## Doctoral Thesis

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- Modeling Road Roughness with Conditional Random Fields (2016)
- A vehicles fatigue damage is a highly relevant figure in the complete vehicle design process. Long term observations and statistical experiments help to determine the influence of differnt parts of the vehicle, the driver and the surrounding environment. This work is focussing on modeling one of the most important influence factors of the environment: road roughness. The quality of the road is highly dependant on several surrounding factors which can be used to create mathematical models. Such models can be used for the extrapolation of information and an estimation of the environment for statistical studies. The target quantity we focus on in this work ist the discrete International Roughness Index or discrete IRI. The class of models we use and evaluate is a discriminative classification model called Conditional Random Field. We develop a suitable model specification and show new variants of stochastic optimizations to train the model efficiently. The model is also applied to simulated and real world data to show the strengths of our approach.

- Dual-Pivot Quicksort and Beyond: Analysis of Multiway Partitioning and Its Practical Potential (2016)
- Multiway Quicksort, i.e., partitioning the input in one step around several pivots, has received much attention since Java 7’s runtime library uses a new dual-pivot method that outperforms by far the old Quicksort implementation. The success of dual-pivot Quicksort is most likely due to more efficient usage of the memory hierarchy, which gives reason to believe that further improvements are possible with multiway Quicksort. In this dissertation, I conduct a mathematical average-case analysis of multiway Quicksort including the important optimization to choose pivots from a sample of the input. I propose a parametric template algorithm that covers all practically relevant partitioning methods as special cases, and analyze this method in full generality. This allows me to analytically investigate in depth what effect the parameters of the generic Quicksort have on its performance. To model the memory-hierarchy costs, I also analyze the expected number of scanned elements, a measure for the amount of data transferred from memory that is known to also approximate the number of cache misses very well. The analysis unifies previous analyses of particular Quicksort variants under particular cost measures in one generic framework. A main result is that multiway partitioning can reduce the number of scanned elements significantly, while it does not save many key comparisons; this explains why the earlier studies of multiway Quicksort did not find it promising. A highlight of this dissertation is the extension of the analysis to inputs with equal keys. I give the first analysis of Quicksort with pivot sampling and multiway partitioning on an input model with equal keys.

- Signature Standard Bases over Principal Ideal Rings (2016)
- By using Gröbner bases of ideals of polynomial algebras over a field, many implemented algorithms manage to give exciting examples and counter examples in Commutative Algebra and Algebraic Geometry. Part A of this thesis will focus on extending the concept of Gröbner bases and Standard bases for polynomial algebras over the ring of integers and its factors \(\mathbb{Z}_m[x]\). Moreover we implemented two algorithms for this case in Singular which use different approaches in detecting useless computations, the classical Buchberger algorithm and a F5 signature based algorithm. Part B includes two algorithms that compute the graded Hilbert depth of a graded module over a polynomial algebra \(R\) over a field, as well as the depth and the multigraded Stanley depth of a factor of monomial ideals of \(R\). The two algorithms provide faster computations and examples that lead B. Ichim and A. Zarojanu to a counter example of a question of J. Herzog. A. Duval, B. Goeckner, C. Klivans and J. Martin have recently discovered a counter example for the Stanley Conjecture. We prove in this thesis that the Stanley Conjecture holds in some special cases. Part D explores the General Neron Desingularization in the frame of Noetherian local domains of dimension 1. We have constructed and implemented in Singular and algorithm that computes a strong Artin Approximation for Cohen-Macaulay local rings of dimension 1.

- Integrating Security Concerns into Safety Analysis of Embedded Systems Using Component Fault Trees (2016)
- Nowadays, almost every newly developed system contains embedded systems for controlling system functions. An embedded system perceives its environment via sensors, and interacts with it using actuators such as motors. For systems that might damage their environment by faulty behavior usually a safety analysis is performed. Security properties of embedded systems are usually not analyzed at all. New developments in the area of Industry 4.0 and Internet of Things lead to more and more networking of embedded systems. Thereby, new causes for system failures emerge: Vulnerabilities in software and communication components might be exploited by attackers to obtain control over a system. By targeted actions a system may also be brought into a critical state in which it might harm itself or its environment. Examples for such vulnerabilities, and also successful attacks, became known over the last few years. For this reason, in embedded systems safety as well as security has to be analyzed at least as far as it may cause safety critical failures of system components. The goal of this thesis is to describe in one model how vulnerabilities from the security point of view might influence the safety of a system. The focus lies on safety analysis of systems, so the safety analysis is extended to encompass security problems that may have an effect on the safety of a system. Component Fault Trees are very well suited to examine causes of a failure and to find failure scenarios composed of combinations of faults. A Component Fault Tree of an analyzed system is extended by additional Basic Events that may be caused by targeted attacks. Qualitative and quantitative analyses are extended to take the additional security events into account. Thereby, causes of failures that are based on safety as well as security problems may be found. Quantitative or at least semi-quantitative analyses allow to evaluate security measures more detailed, and to justify the need of such. The approach was applied to several example systems: The safety chain of the off-road robot RAVON, an adaptive cruise control, a smart farming scenario, and a model of a generic infusion pump were analyzed. The result of all example analyses was that additional failure causes were found which would not have been detected in traditional Component Fault Trees. In the analyses also failure scenarios were found that are caused solely by attacks, and that are not depending on failures of system components. These are especially critical scenarios which should not happen in this way, as they are not found in a classical safety analysis. Thus the approach shows its additional benefit to a safety analysis which is achieved by the application of established techniques with only little additional effort.

