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In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.

In this thesis we integrate discrete dividends into the stock model, estimate
future outstanding dividend payments and solve different portfolio optimization
problems. Therefore, we discuss three well-known stock models, including
discrete dividend payments and evolve a model, which also takes early
announcement into account.
In order to estimate the future outstanding dividend payments, we develop a
general estimation framework. First, we investigate a model-free, no-arbitrage
methodology, which is based on the put-call parity for European options. Our
approach integrates all available option market data and simultaneously calculates
the market-implied discount curve. We illustrate our method using stocks
of European blue-chip companies and show within a statistical assessment that
the estimate performs well in practice.
As American options are more common, we additionally develop a methodology,
which is based on market prices of American at-the-money options.
This method relies on a linear combination of no-arbitrage bounds of the dividends,
where the corresponding optimal weight is determined via a historical
least squares estimation using realized dividends. We demonstrate our method
using all Dow Jones Industrial Average constituents and provide a robustness
check with respect to the used discount factor. Furthermore, we backtest our
results against the method using European options and against a so called
simple estimate.
In the last part of the thesis we solve the terminal wealth portfolio optimization
problem for a dividend paying stock. In the case of the logarithmic utility
function, we show that the optimal strategy is not a constant anymore but
connected to the Merton strategy. Additionally, we solve a special optimal
consumption problem, where the investor is only allowed to consume dividends.
We show that this problem can be reduced to the before solved terminal wealth
problem.

Embedded reactive systems underpin various safety-critical applications wherein they interact with other systems and the environment with limited or even no human supervision. Therefore, design errors that violate essential system specifications can lead to severe unacceptable damages. For this reason, formal verification of such systems in their physical environment is of high interest. Synchronous programs are typically used to represent embedded reactive systems while hybrid systems serve to model discrete reactive system in a continuous environment. As such, both synchronous programs and hybrid systems play important roles in the model-based design of embedded reactive systems. This thesis develops induction-based techniques for safety property verification of synchronous and hybrid programs. The imperative synchronous language Quartz and its hybrid systems’ extensions are used to sustain the findings.
Deductive techniques for software verification typically use Hoare calculus. In this context, Verification Condition Generation (VCG) is used to apply Hoare calculus rules to a program whose statements are annotated with pre- and postconditions so that the validity of an obtained Verification Condition (VC) implies correctness of a given proof goal. Due to the abstraction of macro steps, Hoare calculus cannot directly generate VCs of synchronous programs unless it handles additional label variables or goto statements. As a first contribution, Floyd’s induction-based approach is employed to generate VCs for synchronous and hybrid programs. Five VCG methods are introduced that use inductive assertions to decompose the overall proof goal. Given the right assertions, the procedure can automatically generate a set of VCs that can then be checked by SMT solvers or automated theorem provers. The methods are proved sound and relatively complete, provided that the underlying assertion language is expressive enough. They can be applied to any program with a state-based semantics.
Property Directed Reachability (PDR) is an efficient method for synchronous hardware circuit verification based on induction rather than fixpoint computation. Crucial steps of the PDR method consist of deciding about the reachability of Counterexamples to Induction (CTIs) and generalizing them to clauses that cover as many unreachable states as possible. The thesis demonstrates that PDR becomes more efficient for imperative synchronous programs when using the distinction between the control- and dataflow. Before calling the PDR method, it is possible to derive additional program control-flow information that can be added to the transition relation such that less CTIs will be generated. Two methods to compute additional control-flow information are presented that differ in how precisely they approximate the reachable control-flow states and, consequently, in their required runtime. After calling the PDR method, the CTI identification work is reduced to its control-flow part and to checking whether the obtained control-flow states are unreachable in the corresponding extended finite state machine of the program. If so, all states of the transition system that refer to the same program locations can be excluded, which significantly increases the performance of PDR.

