quantum probability theory. We distinguish universal products according to their behaviour on elements of length two, calling them (r,s)-universal products with complex parameters r and s respectively. In case r and s equal 1, Muraki was able to show that there exist exactly five universal products (Muraki’s five). For r equals s nonzero we get five one parameter families (q-Muraki’s five). We prove that in the case r not equal to s the (r,s)-product, a two parameter deformation of the Boolean product, is the only universal product satisfying our set of axioms. The corresponding independence is called (r,s)-independence. 2. Dual pairs and GNS construction: By use of the GNS construction, one can associate a product of representations with every positive universal product. Since the (r,s)-product does not preserve positivity, we need a substitute for the usual GNS construction for states on involutive algebras. In joint work with M. Gerhold, the product of representations associated with the (r,s)-product was determined, whereby we considered representations on dual pairs instead of Hilbert spaces. This product of representations is - as we could show - essentially different from the Boolean product. 3. Reduction and quantum Lévy processes: U. Franz introduced a category theoretical concept which allows a reduction of the Boolean, monotone and antimonotone independence to the tensor independence. This existing reduction could be modified in order to apply to the (r,s)-independence. Quantum Lévy processes with (r,s)-independent increments can, in analogy with the tensor case, be realized as solutions of quantum stochastic differential equations. To prove this theorem, the previously mentioned reduction principle in the sense of U. Franz and a generalization of M. Schürmann’s theory for symmetric Fock spaces over dual pairs are used. As the main result, we obtain the realization of every (r,s)-Lévy process as solution of a quantum stochastic differential equation. When one, more generally, defines Lévy processes in a categorial way using U. Franz’s definition of independence for tensor categories with inclusions, compatibility of the inclusions with the tensor category structure plays an important role. For this thesis such a compatibility condition was formulated and proved to be equivalent to the characterization proposed by M. Gerhold. 4. Limit distributions: We work with so-called dual semigroups in the sense of D. V. Voiculescu (comonoids in the tensor category of algebras with free product). The polynomial algebra with primitive comultiplication is an example for such a dual semigroup. We use a "weakened" reduction which we call reduction of convolution and which essentially consists of a cotensor functor constructed from the symmetric tensor algebra. It turns dual semigroups into commutative bialgebras and also translates the convolution exponentials. This method, which can be nicely described in the categorial language, allows us to formulate central limit theorems for the (r,s)-independence and to calculate the correponding limit distributions (convergence in moments). We calculate the moments appearing in the central limit theorem for the (r,s)-product: The even moments are homogeneous polynomials in r and s with the Eulerian numbers as coefficients; the odd moments vanish. The moment sequence that we get from the central limit theorem for an arbitrary universal product is the moment sequence of a probability measure on the real line if and only if r equals s greater or equal to 1. In this case we present an explicit formula for the probability measure.
In der Dissertation haben wir uns mit dem numerischen Lösen von unbeschränkten Optimalsteuerungsproblemen beschäftigt. Dabei war das Ziel der Arbeit die Homotopie-Methode von Costanza zu untersuchen, kritisch zu hinterfragen und sie zu erweitern. Dazu haben wir zuerst Optimalsteuerungsprobleme untersucht und Resultate aus der Funktionalanalysis zitiert, die wir benötigen, um notwendige Bedingungen für ein unbeschränktes Optimalsteuerungsproblem herzuleiten. Die zentrale Idee dabei ist, dass wir ein äquivalentes, infinites Optimierungsproblem aufstellen und für dieses die notwendigen Bedingungen herleiten und beweisen. Die erhaltenen Resultate haben wir dann auf unbeschränkte Optimalsteuerungsprobleme übertragen. Ziel des Ansatzes ist es, die unbekannten Anfangs- und Endwerte der Zustände und Adjungierten in Abhängigkeit von frei wählbaren Parametern zu berechnen, so dass nur noch ein reines Anfangs- oder Endwertproblem gelöst werden muss, welches numerisch einfacher zu handhaben ist. Dabei stellte sich im Verlauf der Arbeit heraus, dass Costanzas Ansatz nicht allgemeingültig ist und nur auf spezielle Fälle angewendet werden kann. Wir haben den ursprünglichen Ansatz neu hergeleitet und an den kritischen Stellen angepasst, so dass dieser beispielunabhängig benutzt werden kann. Danach haben wir uns mit der numerische Umsetzung unseres Ansatzes befasst. Zum