- A Phase Field Model for the Evolution of Martensite Microstructures in Metastable Austenites (2016)
- This thesis is concerned with a phase field model for martensitic transformations in metastable austenitic steels. Within the phase field approach an order parameter is introduced to indicate whether the present phase is austenite or martensite. The evolving microstructure is described by the evolution of the order parameter, which is assumed to follow the time-dependent Ginzburg-Landau equation. The elastic phase field model is enhanced in two different ways to take further phenomena into account. First, dislocation movement is considered by a crystal plasticity setting. Second, the elastic model for martensitic transformations is combined with a phase field model for fracture. Finite element simulations are used to study the single effects separately which contribute to the microstructure formation.

- Cyanobacterial lichenized fungi and there photobionts in Vietnam (2016)
- The biodiversity of the cyanobacterial lichen flora of Vietnam is chronically understudied. Previous studies often neglected the lichens that inhabit lowlands especially outcrops and sand dunes that are common habitats in Vietnam. A cyanolichen collection was gathered from lowlands of central and southern Vietnam to study their diversity and distribution. At the same time, cultured photobionts from those lichens were used for olyphasic taxonomic approach. A total of 66 cyanolichens were recorded from lowland regions in central and southern of Vietnam, doubles the number of cyanolichens for Vietnam. 80% of them are new records for Vietnam in which a new species Pyrenopsis melanophthalma and two new unidentified lichinacean taxa were described. A notably floristic segregation by habitats was indicated in the communities. Saxicolous Lichinales dominated in coastal outcrops that corresponded to 56% of lichen species richness. Lecanoralean cyanolichens and basidiolichens were found in the lowland forests. Precipitation correlated negatively to species richness in this study, indicating a competitive relationship. Eleven cyanobacterial strains including 8 baeocyte-forming members of the genus Chroococcidiopsis and 3 heterocyte-forming species of the genera Nostoc and Scytonema were successfully isolated from lichens. Phylogenetic and morphological analyses indicated that Chroococcidiopsis was the unique photobiont in Peltula. New mophological characters were found in two Chroococcidiopsis strains: (1) the purple content of cells in one photobiont strain that was isolated from a new lichinacean taxon, and (2) the pseudofilamentous feature by binary division from a strain that was isolated from Porocyphus dimorphus. With respect to heterocyte-forming cyanobiont, Scytonema was confirmed as the photobiont in the ascolichen Heppia lutosa applying the polyphasic method. The genus Scytonema in the basidiolichens Cyphellostereum was morphologically examinated in lichen thalli. For the first time the intracellular haustorial system of basidiolichen genus Cyphellostereum was noted and investigated. Phylogenetic analysis of photobiont strains Nostoc from Pannaria tavaresii and Parmeliella brisbanensis indicated that a high selectivity occurred in Parmeliella brisbanensis that were from different regions of the world, while low photobiont selectivity occurred among Pannaria tavaresii samples from different geographical regions. The herewith presented dissertation is therefore an important contribution to the lichen flora of Vietnam and a significant improvement of the actual knowledge about cyanolichens in this country.

- Morphology and Morphology Formation of Injection Molded PP-based Nanocomposites (2016)
- The mechanical properties of semi-crystalline polymers depend extremely on their morphology, which is dependent on the crystallization during processing. The aim of this research is to determine the effect of various nanoparticles on morphology formation and tensile mechanical properties of polypropylene under conditions relevant in polymer processing and to contribute ultimately to the understanding of this influence. Based on the thermal analyses of samples during fast cooling, it is found that the presence of nanoparticle enhances the overall crystallization process of PP. The results suggest that an increase of the nucleation density/rate is a dominant process that controls the crystallization process of PP in this work, which can help to reduce the cycle time in the injection process. Moreover, the analysis of melting behaviors obtained after each undercooling reveals that crystal perfection increases significantly with the incorporation of TiO2 nanoparticles, while it is not influenced by the SiO2 nanoparticles. This work also comprises an analysis of the influence of nanoparticles on the microstructure of injection-molded parts. The results clearly show multi-layers along the wall thickness. The spherulite size and the degree of crystallinity continuously decrease from the center to the edge. Generally both the spherulite size and the degree of crystallinity decrease with higher the SiO2 loading. In contrast, an increase in the degree of crystallinity with an increasing TiO2 nanoparticle loading was detected. The tensile properties exhibit a tendency to increase in the tensile strength as the core is reached. The tensile strength decreases with the addition of nanoparticles, while the elongation at break of nanoparticle-filled PP decreases from the skin to the core. With increasing TiO2 loading, the elongation at break decreases.