Grape powdery mildew, Erysiphe necator, is one of the most significant plant pathogens, which affects grape growing regions world-wide. Because of its short generation time and the production of large amounts of conidia throughout the season, E. necator is classified as a moderate to high risk pathogen with respect to the development of fungicide resistance. The number of fungicidal mode of actions available to control powdery mildew is limited and for some of them resistances are already known. Aryl-phenyl-ketones (APKs), represented by metrafenone and pyriofenone, and succinate-dehydrogenase inhibitors (SDHIs), composed of numerous active ingredients, are two important fungicide classes used for the control of E. necator. Over the period 2014 to 2016, the emergence and development of metrafenone and SDHI resistant E. necator isolates in Europe was followed and evaluated. The distribution of resistant isolates was thereby strongly dependent on the European region. Whereas the north-western part is still predominantly sensitive, samples from east European countries showed higher resistance frequencies.
Classical sensitivity tests with obligate biotrophs can be challenging regarding sampling, transport and especially the maintenance of the living strains. Whenever possible, molecular genetic methods are preferred for a more efficient monitoring. Such methods require the knowledge of the resistance mechanisms. The exact molecular target and the resistance mechanism of metrafenone is still unknown. Whole genome sequencing of metrafenone sensitive and resistant wheat powdery mildew isolates, as well as adapted laboratory mutants of Aspergillus nidulans, where performed with the aim to identify proteins potentially linked to the mode of action or which contribute to metrafenone resistance. Based on comparative SNP analysis, four proteins potentially associated with metrafenone resistance were identified, but validation studies could not confirm their role in metrafenone resistance. In contrast to APKs, the mode of action of SDHIs is well understood. Sequencing of the sdh-genes of less sensitive E. necator isolates identified four different target-site mutations, the B-H242R, B-I244V, C-G169D and C-G169S, in sdhB and sdhC, respectively. Based on this information it was possible to develop molecular genetic monitoring methods for the mutations B-H242R and C-G169D. In 2016, the B-H242R was thereby identified as by far the most frequent mutation. Depending on the analysed SDH compound and the sdh-genotype, different sensitivities were observed and revealed a complex cross-resistance pattern.
Growth competition assays without selection pressure, with mixtures of sensitive and resistant E. necator isolates, were performed to determine potential fitness costs associated with fungicide resistance. With the experimental setups used, a clear fitness disadvantage associated with metrafenone resistance was not identified, although a strong variability of fitness was observed among the tested resistant E. necator isolates. For isolates with a reduced sensitivity towards SDHIs, associated fitness costs were dependent on the sdh-genotype analysed. Competition tests with the B-H242R genotypes gave evidence that there are no fitness costs associated with this mutation. In contrast, the C-G169D genotypes were less competitive, indicating a restricted fitness compared to the tested sensitive partners. Competition assays of field isolates, which exhibited several resistances towards different fungicide classes, indicated that there are no fitness costs associated with a multiple resistant phenotype in E. necator. Overall, these results clearly indicate the importance to analyse a representative number of isolates with sensitive and resistant phenotypes.