Lösen der gewöhnlichen Differentialgleichungssysteme mit gegebenen Anfangswerten benutzten wir ein in MATLAB implementiertes, explizites Runge-Kutta-Verfahren mit Schrittweitensteuerung. Ein wichtiger Punkt dabei war die Approximation der Jacobi-Matrix der Zustands- und Adjungiertengleichungen mit Hilfe von Complex step differentiation. Diese liefert schnellere und stabilere Approximationen an die ersten Ableitungen als z.B. der zentrale Differenzenquotient, da bei diesem numerische Auslöschung auftreten kann. Weiterhin haben wir direkte und indirekte Verfahren genannt, die man zum Lösen von Optimalsteuerungsproblemen benutzen kann, um die Genauigkeit unseres Ansatzes zu überprüfen. Im letzten Kapitel haben wir unseren Ansatz an verschiedenen Beispielen getestet. Dabei haben wir zuerst unbeschränkte Optimalsteuerungsprobleme betrachtet, die alle sehr gut gelöst wurden. Dessen numerische Lösung wurde effizient und mit hoher Genauigkeit berechnet. Dies ist insbesondere bemerkenswert, da man mit anderen Ansätzen oft eine gute Startlösung benötigt, damit die jeweiligen Verfahren konvergieren. Abschließend haben wir Beispiele für beschränkte Optimalsteuerungsprobleme betrachtet. Diese haben wir mit unbeschränkten Optimalsteuerungsproblemen approximiert, wobei wir in dem Integranden eine Straffunktion eingeführt haben, die mit dem Parameter S gewichtet wurde. Somit konnten wir unter Anwendung unseres erweiterten Ansatzes die ursprünglichen Probleme gut approximieren und für hinreichend große S waren die Lösungen der unbeschränkten und beschränkten Probleme im numerischen Sinne identisch. Dabei unterschied sich in den Beispielen, wie groß das S gewählt werden muss, um eine gute Näherung zu erhalten.
Independence is a basic concept of probability theory and statistics. In a lot of fields of sciences, dependency of different variables is gained lots of attention from scientists. A measure, named information dependency, is proposed to express the dependency of a group of random variables. This measure is defined as the Kullback-Leibler divergence of a joint distribution with respect to a product-marginal distribution of these random variables. In the bivariate case, this measure is known as mutual information of two random variables. Thus, the measure information dependency has a strong relationship with the Information Theory. The thesis aims to give a thorough study of the information dependency from both mathematical and practical viewpoints. Concretely, we would like to research three following problems: 1. Proving that the information dependency is a useful tool to express the dependency of a group of random variables by comparing it with other measures of dependency. 2. Studying the methods to estimate the information dependency based on the samples of a group of random variables. 3. Investigating how the Independent Component Analysis problem, an interesting problem in statistics, can be solved using information dependency.
Convolutional Neural Network-based image classification models are the current state-of-the-art for solving image classification problems. However, obtaining and using such a model to solve a specific image classification problem presents several challenges in practice. To train the model, we need to find good hyperparameter values for training, such as initial model weights or learning rate. However, finding these values is usually a non-trivial process. Another problem is that the training data used for model training is often class-imbalanced in practice. This usually has a negative impact on model training. However, not only is it challenging to obtain a Convolutional Neural Network-based model, but also to use the model after model training. After training, the model might be applied to images that were drawn from a data distribution that is different from the data distribution the training data was drawn from. These images are typically referred to as out-of-distribution samples. Unfortunately, Convolutional Neural Network-based image classification models typically fail to predict the correct class for out-of-distribution samples without warning, which is problematic when such a model is used for safety-critical applications. In my work, I examined whether information from the layers of a Convolutional Neural Network-based image classification model (pixels and activations) can be used to address all of these issues. As a result, I suggest a method for initializing the model weights based on image patches, a method for balancing a class-imbalanced dataset based on layer activations, and a method for detecting out-of-distribution samples, which is also based on layer activations. To test the proposed methods, I conducted extensive experiments using different datasets. My experiments showed that layer information (pixels and activations) can indeed be used to address all of the aforementioned challenges when training and using Convolutional Neural Network-based image classification models.