- Worst-Case Performance Analysis of Feed-Forward Networks – An Efficient and Accurate Network Calculus (2016)
- Distributed systems are omnipresent nowadays and networking them is fundamental for the continuous dissemination and thus availability of data. Provision of data in real-time is one of the most important non-functional aspects that safety-critical networks must guarantee. Formal verification of data communication against worst-case deadline requirements is key to certification of emerging x-by-wire systems. Verification allows aircraft to take off, cars to steer by wire, and safety-critical industrial facilities to operate. Therefore, different methodologies for worst-case modeling and analysis of real-time systems have been established. Among them is deterministic Network Calculus (NC), a versatile technique that is applicable across multiple domains such as packet switching, task scheduling, system on chip, software-defined networking, data center networking and network virtualization. NC is a methodology to derive deterministic bounds on two crucial performance metrics of communication systems: (a) the end-to-end delay data flows experience and (b) the buffer space required by a server to queue all incoming data. NC has already seen application in the industry, for instance, basic results have been used to certify the backbone network of the Airbus A380 aircraft. The NC methodology for worst-case performance analysis of distributed real-time systems consists of two branches. Both share the NC network model but diverge regarding their respective derivation of performance bounds, i.e., their analysis principle. NC was created as a deterministic system theory for queueing analysis and its operations were later cast in a (min,+)-algebraic framework. This branch is known as algebraic Network Calculus (algNC). While algNC can efficiently compute bounds on delay and backlog, the algebraic manipulations do not allow NC to attain the most accurate bounds achievable for the given network model. These tight performance bounds can only be attained with the other, newly established branch of NC, the optimization-based analysis (optNC). However, the only optNC analysis that can currently derive tight bounds was proven to be computationally infeasible even for the analysis of moderately sized networks other than simple sequences of servers. This thesis makes various contributions in the area of algNC: accuracy within the existing framework is improved, distributivity of the sensor network calculus analysis is established, and most significantly the algNC is extended with optimization principles. They allow algNC to derive performance bounds that are competitive with optNC. Moreover, the computational efficiency of the new NC approach is improved such that this thesis presents the first NC analysis that is both accurate and computationally feasible at the same time. It allows NC to scale to larger, more complex systems that require formal verification of their real-time capabilities.

- Gröbner Bases over Extention Fields of \(\mathbb{Q}\) (2016)
- Gröbner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field. In this thesis, we consider Gröbner bases computations over two types of coefficient fields. First, consider a simple extension \(K=\mathbb{Q}(\alpha)\) of \(\mathbb{Q}\), where \(\alpha\) is an algebraic number, and let \(f\in \mathbb{Q}[t]\) be the minimal polynomial of \(\alpha\). Second, let \(K'\) be the algebraic function field over \(\mathbb{Q}\) with transcendental parameters \(t_1,\ldots,t_m\), that is, \(K' = \mathbb{Q}(t_1,\ldots,t_m)\). In particular, we present efficient algorithms for computing Gröbner bases over \(K\) and \(K'\). Moreover, we present an efficient method for computing syzygy modules over \(K\). To compute Gröbner bases over \(K\), starting from the ideas of Noro [35], we proceed by joining \(f\) to the ideal to be considered, adding \(t\) as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,4,27], that is, by inferring information in characteristic zero from information in characteristic \(p > 0\). For suitable primes \(p\), the minimal polynomial \(f\) is reducible over \(\mathbb{F}_p\). This allows us to apply modular methods once again, on a second level, with respect to the modular factors of \(f\). The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over \(K\). On the other hand, to compute Gröbner bases over \(K'\), our new algorithm first specializes the parameters \(t_1,\ldots,t_m\) to reduce the problem from \(K'[x_1,\ldots,x_n]\) to \(\mathbb{Q}[x_1,\ldots,x_n]\). The algorithm then computes a set of Gröbner bases of specialized ideals. From this set of Gröbner bases with coefficients in \(\mathbb{Q}\), it obtains a Gröbner basis of the input ideal using sparse multivariate rational interpolation. At current state, these algorithms are probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods.

- Interest Rate Modeling - The Potential Approach and Multi-Curve Potential Models (2016)
- This thesis is concerned with interest rate modeling by means of the potential approach. The contribution of this work is twofold. First, by making use of the potential approach and the theory of affine Markov processes, we develop a general class of rational models to the term structure of interest rates which we refer to as "the affine rational potential model". These models feature positive interest rates and analytical pricing formulae for zero-coupon bonds, caps, swaptions, and European currency options. We present some concrete models to illustrate the scope of the affine rational potential model and calibrate a model specification to real-world market data. Second, we develop a general family of "multi-curve potential models" for post-crisis interest rates. Our models feature positive stochastic basis spreads, positive term structures, and analytic pricing formulae for interest rate derivatives. This modeling framework is also flexible enough to accommodate negative interest rates and positive basis spreads.