The screening of metagenomic datasets led to the identification of new phage-derived members of the heme oxygenase and the ferredoxin-dependent bilin reductase enzyme families.
The novel bilin biosynthesis genes were shown to form mini-cassettes on metagenomic scaffolds and further form distinct clusters in phylogenetic analyses (Ledermann et al., 2016). In this project, it was demonstrated that the discovered sequences actually encode for active enzymes. The biochemical characterization of a member of the heme oxygenases (ΦHemO) revealed that it possesses a regiospecificity for the α-methine bridge in the cleavage of the heme macrocycle. The reaction product biliverdin IXα was shown to function as the substrate for the novel ferredoxin-dependent bilin reductases (PcyX reductases), which catalyze its reduction to PEB via the intermediate 15,16-DHBV. While it was demonstrated that ΦPcyX, a phage-derived member of the PcyX reductases, is an active enzyme, it also became clear that the rate of the reaction is highly dependent on the employed redox partner. It turned out that the ferredoxin from the cyanophage P-SSM2 is to date the most suitable redox partner for the reductases of the PcyX group. Furthermore, the solution of the ΦPcyX crystal structure revealed that it adopts an α/β/α-sandwich fold, typical for the FDBR-family. Activity assays and subsequent HPLC analyses with different variants of the ΦPcyX protein demonstrated that, despite their similarity, PcyX and PcyA reductases must act via different reaction mechanisms.
Another part of this project focused on the biochemical characterization of the FDBR KflaHY2 from the streptophyte alga Klebsormidium flaccidum. Experiments with recombinant KflaHY2 showed that it is an active FDBR which produces 3(Z)-PCB as the main reaction product, like it can be found in reductases of the PcyA group. Moreover, it was shown that under the employed assay conditions the reaction of BV to PCB proceeds in two different ways: Both 3(Z)-PΦB and 18¹,18²-DHBV occur as intermediates. Activity assays with the purified intermediates yielded PCB. Hence, both compounds are suitable substrates for KflaHY2.
The results of this work highlight the importance of the biochemical experiments, as catalytic activity cannot solely be predicted by sequence analysis.

In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.

A popular model for the locations of fibres or grains in composite materials
is the inhomogeneous Poisson process in dimension 3. Its local intensity function
may be estimated non-parametrically by local smoothing, e.g. by kernel
estimates. They crucially depend on the choice of bandwidths as tuning parameters
controlling the smoothness of the resulting function estimate. In this
thesis, we propose a fast algorithm for learning suitable global and local bandwidths
from the data. It is well-known, that intensity estimation is closely
related to probability density estimation. As a by-product of our study, we
show that the difference is asymptotically negligible regarding the choice of
good bandwidths, and, hence, we focus on density estimation.
There are quite a number of data-driven bandwidth selection methods for
kernel density estimates. cross-validation is a popular one and frequently proposed
to estimate the optimal bandwidth. However, if the sample size is very
large, it becomes computational expensive. In material science, in particular,
it is very common to have several thousand up to several million points.
Another type of bandwidth selection is a solve-the-equation plug-in approach
which involves replacing the unknown quantities in the asymptotically optimal
bandwidth formula by their estimates.
In this thesis, we develop such an iterative fast plug-in algorithm for estimating
the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed
asymptotics of the estimators of the intensity function and of its second
derivatives and integrals of second derivatives which appear in the formulae
for asymptotically optimal bandwidths. These asymptotics are utilised to determine
the exact number of iteration steps and some tuning parameters. For
both global and local case, fewer than 10 iterations suffice. Simulation studies
show that the estimated intensity by local bandwidth can better indicate
the variation of local intensity than that by global bandwidth. Finally, the
algorithm is applied to two real data sets from test bodies of fibre-reinforced
high-performance concrete, clearly showing some inhomogeneity of the fibre
intensity.

A fast numerical method for an advanced electro-chemo-mechanical model is developed which is able to capture phase separation processes in porous materials. This method is applied to simulate lithium-ion battery cells, where the complex microstructure of the electrodes is fully resolved. The intercalation of ions into the popular cathode material LFP leads to a separation into lithium-rich and lithium-poor phases. The large concentration gradients result in high mechanical stresses. A phase-field method applying the Cahn-Hilliard equation is used to describe the diffusion. For the sake of simplicity, the linear elastic case is considered. Numerical tests for fully resolved three-dimensional granular microstructures are discussed in detail.