Die vorliegende Arbeit beschäftigt sich mit der numerischen Lösung von Optimalsteuerungsproblemen. Dazu wird das Maximumprinzip verwendet, dessen Anwendung auf ein Mehrpunktrandwertproblem führt. Die Aufgabe bestand nun darin, ein Programmpaket zu entwickeln, mit dem solche Mehrpunktrandwertprobleme mit der Mehrzielmethode numerisch gelöst werden können. Dabei wurden verschiedene Anforderungen an das zu entwickelnde Programm gestellt, die bereits existierende Programmpakete nicht oder nur eingeschränkt erfüllen. Die Bedienung soll durch die Verwendung einer grafischen Oberfläche intuitiver und komfortabler gestaltet werden. Ein weiteres Ziel besteht in der Problemunabhängigkeit des Quellcodes, sodass der Quellcode unangetastet bleiben kann. Außerdem sollen für die Benutzung des Programms keine Programmierkenntnisse notwendig sein. Der Funktionsumfang soll im Vergleich zu bestehenden Implementierungen erweitert werden, um die Möglichkeiten der Mehrzielmethode besser ausnutzen sowie die Methoden an das jeweilige zu lösende Problem anpassen zu können. Zunächst werden theoretische Grundlagen der optimalen Steuerung und des Maximumprinzips beschrieben. Die Mehrzielmethode wird vorgestellt und erweitert, sodass mit dieser auch Mehrpunktrandwertprobleme gelöst werden können. Ferner wird auf die Umsetzung der weiteren verwendeten mathematischen Methoden eingegangen. Dazu gehören das Newtonverfahren inklusive Dämpfung und Broydenupdate, verschiedenene Anfangswertproblemlöser (Dormand-Prince- und Rosenbrock-Typ-Verfahren) und die Singulärwertzerlegung, mit der die linearen Gleichungsssysteme gelöst werden. Außerdem werden die Komponenten und Funktionen des Programmpakets beschrieben, beispielsweise die Entwicklung der grafischen Oberfläche. Um das Einlesen der Daten eines Optimalsteuerungsproblems aus der grafischen Oberfläche in das Programm zu ermöglichen, wurde ein Parser verwendet. Die Software enthält Funktionen zur Erstellung von Plots und dem Export von Problemdaten in ein PDF-Dokument. Des Weiteren wird beschrieben, inwieweit die implementierten Verfahren an die Anforderungen eines spezifischen Optimalsteuerungsproblems angepasst werden können. Abschließend werden vier in ihrer Gestalt und ihrem Schwierigkeitsgrad sehr verschiedene Optimalsteuerungsprobleme beispielhaft gelöst. Dazu gehören beispielsweise das als Optimalsteuerungsproblem formulierte Brachistochrone- sowie das Min-Energy-Problem. Anhand der Lösung des Rayleigh-Problems wird gezeigt, wie man die zur Verfügung gestellten Optionen des Programmpakets sinnvoll nutzen kann, um eine Lösung zu bestimmen, die ein aussichtsreicher Kandidat für eine optimale Lösung ist. Abschließend wird ein Wiedereintrittsproblem einer Raumkapsel in die Erdumlaufbahn betrachtet, welches eine besondere Herausforderung darstellt, da das Differenzialgleichungssystem sehr empfindlich reagiert und Lösungen nur für einen kleinen Bereich von Startwerten existieren.
Self-affine tiles and fractals are known as examples in analysis and topology, as models of quasicrystals and biological growth, as unit intervals of generalized number systems, and as attractors of dynamical systems. The author has implemented a software which can find new examples and handle big databases of self-affine fractals. This thesis establishes the algebraic foundation of the algorithms of the IFStile package. Lifting and projection of algebraic and rational iterated function systems and many properties of the resulting attractors are discussed.
Twisted topological K-theory is a twisted version of topological K-theory in the sense of twisted generalized cohomology theories. It was pioneered by Donavan and Karoubi in 1970 where they used bundles of central simple graded algebras to model twists of K-theory. By the end of the last century physicists realised that D-brane charges in the field of string theory may be studied in terms of twisted K-theory. This rekindled interest in the topic lead to a wave of new models for the twists and new ways to realize the respective twisted K-theory groups. The state-of-the-art models today use bundles of projective unitary operators on separable Hilbert spaces as twists and K-groups are modeled by homotopy classes of sections of certain bundles of Fredholm operators. From a physics perspective these treatments are not optimal yet: they are intrinsically infinite-dimensional and these models do not immediately allow the inclusion of differential data like forms and connections.
In this thesis we introduce the 2-stack of k-algebra gerbes. Objects, 1-morphisms and 2-morphisms consist of finite-dimensional geometric data simultaneously generalizing bundle gerbes and bundles of central simple graded k-algebras for k either the field of real numbers or the field of complex numbers. We construct an explicit isomorphism from equivalence classes of k-algebra gerbes over a space X to the full set of twists of real K-theory and complex K-theory respectively. Further, we model relative twisted K-groups for compact spaces X and closed subspaces Y twisted by algebra gerbes. These groups are modeled directly in terms of 1-morphisms and 2-morphisms of algebra gerbes over X. We exhibit a relation to the K-groups introduced by Donavan and Karoubi and we translate their fundamental isomorphism -- an isomorphism relating K-groups over Thom spaces with K-groups twisted by Clifford algebra bundles -- to the new setting. With the help of this fundamental isomorphism we construct an explicit Thom isomorphism and explicit pushforward homomorphisms for smooth maps between compact manifolds, without requiring these maps to be K-oriented. Further -- in order to treat K-groups for non-torsion twists -- we implement a geometric cocycle model, inspired by a related geometric cycle model developed by Baum and Douglas for K-homology in 1982, and construct an assembly map for this model.
As the tree of life is populated with sequenced genomes ever more densely, the new challenge is the accurate and consistent annotation of entire clades of genomes. In my dissertation, I address this problem with a new approach to comparative gene finding that takes a multiple genome alignment of closely related species and simultaneously predicts the location and structure of protein-coding genes in all input genomes, thereby exploiting negative selection and sequence conservation. The model prefers potential gene structures in the different genomes that are in agreement with each other, or—if not—where the exon gains and losses are plausible given the species tree. The multi-species gene finding problem is formulated as a binary labeling problem on a graph. The resulting optimization problem is NP hard, but can be efficiently approximated using a subgradient-based dual decomposition approach.
I tested the novel approach on whole-genome alignments of 12 vertebrate and 12 Drosophila species. The accuracy was evaluated for human, mouse and Drosophila melanogaster and compared to competing methods. Results suggest that the new method is well-suited for annotation of a large number of genomes of closely related species within a clade, in particular, when RNA-Seq data are available for many of the genomes. The transfer of existing annotations from one genome to another via the genome alignment is more accurate than previous approaches that are based on protein-spliced alignments, when the genomes are at close to medium distances. The method is implemented in C++ as part of the gene finder AUGUSTUS.
In dieser Dissertation wird eine Problemstellung der Optimalen Steuerung aus dem Bereich der Linearen Elastizitätstheorie dargelegt und gelöst. Die Dissertation gliedert sich in die folgenden Schwerpunkte: Modellierung der Problemstellung, Formulierung der Optimalsteuerungsprobleme für den zeitunabhängigen (stationären) bzw. zeitabhängigen (instationären) Problem, die Herleitung der notwendigen Bedingungen für eine ermittelte optimale Lösung und die Berechnung von numerischen Lösungen des stationären bzw. instationären Problems sowie deren Überprüfung der Erfüllung der notwendigen Bedingungen. In der Modellierung werden Gleichungen zur Bestimmung der Deformation (Auslenkung) einer Zylinderschale unter rotations-symmetrischer Krafteinwirkung aus Grundgleichungen der Mechanik (Kräftegleichgewicht, Impulserhaltungssatz) hergeleitet. Bei dieser Herleitung werden die Hypothesen von Mindlin und Reissner verwendet und die spezielle Geometrie der Zylinderschale berücksichtigt. Die Dissertation erbringt den Nachweis der Existenz einer Lösung der modellierten Gleichungen im schwachen Sinne, d.h. für Lösungen in Sobolev-Räumen. Für die Formulierung der Optimalsteuerungsprobleme für den stationären und instationären Fall für Praxis relevante Problemstellungen setzen wir das Volumen des Zylinderrohres als konstant voraus (Volumenbedingung). Die Zielstellung der Optimalsteerungsprobleme besteht darin eine optimale Dicke zu bestimmen, welche die integrale Deformation (Auslenkung) der Zylinderschale (im instationären Fall zu einer ausgezeichneten Zeit) minimiert. Eine optimale Lösung (optimale Dicke) muss die notwendige Bedingung erster Ordnung (Variationsungleichung) für alle zulässigen Dicken, welche auch der Volumenbedingung genügen, erfüllen. Die Herleitung der konkreten Form dieser notwendigen Bedingungen für den stationären bzw. für die instationären Fälle wird in der Dissertation dargelegt. Durch die Verwendung der zugehörigen adjungierten Zustände können die notwendigen Bedingungen effizienter formuliert werden. Zur Berechnung einer Lösung der Gleichungen im stationären Fall bzw. in den instationären Fällen wurde die Finite Elemente Methode bzw. die Rothe-Methode im zeitabhängigen Fall verwendet, wobei die Lösungsräume exakt berücksichtigt werden. Das Optimierungsproblem wird diskretisiert und mit fmincon aus der Optimization-Toolbox von Matlab gelöst. Die damit berechneten diskreten optimalen Lösungen (optimale Dicke) für die einzelnen Problemstellungen werden auf die Erfüllung der notwendigen Bedingungen getestet. Die Dissertation wird durch viele Beispiel-Rechnungen abgerundet und deren Lösungen in grafischer Form präsentiert.