Due to the steadily growing flood of data, the appropriate use of visualizations for efficient data analysis is as important today as it has never been before. In many application domains, the data flood is based on processes that can be represented by node-link diagrams. Within such a diagram, nodes may represent intermediate results (or products), system states (or snapshots), milestones or real (and possibly georeferenced) objects, while links (edges) can embody transition conditions, transformation processes or real physical connections. Inspired by the engineering sciences application domain and the research project “SinOptiKom: Cross-sectoral optimization of transformation processes in municipal infrastructures in rural areas”, a platform for the analysis of transformation processes has been researched and developed based on a geographic information system (GIS). Caused by the increased amount of available and interesting data, a particular challenge is the simultaneous visualization of several visible attributes within one single diagram instead of using multiple ones. Therefore, two approaches have been developed, which utilize the available space between nodes in a diagram to display additional information.
Motivated by the necessity of appropriate result communication with various stakeholders, a concept for a universal, dashboard-based analysis platform has been developed. This web-based approach is conceptually capable of displaying data from various data sources and has been supplemented by collaboration possibilities such as sharing, annotating and presenting features.
In order to demonstrate the applicability and usability of newly developed applications, visualizations or user interfaces, extensive evaluations with human users are often inevitable. To reduce the complexity and the effort for conducting an evaluation, the browser-based evaluation framework (BREF) has been designed and implemented. Through its universal and flexible character, virtually any visualization or interaction running in the browser can be evaluated with BREF without any additional application (except for a modern web browser) on the target device. BREF has already proved itself in a wide range of application areas during the development and has since grown into a comprehensive evaluation tool.

The research problem is that the land-use (re-)planning process in the existing Egyptian cities
does not attain sustainability. This is because of the unfulfillment of essential principles within
their land-use structures, lack of harmony between the added and old parts in the cities, and
other reasons. This leads to the need for developing an assessment system, which is a
computational spatial planning support system-SPSS. This SPSS is used for identifying the
degree of sustainability attainment in land-uses plans, predicting probable problems, and
suggesting modifications in the evaluated plans.
The main goal is to design the SPSS for supporting sustainability in the Egyptian cities. The
secondary goals are: studying the Egyptian planning and administrative systems for designing
the technical and administrative frameworks for the SPSS, the development of an assessment
model from the SPSS for assessing sustainability in land-use structures of urban areas, as well
as the identification of the improvements required in the model and the recommendations for
developing the SPSS.
The theoretical part aims to design each of the administrative and technical frameworks of the
SPSS. This requires studying each of the main planning approaches, the sustainability in urban
land-use planning, and the significance of using efficient assessment tools for evaluating the
sustainability in this process. The added value of the planning support systems-PSSs for
planning and their role in supporting sustainability attainment in urban land-use planning are
discussed. Then, a group of previous examples in the sustainability assessment from various
countries (developed and developing countries) are selected, which have used various
assessment tools. This is to extract some learned lessons to be guides for the SPSS. And so,
the comprehensive technical framework for the SPSS is designed, which includes the suggested
methods and techniques that perform various stages of the assessment process.
The Egyptian context is studied regarding the planning and administration systems within the
Egyptian cities, as well as the spatial and administrative problems facing the sustainable
development. And so, the administrative framework for the SPSS is identified, which includes
the entities that should be involved in the assessment process.
The empirical part focuses on the design of a selected assessment model from the
comprehensive technical framework of the SPSS to be established as a minimized version from
it. This model is programmed in the form of a new toolbox within the ArcGIS™ software through
geoscripting using Python programming language to be applied for assessing the sustainability
attainment in the land-use structure of urban areas. The required assessing criteria for the model
specialized for the Egyptian and German cities are identified, for applying it on German and
Egyptian study areas.
The conclusions regarding each of PSSs, the Egyptian local administration and planning
systems, sustainability attainment in the land-use planning process in Egyptian Cities, as well as
the proposed SPSS and the developed toolbox are drawn. The recommendations are regarding
each of challenges facing the development and application of PSSs, the Egyptian local
administration and planning systems, the spatial problems in Egyptian cities, the establishment
of the SPSS, and the application of the toolbox. The future agenda is in the fields of sustainable urban land-use planning, planning support science, and the development process in the
Egyptian cities.